hypothesis theory of gravity

Einstein's Theory of Gravitation

Our modern understanding of gravity comes from Albert Einstein’s theory of general relativity, which stands as one of the best-tested theories in science. General relativity predicted many phenomena years before they were observed, including black holes , gravitational waves , gravitational lensing , the expansion of the universe , and the different rates clocks run in a gravitational field. Today, researchers continue to test the theory’s predictions for a better understanding of how gravity works.

Center for Astrophysics | Harvard & Smithsonian astrophysicists research the predictions of general relativity in many ways:

Capturing the first image of a supermassive black hole using the Event Horizon Telescope (EHT). This image of the black hole at the center of the nearby galaxy M87 reveals how gravitation affects the matter in orbit and the light that material emits, providing a novel test of general relativity in a regime where gravity is very strong. CfA Plays Central Role In Capturing Landmark Black Hole Image

Using gravitational lensing to search for the earliest galaxies in the universe. While they’re too faint to be seen directly, closer-by galaxies and clusters sometimes magnify their light, allowing us to learn about the ancestors of the Milky Way and other modern galaxies. Discovering Distant Radio Galaxies via Gravitational Lensing

Reconstructing the location of most of the mass in the universe using gravitational lensing. Next-generation observatories like the Large Synoptic Survey Telescope (LSST) will provide a census of millions of galaxies from their gravitational distortions. Mapping Dark Matter

Performing follow-up observations of gravitational wave events, to confirm the nature of the source. Collisions between neutron stars produce a lot of light in the form of short duration gamma ray bursts in addition to gravitational waves. Astronomers observed such a collision in 2017 using Dark Energy Camera on the Blanco Telescope in Chile, providing complementary data to the observation from LIGO. Astronomers See Light Show Associated With Gravitational Waves

Studying gravitational wave sources that aren’t visible to LIGO, but will be to future gravitational observatories. Using visible light telescopes, astronomers have observed that white dwarf binaries are relatively common in the galaxy, and some of them are in sufficiently tight orbits to be emitters of gravitational waves. One pair in particular orbits every 12.75 minutes, which will make it the strongest source for the future Laser Interferometer Space Antenna (LISA). Space-Warping White Dwarfs Produce Gravitational Waves

Determining whether black holes are actually what GR predicts. While GR is very clear that black holes exist, alternative theories propose different objects that behave in different ways. The challenge is that black holes appear very small in our telescopes, so it’s hard to observe their behavior. However, researchers have ruled out a number of alternative explanations, based on many observations of black holes. Do Stars Fall Quietly into Black Holes, or Crash into Something Utterly Unknown?

Testing general relativity’s prediction about the shape of a black hole. The Event Horizon Telescope is designed to take a picture of the “shadow” of the Milky Way’s supermassive black hole, which is the dark region through which no light passes. The shape of this shadow is predicted by GR, so the EHT will provide the first precision measurement of a fundamental property of a black hole. Event Horizon Telescope Reveals Magnetic Fields at Milky Way's Central Black Hole

A Century of Relativity

Albert Einstein published his full theory of general relativity in 1915, followed by a flurry of research papers by Einstein and others exploring the predictions of the theory. In general relativity (GR), concentrations of mass and energy curve the structure of spacetime, affecting the motion of anything passing near — including light. The theory explained the anomalous orbit of Mercury, but the first major triumph came in 1919 when Arthur Eddington and his colleagues measured the influence of the Sun’s gravity on light from stars during a total solar eclipse.

Physicists made many exotic predictions using general relativity. The bending of light around the Sun is small, but researchers realized the effect would be much larger for galaxies, to the point where gravity would form images of more distant objects — the phenomenon now called gravitational lensing. GR also predicted the existence of black holes: objects with gravity so intense that nothing getting too close can escape again, not even light.

General relativity showed that gravitation has a speed, which is the same as the speed of light. Catastrophic events like collisions between black holes or neutron stars produce gravitational waves. Researchers finally detected these waves in 2015 using the Laser Interferometer Gravitational Observatory (LIGO), a sensitive laboratory that took decades to develop.

For many aspects of astronomy — the motion of planets around stars, the structure of galaxies, etc. — researchers don’t need to use general relativity. However, in places where gravity is strong, and to describe the structure of the universe itself, GR is necessary. For that reason, researchers continue to use GR and probe its limits.

Black holes are extremely common in the universe. Stellar-mass black holes, the remnants of massive stars that exploded, are sometimes the source of powerful X-ray emissions when they are in binary systems with stars. In addition, nearly every galaxy harbors a supermassive black hole at its center, some of which produce powerful jets of matter visible from across the universe. GR is essential to understanding how these objects become so bright, as well as studying how black holes form and grow. The Event Horizon Telescope (EHT) is a world-spanning array of observatories that captured the first image of a supermassive black hole, providing a new arena for testing GR’s predictions.

Gravitational waves are a new branch of astronomy, providing a complementary way to study astrophysical systems to the standard light-based observations. Researchers use GR to provide “templates” of many possible gravitational wave signals, which is how they identify the source and its properties. Gravitational wave astronomy combines with light-based astronomy to characterize some of the most extreme events in the cosmos: collisions of black holes and neutron stars .

Astronomers use gravitational lensing to locate some of the earliest galaxies in the universe, which are too faint to be seen without the magnification provided by gravity. In addition, the distortion created by lensing allows researchers to study dark matter , and map the structure of the universe on the largest scales.

Not long after Einstein published GR, researchers realized the theory predicts that the universe changes in time. Observations in the 1920s found that prediction was true: the universe is expanding, with galaxies moving away from each other. Using GR, cosmologists found the cosmos had a beginning, and was once hotter and denser than it is today. GR provides the mathematical framework for describing the structure and evolution of the universe from its beginnings 13.8 billion years ago, and into the future.

This artist’s illustration depicts two merging neutron stars and the gravitational waves they emit. As the LIGO and Virgo gravitational wave observatories have confirmed, collisions of black holes and neutron stars emit enough gravitational waves to be seen billions of light-years away.

• Why do we need an extremely large telescope like the Giant Magellan Telescope?
• What happens to space time when cosmic objects collide?
• Black Holes
• Gravitational Lensing
• Gravitational Waves
• Very Long Baseline Interferometry
• High Energy Astrophysics
• Optical and Infrared Astronomy
• Theoretical Astrophysics

Related News

M87* one year later: proof of a persistent black hole shadow, unveiling black hole spins using polarized radio glasses, the giant magellan telescope’s final mirror fabrication begins, new horizons in physics breakthrough prize awarded to cfa astrophysicist, cfa selects contractor for next generation event horizon telescope antennas, sheperd doeleman awarded the 2023 georges lemaître international prize, 'the dawn of a new era in astronomy', astronomers reveal first image of the black hole at the heart of our galaxy, bringing black holes to light, connecting the dots: from black hole theory to actual images, physics of the primordial universe, sensing the dynamic universe, castles survey, telescopes and instruments, event horizon telescope (eht), giant magellan telescope, the greenland telescope.

Subscribe or renew today

Every print subscription comes with full digital access

Science News

Einstein’s genius changed science’s perception of gravity.

General relativity has grown more important than it was in Einstein's day

GETTING A GRIP ON GRAVITY   Einstein's general theory of relativity explains gravity as a distortion of space (or more precisely, spacetime) caused by the presence of matter or energy. A massive object generates a gravitational field by warping the geometry of the surrounding spacetime. 

Nicolle Rager Fuller

Share this:

By Tom Siegfried

October 4, 2015 at 5:30 am

Albert Einstein opened humankind’s eyes to the universe.

Before Einstein, space seemed featureless and changeless, as Isaac Newton had defined it two centuries earlier. And time, Newton declared, flowed at its own pace, oblivious to the clocks that measured it. But Einstein looked at space and time and saw a single dynamic stage — spacetime — on which matter and energy strutted, generating sound and fury, signifying gravity.

Newton’s law of gravity had united the earthly physics of falling apples with the cosmic dances of planets and stars. But he couldn’t explain how, and he famously refused to try. It took an Einstein to figure out gravity’s true modus operandi. Gravity, Einstein showed, did not just make what goes up always come down. Gravity made the universe go ’round.

Gravity’s secrets succumbed to Einstein’s general theory of relativity, unveiled in a series of papers submitted a century ago this November to the Prussian Academy in Berlin. A decade earlier, his special theory of relativity had merged matter with energy while implying the unity of space and time (soon to be christened as spacetime). After years of struggle, Einstein succeeded in showing that matter and spacetime mutually interact to mimic Newton’s naïve idea that masses attract each other. Gravity, said Einstein, actually moved matter along the curving pathways embodied in spacetime — paths imprinted by mass and energy themselves. As expressed decades later by the physicist John Archibald Wheeler, mass grips spacetime, telling it how to curve, and spacetime grips mass, telling it how to move.

Einstein’s theory explained a famous observation that Newtonian gravity could not: a subtlety in the orbit of the planet Mercury. And his equations implied further slight deviations from Newtonian calculations. Over the last century, general relativity’s predictions have been repeatedly verified by modern precision measurements. To physicists today, general relativity and gravity are essentially synonyms.

Timeline: Einstein and relativity

But general relativity is about more than just understanding gravity. It’s about explaining the totality of existence. General relativity inspired a new vision of the entire fabric of the cosmos. From general relativity flowed the realization that the universe is expanding, that it contains spacetime bottomless pits called black holes, that it is traversed by ripples in space triggered by cataclysmic collisions.  

“The implications for the further reaches of the universe were more surprising than even Einstein ever realized,” physicist Stephen Hawking has written.

General relativity explains how the universe can obey physical laws that apply to any form of motion. It’s at the heart of identifying and investigating crucial questions about space and time, existence and reality. And its implications are not limited to esoteric concerns on cosmic scales — it has its down-to-Earth impacts as well. Without general relativity, for instance, GPS devices would be worthless. Satellite signals designed to keep your car on the right road would be off by miles if not corrected for the effects predicted by Einstein’s math.

Gravitation revolution

On his road to general relativity, Einstein himself took many wrong turns. From 1907 to 1914 he struggled with what the physicist Abraham Pais called “one of the hardest problems of the century” — explaining gravity in a way that permits the laws of nature to be the same for all observers, no matter how they are moving. Einstein had to learn new math and jettison common prejudices, such as the universal belief that Euclidean geometry described reality accurately. He struggled with distractions, both in his personal life and in physics problems posed by quantum theory. And he found that nature stubbornly refused to cooperate. By 1914 he had essentially given up, believing that a partially successful attempt — a sort of general relativity lite — was the best that nature would allow.

But then somehow Einstein’s brain rebooted. His theory began to solidify, and he swiftly composed four papers, one a week, during November 1915. By the last paper he had finally found the decisive equation that launched his gravitation revolution. 

Four years later, general relativity made Einstein himself a celebrity. If gravity curves space, he had realized early on, a light beam passing near a massive object (say, the sun) would be deflected from its course. That deflection would shift the apparent position of the source of the light (say, a distant star). During a solar eclipse, such a shift could be photographed and measured. Such measurements, made during eclipse expeditions in 1919, confirmed Einstein’s calculation. Even without Twitter to spread the word, Einstein’s triumph sparked a media sensation.

“Lights all askew in the heavens, men of science more or less agog,” proclaimed one of the most famous newspaper headlines in science history, in the Nov. 10 New York Times . And from the Times of London on November 7: “Revolution in Science, New Theory of the Universe, Newtonian Ideas Overthrown.”

Einstein became a legend, his name forevermore synonymous with genius.

As it turned out, some bending of light would have been expected even with Newtonian gravity, as Johann von Soldner had calculated (unknown to Einstein) more than a century earlier. But Einstein predicted precisely twice as much bending as von Soldner had. And although the earliest measurements were crude, they were much closer to Einstein’s predictions than Newton’s. In subsequent eclipses, Einstein’s calculation has been repeatedly confirmed. Gravity deflects light just as general relativity requires.

General relativity’s light-bending effect proved valuable for much more than affirming Einstein’s theory. By bending light, masses act like a lens; such “gravitational lensing” alters the apparent position of a distant object, creating multiple images of it, or (if the images overlap) appearing to brighten it. Such effects can be used to probe matter’s distribution in space or detect the presence of unseen masses.

“Since the discovery of the first gravitational lens, the phenomenon has been exploited to map the distribution of mass around galaxies and clusters, and to search for dark matter, dark energy, compact objects, and extrasolar planets,” physicist Clifford Will notes in a recent paper .

Gravitational lensing was first observed in 1979, but Einstein had suspected its possibility in 1912, even before his theory had been completed. Only in 1936 did he publish a paper about it, Will notes, “primarily, it seems, to get a Czech electrical engineer named Rudi Mandl to stop pestering him about it.” Will, of the University of Florida in Gainesville, does not mention that Mandl had first approached Science News Letter ( Science News ’ predecessor) with the gravitational lens idea; the magazine paid his expenses to visit Einstein, who then agreed to do the calculation that Mandl had suggested ( SNL: 12/19/36, p. 388 ). Einstein’s paper, published in Science, suggested that the effect would never be noticed. But modern astronomical explorations proved the contrary.

Einstein was also ambivalent about other consequences of general relativity. In 1916, for instance, he raised the possibility of gravitational radiation — waves rippling through spacetime after a massive body abruptly changes its motion, as when colliding with another mass. Such waves should exist, Einstein reasoned, because general relativity required gravity’s influence to propagate at the speed of light (whereas Newton’s gravity transmitted itself instantaneously). But later Einstein changed his mind. In 1936, he and Nathan Rosen submitted a paper arguing that such waves did not exist after all. But their paper contained a mistake. Today, gravitational waves’ reality has been established convincingly by indirect methods, and experiments to detect them directly are under way (see “Magnifying the Cosmos”).

Astronomers have already detected another offspring of general relativity, black holes, throughout the cosmos. But Einstein didn’t believe they would exist, either.

Explaining Mercury’s odd orbit

Black holes’ existence had been foreshadowed only weeks after Einstein presented his general relativity papers to the Prussian Academy. Karl Schwarzschild, a German astronomer serving in World War I on the Russian front, worked out solutions to Einstein’s complicated equations for the spacetime geometry around a massive sphere. It was the first mathematical step toward describing black holes in space. But Schwarzschild didn’t pursue the topic; he died a few months later from a skin disease. Not until the late 1960s did black holes emerge as general relativity’s most prominent advertisement, stimulating both science and the popular imagination. They became the most glamorous product of a theory conceived by science’s best imaginer.

Falling freely

Einstein’s imagination gave birth to general relativity’s core idea as he gazed out his office window while he was supposed to be evaluating patents. “All of a sudden I was struck by a thought,” Einstein later said. “If a person falls freely, he will certainly not feel his own weight.”

It was 1907, two years after his special theory of relativity had rewritten textbook notions about time and motion. 

Special relativity showed that the laws of nature don’t depend on how you are moving, as long as it’s uniform motion — constant speed in a straight line. But in real life, objects and people move in all sorts of nonuniform ways. (Let the air out of a balloon, for instance.) Even some “simple” motions, like the rotation of a sphere or orbit of a planet, are nonuniform, as they constantly change direction and are therefore accelerating. Einstein wanted to extend relativity to all forms of accelerated motions. But he didn’t know how.

Then his happy thought in the patent office raised hope. A person falling freely accelerates toward the ground because of gravity but feels no force (until impact). Therefore, Einstein realized, gravity and acceleration are two sides of a coin. The upward thrust of an accelerating rocket ship pins the occupants to the floor just as the gravitational pull of the Earth keeps your feet on the ground. This acceleration-gravity equivalence explained a curious Newtonian coincidence: A body’s mass (its inertial resistance to changes in motion) is equal to its weight (or gravitational mass), its response to gravity. Einstein built special relativity on the principle that light’s velocity was constant; he suspected that general relativity could be built on the principle that inertial and gravitational mass are equivalent. If he succeeded, it would mean that nature’s laws could be the same for all forms of motion.

At first progress was slow. Then a key clue emerged in 1908, when the mathematician Hermann Minkowski showed how special relativity required the merger of space with time ( SN: 9/13/08, p. 26 ). In special relativity, measures of space or time differ for different observers. But Minkowski showed that space and time combined — spacetime — yielded a mathematical description of events that all observers could agree on. Any event’s location could be specified by a set of space and time coordinates.

Bending light

One of the first predictions of general relativity to be tested involved the bending of light. Because a massive body, such as a star, warps spacetime around it, a light beam passing nearby should be deflected from a straight-line path. From Earth, light from a distant star passing near the sun would be bent in such a way as to alter the apparent position of the distant star. In 1919, astronomers photographed stars near the sun during a solar eclipse. Comparing the positions of the stars in those photos with nighttime photos taken earlier showed that the stars’ positions did appear to shift by about the amount Einstein predicted. Actually, calculations based on Newton’s gravity also predicted light bending, but only by half as much as Einstein calculated. General relativity requires twice as much bending because the light ray is bent while passing through space already curved to begin with.

Establishing such coordinates requires a frame of reference, or origin point. Different observers will choose different origins. So if nature’s laws are the same for everybody, then any one person’s set of coordinates should in some sense be equivalent to anybody else’s. Einstein’s quest, then, became an effort to find the formula for transforming any one coordinate system into any other, while maintaining the equivalence between gravity and acceleration.

By 1912, Einstein realized that his goal would require abandoning Euclidean geometry. Real space, he realized, could not conform to the idealized lines and angles of the textbooks. Gravity distorted the coordinates, just as a grid of straight lines on a rubber sheet would curve if you placed a heavy cannonball on it.

But Einstein did not possess the mathematical skills to cope with non-Euclidean geometry. Fortunately, his college friend Marcel Grossmann, an accomplished mathematician, was eager to help. Familiar with 19th century–mathematician Bernhard Riemann’s work on the math for describing curved surfaces, Grossmann helped Einstein produce the outline ( Entwurf in German) of a new gravity theory. It had one drawback, though — it worked for some coordinate systems, but not all possible systems. Einstein was dismayed, writing to the physicist Hendrik Lorentz in August 1913 that “there are still such major snags in the thing that my confidence in the admissibility of the theory is still shaky.” If acceleration is equivalent to a gravitational field, Einstein noted, every kind of acceleration should be describable by the equations for gravity. If not, “the theory refutes its own starting point; then it has no foundation whatsoever.”

Two days later, Einstein seemed much happier, writing to Lorentz that the Entwurf theory’s deficiency had been solved. In November, Einstein described that solution in a letter to physicist Paul Ehrenfest, asserting that equations describing all accelerations simply couldn’t exist; some coordinate systems get special status in order to preserve the law of energy-momentum conservation. That made his original goal impossible. But Einstein seemed satisfied that he had done the best that nature would permit.

“Can there be anything more beautiful than this, that the necessary specialization follows from the conservation laws?” Einstein asked Ehrenfest. It turned out that there actually was something more beautiful. But to find it, Einstein had to move to Berlin.

Breakthrough in Berlin

Einstein’s life until then had been peripatetic. Born in Ulm, Germany, in 1879, he moved when an infant to Munich and then as a teenager to Milan, Italy, having dropped out of high school. He went back to school in Switzerland, eventually graduating from college in Zurich. Failing to find an academic job, he joined the Swiss patent office in 1902, and the next year married Mileva Marić.

During his years at the patent office, Einstein produced an explosive output of papers poking holes in traditional physics, including reports on his special theory of relativity and Nobel Prize–winning work on quantum physics. Eventually those papers led to sufficient recognition in the physics world to get a faculty appointment in Prague. But at the first chance he returned to Zurich, where Grossmann now taught math. There Einstein and Grossmann developed the Entwurf theory. And then Berlin, the pinnacle of German (and world) physics, called. Prominent physicists at the university there made Einstein an offer he couldn’t refuse: no teaching responsibilities. As he wrote to Lorentz: “I could not resist the temptation to accept a position in which I am relieved of all responsibilities so that I can give myself over completely to rumination.” 

Story continues below box

The Einstein field equations

Because mass and energy distort the shape of spacetime, the Euclidean geometry of standard textbooks can’t accurately describe it. Einstein’s general relativity uses more complicated math built on the non-Euclidean geometry devised in the 19th century by Bernhard Riemann. With help from his friend Marcel Grossmann, Einstein adopted further advances by the mathematicians Gregorio Ricci-Curbastro, Tullio Levi-Civita and Elwin Christoffel to describe spacetime geometry in terms of mathematical expressions called tensors. Tensors are a lot like vectors — quantities, such as velocity, composed of two components (in velocity’s case, speed and direction). Tensors are similar but can encompass more than just two components. Einstein used tensors to develop his equation describing the gravitational field, known as the Einstein field equation.

The Einstein Field Equation

Gμν = 8πtμν.

On the left side of the equation is a tensor describing the geometry of spacetime — the gravitational field. On the right is the tensor describing the matter and energy density — the source of the gravitational field. The equation shows that spacetime geometry equals mass-energy density when adjusted with the proper units and numerical constants. (Actually, the equation stands for a set of multiple equations owing to the complexity of tensors. So experts usually speak of the Einstein field equations, plural.)

The Einstein Field Equation with Lambda

Gμν + λgμν = 8πtμν.

When he applied this equation to the entire universe, Einstein found that the universe would be unstable, easily disturbed into a state in which spacetime would be either expanding or collapsing. So he added a term that came to be called the cosmological constant, symbolized by the Greek letter lambda. It represents a constant amount of energy density throughout space that would supposedly keep the universe stable and changeless. Later, evidence that the universe was indeed expanding led Einstein to renounce lambda. But it has been revived by modern cosmologists to explain the apparent increase in the universe’s rate of expansion that was discovered in the late 1990s.

Throughout all these moves, Einstein’s personal life deteriorated. Growing apart from Mileva, he pursued a relationship with his cousin Elsa. And Mileva did not relish the idea of life in Berlin. In July 1914 she took their two sons back to Zurich, and Einstein was left in Berlin free to ruminate about general relativity to his heart’s content. During the following year his freedom nourished a fertile few months in which he saw a new path to success. In mid-1915 he saw that there was a way to make relativity truly general. Rather than imposing energy-momentum conservation on the equations, he worked on devising equations that would impose the conservation law on the universe. 

Einstein now applied the full force of all the mathematical wizardry he had mastered in the preceding years, adopting embellishments of Riemann’s math called tensors. By November 1915, Einstein sensed victory. On November 4, he submitted a paper on general relativity to the Prussian Academy, following up with an addendum November 11. The next week, he presented yet another new paper, this time in a lecture to the academy, showing how general relativity’s spacetime curvature solved the outstanding gravitational problem in all of physics, the anomaly in the orbit of Mercury. During the following week, he finally found the correct form for the equations describing the gravitational field, presenting his result on November 25. Einstein’s quest was completed. General relativity worked.

Universal implications

Einstein quickly realized (or had known all along) that his new theory of gravity was really a theory of the cosmos. In 1917, he wrote a famous paper applying general relativity to the universe as a whole. Today that paper stands as the foundation for modern cosmology. But at the time, Einstein was troubled — his equations implied an unstable universe, either growing or collapsing. In those days, the universe was supposed to be eternal, everlasting and changeless. So Einstein altered his equation, adding a factor called the cosmological constant, representing a constant energy density in space that kept the universe static.

Others didn’t buy it. Alexander Friedmann, a Russian meteorologist-mathematician, developed a description of an expanding or contracting universe from Einstein’s original equations. Einstein first thought Friedmann to be in error, but then relented, although still viewing the “expanding universe” as of mathematical interest only. But a few years later, when Edwin Hubble’s analysis of light from distant galaxies confirmed the universe’s expansion, Einstein gave in. Despite his own reluctance to accept it, Einstein’s general relativity math did in fact imply what Wheeler later called the “most dramatic prediction that science has ever made” — the expansion of the universe.

Today, Einstein’s cosmological constant has been revived. Rather than preventing the universe from collapse, the vacuum energy it describes can explain why the universe now expands at an accelerating pace. General relativity, cosmological constant and all, today forms the core science for analyzing the history of the universe and forecasting its future.

But apart from its use in cosmology, general relativity was not widely applied to scientific problems in its first four decades. For the most part, general relativity languished in departments of mathematics, rarely studied in physics.

Gravitational collapse

At the time, nobody paid any attention. But in the 1960s, newfound astrophysical anomalies suggested that gravitational collapse was at work in the cosmos, and Oppenheimer and Snyder’s idea was revived as what came to be known as black holes. Famous for swallowing anything they encounter and allowing nothing to escape, black holes are probably the most bizarre astrophysical consequences of general relativity. Small black holes have been detected throughout space and supermassive black holes reside in the cores of most galaxies.

More recently black holes (schematic of one shown) have been used as thought-experiment laboratories for investigating several outstanding mysteries about the nature of space and time.

“General relativity was essentially a dead subject for a long time,” says the prominent relativity theorist Wolfgang Rindler of the University of Texas at Dallas. “People considered it a dormant science.”

Only after Einstein died in 1955 did general relativity come to life. About that time Wheeler, at Princeton University, began a program to explore its implications and train students to pursue them. By the early 1960s, new astronomical phenomena demanded explanations that Newtonian physics could not provide, and general relativity was poised for its renaissance. In the decades that followed, general relativity proved crucial for describing all sorts of celestial phenomena. At the same time, physicists devised ever more precise tests of its predictions, and Einstein passed them all. As Will has noted, “It is remarkable that this theory, born 100 years ago out of almost pure thought, has managed to survive every test.” 

Even harder than finding the equations for general relativity is explaining how, out of almost pure thought, Einstein did it. Science historian Gerald Holton once remarked, while describing Einstein in the context of defining scientific genius, that “there is a mutual mapping of the mind and lifestyle of this scientist, and of the laws of nature.” Einstein himself ascribed his success to discovering the deep relationship between mathematics and the physical world.

In creating general relativity, Einstein’s path was forked; he had to envision physical processes governing matter, space and time, while at the same time formulating abstract mathematical expressions corresponding to that reality. As a student, Einstein testified, he neglected mathematics. His intuition was not strong enough to guide him to the most profound of math’s many subfields. But in the physical realm of natural phenomena, “I soon learned to scent out that which was able to lead to fundamentals and to turn aside from … the multitude of things which clutter up the mind and divert it from the essential.” At first he didn’t realize that “a more profound knowledge of the basic principles of physics is tied up with the most intricate mathematical methods.” He learned that from his pursuit of general relativity.

In the end, Einstein’s mind produced a mathematical theory so rich that it entailed unsuspected cosmic novelties. Fantastic physical phenomena were first discovered not through the lenses of telescopes, but within the squiggles Einstein had scratched out on paper to make the world make sense — to him. And now physical nature makes sense to modern science only because of Einstein’s insights.

“Einstein’s ideas,” his friend the physicist Max Born wrote over half a century ago, “have given the physical sciences the impetus which has liberated them from outdated philosophical doctrine, and made them one of the decisive factors in the modern world of man.”

Gravity’s pull on time

General relativity and GPS

This article appears in the October 17, 2015, Science News with the headline, “Getting a grip on gravity: Einstein’s genius reconstructed science’s perception of the cosmos.”

More Stories from Science News on Astronomy

Separating science fact from fiction in Netflix’s ‘3 Body Problem’ 

How a sugar acid crucial for life could have formed in interstellar clouds

How a 19th century astronomer can help you watch the total solar eclipse

A new image reveals magnetic fields around our galaxy’s central black hole

Did the James Webb telescope ‘break the universe’? Maybe not

JWST spies hints of a neutron star left behind by supernova 1987A

How to build an internet on Mars

Astronomers have snapped a new photo of the black hole in galaxy M87

Subscribers, enter your e-mail address for full access to the Science News archives and digital editions.

Not a subscriber? Become one now .

13 Gravitation

13.7 einstein’s theory of gravity, learning objectives.

By the end of this section, you will be able to:

• Describe how the theory of general relativity approaches gravitation
• Explain the principle of equivalence
• Calculate the Schwarzschild radius of an object
• Summarize the evidence for black holes

Newton’s law of universal gravitation accurately predicts much of what we see within our solar system. Indeed, only Newton’s laws have been needed to accurately send every space vehicle on its journey. The paths of Earth-crossing asteroids, and most other celestial objects, can be accurately determined solely with Newton’s laws. Nevertheless, many phenomena have shown a discrepancy from what Newton’s laws predict, including the orbit of Mercury and the effect that gravity has on light. In this section, we examine a different way of envisioning gravitation.

A Revolution in Perspective

In 1905, Albert Einstein published his theory of special relativity. This theory is discussed in great detail in Relativity in the third volume of this text, so we say only a few words here. In this theory, no motion can exceed the speed of light—it is the speed limit of the Universe. This simple fact has been verified in countless experiments. However, it has incredible consequences—space and time are no longer absolute. Two people moving relative to one another do not agree on the length of objects or the passage of time. Almost all of the mechanics you learned in previous chapters, while remarkably accurate even for speeds of many thousands of miles per second, begin to fail when approaching the speed of light.

This speed limit on the Universe was also a challenge to the inherent assumption in Newton’s law of gravitation that gravity is an action-at-a-distance force . That is, without physical contact, any change in the position of one mass is instantly communicated to all other masses. This assumption does not come from any first principle, as Newton’s theory simply does not address the question. (The same was believed of electromagnetic forces, as well. It is fair to say that most scientists were not completely comfortable with the action-at-a-distance concept.)

A second assumption also appears in Newton’s law of gravitation (Equation) . The masses are assumed to be exactly the same as those used in Newton’s second law, [latex] \overset{\to }{F}=m\overset{\to }{a} [/latex]. We made that assumption in many of our derivations in this chapter. Again, there is no underlying principle that this must be, but experimental results are consistent with this assumption. In Einstein’s subsequent theory of general relativity (1916), both of these issues were addressed. His theory was a theory of space-time geometry and how mass (and acceleration) distort and interact with that space-time . It was not a theory of gravitational forces. The mathematics of the general theory is beyond the scope of this text, but we can look at some underlying principles and their consequences.

The Principle of Equivalence

Einstein came to his general theory in part by wondering why someone who was free falling did not feel his or her weight. Indeed, it is common to speak of astronauts orbiting Earth as being weightless, despite the fact that Earth’s gravity is still quite strong there. In Einstein’s general theory, there is no difference between free fall and being weightless. This is called the principle of equivalence . The equally surprising corollary to this is that there is no difference between a uniform gravitational field and a uniform acceleration in the absence of gravity. Let’s focus on this last statement. Although a perfectly uniform gravitational field is not feasible, we can approximate it very well.

Within a reasonably sized laboratory on Earth, the gravitational field [latex] \overset{\to }{g} [/latex] is essentially uniform. The corollary states that any physical experiments performed there have the identical results as those done in a laboratory accelerating at [latex] \overset{\to }{a}=\overset{\to }{g} [/latex] in deep space, well away from all other masses. (Figure) illustrates the concept.

Figure 13.28 According to the principle of equivalence, the results of all experiments performed in a laboratory in a uniform gravitational field are identical to the results of the same experiments performed in a uniformly accelerating laboratory.

How can these two apparently fundamentally different situations be the same? The answer is that gravitation is not a force between two objects but is the result of each object responding to the effect that the other has on the space-time surrounding it. A uniform gravitational field and a uniform acceleration have exactly the same effect on space-time.

A Geometric Theory of Gravity

Euclidian geometry assumes a “flat” space in which, among the most commonly known attributes, a straight line is the shortest distance between two points, the sum of the angles of all triangles must be 180 degrees, and parallel lines never intersect. Non-Euclidean geometry was not seriously investigated until the nineteenth century, so it is not surprising that Euclidean space is inherently assumed in all of Newton’s laws.

The general theory of relativity challenges this long-held assumption. Only empty space is flat. The presence of mass—or energy, since relativity does not distinguish between the two—distorts or curves space and time, or space-time, around it. The motion of any other mass is simply a response to this curved space-time. (Figure) is a two-dimensional representation of a smaller mass orbiting in response to the distorted space created by the presence of a larger mass. In a more precise but confusing picture, we would also see space distorted by the orbiting mass, and both masses would be in motion in response to the total distortion of space. Note that the figure is a representation to help visualize the concept. These are distortions in our three-dimensional space and time. We do not see them as we would a dimple on a ball. We see the distortion only by careful measurements of the motion of objects and light as they move through space.

Figure 13.29 A smaller mass orbiting in the distorted space-time of a larger mass. In fact, all mass or energy distorts space-time.

For weak gravitational fields, the results of general relativity do not differ significantly from Newton’s law of gravitation. But for intense gravitational fields, the results diverge, and general relativity has been shown to predict the correct results. Even in our Sun’s relatively weak gravitational field at the distance of Mercury’s orbit, we can observe the effect. Starting in the mid-1800s, Mercury’s elliptical orbit has been carefully measured. However, although it is elliptical, its motion is complicated by the fact that the perihelion position of the ellipse slowly advances. Most of the advance is due to the gravitational pull of other planets, but a small portion of that advancement could not be accounted for by Newton’s law. At one time, there was even a search for a “companion” planet that would explain the discrepancy. But general relativity correctly predicts the measurements. Since then, many measurements, such as the deflection of light of distant objects by the Sun, have verified that general relativity correctly predicts the observations.

We close this discussion with one final comment. We have often referred to distortions of space-time or distortions in both space and time. In both special and general relativity, the dimension of time has equal footing with each spatial dimension (differing in its place in both theories only by an ultimately unimportant scaling factor). Near a very large mass, not only is the nearby space “stretched out,” but time is dilated or “slowed.” We discuss these effects more in the next section.

Black Holes

Einstein’s theory of gravitation is expressed in one deceptively simple-looking tensor equation (tensors are a generalization of scalars and vectors), which expresses how a mass determines the curvature of space-time around it. The solutions to that equation yield one of the most fascinating predictions: the black hole . The prediction is that if an object is sufficiently dense, it will collapse in upon itself and be surrounded by an event horizon from which nothing can escape. The name “black hole,” which was coined by astronomer John Wheeler in 1969, refers to the fact that light cannot escape such an object. Karl Schwarzschild was the first person to note this phenomenon in 1916, but at that time, it was considered mostly to be a mathematical curiosity.

Surprisingly, the idea of a massive body from which light cannot escape dates back to the late 1700s. Independently, John Michell and Pierre Simon Laplace used Newton’s law of gravitation to show that light leaving the surface of a star with enough mass could not escape. Their work was based on the fact that the speed of light had been measured by Ole Roemer in 1676. He noted discrepancies in the data for the orbital period of the moon Io about Jupiter. Roemer realized that the difference arose from the relative positions of Earth and Jupiter at different times and that he could find the speed of light from that difference. Michell and Laplace both realized that since light had a finite speed, there could be a star massive enough that the escape speed from its surface could exceed that speed. Hence, light always would fall back to the star. Oddly, observers far enough away from the very largest stars would not be able see them, yet they could see a smaller star from the same distance.

Recall that in Gravitational Potential Energy and Total Energy , we found that the escape speed, given by (Figure) , is independent of the mass of the object escaping. Even though the nature of light was not fully understood at the time, the mass of light, if it had any, was not relevant. Hence, (Figure) should be valid for light. Substituting c , the speed of light, for the escape velocity, we have

Thus, we only need values for R and M such that the escape velocity exceeds c , and then light will not be able to escape. Michell posited that if a star had the density of our Sun and a radius that extended just beyond the orbit of Mars, then light would not be able to escape from its surface. He also conjectured that we would still be able to detect such a star from the gravitational effect it would have on objects around it. This was an insightful conclusion, as this is precisely how we infer the existence of such objects today. While we have yet to, and may never, visit a black hole, the circumstantial evidence for them has become so compelling that few astronomers doubt their existence.

Before we examine some of that evidence, we turn our attention back to Schwarzschild’s solution to the tensor equation from general relativity. In that solution arises a critical radius, now called the Schwarzschild radius [latex] ({R}_{\text{S}}) [/latex]. For any mass M , if that mass were compressed to the extent that its radius becomes less than the Schwarzschild radius, then the mass will collapse to a singularity, and anything that passes inside that radius cannot escape. Once inside [latex] {R}_{\text{S}} [/latex], the arrow of time takes all things to the singularity. (In a broad mathematical sense, a singularity is where the value of a function goes to infinity. In this case, it is a point in space of zero volume with a finite mass. Hence, the mass density and gravitational energy become infinite.) The Schwarzschild radius is given by

If you look at our escape velocity equation with [latex] {v}_{\text{esc}}^{}=c [/latex], you will notice that it gives precisely this result. But that is merely a fortuitous accident caused by several incorrect assumptions. One of these assumptions is the use of the incorrect classical expression for the kinetic energy for light. Just how dense does an object have to be in order to turn into a black hole?

Calculating the Schwarzschild Radius

Calculate the Schwarzschild radius for both the Sun and Earth. Compare the density of the nucleus of an atom to the density required to compress Earth’s mass uniformly to its Schwarzschild radius. The density of a nucleus is about [latex] 2.3\,×\,{10}^{17}\,{\text{kg/m}}^{3} [/latex].

We use (Figure) for this calculation. We need only the masses of Earth and the Sun, which we obtain from the astronomical data given in Appendix D .

Substituting the mass of the Sun, we have

This is a diameter of only about 6 km. If we use the mass of Earth, we get [latex] {R}_{\text{S}}=8.85\,×\,{10}^{-3}\,\text{m} [/latex]. This is a diameter of less than 2 cm! If we pack Earth’s mass into a sphere with the radius [latex] {R}_{\text{S}}=8.85\,×\,{10}^{-3}\,\text{m} [/latex], we get a density of

Significance

A neutron star is the most compact object known—outside of a black hole itself. The neutron star is composed of neutrons, with the density of an atomic nucleus, and, like many black holes, is believed to be the remnant of a supernova—a star that explodes at the end of its lifetime. To create a black hole from Earth, we would have to compress it to a density thirteen orders of magnitude greater than that of a neutron star. This process would require unimaginable force. There is no known mechanism that could cause an Earth-sized object to become a black hole. For the Sun, you should be able to show that it would have to be compressed to a density only about 80 times that of a nucleus. (Note: Once the mass is compressed within its Schwarzschild radius, general relativity dictates that it will collapse to a singularity. These calculations merely show the density we must achieve to initiate that collapse.)

Check Your Understanding

Consider the density required to make Earth a black hole compared to that required for the Sun. What conclusion can you draw from this comparison about what would be required to create a black hole? Would you expect the Universe to have many black holes with small mass?

The event horizon

The Schwarzschild radius is also called the event horizon of a black hole. We noted that both space and time are stretched near massive objects, such as black holes. (Figure) illustrates that effect on space. The distortion caused by our Sun is actually quite small, and the diagram is exaggerated for clarity. Consider the neutron star, described in (Figure) . Although the distortion of space-time at the surface of a neutron star is very high, the radius is still larger than its Schwarzschild radius. Objects could still escape from its surface.

However, if a neutron star gains additional mass, it would eventually collapse, shrinking beyond the Schwarzschild radius. Once that happens, the entire mass would be pulled, inevitably, to a singularity. In the diagram, space is stretched to infinity. Time is also stretched to infinity. As objects fall toward the event horizon, we see them approaching ever more slowly, but never reaching the event horizon. As outside observers, we never see objects pass through the event horizon—effectively, time is stretched to a stop.

Visit this site to view an animated example of these spatial distortions.

Figure 13.30 The space distortion becomes more noticeable around increasingly larger masses. Once the mass density reaches a critical level, a black hole forms and the fabric of space-time is torn. The curvature of space is greatest at the surface of each of the first three objects shown and is finite. The curvature then decreases (not shown) to zero as you move to the center of the object. But the black hole is different. The curvature becomes infinite: The surface has collapsed to a singularity, and the cone extends to infinity. (Note: These diagrams are not to any scale.)

The evidence for black holes

Not until the 1960s, when the first neutron star was discovered, did interest in the existence of black holes become renewed. Evidence for black holes is based upon several types of observations, such as radiation analysis of X-ray binaries, gravitational lensing of the light from distant galaxies, and the motion of visible objects around invisible partners. We will focus on these later observations as they relate to what we have learned in this chapter. Although light cannot escape from a black hole for us to see, we can nevertheless see the gravitational effect of the black hole on surrounding masses.

The closest, and perhaps most dramatic, evidence for a black hole is at the center of our Milky Way galaxy. The UCLA Galactic Group, using data obtained by the W. M. Keck telescopes, has determined the orbits of several stars near the center of our galaxy. Some of that data is shown in (Figure) . The orbits of two stars are highlighted. From measurements of the periods and sizes of their orbits, it is estimated that they are orbiting a mass of approximately 4 million solar masses. Note that the mass must reside in the region created by the intersection of the ellipses of the stars. The region in which that mass must reside would fit inside the orbit of Mercury—yet nothing is seen there in the visible spectrum.

Figure 13.31 Paths of stars orbiting about a mass at the center of our Milky Way galaxy. From their motion, it is estimated that a black hole of about 4 million solar masses resides at the center. (credit: UCLA Galactic Center Group – W.M. Keck Observatory Laser Team)

The physics of stellar creation and evolution is well established. The ultimate source of energy that makes stars shine is the self-gravitational energy that triggers fusion. The general behavior is that the more massive a star, the brighter it shines and the shorter it lives. The logical inference is that a mass that is 4 million times the mass of our Sun, confined to a very small region, and that cannot be seen, has no viable interpretation other than a black hole. Extragalactic observations strongly suggest that black holes are common at the center of galaxies.

Visit the UCLA Galactic Center Group main page for information on X-ray binaries and gravitational lensing. Visit this page to view a three-dimensional visualization of the stars orbiting near the center of our galaxy, where the animation is near the bottom of the page.

Dark matter

Stars orbiting near the very heart of our galaxy provide strong evidence for a black hole there, but the orbits of stars far from the center suggest another intriguing phenomenon that is observed indirectly as well. Recall from Gravitation Near Earth’s Surface that we can consider the mass for spherical objects to be located at a point at the center for calculating their gravitational effects on other masses. Similarly, we can treat the total mass that lies within the orbit of any star in our galaxy as being located at the center of the Milky Way disc. We can estimate that mass from counting the visible stars and include in our estimate the mass of the black hole at the center as well.

But when we do that, we find the orbital speed of the stars is far too fast to be caused by that amount of matter. (Figure) shows the orbital velocities of stars as a function of their distance from the center of the Milky Way. The blue line represents the velocities we would expect from our estimates of the mass, whereas the green curve is what we get from direct measurements. Apparently, there is a lot of matter we don’t see, estimated to be about five times as much as what we do see, so it has been dubbed dark matter . Furthermore, the velocity profile does not follow what we expect from the observed distribution of visible stars. Not only is the estimate of the total mass inconsistent with the data, but the expected distribution is inconsistent as well. And this phenomenon is not restricted to our galaxy, but seems to be a feature of all galaxies. In fact, the issue was first noted in the 1930s when galaxies within clusters were measured to be orbiting about the center of mass of those clusters faster than they should based upon visible mass estimates.

Figure 13.32 The blue curve shows the expected orbital velocity of stars in the Milky Way based upon the visible stars we can see. The green curve shows that the actually velocities are higher, suggesting additional matter that cannot be seen. (credit: modification of work by Matthew Newby)

There are two prevailing ideas of what this matter could be—WIMPs and MACHOs. WIMPs stands for weakly interacting massive particles. These particles (neutrinos are one example) interact very weakly with ordinary matter and, hence, are very difficult to detect directly. MACHOs stands for massive compact halo objects, which are composed of ordinary baryonic matter, such as neutrons and protons. There are unresolved issues with both of these ideas, and far more research will be needed to solve the mystery.

• According to the theory of general relativity, gravity is the result of distortions in space-time created by mass and energy.
• The principle of equivalence states that that both mass and acceleration distort space-time and are indistinguishable in comparable circumstances.
• Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass.
• Evidence for the existence of black holes is still circumstantial, but the amount of that evidence is overwhelming.

Key Equations

Conceptual questions.

The principle of equivalence states that all experiments done in a lab in a uniform gravitational field cannot be distinguished from those done in a lab that is not in a gravitational field but is uniformly accelerating. For the latter case, consider what happens to a laser beam at some height shot perfectly horizontally to the floor, across the accelerating lab. (View this from a nonaccelerating frame outside the lab.) Relative to the height of the laser, where will the laser beam hit the far wall? What does this say about the effect of a gravitational field on light? Does the fact that light has no mass make any difference to the argument?

The laser beam will hit the far wall at a lower elevation than it left, as the floor is accelerating upward. Relative to the lab, the laser beam “falls.” So we would expect this to happen in a gravitational field. The mass of light, or even an object with mass, is not relevant.

As a person approaches the Schwarzschild radius of a black hole, outside observers see all the processes of that person (their clocks, their heart rate, etc.) slowing down, and coming to a halt as they reach the Schwarzschild radius. (The person falling into the black hole sees their own processes unaffected.) But the speed of light is the same everywhere for all observers. What does this say about space as you approach the black hole?

What is the Schwarzschild radius for the black hole at the center of our galaxy if it has the mass of 4 million solar masses?

[latex] 1.19\,×\,{10}^{7}\text{km} [/latex]

What would be the Schwarzschild radius, in light years, if our Milky Way galaxy of 100 billion stars collapsed into a black hole? Compare this to our distance from the center, about 13,000 light years.

Additional Problems

A neutron star is a cold, collapsed star with nuclear density. A particular neutron star has a mass twice that of our Sun with a radius of 12.0 km. (a) What would be the weight of a 100-kg astronaut on standing on its surface? (b) What does this tell us about landing on a neutron star?

(a) How far from the center of Earth would the net gravitational force of Earth and the Moon on an object be zero? (b) Setting the magnitudes of the forces equal should result in two answers from the quadratic. Do you understand why there are two positions, but only one where the net force is zero?

How far from the center of the Sun would the net gravitational force of Earth and the Sun on a spaceship be zero?

[latex] 1.49\,×\,{10}^{8}\text{km} [/latex]

Calculate the values of g at Earth’s surface for the following changes in Earth’s properties: (a) its mass is doubled and its radius is halved; (b) its mass density is doubled and its radius is unchanged; (c) its mass density is halved and its mass is unchanged.

Suppose you can communicate with the inhabitants of a planet in another solar system. They tell you that on their planet, whose diameter and mass are [latex] 5.0\,×\,{10}^{3}\,\text{km} [/latex] and [latex] 3.6\,×\,{10}^{23}\,\text{kg} [/latex], respectively, the record for the high jump is 2.0 m. Given that this record is close to 2.4 m on Earth, what would you conclude about your extraterrestrial friends’ jumping ability?

The value of g for this planet is 2.4 m/s 2 , which is about one-fourth that of Earth. So they are weak high jumpers.

(a) Suppose that your measured weight at the equator is one-half your measured weight at the pole on a planet whose mass and diameter are equal to those of Earth. What is the rotational period of the planet? (b) Would you need to take the shape of this planet into account?

A body of mass 100 kg is weighed at the North Pole and at the equator with a spring scale. What is the scale reading at these two points? Assume that [latex] g=9.83\,{\text{m/s}}^{2} [/latex] at the pole.

At the North Pole, 983 N; at the equator, 980 N

Find the speed needed to escape from the solar system starting from the surface of Earth. Assume there are no other bodies involved and do not account for the fact that Earth is moving in its orbit. [ Hint: (Figure) does not apply. Use (Figure) and include the potential energy of both Earth and the Sun.

Consider the previous problem and include the fact that Earth has an orbital speed about the Sun of 29.8 km/s. (a) What speed relative to Earth would be needed and in what direction should you leave Earth? (b) What will be the shape of the trajectory?

a. The escape velocity is still 43.6 km/s. By launching from Earth in the direction of Earth’s tangential velocity, you need [latex] 43.4-29.8=13.8\,\text{km/s} [/latex] relative to Earth. b. The total energy is zero and the trajectory is a parabola.

A comet is observed 1.50 AU from the Sun with a speed of 24.3 km/s. Is this comet in a bound or unbound orbit?

An asteroid has speed 15.5 km/s when it is located 2.00 AU from the sun. At its closest approach, it is 0.400 AU from the Sun. What is its speed at that point?

Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a) Calculate the speed of a satellite in an orbit 900 km above Earth’s surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite’s orbit at an angle of [latex] 90\text{°} [/latex]. What is the velocity of the rivet relative to the satellite just before striking it? (c) If its mass is 0.500 g, and it comes to rest inside the satellite, how much energy in joules is generated by the collision? (Assume the satellite’s velocity does not change appreciably, because its mass is much greater than the rivet’s.)

A satellite of mass 1000 kg is in circular orbit about Earth. The radius of the orbit of the satellite is equal to two times the radius of Earth. (a) How far away is the satellite? (b) Find the kinetic, potential, and total energies of the satellite.

a. [latex] 1.3\,×\,{10}^{7}\,\text{m} [/latex]; b. [latex] 1.56\,×\,{10}^{10}\,\text{J} [/latex]; [latex] \text{−}3.12\,×\,{10}^{10}\,\text{J} [/latex]; [latex] -1.56\,×\,{10}^{10}\,\text{J} [/latex]

After Ceres was promoted to a dwarf planet, we now recognize the largest known asteroid to be Vesta, with a mass of [latex] 2.67\,×\,{10}^{20}\,\text{kg} [/latex] and a diameter ranging from 578 km to 458 km. Assuming that Vesta is spherical with radius 520 km, find the approximate escape velocity from its surface.

(a) Using the data in the previous problem for the asteroid Vesta, what would be the orbital period for a space probe in a circular orbit of 10.0 km from its surface? (b) Why is this calculation marginally useful at best?

a. [latex] 6.24\,×\,{10}^{3}\,\text{s} [/latex] or about 1.7 hours. This was using the 520 km average diameter. b. Vesta is clearly not very spherical, so you would need to be above the largest dimension, nearly 580 km. More importantly, the nonspherical nature would disturb the orbit very quickly, so this calculation would not be very accurate even for one orbit.

What is the orbital velocity of our solar system about the center of the Milky Way? Assume that the mass within a sphere of radius equal to our distance away from the center is about a 100 billion solar masses. Our distance from the center is 27,000 light years.

(a) Using the information in the previous problem, what velocity do you need to escape the Milky Way galaxy from our present position? (b) Would you need to accelerate a spaceship to this speed relative to Earth?

a. 323 km/s; b. No, you need only the difference between the solar system’s orbital speed and escape speed, so about [latex] 323-228=95\,\text{km/s} [/latex].

Circular orbits in (Figure) for conic sections must have eccentricity zero. From this, and using Newton’s second law applied to centripetal acceleration, show that the value of [latex] \alpha [/latex] in (Figure) is given by [latex] \alpha =\frac{{L}^{2}}{\text{G}M{m}^{2}} [/latex] where L is the angular momentum of the orbiting body. The value of [latex] \alpha [/latex] is constant and given by this expression regardless of the type of orbit.

Show that for eccentricity equal to one in (Figure) for conic sections, the path is a parabola. Do this by substituting Cartesian coordinates, x and y , for the polar coordinates, r and [latex] \theta [/latex], and showing that it has the general form for a parabola, [latex] x=a{y}^{2}+by+c [/latex].

Setting [latex] e=1 [/latex], we have [latex] \frac{\alpha }{r}=1+\text{cos}\theta \to \alpha =r+r\text{cos}\theta =r+x [/latex]; hence, [latex] {r}^{2}={x}^{2}+{y}^{2}={(\alpha -x)}^{2} [/latex]. Expand and collect to show [latex] x=\frac{1}{-2\alpha }\,{y}^{2}+\frac{\alpha }{2} [/latex].

Using the technique shown in Satellite Orbits and Energy , show that two masses [latex] {m}_{1} [/latex] and [latex] {m}_{2} [/latex] in circular orbits about their common center of mass, will have total energy [latex] E=K+E={K}_{1}+{K}_{2}-\frac{G{m}_{1}{m}_{2}}{{r}^{}}=-\frac{G{m}_{1}{m}_{2}}{2{r}^{}} [/latex]. We have shown the kinetic energy of both masses explicitly. ( Hint: The masses orbit at radii [latex] {r}_{1} [/latex] and [latex] {r}_{2} [/latex], respectively, where [latex] r={r}_{1}+{r}_{2} [/latex]. Be sure not to confuse the radius needed for centripetal acceleration with that for the gravitational force.)

Given the perihelion distance, p , and aphelion distance, q , for an elliptical orbit, show that the velocity at perihelion, [latex] {v}_{p} [/latex], is given by [latex] {v}_{p}=\sqrt{\frac{2G{M}_{\text{Sun}}}{(q+p)}\,\frac{q}{p}} [/latex]. ( Hint: Use conservation of angular momentum to relate [latex] {v}_{p} [/latex] and [latex] {v}_{q} [/latex], and then substitute into the conservation of energy equation.)

Substitute directly into the energy equation using [latex] p{v}_{p}=q{v}_{q} [/latex] from conservation of angular momentum, and solve for [latex] {v}_{p} [/latex].

Comet P/1999 R1 has a perihelion of 0.0570 AU and aphelion of 4.99 AU. Using the results of the previous problem, find its speed at aphelion. ( Hint: The expression is for the perihelion. Use symmetry to rewrite the expression for aphelion.)

Challenge Problems

A tunnel is dug through the center of a perfectly spherical and airless planet of radius R . Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit at the surface.

[latex] g=\frac{4}{3}\,G\rho \pi r\to F=mg=[\frac{4}{3}\,Gm\rho \pi ]\,r [/latex], and from [latex] F=m\,\frac{{d}^{2}r}{d{t}^{2}} [/latex], we get [latex] \frac{{d}^{2}r}{d{t}^{2}}=[\frac{4}{3}\,G\rho \pi ]\,r [/latex] where the first term is [latex] {\omega }^{2} [/latex]. Then [latex] T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{3}{4G\rho \pi }} [/latex] and if we substitute [latex] \rho =\frac{M}{4\text{/}3\pi {R}^{3}} [/latex], we get the same expression as for the period of orbit R .

Following the technique used in Gravitation Near Earth’s Surface , find the value of g as a function of the radius r from the center of a spherical shell planet of constant density [latex] \rho [/latex] with inner and outer radii [latex] {R}_{\text{in}} [/latex] and [latex] {R}_{\text{out}} [/latex] . Find g for both [latex] {R}_{\text{in}}<r<{R}_{\text{out}} [/latex] and for [latex] r<{R}_{\text{in}} [/latex]. Assuming the inside of the shell is kept airless, describe travel inside the spherical shell planet.

Show that the areal velocity for a circular orbit of radius r about a mass M is [latex] \frac{\text{Δ}A}{\text{Δ}t}=\frac{1}{2}\sqrt{GMr} [/latex]. Does your expression give the correct value for Earth’s areal velocity about the Sun?

Using the mass of the Sun and Earth’s orbital radius, the equation gives [latex] 2.24\,×\,{10}^{15}{\text{m}}^{2}\text{/s} [/latex]. The value of [latex] \pi {R}_{\text{ES}}^{2}\text{/}(1\,\text{year}) [/latex] gives the same value.

Show that the period of orbit for two masses, [latex] {m}_{1} [/latex] and [latex] {m}_{2} [/latex], in circular orbits of radii [latex] {r}_{1} [/latex] and [latex] {r}_{2} [/latex], respectively, about their common center-of-mass, is given by [latex] T=2\pi \sqrt{\frac{{r}^{3}}{G({m}_{1}+{m}_{2})}}\,\text{where}\,r={r}_{1}+{r}_{2} [/latex]. ( Hint: The masses orbit at radii [latex] {r}_{1} [/latex] and [latex] {r}_{2} [/latex], respectively where [latex] r={r}_{1}+{r}_{2} [/latex]. Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expressions for the kinetic energy.)

Show that for small changes in height h , such that [latex] h\text{<}\,\text{<}{\text{R}}_{\text{E}} [/latex], (Figure) reduces to the expression [latex] \text{Δ}U=m\text{g}h [/latex].

[latex] \text{Δ}U={U}_{f}-{U}_{i}=-\frac{G{M}_{\text{E}}m}{{r}_{f}}+\frac{G{M}_{\text{E}}m}{{r}_{i}}=G{M}_{\text{E}}m(\frac{{r}_{f}-{r}_{i}}{{r}_{f}{r}_{i}}) [/latex] where [latex] h={r}_{f}-{r}_{i} [/latex]. If [latex] h\text{<}\,\text{<}{\text{R}}_{\text{E}} [/latex], then [latex] {r}_{f}{r}_{i}\approx {R}_{\text{E}}^{2} [/latex], and upon substitution, we have

[latex] \text{Δ}U=G{M}_{\text{E}}m(\frac{h}{{R}_{\text{E}}^{2}})=m(\frac{G{M}_{\text{E}}}{{R}_{\text{E}}^{2}})h [/latex] where we recognize the expression with the parenthesis as the definition of g .

Using (Figure) , carefully sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass, m . Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal acceleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle [latex] \epsilon [/latex], defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. [latex] \text{tan}(\lambda +\epsilon )=\frac{g}{(g-{\omega }^{2}{R}_{\text{E}})}\text{tan}\lambda [/latex], where [latex] \omega [/latex] is the angular velocity of Earth.

(a) Show that tidal force on a small object of mass m , defined as the difference in the gravitational force that would be exerted on m at a distance at the near and the far side of the object, due to the gravitation at a distance R from M , is given by [latex] {F}_{\text{tidal}}=\frac{2GMm}{{R}^{3}}\text{Δ}r [/latex] where [latex] \text{Δ}r [/latex] is the distance between the near and far side and [latex] \text{Δ}r\text{<}\,\text{<}R [/latex]. (b) Assume you are falling feet first into the black hole at the center of our galaxy. It has mass of 4 million solar masses. What would be the difference between the force at your head and your feet at the Schwarzschild radius (event horizon)? Assume your feet and head each have mass 5.0 kg and are 2.0 m apart. Would you survive passing through the event horizon?

a. Find the difference in force,

[latex] {F}_{\text{tidal}}==\frac{2GMm}{{R}^{3}}\text{Δ}r [/latex];

b. For the case given, using the Schwarzschild radius from a previous problem, we have a tidal force of [latex] 9.5\,×\,{10}^{-3}\,\text{N} [/latex]. This won’t even be noticed!

Find the Hohmann transfer velocities, [latex] \text{Δ}{v}_{\text{EllipseEarth}}^{} [/latex] and [latex] \text{Δ}{v}_{\text{EllipseMars}}^{} [/latex], needed for a trip to Mars. Use (Figure) to find the circular orbital velocities for Earth and Mars. Using (Figure) and the total energy of the ellipse (with semi-major axis a ), given by [latex] E=-\frac{Gm{M}_{\text{s}}}{2{a}^{}} [/latex], find the velocities at Earth (perihelion) and at Mars (aphelion) required to be on the transfer ellipse. The difference, [latex] \text{Δ}v [/latex], at each point is the velocity boost or transfer velocity needed.

• OpenStax University Physics. Authored by : OpenStax CNX. Located at : https://cnx.org/contents/[email protected]:Gofkr9Oy@15 . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Privacy Policy

• The Magazine
• Stay Curious
• The Sciences
• Environment
• Planet Earth

How to Understand Einstein's Theory of Gravity

Einstein's general relativity may be complicated, but it's our best way of understanding the universe..

An astronaut wakes up in a spaceship, with no memory of how she got there. Sitting alone in a chair, she wonders: “Where in the universe am I?”

The ship has no windows. Its instruments are dead. The only clue is the push of the chair against her body. Phew, there’s gravity, she thinks. Her vessel must still be on Earth.

But then a second possibility occurs to her. The ship could be accelerating through space, pressing her into the seat like a race car picking up speed. From inside the vessel, there is — terrifyingly — no way to tell.

This spacefarer’s dilemma would have been familiar to Albert Einstein. His 1915 general theory of relativity built on the notion that gravity and acceleration are not just easily confused, but are one and the same. This equivalence, “the happiest thought” of Einstein’s life, was his starting point for redefining gravity.

General relativity grew out of Einstein’s theory of special relativity, which describes how the speed of light (in a vacuum) can always be constant.

According to relativity, anything that can happen inside of a box picking up speed — i.e., accelerating — also happens in the presence of gravity. Imagine, for example, a horizontal laser inside an elevator that’s accelerating upward. As the light travels sideways, the elevator rises, causing the beam to strike a spot on the wall slightly lower than where it started. If the elevator accelerates quickly enough, the beam visibly bends toward the floor.

Einstein showed the same thing happens to a beam inside a stationary elevator within a powerful gravitational field; the gravity bends the light. Similarly, he expected a beam of starlight should bend when passing through the sun’s gravity. This prediction proved correct when the stars moved during the 1919 solar eclipse.

Relativity describes why a clock on a satellite ticks a few dozen microseconds faster than a clock on Earth; without accounting for that discrepancy, GPS technologies wouldn’t work.

To link acceleration and gravity in this way, Einstein overthrew one of his own heroes: Isaac Newton. You may have learned that Newton described gravity as a force, an invisible rubber band that pulls together objects with mass. Newton’s math did a good job at predicting how everything from projectiles to planets moved — but it kept gravity separate from acceleration.

Einstein argued that gravity isn’t a force at all. He described it as a curvature of time and space caused by mass and energy. Confused? The German physicist was, too, and he struggled with the theory for nearly a decade. He got help from mathematician Marcel Grossmann, an old friend who shared his notes when a young Einstein skipped class.

Their math, laid down in 10 equations, explained how gravity could move around objects via a warped reality, accelerating without ever feeling any mysterious Newtonian forces.

The Relative Basics

The main takeaways behind Einstein’s general theory of relativity:

1. Time and space are neither flat nor fixed; they are curved and distorted by mass and energy.

2. Gravity is not a force, but rather a distortion of time and space.

3. The effects of gravity are indistinguishable from the effects of acceleration, over a small space.

Einstein’s Peculiar Predictions

Relativity makes numerous bizarre predictions, many of them experimentally verified. They only seem bizarre because we don’t notice them in our daily lives — we live, for the most part, in Newton’s reality. But beyond that lies Einstein’s universe, where gravity bends space and time to its will. Here are some of the theory’s strangest side effects:

Gravity literally slows down time. Waves of light emitted by stars stretch out because of this time bending, and objects closer to a massive object age more slowly. Super-precise clocks, which tick according to the vibrations of atoms, have verified that gravity alters time’s flow.

Satellites have shown that rotating celestial bodies twirl the fabric of the cosmos around themselves, like honey twisted by a spoon, affecting the motion of gyroscopes.

One prediction solved a long-standing dilemma, a weird wobble in Mercury’s orbit that Newton’s math couldn’t account for. (Astronomers had initially blamed a hidden planet called Vulcan.) Relativity explained the wonky orbit in terms of the warping of space by the sun’s powerful gravity.

Tiny ripples in reality, caused by colliding black holes, have tripped sensors in highly sensitive instruments buried underground on Earth.

This story originally appeared in print as "It's All Relative."

Already a subscriber?

Register or Log In

Keep reading for as low as $1.99!

Sign up for our weekly science updates.

Save up to 40% off the cover price when you subscribe to Discover magazine.

7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity

Section learning objectives.

By the end of this section, you will be able to do the following:

• Explain Newton’s law of universal gravitation and compare it to Einstein’s theory of general relativity
• Perform calculations using Newton’s law of universal gravitation

Teacher Support

The learning objectives in this section will help your students master the following standards:

• (D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects;
• (A) research and describe the historical development of the concepts of gravitational, electromagnetic, weak nuclear, and strong nuclear forces;
• (B) describe and calculate how the magnitude of the gravitational force between two objects depends on their masses and the distance between their centers.

Section Key Terms

In this section, students will apply Newton’s law of universal gravitation to objects close at hand and far off in the depths of the solar system.

[BL] [OL] Compare the contributions of Kepler, Newton, and Einstein. Place them historically with dates.

[AL] Ask if anyone knows the difference between special relativity and general relativity. Special relativity is a theory of spacetime and applies to observers moving at constant velocity. General relativity is a theory of gravity and applies to observers that are accelerating. General relativity is broader and includes special relativity, which was published first.

Concepts Related to Newton’s Law of Universal Gravitation

Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. See Figure 7.7 . But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner, Galileo Galilei, had contended that falling bodies and planetary motions had the same cause. Some of Newton’s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph. It had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose an explanation of the mechanism that caused them to follow these paths and not others.

[BL] [OL] Ask students if it really is obvious why all things fall straight down. Ask them to back up their reasons. Ask if the name Halley rings a bell.

[OL] [AL] Ask if anyone thinks it is strange or even mysterious that a force can act at a distance across empty space. Ask the students to compare and contrast gravitational force with magnetic and electrostatic forces. Note how much force at a distance is like magic or having superpowers.

The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Expressed in modern language, Newton’s universal law of gravitation states that every object in the universe attracts every other object with a force that is directed along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This attraction is illustrated by Figure 7.8 .

For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law of gravitation is

where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant . G is a universal constant, meaning that it is thought to be the same everywhere in the universe. It has been measured experimentally to be G = 6.673 × 10 − 11 N ⋅ m 2 /kg 2 G = 6.673 × 10 − 11 N ⋅ m 2 /kg 2 .

If a person has a mass of 60.0 kg, what would be the force of gravitational attraction on him at Earth’s surface? G is given above, Earth’s mass M is 5.97 × 10 24 kg, and the radius r of Earth is 6.38 × 10 6 m. Putting these values into Newton’s universal law of gravitation gives

We can check this result with the relationship: F = m g = ( 60 kg ) ( 9.8 m/s 2 ) = 588 N F = m g = ( 60 kg ) ( 9.8 m/s 2 ) = 588 N

You may remember that g , the acceleration due to gravity, is another important constant related to gravity. By substituting g for a in the equation for Newton’s second law of motion we get F = m g F = m g . Combining this with the equation for universal gravitation gives

Cancelling the mass m on both sides of the equation and filling in the values for the gravitational constant and mass and radius of the Earth, gives the value of g, which may look familiar.

This is a good point to recall the difference between mass and weight. Mass is the amount of matter in an object; weight is the force of attraction between the mass within two objects. Weight can change because g is different on every moon and planet. An object’s mass m does not change but its weight m g can.

[BL] [OL] Be sure no one is confusing G with g .

[AL] Ask if anyone can explain why G is a universal constant that applies anywhere in the universe. Have them discuss the idea that the laws of physics are the same everywhere and that, at one time, people were not so sure about this. Emphasize that g is not a universal constant.

Virtual Physics

Gravity and orbits.

Move the sun, Earth, moon and space station in this simulation to see how it affects their gravitational forces and orbital paths. Visualize the sizes and distances between different heavenly bodies. Turn off gravity to see what would happen without it!

• The Moon is not affected by the gravitational field of the Sun.
• The Moon is not affected by the gravitational field of the Earth.
• The Moon is affected by the gravitational fields of both the Earth and the Sun, which are always additive.
• The moon is affected by the gravitational fields of both the Earth and the Sun, which are sometimes additive and sometimes opposite.

This is a good animation of the Earth-Moon-Sun system. Have the students try all of the buttons. This will show the paths of the Earth and the moon separately and together. Explain the gravitational force and velocity vectors. Point out the interesting shape of the moon’s path around the sun. Explain that the velocity vector of the moon changes because sometimes the moon is traveling in the direction of Earth’s orbit and sometimes it is traveling in the opposite direction.

Take-Home Experiment: Falling Objects

In this activity you will study the effects of mass and air resistance on the acceleration of falling objects. Make predictions (hypotheses) about the outcome of this experiment. Write them down to compare later with results.

• Four sheets of 8 - 1 / 2 × 11 8 - 1 / 2 × 11 -inch paper
• Crumple one up into a small ball.
• Leave one uncrumpled.
• Take the other two and crumple them up together, so that they make a ball of exactly twice the mass of the other crumpled ball.
• Compare crumpled one-paper ball with crumpled two-paper ball.
• Compare crumpled one-paper ball with uncrumpled paper.
• Some objects fall faster because of air resistance, which acts in the direction of the motion of the object and exerts more force on objects with less surface area.
• Some objects fall faster because of air resistance, which acts in the direction opposite the motion of the object and exerts more force on objects with less surface area.
• Some objects fall faster because of air resistance, which acts in the direction of motion of the object and exerts more force on objects with more surface area.
• Some objects fall faster because of air resistance, which acts in the direction opposite the motion of the object and exerts more force on objects with more surface area.

Ask for predictions (hypotheses) about the outcome of this experiment. Have the students write them down to compare later with results.

It is possible to derive Kepler’s third law from Newton’s law of universal gravitation. Applying Newton’s second law of motion to angular motion gives an expression for centripetal force, which can be equated to the expression for force in the universal gravitation equation. This expression can be manipulated to produce the equation for Kepler’s third law. We saw earlier that the expression r 3 /T 2 is a constant for satellites orbiting the same massive object. The derivation of Kepler’s third law from Newton’s law of universal gravitation and Newton’s second law of motion yields that constant:

where M is the mass of the central body about which the satellites orbit (for example, the sun in our solar system). The usefulness of this equation will be seen later.

[OL] This equation illustrates the difference between Kepler’s and Newton’s work. Ask the students to explain why this is so.

[AL] Ask the students what the attraction would be between two 10 kg balls separated by a distance of 1.0 m. Could they feel it? Later, ask them to calculate it after they have done some similar calculations. Solution:

The universal gravitational constant G is determined experimentally. This determination was first done accurately in 1798 by English scientist Henry Cavendish (1731–1810), more than 100 years after Newton published his universal law of gravitation. The measurement of G is very basic and important because it determines the strength of one of the four forces in nature. Cavendish’s experiment was very difficult because he measured the tiny gravitational attraction between two ordinary-sized masses (tens of kilograms at most) by using an apparatus like that in Figure 7.9 . Remarkably, his value for G differs by less than 1% from the modern value.

Einstein’s Theory of General Relativity

Einstein’s theory of general relativity explained some interesting properties of gravity not covered by Newton’s theory. Einstein based his theory on the postulate that acceleration and gravity have the same effect and cannot be distinguished from each other. He concluded that light must fall in both a gravitational field and in an accelerating reference frame. Figure 7.10 shows this effect (greatly exaggerated) in an accelerating elevator. In Figure 7.10 (a) , the elevator accelerates upward in zero gravity. In Figure 7.10 (b) , the room is not accelerating but is subject to gravity. The effect on light is the same: it “falls” downward in both situations. The person in the elevator cannot tell whether the elevator is accelerating in zero gravity or is stationary and subject to gravity. Thus, gravity affects the path of light, even though we think of gravity as acting between masses, while photons are massless.

[BL] [OL] Ask the students to discuss the postulate. Can they relate the identity of gravity and acceleration to experience?

Einstein’s theory of general relativity got its first verification in 1919 when starlight passing near the sun was observed during a solar eclipse. (See Figure 7.11 .) During an eclipse, the sky is darkened and we can briefly see stars. Those on a line of sight nearest the sun should have a shift in their apparent positions. Not only was this shift observed, but it agreed with Einstein’s predictions well within experimental uncertainties. This discovery created a scientific and public sensation. Einstein was now a folk hero as well as a very great scientist. The bending of light by matter is equivalent to a bending of space itself, with light following the curve. This is another radical change in our concept of space and time. It is also another connection that any particle with mass or energy (e.g., massless photons) is affected by gravity.

To summarize the two views of gravity, Newton envisioned gravity as a tug of war along the line connecting any two objects in the universe. In contrast, Einstein envisioned gravity as a bending of space-time by mass.

Boundless Physics

Nasa gravity probe b.

NASA’s Gravity Probe B (GP-B) mission has confirmed two key predictions derived from Albert Einstein’s general theory of relativity. The probe, shown in Figure 7.12 was launched in 2004. It carried four ultra-precise gyroscopes designed to measure two effects hypothesized by Einstein’s theory:

• The geodetic effect , which is the warping of space and time by the gravitational field of a massive body (in this case, Earth)
• The frame-dragging effect , which is the amount by which a spinning object pulls space and time with it as it rotates

Both effects were measured with unprecedented precision. This was done by pointing the gyroscopes at a single star while orbiting Earth in a polar orbit. As predicted by relativity theory, the gyroscopes experienced very small, but measureable, changes in the direction of their spin caused by the pull of Earth’s gravity.

The principle investigator suggested imagining Earth spinning in honey. As Earth rotates it drags space and time with it as it would a surrounding sea of honey.

• Gravity has no effect on the space-time continuum, and gravity only affects the motion of light.
• The space-time continuum is distorted by gravity, and gravity has no effect on the motion of light.
• Gravity has no effect on either the space-time continuum or on the motion of light.
• The space-time continuum is distorted by gravity, and gravity affects the motion of light.

Explain that it is very exciting when a prediction of relativity theory is tested successfully. Some of the predictions were in doubt because they sounded so bizarre.

Calculations Based on Newton’s Law of Universal Gravitation

Tips for success.

When performing calculations using the equations in this chapter, use units of kilograms for mass, meters for distances, newtons for force, and seconds for time.

The mass of an object is constant, but its weight varies with the strength of the gravitational field. This means the value of g varies from place to place in the universe. The relationship between force, mass, and acceleration from the second law of motion can be written in terms of g .

In this case, the force is the weight of the object, which is caused by the gravitational attraction of the planet or moon on which the object is located. We can use this expression to compare weights of an object on different moons and planets.

[BL] Check to make sure students are clear about the distinction between mass and weight.

[OL] Recall the antics of astronauts of on the moon performed to illustrate the effect of a different value for g .

Watch Physics

Mass and weight clarification.

This video shows the mathematical basis of the relationship between mass and weight. The distinction between mass and weight are clearly explained. The mathematical relationship between mass and weight are shown mathematically in terms of the equation for Newton’s law of universal gravitation and in terms of his second law of motion.

Grasp Check

Would you have the same mass on the moon as you do on Earth? Would you have the same weight?

• You would weigh more on the moon than on Earth because gravity on the moon is stronger than gravity on Earth.
• You would weigh less on the moon than on Earth because gravity on the moon is weaker than gravity on Earth.
• You would weigh less on the moon than on Earth because gravity on the moon is stronger than gravity on Earth.
• You would weigh more on the moon than on Earth because gravity on the moon is weaker than gravity on Earth.

This may be a rather long-winded explanation of the mass-weight distinction, but it should drive home the point.

Two equations involving the gravitational constant, G , are often useful. The first is Newton’s equation, F = G m M r 2 F = G m M r 2 . Several of the values in this equation are either constants or easily obtainable. F is often the weight of an object on the surface of a large object with mass M , which is usually known. The mass of the smaller object, m , is often known, and G is a universal constant with the same value anywhere in the universe. This equation can be used to solve problems involving an object on or orbiting Earth or other massive celestial object. Sometimes it is helpful to equate the right-hand side of the equation to m g and cancel the m on both sides.

The equation r 3 T 2 = G M 4 π 2 r 3 T 2 = G M 4 π 2 is also useful for problems involving objects in orbit. Note that there is no need to know the mass of the object. Often, we know the radius r or the period T and want to find the other. If these are both known, we can use the equation to calculate the mass of a planet or star.

This video demonstrates calculations involving Newton’s universal law of gravitation.

• g and G are both unchanging constants but have different units.
• G is a universal constant that relates force with a pair of masses at a distance, while g relates force with mass and varies with location.
• g describes acceleration while G describes gravitational force.
• g describes gravitational force while G describes acceleration.

This video is a thorough demonstration of many of the calculations to be learned in this subsection.

Worked Example

Change in g.

The value of g on the planet Mars is 3.71 m/s 2 . If you have a mass of 60.0 kg on Earth, what would be your mass on Mars? What would be your weight on Mars?

Weight equals acceleration due to gravity times mass: W = m g W = m g . An object’s mass is constant. Call acceleration due to gravity on Mars g M and weight on Mars W M .

Mass on Mars would be the same, 60 kg.

The value of g on any planet depends on the mass of the planet and the distance from its center. If the material below the surface varies from point to point, the value of g will also vary slightly.

This is a typical mass-weight calculation.

Earth’s g at the Moon

Find the acceleration due to Earth’s gravity at the distance of the moon.

Express the force of gravity in terms of g .

Combine with the equation for universal gravitation.

Cancel m and substitute.

The value of g for the moon is 1.62 m/s 2 . Comparing this value to the answer, we see that Earth’s gravitational influence on an object on the moon’s surface would be insignificant.

[BL] [OL] Review the meanings of all the symbols in these equations: F , G , m , M , r , T , and π π .

[OL] [AL] Have the students memorize the values of G , g , and π to three significant figures.

Practice Problems

• 5.94 × 10 17 kg
• 5.94 × 10 24 kg
• 9.36 × 10 17 kg
• 9.36 × 10 24 kg

Check Your Understanding

• He gave an exact mathematical form for the theory.
• He added a correction term to a previously existing formula.
• Newton found the value of the universal gravitational constant.
• Newton showed that gravitational force is always attractive.
• Gravitational force between two objects is directly proportional to the sum of the squares of their masses and inversely proportional to the square of the distance between them.
• Gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Gravitational force between two objects is directly proportional to the sum of the squares of their masses and inversely proportional to the distance between them.
• Gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the distance between them.

Newton’s law of universal gravitation explains the paths of what?

• A charged particle
• A ball rolling on a plane surface
• A planet moving around the sun
• A stone tied to a string and whirled at constant speed in a horizontal circle

Use the Check Your Answers questions to assess whether students master the learning objectives for this section. If students are struggling with a specific objective, the Check Your Answers will help identify which objective is causing the problem and direct students to the relevant content.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute Texas Education Agency (TEA). The original material is available at: https://www.texasgateway.org/book/tea-physics . Changes were made to the original material, including updates to art, structure, and other content updates.

Access for free at https://openstax.org/books/physics/pages/1-introduction

• Authors: Paul Peter Urone, Roger Hinrichs
• Publisher/website: OpenStax
• Book title: Physics
• Publication date: Mar 26, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/physics/pages/1-introduction
• Section URL: https://openstax.org/books/physics/pages/7-2-newtons-law-of-universal-gravitation-and-einsteins-theory-of-general-relativity

© Jan 19, 2024 Texas Education Agency (TEA). The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

9.2: Newton’s Universal Theory of Gravity

• Last updated
• Save as PDF
• Page ID 89683

Newton supposedly gained insight into the gravitational force by observing an apple falling from a tree and concluding that if it is the same force that makes apples fall at sea level and at the top of a mountain, perhaps that force can be exerted all the way up to the moon. It is rather remarkable that Newton was able to make the connection between falling apples and the motion of the moon around the Earth to find a single theory to describe both situations.

We should be clear that the theory of gravity is a different theory than Newton’s “Laws of Motion” (Newton’s Three Laws). The Laws of Motion introduce the concepts of force and inertial mass, and prescribe how to use those concepts in order to model motion using kinematics. Newton’s Universal Theory of Gravity is a theory that describes the force of gravity that two bodies with (gravitational) mass exert on each other.

Newton’s Universal Theory of Gravity states that if two bodies, with masses \(M_1\) and \(M_2\) , located at positions \(\vec r_1\) and \(\vec r_2\) , respectively, are separated by a distance, \(r\) , then \(M_2\) will exert an attractive force on \(M_1\) , \(\vec F_{12}\) , given by:

\[\begin{aligned} \vec F_{12}=-G\frac{M_1M_2}{r^2}\hat r_{21}\end{aligned}\]

where \(\hat r_{21}\) is the unit vector from \(M_2\) to \(M_1\) :

\[\begin{aligned} \vec r_{21} &= \vec r_2 - \vec r_1\\[4pt] \hat r_{21} &= \frac{1}{r} \vec r_{21}\end{aligned}\]

as shown in Figure \(\PageIndex{1}\) . \(\vec F_{12}\) should be read as “the force on body 1 from body 2”. \(G=6.67\times 10^{-11}\text{Nm}^{2}/\text{kg}^{2}\) is Newton’s Universal Constant of Gravity. It should be noted that Newton’s theory for the force of gravity written in this form only applies to either point masses (separated by a distance \(r\) ) or spherical bodies whose centers are separated by a distance \(r\) that is larger than the radius of either sphere.

Originally, Newton argued that the force of gravity would be proportional to the masses of the bodies, and inversely proportional to the square of the distance between them:

\[\begin{aligned} F_{12}\propto \frac{M_1M_2}{r^2}\end{aligned}\]

and \(G\) is simply the constant of proportionality.

Because of Newton’s Third Law, body 1 exerts a force on body 2 that is equal in magnitude but opposite in direction:

\[\begin{aligned} \vec F_{12} = -\vec F_{21}\end{aligned}\]

Example \(\PageIndex{1}\)

Calculate the magnitude of the force of gravity between yourself and a person standing \(50\text{cm}\) from you and compare that to your weight at the surface of the Earth (the force of gravity exerted by the Earth on you).

If we assume that the two people have a mass of \(50\text{kg}\) , the force of gravity exerted by one on the other, if they are separated by \(50\text{cm}\) , is given by:

\[\begin{aligned} F=G\frac{M_1M_2}{r^2}=(6.67\times 10^{-11}\text{Nm}^{2}/\text{kg}^{2})\frac{(50\text{kg})(50\text{kg})}{(0.5\text{m})^2}=6.67\times 10^{-7}\text{N}\end{aligned}\]

This is a very small force, compared to their weight, \(F_g\) :

\[\begin{aligned} F_g=M_1g=(50\text{kg})(9.8\text{N/kg})=490\text{N}\end{aligned}\]

which is approximately 700 million times bigger.

The force of gravity is a very weak force when compared to other forces in Nature, such as the electric force between two charges. Newton’s Universal Constant of Gravity is very small, so the force of gravity between two objects is very small unless either of the masses involved are very large, or the distance between them is very small. In general, when modeling the motion of objects on the Earth, it is generally safe to ignore the forces of gravity between objects and only include their weight (the force of gravity from the Earth).

Exercise \(\PageIndex{4}\)

The radius of the Earth is \(6371\) km. What is the order of magnitude of the Earth’s mass?

• \(10^{24}\text{kg}\)
• \(10^{18}\text{kg}\)
• \(10^{19}\text{kg}\)
• \(10^{21}\text{kg}\)

Not enough information.

Example \(\PageIndex{2}\)

Determine the constant in Kepler’s Third Law for planets orbiting the Sun, namely the value of the ratio:

\[\begin{aligned} \frac{s^3}{T^2}\end{aligned}\]

where \(s\) is the semi-major axis and \(T\) is the orbital period.

Since Kepler’s Third Law holds for any body orbiting the Sun, we can easily determine the ratio by considering a planet of mass \(m\) in a circular orbit of radius \(R\) centerd about the Sun. The semi-major axis of the orbit is equal to the radius of the orbit for a circular orbit ( \(s=R\) ).

If the planet is in a circular orbit about the Sun, its speed, \(v\) , will be constant, by Kepler’s Second Law, and it will thus be executing uniform circular motion. The only force exerted on the planet is the force of gravity exerted by the Sun. Thus the force of gravity must be equal to the mass of the planet times its radial (centripetal) acceleration, \(a_R\) , which is given by:

\[\begin{aligned} a_R=\frac{v^2}{R}\end{aligned}\]

Newton’s Second Law for the planet can be written as:

\[\begin{aligned} \sum F = F_g &= ma_R\\[4pt] G\frac{Mm}{R^2}&=m\frac{v^2}{R}\\[4pt] G\frac{M}{R}&=v^2\end{aligned}\]

where \(M\) is the mass of the Sun. The speed of the planet is given by the circumference of the orbit divided by the orbital period \(T\) , since it is constant:

\[\begin{aligned} v=\frac{2\pi R}{T}\end{aligned}\]

Re-arranging the equation from Newton’s Second Law:

\[\begin{aligned} G\frac{M}{R}&=v^2\\[4pt] G\frac{M}{R}&=\frac{4\pi^2 R^2}{T^2}\\[4pt] \therefore \frac{R^3}{T^2}&=G\frac{M}{4\pi^2}\end{aligned}\]

Thus, we find that the ratio of the cube of the orbital radius to the period squared is a constant that depends only on the mass of the Sun, which will then be the same for all planets (as it does not depend on, say, the mass of the planet that we considered).

The relation above can, for example, be used to determine the mass of the Sun, since we can use geometry to determine the semi-major axis for the orbit of a planet, as Kepler did with the data from Tycho Brahe.

Example \(\PageIndex{3}\)

The acceleration due to Earth’s gravity depends on the force that the Earth exerts on an object. Using the Earth’s mass and radius, determine the acceleration of falling objects due to Earth’s gravity at the surface of the Earth. Also, determine the altitude where the acceleration due to Earth’s gravity is half of that at the Earth’s surface.

We can find the acceleration due to Earth’s gravity by determining the acceleration of a mass \(m\) upon which gravity is the only acting force. In other words, we model an object that is in free-fall a distance \(R\) away from the center of the Earth. Newton’s Second Law can be used in one dimension (corresponding to the direction that connects the falling mass to the center of the Earth):

\[\begin{aligned} \sum F &= G\frac{Mm}{R^2}=ma\\[4pt] \therefore a&=G\frac{M}{R^2}\end{aligned}\]

where \(M=5.97\times 10^{24}\text{kg}\) is the mass of the Earth. At the surface of the Earth, \(R=R_\oplus=6.371\times 10^{6}\text{m}\) :

\[\begin{aligned} a&=G\frac{M}{R_\oplus^2}=(6.67\times 10^{-11}\text{Nm}^{2}/\text{kg}^{2})\frac{(5.97\times 10^{24}\text{kg})}{(6.371\times 10^{6}\text{m})^2}\\[4pt] &=9.81\text{m/s}^{2}\end{aligned}\]

which, of course, is the value of \(g\) that we have been using so far for the acceleration due to gravity near the surface of the Earth. To find the altitude at which this is reduced by half, we first find the value of \(R\) that corresponds to this acceleration:

\[\begin{aligned} \frac{1}{2}G\frac{M}{R_\oplus^2}&=G\frac{M}{R^2}\\[4pt] \therefore R &=\sqrt{2}R_\oplus = 9.0\times 10^{6}\text{m}\end{aligned}\]

which corresponds to an altitude of \(h=R-R_\oplus=2640\text{km}\) , well above the Earth’s atmosphere.

The acceleration of falling objects decreases as one moves further from the center of the Earth. It is thus an approximation to assume that \(g\) is a constant, although in most cases this is a very good approximation. In practice, the value of \(g\) will depend both on the distance from the center of the Earth and the composition (density) of the material in the Earth’s crust below where \(g\) is being measured. Precise measurements of \(g\) have bee used to determine the composition of the Earth’s crust and to search for mineral and oil deposits.

Weight and apparent weight

You have probably seen images of astronauts floating around the International Space Station (ISS) or other orbiting vessels, and heard of the term “weightlessness” to describe their motion. The ISS is in orbit at an altitude of approximately \(400\text{km}\) , where the force of Earth’s gravity is far from negligible (in Example 9.2.3 we showed that one needs to go to an altitude of \(2640\text{km}\) for the force to be reduced by half of that at the surface of the Earth). The contradiction between being weightless and the fact that weight is not zero is resolved by understanding that the popular term “weightless” is imprecise from a physics perspective.

The correct term to use from a physics perspective is to say that the apparent weight of the astronauts is zero when they are floating around. Weight is the magnitude of the force of gravity exerted by the Earth. Apparent weight is, for example, the force that one measures when standing on a spring scale, which is equal to the normal force exerted by the spring scale on the person. Apparent weight could also be determined by the tension in a string from which a person is suspended. The apparent weight is the sum of the forces exerted on a person minus the force of gravity. If gravity is the only force exerted on a person (or object), that person’s apparent weight is zero, which is what is popularly called being weightless.

One way to feel weightless is when you are in free-fall (e.g. the first few seconds of a parachute jump from an airplane). One can think of being in orbit as continuously falling towards the center of the Earth, but with an initial velocity in a direction such that you never actually collide with the Earth. The feeling of weightlessness will occur any time that the only force exerted on you is the force of gravity. If you are in a spacecraft in any orbit and the only force on the spacecraft is from gravity (i.e. no rockets or wings are exerting any forces), then everything in the spacecraft will have the same acceleration, since gravity is the only force acting on anything in the spacecraft, and it will appear that everything is just floating. To an outside observer, it would be clear that the spacecraft and its contents are all accelerating.

Effects of Earth’s rotation

Earth’s rotation affects the apparent weight of objects near the Earth’s surface. Consider a person standing on a spring scale at the North pole (top free-body diagram in Figure \(\PageIndex{2}\) ). The only two forces exerted on the person are their weight, \(\vec F_g\) , and the normal force, \(\vec N\) , exerted by the spring scale. Since the person is not accelerating, the normal force and the weight have the same magnitude and opposite directions. The scale will thus read the actual weight of the person 1 .

Consider, instead, a person standing on a spring scale at the equator (Figure \(\PageIndex{2}\)). That person is accelerating because they are undergoing uniform circular motion as they rotate along with the Earth. Again, the only forces acting on the person are their weight and the normal force exerted by the scale. The sum of the forces must now be equal to the person’s mass, \(m\) , times the radial acceleration, \(a_r\) , that is necessary for them to follow the surface of the Earth as the Earth rotates about its axis. Newton’s Second Law allows us to find the magnitude of the normal force acting on the person:

\[\begin{aligned} \sum F &= F_g-N=ma_r=m\frac{v^2}{R}\\[4pt] \therefore N &= F_g - m\frac{v^2}{R}\\[4pt] &=G\frac{Mm}{R^2} - m\frac{v^2}{R}\\[4pt] &=m\left(G\frac{M}{R^2} - \frac{v^2}{R} \right)\\[4pt] &=m\left(g - \frac{v^2}{R} \right)\end{aligned}\]

where \(M\) is the mass of the Earth, \(R\) is the radius of the Earth, and \(v\) is the speed at the surface of the Earth due to the Earth’s rotation. In the last line, we used the result from Example 9.2.3 where we determined the value of \(g\) in terms of the mass and radius of the Earth.

We see that the normal force is reduced compared to what it would be if the Earth were not rotating ( \(v=0\) ) or if one is standing at one of the poles. Your apparent weight, which you can measure by standing on a spring scale, is thus smaller at the equator than it is at the poles. The quantity in parenthesis can be thought of as a modified or “effective” value of \(g\) at the equator.

Exercise \(\PageIndex{5}\)

What is the effective value of \(g\) at the equator?

• \(9.80 \text{m/s}^2\)
• \(9.78 \text{m/s}^2\)
• \(9.70 \text{m/s}^2\)
• \(9.51 \text{m/s}^2\)

If you are circling the Earth a distance \(R\) from the center of the Earth at a constant speed \(v\) , it is possible for your apparent weight to be zero. Imagine standing on a scale in an aircraft that is circling the Earth and measuring your apparent weight with the spring scale. As the speed of the aircraft increases, your apparent weight, \(N\) , decreases according to the formula that we just found:

\[\begin{aligned} N=m\left(G\frac{M}{R^2} - \frac{v^2}{R} \right)\end{aligned}\]

At a certain speed, \(v\) , your apparent weight is zero and you feel weightless:

\[\begin{aligned} G\frac{M}{R^2} &= \frac{v^2}{R}\\[4pt] \therefore v&= \sqrt{G\frac{M}{R} }\end{aligned}\]

This speed corresponds to a centripetal acceleration that is exactly equal to that due to gravity. This makes sense, since gravity is the only force that is acting on you (the normal force is now zero), which is exactly what we call being in orbit.

Earth’s rotation has some interesting consequences for stationary objects. At any position on Earth that is not at the equator or the poles, the sum of the forces on any stationary object (meaning stationary relative to the Earth’s surface) cannot be zero. This is because the object must rotate along with the Earth, so the net force on the object must point toward the center of the circle about which that location on Earth is rotating.

Take, for example, a plumb line, which consists of a mass hanging from a string. The two forces acting on the mass are gravity and tension. The force of gravity must point towards the center of the Earth. We would expect that the force of tension, and therefore the string, would point directly away from the center of the Earth. However, we find that if the plumb line is located at some angle \(\theta\) from the equator (but not at the equator or poles), as in Figure \(\PageIndex{3}\), then the string will point slightly away from the center of the Earth. In order for the mass to remain stationary relative to the ground, it must rotate along with the Earth (radius \(R\) ) around a circle of radius \(R\cos\theta\) . Thus, the tension from the string cannot point away from the center of the Earth, because the net force must point towards the center of the circle of radius \(R\cos\theta\) .

Exercise \(\PageIndex{6}\)

You cut the string of the plumb line. Where does the mass land relative to its initial latitude (the angle \(\theta\) in Figure \(\PageIndex{3}\))?

• At the same latitude.
• Closer to the nearest pole.
• Closer to the equator.

The gravitational field

The gravitational force exerted on a mass \(m\) by a mass \(M\) can be written as:

\[\begin{aligned} \vec F(\vec r) = -G\frac{Mm}{r^2}\hat r\end{aligned}\]

if we define a coordinate system with the origin located at the center of mass \(M\) so that \(\vec r\) is the position of \(m\) relative to \(M\) . We can define the “gravitational field”, \(\vec g(\vec r)\) , at position, \(\vec r\) , due to the presence of mass \(M\) as the gravitational force per unit mass exerted by \(M\) :

\[\vec g(\vec r)=\frac{\vec F(\vec r)}{m}=-G\frac{Mm}{r^{2}}\hat r\]

The word “field” is just a mathematical term for a function that depends on position. Since \(\vec g(\vec r)\) is a vector, we would refer to it as a “vector field”.

Defining the gravitational field makes it easy to calculate the force of gravity from \(M\) on any mass \(m\) :

\[\begin{aligned} \vec F_g = m\vec g(\vec r)\end{aligned}\]

At the surface of the Earth, the magnitude of the gravitational field is given by:

\[\begin{aligned} g(R_\oplus)=\frac{GM}{R_\oplus^2}=9.81\text{N/kg}\end{aligned}\]

where \(R_\oplus\) is the radius of the Earth. Of course, this also corresponds to the acceleration of objects in free-fall near the surface of the Earth, which we can find from Newton’s Second Law:

\[\begin{aligned} \sum \vec F &= \vec F_g = m\vec a\\[4pt] m\vec g(R_\oplus)&= m\vec a\\[4pt] \therefore \vec a &= \vec g(R_\oplus)\end{aligned}\]

but we see here why it more precise to refer to \(g\) as the “magnitude of the gravitational field at the surface of the Earth” rather than “the acceleration due to Earth’s gravity”. It is also worth noting that the two are only equal if the gravitational mass (on the left of the equation in the second line) is the same as the inertial mass (on the right of the equation). The gravitational mass is the mass that appears in the gravitational force defined by Newton, whereas the inertial mass is the mass that appears with the acceleration in Newton’s Second Law.

Exercise \(\PageIndex{7}\)

Two small objects with different masses, \(m_1\) and \(m_2\) , are located a distance \(r\) from a nearby star. What can you say about the magnitude of the gravitational field and the magnitude of the gravitational force on \(m_1\) and \(m_2\) ?

• The field is different and the forces are different.
• The field is different but the forces are the same.
• The field is the same but the forces are different.
• The field is the same and the forces are the same.

Suppose that there are two large mass bodies, \(M_1\) and \(M_2\) , and a smaller mass body, \(m\) . We can calculate the net gravitational force on \(m\) by summing the gravitational force vectors from \(M_1\) and \(M_2\) separately. If the gravitational fields from \(M_1\) and \(M_2\) are given by \(\vec g_1(\vec r)\) and \(\vec g_2(\vec r)\) , respectively, then the total gravitational force on \(m\) is given by:

\[\begin{aligned} \vec F &= m\vec g_1(\vec r) + m\vec g_2(\vec r)=m(\vec g_1(\vec r)+\vec g_2(\vec r))\\[4pt] &=m \vec g(\vec r)\end{aligned}\]

where we have introduced the total gravitational field:

\[\begin{aligned} \vec g(\vec r) = \vec g_1(\vec r)+\vec g_2(\vec r)\end{aligned}\]

In other words, if there are multiple bodies that result in a non-negligible force of gravity, we can calculate their gravitational fields independently and sum them together to define a net gravitational field, \(\vec g(\vec r)\) , that models the net force of gravity from all of the bodies. The net gravitational force on a new body of mass \(m'\) is then simply given by \(m'\vec g(\vec r)\) , and we do not need to add any more vectors together. For example, when calculating the motion of satellites that can be influenced by the force of gravity from both the Earth and the Moon, we simply need to calculate the net gravitational field from the Earth and Moon, and the motion of any satellite can then be modeled using that net gravitational field.

Exercise \(\PageIndex{8}\)

T here are two planets with equal mass located a distance \(d\) apart. Position \(A\) is located midway between the two planets. Position \(B\) is located a distance \(d/2\) from one of the planets, in the position shown in Figure \(\PageIndex{4}\). How does the field at \(A\) compare to the field at \(B\) ?

• The magnitude of the field is the same at \(A\) and \(B\) .
• The magnitude of the field is greater at \(A\) than at \(B\) .
• The magnitude of the field is greater at \(B\) than at \(A\) .

Gauss’ Law

Newton’s Universal Theory of Gravity postulates that the force of gravity between two bodies decreases as the squared of the distance between those two bodies. Using the terminology of a field, we would say that the strength of the gravitational field from an object decreases as the inverse of the square of the distance to that object. This is an example of a what we generally call an “inverse-square law”. The electric force between two charges is also given by an inverse-square law, and we now understand that these forces behave as if they were “transmitted” by waves or particles.

If a force is given by an inverse-square law, then Gauss’ Law gives a way to determine the strength of the field that is associated with that force. In the case of gravity, Gauss’ Law states that:

\[\begin{aligned} \oint \vec g(\vec r) \cdot d\vec A = 4\pi G M^{enc}\end{aligned}\]

where the integral on the left is an integral over a “closed surface” that completely surrounds the mass for which we want to determine the gravitational field. To evaluate the integral, imagine taking a closed surface, \(S\) , that surrounds the mass. The vector \(d\vec A\) is defined as the vector that goes with a small element of that surface and points outwards from the closed surface. The magnitude of the vector is equal to the area of that small surface ( \(dA\) ), as illustrated in Figure \(\PageIndex{5}\). You can then take the scalar product of \(d\vec A\) with the gravitational field, \(\vec g(\vec r)\) , at that point on the surface. If you sum all of those scalar products, you get the value of the integral on the left. Gauss’ Law states that the value of that integral is equal \(4 \pi G\) times the total mass that is enclosed by the surface.

Olivia's Thoughts

If you want to know if a surface is closed, ask yourself if you could put water inside the surface and not be worried about it spilling out. For example, if you put water in a sphere or a cube , the water would not spill out even if you shook it around, so they are closed surfaces. A flat square is an open surface because there is no “inside” to even put the water in. A bowl is an open surface because, though you can put water in it, the water could spill out.

We will go into more detail about Gauss’ Law when we cover electromagnetism, but it is worth seeing how to use it in a simple scenario. Figure \(\PageIndex{5}\) shows a spherical body of mass \(M\) and radius \(R\) for which we would like to determine the value of the gravitational field at a distance \(r\) from the center of the body.

To do so, we draw a “Gaussian surface”, \(S\) , that is a sphere with a radius \(r\) , and centered about the body. At any point on the surface, the area element vector \(d\vec A\) points away from the center of the spherical surface. The gravitational field vector, \(\vec g(\vec r)\) will always point towards the center of the spherical surface, as illustrated. Furthermore, by symmetry, the magnitude of \(\vec g(\vec r)\) is constant along the whole Gaussian surface. Thus, the scalar product \(\vec g(\vec r) \cdot d\vec A=-g(r)dA\) everywhere along the surface (it is negative because the two vectors are anti-parallel). The integral is thus given by:

\[\begin{aligned} \oint \vec g(\vec r) \cdot d\vec A = -g(r)\oint dA \end{aligned}\]

where we factored \(g(r)\) out of the integral, since the magnitude of \(\vec g(\vec r)\) is constant for all of the area elements \(dA\) on the sphere. Remember that an integral is a sum. The integral \(\oint dA\) thus means “sum all of the area elements \(dA\) over the entire surface \(S\) ”. Thus, the integral is the total area of the spherical surface \(S\) 2 :

\[\begin{aligned} \oint \vec g(\vec r) \

d\vec A = -g(r)\oint dA =-g(r)(4\pi r^2)\end{aligned}\]

Inserting this into Gauss’ Law, we find:

\[\begin{aligned} \oint \vec g(\vec r) \cdot d\vec A &= 4\pi G M^{enc}\\[4pt] -g(r)(4\pi r^2) &= 4\pi G M^{enc}\\[4pt] \therefore g(r) &= - \frac{GM}{r^2}\end{aligned}\]

where \(M^{enc}=M\) is the total mass enclosed by the Gaussian surface (in this case, the entire mass \(M\) is enclosed). This is of course the result that we expected and obtained earlier from Newton’s formulation. Note that Gauss’ Law is only easy to use if the system is highly symmetric (e.g. spherically symmetric), and that it does not give the direction of the field vector, which must be obtained from symmetry arguments.

Here’s an analogy to describe Gauss’s Law for gravity: A famous celebrity is doing an event, and they attract a certain number of fans who want to get as close to the celebrity as possible. You put up a barricade around the celebrity. The gravitational field is represented by how crowded it is somewhere along the barricade. If a second celebrity is at the event, they will attract their own fans, so there will be more people around the barricade. The number of celebrities is kind of like the enclosed mass \(M^{enc}\) .

A photographer is coming to the event, and you told him to stand at some location that is a distance \(r\) from the celebrities. The photographer wants to know how crowded it will be when he is standing behind the barricade at that location. Gauss’s law gives us a way to figure this out. If you know which celebrities are at the event ( \(M^{enc}\) ), you can determine how many people will be there (this is like finding \(4\pi GM^{enc}\) ). Then, if you can build a barricade such that the fans are evenly distributed around it, and you know how long that barricade is ( \(\oint dA\) ), you can easily calculate how crowded it will be at some point along the barricade (you can just divide the number of people by the length of the barricade). The barricade represents our Gaussian surface and, like a Gaussian surface, it can be whatever shape we want as long as it encloses the celebrities and passes through the point we are interested in. If we want to make sure the people are spread out evenly, the shape of the barricade is going to depend on the specific case. Let’s take the example of our single spherical body. This is analagous to having one celebrity at the event.

Figure \(\PageIndex{6}\) shows two possible barricades we could build. Although we can technically build the barricade on the left, it doesn’t help us because the areas closer to the celebrity will be more crowded. Instead, we want to build the barricade on the right, which is a circle of radius \(r\) , because the fans are evenly spread out. This is why we use a spherical Gaussian surface when we’re considering the field due to a spherical body - at any point a distance \(r\) from the body, the field will be the same. (Note: Remember that, unlike the barricade, the Gaussian surface isn’t a physical thing, so it won’t affect the gravitational field. It is just a mathematical tool that allows us to take advantage of what the field already looks like.)

We can also use Gauss’ Law to determine the gravitational field inside of the body of mass \(M\) and radius \(R\) . This is illustrated in Figure \(\PageIndex{7}\), which shows a spherical Gaussian surface of radius \(r\) that is inside of the body of mass \(M\) .

The gravitational field inside of the body of mass \(M\) is also symmetric and constant in magnitude across the whole surface, so that the integral is the same as before:

\[\begin{aligned} \oint \vec g(\vec r) \cdot d\vec A=-g(r)(4\pi r^2)\end{aligned}\]

However, in order to use Gauss’ Law, we need to determine the mass of the body that is enclosed within the spherical surface, which will be less than \(M\) . If we assume that the mass density, \(\rho\) , of the object is constant (the body is made of a uniform material), then the density is simply the mass of the object over its volume:

\[\begin{aligned} \rho = \frac{M}{\frac{4}{3}\pi R^3}\end{aligned}\]

The amount of mass enclosed by the spherical surface of radius \(r\) is the density multiplied by the volume of a sphere of radius \(r\) :

\[\begin{aligned} M^{enc} = \rho \frac{4}{3}\pi r^3 = M\frac{r^3}{R^3}\end{aligned}\]

Applying Gauss’ Law, we can now find the magnitude of the gravitational field inside of the spherical body at a distance \(r\) from the center:

\[\begin{aligned} \oint \vec g(\vec r) \cdot d\vec A &= 4\pi G M^{enc}\\[4pt] -g(r)(4\pi r^2) &= 4\pi G M\frac{r^3}{R^3}\\[4pt] \therefore g(r) &= - \frac{G M}{R^3}r\end{aligned}\]

And we find that, inside a uniform spherical body of mass \(M\) , the gravitational field increases linearly with radius as one moves out from the center. At the center of the body, the gravitational field is zero.

Exercise \(\PageIndex{9}\)

What can you say about the magnitude of the gravitational field inside a spherical shell of mass \(M\) ?

• It increases as you move out from the center of the spherical shell.
• It decreases as you move out from the center of the spherical shell.
• It is equal to zero.
• It is nonzero and constant in magnitude.

1. The weight that is displayed on the scale is equal in magnitude to the normal force exerted by the scale on the person. It is the reaction force to that normal force.

2. The surface area of a sphere of radius \(r\) is \(4πr^{2}\).

Forgot password? New user? Sign up

Existing user? Log in

General Relativity

Already have an account? Log in here.

General relativity is Einstein's theory of gravity , in which gravitational forces are presented as a consequence of the curvature of spacetime . In general relativity, objects moving under gravitational attraction are merely flowing along the "paths of least resistance" in a curved, non-Euclidean space. The amount that spacetime curves depends on the matter and energy present in the spacetime, as summarized by a famous quote by the physicist John Archibald Wheeler:

\[``\textrm{Spacetime tells matter how to move; matter tells spacetime how to curve}."\]

Planet curving the nearby spacetime, depicted as the bending of a two-dimensional mesh [1] .

This statement is summarized in the two central equations of general relativity:

\[\begin{align} G_{\mu \nu} &= \frac{8 \pi G}{c^4} T_{\mu \nu}\\\\ \frac{d^2 x^{\mu}}{d \tau^2} + \Gamma^{\mu}_{\alpha \beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau} &= 0. \end{align}\]

The first is actually a set of equations called Einstein's field equations ; the left-hand side encodes the curvature of spacetime while the right-hand side encodes the matter/energy content. The second, called the geodesic equation , governs how the trajectories of objects evolve in a curved spacetime. Below, the mathematics and physical intuition behind these equations will be explained.

The effects of general relativity are most visible in the presence of extremely massive/dense objects such as those found in astronomy and cosmology . These effects include gravitational time dilation, redshifting of light in a gravitational potential, precession of planetary orbits, lensing of light, the existence of black holes , and gravitational waves .

Although general relativity has been enormously successful both in terms of the theory and its experimental verification, extremely technical mathematical inconsistencies have shown that the theory is most likely a low-energy, large length-scale approximation to a more complete theory of "quantum gravity" such as string theory which incorporates the effects of quantum mechanics .

Theoretical and Experimental History of General Relativity

Metrics: an introduction to non-euclidean geometry, parallel transport and the geodesic equation, gravity as geometry: einstein's equation, a brief application: black holes.

The Equivalence Principle

After Einstein's development of special relativity in the early twentieth century, he had successfully fully explained electromagnetism and mechanics in a relativistic framework. It seemed like the only missing piece of the puzzle was gravity. In Newtonian gravitation, the gravitational influences of masses occur instantaneously, in violation of relativity's light-speed limit. Gravity needed revision and incorporation into the relativistic framework.

The theory of general relativity began with another of Einstein's famous Gedankenexperiments . Consider an observer inside a closed room. The observer drops an object, which seems to accelerate as it falls to hit the ground. Einstein's realization was that it is impossible to tell whether the object has accelerated under the influence of gravity or if the object is stationary but the room was on a rocket accelerating upwards, making it seem as if the object traveled towards the floor rather than the floor towards the object. This equivalence of accelerated motion vs. accelerated frames is appropriately termed the equivalence principle . Along with Einstein's idea from special relativity that physics has no preferred coordinate system, it forms the cornerstone of the conceptual foundation of general relativity.

Another, more applicable way of viewing the equivalence principle is as follows: consider a small mass \(m\) acting under the influence of gravity (in the Newtonian limit) from some larger mass \(M\). Then the force on the mass is

\[F_g = ma = \frac{GMm}{r^2} \implies a = \frac{GM}{r^2}.\]

Above, canceling \(m\) on both sides of Newton's second law gave the acceleration due to the gravity of \(M\). But there is no a priori reason why the small \(m\) in \(F=ma\), called the inertial mass , ought to be equal to the \(m\) in \(F_g = \frac{GMm}{r^2}\), called the gravitational mass . Einstein's equivalence principle is a statement of equivalence of the inertial and gravitational masses: the mass due to the acceleration of a frame is the same as the mass due to gravity. In this picture, Einstein reimagined gravity as indistinguishable from accelerated frames, and used these ideas to recast gravity as objects accelerating through curved geometries. In the next decades, Einstein worked with several mathematicians of the era, particularly David Hilbert, in developing a geometric theory of gravity. This theory was what would eventually become general relativity.

Early Predictions and Tests

The equivalence of inertial and gravitational mass led to one of Einstein's first predictions as a result of general relativity: the gravitational redshift of light , in which light loses energy as it climbs out of a gravitational field. Shortly after, in 1916, Einstein proposed three concrete experimental tests of the extensive geometric theory that he had developed over about a decade. The first was the gravitational redshift; the other two were the deflection of light due to the gravity of large masses and the perihelion precession of mercury.

At around the same time, the German physicist Karl Schwarzschild discovered his black hole solution to Einstein's equations, the Schwarzchild metric . Several years later, the Russian physicist Alexander Friedmann and others found solutions that admitted an expanding or contracting universe, leading to modern cosmology and the Big Bang. Convinced the universe was static, Einstein did not accept these solutions, adding a cosmological constant term to his equations to ensure that the universe had to be static. In later years, Einstein famously spoke of regretting this error.

In terms of experimental verification, the British astronomer Sir Arthur Eddington led an astronomical expedition that confirmed the gravitational deflection of light by the sun in 1919. Similar early evidence also came from astronomy: it had been known since the mid-nineteenth century that the axis of Mercury's orbit rotated by a small angle each revolution, the so-called "perihelion precession." Einstein's computation of this rotation in general relativity matched the anomalous angle spectacularly.

Modern Tests and Research

In the modern era of physics, countless other experimental tests of general relativity have been performed, with the theory agreeing spectacularly with experiment.

Einstein's original prediction of gravitational redshift was the last to be confirmed--not until the famous Pound-Rebka experiment in 1959, where the redshifting of gamma rays was measured in a laboratory at Harvard University.

Another well-known later experiment was the Hafele-Keating experiment in 1971, where two American physicists flew with several atomic clocks in commercial airliners around the world twice. The atomic clocks onboard the planes were compared to atomic clocks on the ground and the airborne clocks were found to have experienced a slightly slower passage of time precisely in agreement with gravitational time dilation predicted by general relativity.

The physical consequences of general relativity are in fact quite applicable to everyday life. Gravitational time dilation turns out to affect the times measured by GPS satellites to non-negligible extents. GPS "triangulation" actually requires four satellites: three to identify the position and a fourth to calibrate for the error in timing incurred by gravitational time dilation. The size of this error is significant enough to give incorrect GPS predictions within hours of a satellite launch.

The existence of black holes is one of the major predictions of general relativity. Until recently, black holes had never been observed directly, only indirectly via their gravitational influence on other astronomical bodies. In early 2016, however, it was announced that another prediction of general relativity-- gravitational waves --had been observed from the merger of two inspiraling binary black holes. The resulting direct signal of the black hole merger was observed by scientists at the Laser Interferometry Gravitational-Wave Observatory (LIGO).

Some theoretical problems (as well as many experimental problems) are still open in general relativity. For instance, it is not yet known how to reconcile general relativity with quantum theory in a fully consistent way. Some other technical problems include mathematically proving the stability of certain black hole spacetimes, precision gravitational wave astronomy, and the need for a modification of the theory to account for the gravitational influences of dark matter and dark energy.

Which of the following is the most correct statement of the equivalence principle?

Which of the following experimental signals of general relativity has not been observed as of early 2016?

Note that although it is conventional in general relativity to use a system of units in which the speed of light \(c = 1\), for clarity all factors of \(c\) are included throughout this article.

What is a metric?

The "curvature of spacetime" in general relativity mathematically just means that the distances between objects change in a curved spacetime from what one would expect in Euclidean geometry. For instance, a person living on the surface of a sphere, a curved space, doesn't expect that the shortest path between two points is a straight line. On the surface of a sphere, the paths of shortest length or geodesics are the great circles connecting two opposite poles.

There are a few differences between this sphere example and general relativity. The first is that one usually imagines the sphere as being embedded in some larger space, so that a person is confined to the surface of the sphere but there is some space that is not on the surface. This is not the case in general relativity--rather, the curved space is all there is. One can recognize that a space is curved by what the geodesics look like between two points. If geodesics are not straight lines, then there is some indication that the space is curved.

The other difference is that in GR, it is not just space but rather space time that is curved. This means that not only are the distances between two objects, but also the times between two events. Depending on how close one is to a source of gravitation, the time measured between events may be stretched more or less.

Mathematically, the way that distances and times between events are measured is expressed in an object called a metric . A metric is effectively a matrix that lets one compute dot products between vectors. To demonstrate the purpose of the metric notice that the Pythagorean theorem in Euclidean space can be written as a matrix product:

\[d^2 = x^2 + y^2 + z^2 \iff \begin{pmatrix} x & y & z \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}.\]

In Euclidean space, the metric is the identity matrix--the matrix above between the two coordinate vectors. The matrix above is written as \(\delta_{ij}\), the Kronecker delta \((\)0 if \(i \neq j\), 1 if \( i = j). \) In a general non-Euclidean space, the metric need not be the identity matrix. This is all it means to say a space is curved--the way distances are measured has been somehow warped. A general spatial metric is written as \(g_{ij},\) where the indices \(i\) and \(j\) label the rows and columns of the matrix. Even in Euclidean spaces, the metric need not be the identity, depending on the coordinate system. For instance, in spherical coordinates in Euclidean space, the metric takes the form

\[\begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2 \theta \end{pmatrix}.\]

The Minkowski Metric

In extending the metric from space to spacetime, a fourth dimension must be added. In a flat Euclidean spacetime in Cartesian coordinates, the metric looks like the following:

\[ \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0& 1 \end{pmatrix}.\]

This is called the Minkowski metric , and flat Euclidean spacetime is correspondingly called Minkowski spacetime . Depending on context, sometimes the metric is written so that all components are the negative as what is given above. Often, the Minkowski metric is denoted as \(\eta_{\mu \nu}\) instead of \(g_{\mu \nu}\).

The reason for this strange metric, with its negative component in the time direction, is that it correctly captures the fundamental postulates of special relativity . It is the simplest metric that is invariant under Lorentz transformations .

Consider taking the dot product of the basic coordinate vector \((ct, x, y, z)\) with itself:

\[d^2 = -c^2 t^2 + x^2 + y^2 + z^2.\]

Since the Minkowski metric is invariant under Lorentz transformations, this metric correctly accounts for the fact that the speed of light is \(c\) in all frames. Since the speed of light is \(c\) in some frame, i.e.

\[c^2 = \frac{|\vec{x}|^2}{t^2} = \frac{x^2 + y^2 + z^2}{t^2},\]

\(d = 0\) in that frame. But by invariance of the Minkowski metric, \(d=0\) in all frames, so the speed of light is always \(c\) in all frames.

The Invariant Interval

The way distances are measured can change continuously in general relativity. As a result, the metric is usually defined in terms of quantities that vary infinitesimally, like differentials. The quantity \(d^2\) above is written

\[ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 = -dt^2 + d\vec{x}^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}.\]

The quantity \(ds^2\) is called the invariant interval , since the metric is Lorentz-invariant. In the last equality above, the invariant interval is rewritten in Einstein summation notation , wherein repeated indices are summed over. The quantity \(g_{\mu \nu} dx^{\mu} dx^{\nu}\) describes the dot product of the coordinate vector \(dx^{\mu} = (cdt, dx, dy, dz)\) with itself; the indices \(\mu\) and \(\nu\) label the indices of the vector and the matrix representing the matrix. When discussing spacetimes, the spatial indices \(i\) and \(j\) are usually promoted to these Greek letters.

Often, a general metric is written in terms of the invariant interval \(g_{\mu \nu} dx^{\mu} dx^{\nu}\) since this is more compact than writing out an entire matrix.

Two vectors \(a^i = (1,2,3)\) and \(b^i = (2,3,4)\) live in a space with spatial metric given by the invariant interval

\[ds^2 = 2dx^2 + \frac12 dy^2 + dz^2.\]

Compute the inner product \(a_i b^i = g_{ij} a^i b^j\).

Light travels from spacetime point \(x\) to spacetime point \(y\) in Minkowski space. What is the value of the invariant interval between \(x\) and \(y?\)

Parallel Transport of Vectors

One of the central characteristics of curved spacetimes is that the "parallel transport" of vectors becomes nontrivial. It turns out that this observation leads to much of modern differential geometry and the math of general relativity.

The "parallel transport" of vectors refers to sliding a vector along a curve so that it is always tangent to the curve. In a Euclidean spacetime, this is easy: just follow the direction of the tangent vector at any given point, and the vector will always be tangent. In a curved space, however, it is not so easy. In the below diagram, one can see what goes wrong:

The parallel transport of a tangent vector along a closed loop on the curved surface of a sphere, resulting in an angular defect \(\alpha\) [2] .

In the above diagram, a vector has been parallel transported along the surface of a sphere in a closed loop. The vector starts out parallel to the curve and remains fairly parallel as it follows the tangent vector. As it rounds the top of the loop, where the curvature of the loop is large, however, sliding it along the tangent shifts the direction of the vector. After going around the entire loop, the vector has shifted by an angle of \(\alpha\) with respect to its initial direction, the angular defect of this closed loop.

To fix this problem, one must modify what it means to parallel transport a vector in a curved space. Normally, in a flat space, one would think that a particle freely falling along a straight line would obey the equation

\[\frac{d^2 x^{\mu}}{d\tau^2} = 0,\]

where \(\tau\) is the time measured by the particle and \(x^{\mu} = (ct,\vec{x})\) are the coordinates of the particle. This equation is essentially the statement that \(F = ma = 0\), since effectively \(a = \frac{d^2 x^{\mu}}{d\tau^2}\).

However, this quantity doesn't transform nicely under coordinate transformations. Since behaving well under coordinate transformations is essential in GR, this equation must be modified to the equivalent expression [3] :

\[\frac{d x^{\mu}}{d\tau} \partial_{\mu} \frac{dx^{\nu}}{d\tau} = 0,\]

where \(\partial_{\mu} = \frac{\partial}{\partial x^{\mu}}\) is the usual partial derivative with respect to the coordinate \(x^{\mu}\).

In a flat space, parallel transporting an arbitrary vector \(a^{\nu}\) therefore means that it obeys the equation

\[v^{\mu} \partial_{\mu} a^{\nu} = 0,\]

where \(v^{\mu}\) is the usual tangent vector to the path.

Covariant Derivatives, the Christoffel Connection, and the Geodesic Equation

In a curved space, the derivative \(\partial_{\mu}\) is modified to correctly parallel transport vectors. It is changed to the covariant derivative [3]

\[\nabla_{\mu} a^{\nu} = \partial_{\mu} a^{\nu} + \Gamma^{\nu}_{\mu \lambda} a^{\lambda},\]

where the quantity \(\Gamma^{\nu}_{\mu \lambda}\), called the Christoffel symbol or Christoffel connection , is defined in terms of the metric as

\[\Gamma^{\nu}_{\mu \lambda} = \frac12 g^{\nu \sigma} (\partial_{\mu} g_{\sigma \lambda} + \partial_{\lambda} g_{\mu \sigma} - \partial_{\sigma} g_{\mu \lambda}).\]

This quantity is called a "connection" because it "connects" tangent vectors at two points. It modifies the ordinary partial derivative so that the tangent vectors are correctly adjusted to account for the curvature of the space.

The \(g^{\nu \sigma}\) above with both indices raised are the components of the inverse metric . The inverse metric is equal to the matrix inverse of the metric.

With all of these modifications, the parallel transport of a tangent vector \(v^{\mu}\) \(\big(\)noting that \(v^{\mu} = \frac{\partial x^{\mu}}{\partial \tau}\big) \) is given by the geodesic equation [3]

\[v^{\nu} \nabla_{\nu} v^{\mu} = 0 \iff \frac{d^2 x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\alpha \beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau} = 0.\]

Note that, as always in general relativity, repeated indices above are summed (and therefore can be labeled with whatever letter is desired). Note also that this equation looks a lot like \(F = ma = 0\), except with the modifying term \(\Gamma^{\mu}_{\alpha \beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau}\) capturing the influence of the curvature of spacetime.

Paths \(x^{\mu} (\tau)\) in spacetime that obey the geodesic equation are said to be geodesics . They are the shortest path between two points in a curved spacetime, and are the trajectories that freely falling particles follow when spacetime is curved. Since these trajectories are generally not straight lines when gravitational sources are involved, the effects of gravity are to curve spacetime, changing \(g_{\mu \nu}\) and resultantly altering the trajectories of particles. This is how "spacetime tells matter how to move" in general relativity.

\[ds^2 = r^2 \, d\theta^2 + \dfrac{1}{1+r^2} \sin^2 (\theta) \, d\phi^2\]

A strange metric on a sphere of radius \(r\) is given by the invariant interval described above.

Compute the Christoffel symbol \(\large \Gamma^{\phi}_{\phi \theta}\).

A metric on a two-dimensional space is given by the invariant interval

\[ds^2 = \big(1+y^2\big) dx^2 + \big(1+x^2\big) dy^2.\]

Which of the following gives the \(x\)-component of the geodesic equation for this metric?

Poisson's Equation and the Weak-Field Limit

In the most refined mathematical approach to Newtonian gravity, the acceleration of an object is given in terms of the gravitational potential \(\Phi\) by the equation

\[a = -\nabla \Phi,\]

where \(\nabla\) is the gradient operator. This gravitational potential obeys Poisson's equation [3]

\[\nabla^2 \Phi = 4\pi G \rho,\]

an equation analogous to Gauss's law in electricity and magnetism. In this equation, \(\rho\) is the density of gravitating matter.

Since general relativity should reduce to Newtonian gravitation in the static, slowly-moving, weak gravitation case, a fully general-relativistic equation of gravity ought to reduce to Poisson's equation. Furthermore, the left-hand side ought to be somehow encoded by the metric, since the metric encodes all the effects of curved spacetime and gravity in general relativity. Furthermore, it turns out that in the weak-field limit, only one of the metric components matters and is given by \(g_{00} \approx -(1+2\Phi)\), so the metric is really directly connected to the Newtonian potential in this limit.

The Stress-Energy Tensor

The metric is a matrix, so such an equation also ought to be a matrix equation. This is possible because there is in fact a matrix which encodes all of the information about the matter and energy which gravitates: the stress-energy tensor \(T_{\mu \nu}\). This is a symmetric four-by-four matrix given diagrammatically by

Diagrammatic structure of the matrix representation of the stress-energy tensor

That is, \(T_{00} = \rho\) is the energy density, and the other components give momenta, pressures, and shear stresses of the gravitating matter. Only the upper-right half of the matrix is shown because it is symmetric about the diagonal. Since \(T_{00} = \rho\) is the energy density, it seems reasonable to expect \(T_{\mu \nu}\) to be the right-hand side of an equation of general relativity that will reduce to Poisson's equation.

It turns out that the conservation of energy in general relativity is correctly expressed using the covariant derivative as

\[\nabla_{\mu} T^{\mu \nu} = 0.\]

Curvature and Einstein's Equations

If \(T^{\mu \nu}\) is the right-hand side of an equation of general relativity, therefore, the left-hand side had better also vanish under the covariant derivative. It turns out that there is a combination of second derivatives of the metric for which this covariant derivative property also holds true, the Einstein tensor \(G_{\mu \nu}\):

\[G_{\mu \nu} = R_{\mu \nu} - \frac12 R g_{\mu \nu},\]

where \(R_{\mu \nu}\) is the Ricci tensor and \(R = R^{\lambda}_{\lambda}\), the trace of the Ricci tensor, is called the Ricci scalar . The Ricci tensor is defined in terms of the Riemann curvature tensor , which in turn is defined in terms of the Christoffel symbols defined earlier

\[R^{\rho}_{\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}_{\nu \sigma} - \partial_{\nu} \Gamma^{\rho}_{\mu \sigma} + \Gamma^{\rho}_{\mu \lambda} \Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda} \Gamma^{\lambda}_{\mu \sigma},\]

so that \(R_{\mu \nu} = R^{\lambda}_{\mu \lambda \nu}\) is the partial trace of the Riemann curvature tensor. The Riemann curvature tensor has deep connections to the covariant derivative and parallel transport of vectors, and can also be defined in terms of that language.

The equations above are enough to give the central equation of general relativity as proportionality between \(G_{\mu \nu}\) and \(T_{\mu \nu}\). Demanding that this equation reduces to Poisson's equation of Newtonian gravity in the weak-field limit using \(g_{00} \approx -(1+2\Phi)\) sets the proportionality constant to be \(\frac{8 \pi G}{c^4}\). Thus, by encoding the energy density in a matrix (the stress-energy tensor), and finding a matrix defined in terms of second derivatives of the metric that obeys the same covariant derivative property, one arrives at Einstein's field equations , the central equations of general relativity [3] :

\[G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.\]

Note that this equation holds for all choices of indices \(\mu\) and \(\nu\) and therefore is really a set of equations, not just a single equation.

The definitions and notation of general relativity are quite dense and computing any quantities is extremely intensive. However, this compact and beautiful equation summarizes the second half of Wheeler's quote: "matter tells spacetime how to curve." The stress-energy tensor \(T_{\mu \nu}\) described by the energy content of whatever matter is in the space sets \(G_{\mu \nu}\), a function of the metric \(g_{\mu \nu}\), and thus determines how spacetime curves in response to matter.

The Poincare half-plane model for hyperbolic space puts the following metric on the plane:

\[ds^2 = \frac{1}{y^2} (dx^2 + dy^2).\]

Compute the Ricci scalar \(R\) for this metric in matrix form. Is this a vacuum solution to Einstein's equations?

Give your answer as an \((R,\) Yes/No\()\) pair.

Einstein's Equation in Vacuum

Solving Einstein's equations in general is incredibly difficult, even numerically with the aid of computers. Only a few exact analytic solutions are known for the metric given different stress-energy tensors. The simplest solutions are in vacuum (possible outside a gravitating source): \(T_{\mu \nu} = 0\). In this case, Einstein's equations reduce to the slightly simpler equation (provided the number of dimensions is greater than 2):

\[R_{\mu \nu} = 0. \qquad \text{(Vacuum Einstein Equations)}\]

One obvious solution to this equation is just the Minkowski metric. Since all components are just numbers and not functions of space or time, all derivatives of the Minkowski metric are zero, so all Christoffel symbols vanish, and the curvature vanishes as well.

The Schwarzschild Metric and Black Holes

The Minkowski metric is not a function of space or time, so it is highly symmetric. The next simplest solution of the vacuum Einstein equations is the Schwarzschild metric , which corresponds to the case of spacetime outside a spherically symmetric mass distribution. It is given by the invariant interval in spherical coordinates:

\[ds^2 = -\left(1-\frac{2GM}{rc^2}\right) c^2 dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1} dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2.\]

This metric describes any spherically symmetric mass distribution of mass \(M\), including planets, stars... and black holes! Note that the factor \(1-\frac{2GM}{rc^2}\) above makes the metric become degenerate at \(r_s = \frac{2GM}{c^2}\), the Schwarzschild radius and location of the event horizon of a black hole. A black hole is just a spherically symmetric mass distribution which is sufficiently dense so that \(r_s\) is actually outside the radius of the object. The Schwarzschild radius of Earth, for instance, is only about \(9\) millimeters, deep inside the core of Earth where the Schwarzschild metric no longer applies.

One interesting thing to note is that the above formula implies the existence of gravitational time dilation . An object held fixed at a radius of \(r\) from the center of a spherically symmetric mass distribution experiences the passage of time at a rate adjusted by a factor of \(\sqrt{1-\frac{2GM}{rc^2}}\) compared to an observer at infinity, i.e. slower. As discussed above, this is an effect which has been experimentally confirmed above the surface of Earth.

Black holes are often said to have a "curvature singularity." This seems to contradict the fact that the Schwarzschild metric is a solution to the vacuum Einstein equations since \(R_{\mu \nu} = R = 0\). However, not all components of the Riemann curvature tensor vanish, and the scalar quantity called the Kretschmann scalar for the Schwarzschild metric is given by [3]

\[K = R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma} = \frac{48 G^2 M^2 }{c^4 r^6}.\]

Since this quantity diverges as \(r \to 0\), black holes indeed have a curvature singularity as \(r \to 0\), although it is suspected that classical general relativity will break down before this point, preventing the formation of a singularity.

As \(r \to r_s\), the \(dt^2\) term in the Schwarzschild metric goes to zero. This should be interpreted as saying that an observer far from a black hole watching an object fall in will never see that object fall past the horizon. For, as it approaches the horizon, it appears to stop experiencing the passage of time and the physical distance to the horizon seems to become enormous. A careful analysis will show, however, that an infalling object in classical general relativity experiences nothing unusual as it passes the event horizon.

Time passes more slowly by a factor of \(x\) at plane cruising altitude of \(12000 \text{ m}\) above the earth's surface, compared to the time experienced by an object at infinity. At approximately how many places after the decimal point does \(x\) differ from \(1.000\ldots?\)

\(\) Useful Constants:

• The radius of the earth is \(6.37 \times 10^6 \text{ m}\).
• The speed of light is \(3 \times 10^8 \text{ m}/\text{s}\).
• Newton's gravitational constant is \(6.67 \times 10^{-11} \text{ N}\cdot \text{m}^2 / \text{kg}^2\).
• The mass of the earth is \(5.97\times 10^{24} \text{ kg}\).

SR has claimed that space and time exhibit a particular symmetric pattern. The transformation group is called the Lorentz transformations or velocity transformations. It is the set of linear transformations \[(a^{\mu})'=\sum_{\nu=1}^4 L_{\nu}^{\mu}a^{\nu}.\]

• Mysid, . Spacetime lattice analogy . Retrieved from https://commons.wikimedia.org/w/index.php?curid=45121761
• Antonelli, L. Parallel Transport . Retrieved from https://commons.wikimedia.org/w/index.php?curid=1122750
• Carroll, S. (2004). Spacetime and Geometry: An Introduction to General Relativity . San Francisco: Pearson.

Problem Loading...

Note Loading...

Set Loading...

GRAVITATIONAL THEORIES

Newtonian gravitational theory, the general theory of relativity, scalar tensor theories, other theories.

share this!

December 4, 2023

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

peer-reviewed publication

trusted source

New theory claims to unite Einstein's gravity with quantum mechanics

by University College London

A radical theory that consistently unifies gravity and quantum mechanics while preserving Einstein's classical concept of spacetime has been announced in two papers published simultaneously by UCL (University College London) physicists.

Modern physics is founded upon two pillars: quantum theory on the one hand, which governs the smallest particles in the universe, and Einstein's theory of general relativity on the other, which explains gravity through the bending of spacetime. But these two theories are in contradiction with each other and a reconciliation has remained elusive for over a century.

The prevailing assumption has been that Einstein's theory of gravity must be modified, or "quantized," in order to fit within quantum theory. This is the approach of two leading candidates for a quantum theory of gravity, string theory and loop quantum gravity .

But a new theory, developed by Professor Jonathan Oppenheim (UCL Physics & Astronomy) and laid out in a paper in Physical Review X, challenges that consensus and takes an alternative approach by suggesting that spacetime may be classical—that is, not governed by quantum theory at all.

Instead of modifying spacetime, the theory—dubbed a "postquantum theory of classical gravity"—modifies quantum theory and predicts an intrinsic breakdown in predictability that is mediated by spacetime itself. This results in random and violent fluctuations in spacetime that are larger than envisaged under quantum theory, rendering the apparent weight of objects unpredictable if measured precisely enough.

A second paper, published simultaneously in Nature Communications and led by Professor Oppenheim's former Ph.D. students, looks at some of the consequences of the theory, and proposes an experiment to test it: to measure a mass very precisely to see if its weight appears to fluctuate over time.

For example, the International Bureau of Weights and Measures in France routinely weigh a 1kg mass which used to be the 1kg standard. If the fluctuations in measurements of this 1kg mass are smaller than required for mathematical consistency, the theory can be ruled out.

The outcome of the experiment, or other evidence emerging that would confirm the quantum vs. classical nature of spacetime, is the subject of a 5000:1 odds bet between Professor Oppenheim and Professor Carlo Rovelli and Dr. Geoff Penington —leading proponents of quantum loop gravity and string theory respectively.

For the past five years, the UCL research group has been stress-testing the theory, and exploring its consequences.

Professor Oppenheim said, "Quantum theory and Einstein's theory of general relativity are mathematically incompatible with each other, so it's important to understand how this contradiction is resolved. Should spacetime be quantized, or should we modify quantum theory, or is it something else entirely? Now that we have a consistent fundamental theory in which spacetime does not get quantized, it's anybody's guess."

Co-author Zach Weller-Davies, who as a Ph.D. student at UCL helped develop the experimental proposal and made key contributions to the theory itself, said, "This discovery challenges our understanding of the fundamental nature of gravity but also offers avenues to probe its potential quantum nature.

"We have shown that if spacetime doesn't have a quantum nature, then there must be random fluctuations in the curvature of spacetime which have a particular signature that can be verified experimentally.

"In both quantum gravity and classical gravity, spacetime must be undergoing violent and random fluctuations all around us, but on a scale which we haven't yet been able to detect. But if spacetime is classical, the fluctuations have to be larger than a certain scale, and this scale can be determined by another experiment where we test how long we can put a heavy atom in superposition of being in two different locations."

Co-authors Dr. Carlo Sparaciari and Dr. Barbara Šoda, whose analytical and numerical calculations helped guide the project, expressed hope that these experiments could determine whether the pursuit of a quantum theory of gravity is the right approach.

Dr. Šoda (formerly UCL Physics & Astronomy, now at the Perimeter Institute of Theoretical Physics, Canada) said, "Because gravity is made manifest through the bending of space and time, we can think of the question in terms of whether the rate at which time flows has a quantum nature, or classical nature.

"And testing this is almost as simple as testing whether the weight of a mass is constant, or appears to fluctuate in a particular way."

Dr. Sparaciari (UCL Physics & Astronomy) said, "While the experimental concept is simple, the weighing of the object needs to be carried out with extreme precision.

"But what I find exciting is that starting from very general assumptions, we can prove a clear relationship between two measurable quantities—the scale of the spacetime fluctuations, and how long objects like atoms or apples can be put in quantum superposition of two different locations. We can then determine these two quantities experimentally."

Weller-Davies added, "A delicate interplay must exist if quantum particles such as atoms are able to bend classical spacetime. There must be a fundamental trade-off between the wave nature of atoms, and how large the random fluctuations in spacetime need to be."

The proposal to test whether spacetime is classical by looking for random fluctuations in mass is complementary to another experimental proposal that aims to verify the quantum nature of spacetime by looking for something called "gravitationally mediated entanglement."

Professor Sougato Bose (UCL Physics & Astronomy), who was not involved with the announcement today, but was among those to first propose the entanglement experiment, said, "Experiments to test the nature of spacetime will take a large-scale effort, but they're of huge importance from the perspective of understanding the fundamental laws of nature. I believe these experiments are within reach—these things are difficult to predict, but perhaps we'll know the answer within the next 20 years."

The postquantum theory has implications beyond gravity. The infamous and problematic "measurement postulate" of quantum theory is not needed, since quantum superpositions necessarily localize through their interaction with classical spacetime .

The theory was motivated by Professor Oppenheim's attempt to resolve the black hole information problem. According to standard quantum theory , an object going into a black hole should be radiated back out in some way as information cannot be destroyed, but this violates general relativity , which says you can never know about objects that cross the black hole's event horizon. The new theory allows for information to be destroyed, due to a fundamental breakdown in predictability.

Jonathan Oppenheim et al, Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity, Nature Communications (2023). DOI: 10.1038/s41467-023-43348-2 . www.nature.com/articles/s41467-023-43348-2

Journal information: Physical Review X

Provided by University College London

Explore further

Feedback to editors

Artificial intelligence helps scientists engineer plants to fight climate change

6 hours ago

Ultrasensitive photonic crystal detects single particles down to 50 nanometers

7 hours ago

Scientists map soil RNA to fungal genomes to understand forest ecosystems

8 hours ago

Researchers show it's possible to teach old magnetic cilia new tricks

Mantle heat may have boosted Earth's crust 3 billion years ago

Study suggests that cells possess a hidden communication system

Researcher finds that wood frogs evolved rapidly in response to road salts

Imaging technique shows new details of peptide structures

9 hours ago

Cows' milk particles used for effective oral delivery of drugs

New research confirms plastic production is directly linked to plastic pollution

Relevant physicsforums posts, how to avoid breaking physics with your “what if” question.

Apr 18, 2024

NASA is seeking a faster, cheaper way to bring Mars samples to Earth

Apr 17, 2024

Could you use the moon to reflect sunlight onto a solar sail?

Apr 14, 2024

Biot Savart law gives us magnetic field strength or magnetic flux density?

Why charge density of moving dipole is dependent on time.

Apr 5, 2024

I have a question about energy & ignoring friction losses

Apr 3, 2024

More from Other Physics Topics

Related Stories

Quantum 'magic' could help explain the origin of spacetime

Apr 13, 2023

Quantum collaboration gives new gravity to the mysteries of the universe

Feb 17, 2021

Study proposes combining continuum mechanics with Einstein field equations

Jul 24, 2023

Bridge between quantum mechanics and general relativity still possible

Sep 19, 2019

Physicists propose test of quantum gravity using current technology

Oct 27, 2017

Synthetic black holes radiate like real ones

Nov 11, 2022

Recommended for you

Scientists at the MAJORANA Collaboration look for rule-violating electrons

15 hours ago

Scientists tune the entanglement structure in an array of qubits

10 hours ago

Lead-vacancy centers in diamond as building blocks for large-scale quantum networks

11 hours ago

Light stands still in a deformed crystal

13 hours ago

How light can vaporize water without the need for heat

Apr 23, 2024

Hunting for the elusive: IceCube observes seven potential tau neutrinos

Let us know if there is a problem with our content.

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

• International edition
• Australia edition
• Europe edition

Physicist Claudia de Rham: ‘Gravity connects everything, from a person to a planet’

The scientist on training as a diver, pilot and astronaut in order to understand the true nature of gravity, and what happens at the centre of a black hole

P rof Claudia de Rham is a theoretical physicist at Imperial College London, developing and testing “new models and paradigms” at the intersection of gravity, cosmology and particle physics. She has just published her first book, The Beauty of Falling: A Life in Pursuit of Gravity , charting her lifelong attempt to understand the “true nature” of the force around her, which has seen her train as an astronaut, diver and pilot.

In your book, you say you have been “chasing gravity my entire life”. What does that mean? It’s something we all have in ourselves, this playful relationship with gravity. But to some extent, it has taken on a meaning of its own for me in terms of really trying to go to outer space, to challenge gravity in a slightly different way. And when that hasn’t worked out, then to do that from a more scientific point of view, in terms of understanding the underlying framework behind the models we have for gravity; coming up with new models, and finding ways to see if we can experiment with those.

What is it about gravity that you find so compelling? Gravity is so universal. And I think this is very fundamental, it’s a phenomenon that affects everything, everyone, all the time, everywhere. It does absolutely everything: it connects everything and everyone. We all experience it in the same way, whether we are a person or a planet, a black hole or a balloon, a hammer, a feather, a piece of cheese, or a pumpkin seed. It’s something which is within us without us being able to defeat it. The feeling of weightlessness, the feeling of freefall is what gravity is. It is a complete freedom in itself.

You’ve trained as a diver and a pilot to better understand the fundamental forces around us, and got to the final selection stage of astronaut training for the European Space Agency. How did that journey begin? It started from wanting to chase gravity, in some sense. For me that meant trying to go into outer space and experience gravity inside a different environment, and experiencing this feeling of weightlessness on a deeper level. So once I was able to formulate that in my head as a child, that really became a long-term goal. For such a long period of my life, every single thing I would do was focused on the idea that I wanted to become an astronaut, I wanted to participate in astronaut selection and put myself in the best position for that.

What did astronaut training entail? We were subjected to a battery of tests, most of which were designed to assess our team-based abilities in stressful situations. This included a pretend rescue expedition through the “jungle” – we were asked to plan the risky mission knowing resources were limited, night was coming, and, if we weren’t careful, some of us may not make it back. Another saw us paired up as pretend air traffic controllers in an overcrowded airport, tasked with landing several planes running short of fuel. The twist was that each of us only had access to half of the information, so we had to engage in precise communication and collaboration with our partner to complete the mission. Out of nearly 10,000 applicants to apply, 99.5% had been rejected by the last stage. Only 42 got to this second training stage.

Your hopes were cut short by the discovery of latent tuberculosis during the final selection round, which you describe as “a little souvenir I must have unknowingly brought back with me from my time in Madagascar” from your childhood. Do you wonder what life might have been like had you not fallen ill? I see the candidates going out to space… it’s, of course, a dream. But at the same time, I don’t regret the path my life has taken since then. I think it opened up other opportunities.

How much more is there to learn about gravity that we don’t already know? The beauty with gravity as we know it now is that there is much more to learn about it. We now know that Einstein’s theory of general relativity stops being a consistent and accurate description of what happens in some situations, like very close to the centre of black holes , or the big bang. His theory would seem to suggest that we could measure some physical quantities, like the curvature of spacetime, that would seem to become infinite (we call this singularity), but measuring an infinite value would simply not make sense. In order to understand what takes over when his theory is no longer valid, we may want to ask ourselves questions like “what happens at the very centre of a black hole?” or “what happened at the time of the big bang?” – but if the notions of space and time no longer make sense, then the questions themselves may not make sense either. We don’t even have the language to phrase the questions we have, less still answer them.

What can gravity tell us about the future? What we know at the moment is that the expansion of the universe is accelerating, which sort of was a surprise 25 years ago. What we need to understand is what is causing this accelerated expansion, and whether the laws of gravity that we’re using to describe that are valid. We need to understand this better to [understand] what will happen in the future. Will this accelerated expansion continue for ever? Will it become even more accelerated? Will it stop? We don’t know, because we don’t even know precisely what the origin of this phenomenon is. So the fate of our universe, and the fate of space and time in some sense, will depend on what the driver of this accelerated expansion of the universe is.

Is it frustrating to have so many questions, and so few answers? It’s exciting. It’s little steps. If what you’re trying to do is see if there’s going to be an application of your discoveries for tomorrow, then this is not the field for you. But you do learn: you gain a much deeper appreciation of nature all around you, and it’s these everyday small steps that really lead to the biggest steps in the longer term. Every single small understanding is a breakthrough in itself, because it will allow us to go deeper. In science, even a negative result is a result in itself.

The Beauty of Falling: A Life in Pursuit of Gravity by Claudia de Rham is published by Princeton University Press (£20). To support the Guardian and Observer order your copy at guardianbookshop.com . Delivery charges may apply

• The Observer
• Imperial College London

More on this story

The big idea: are we about to discover a new force of nature?

World’s top cosmologists convene to question conventional view of the universe

‘Wobbly spacetime’ may help resolve contradictory physics theories

‘What the heck is going on?’ Extremely high-energy particle detected falling to Earth

Science Weekly Remembering physicist Peter Higgs – podcast

New 3D cosmic map raises questions over future of universe, scientists say

Scientists may be on brink of discovering fifth force of nature

Dark energy ‘chameleon trap’ wins £100,000 prize for Nottingham scientist

New US lab to create versions of atoms never recorded on Earth

Science Weekly Will the Large Hadron Collider find a new fifth force of nature?

Most viewed.

What Is Gravity, and How Does It Work?

Legend has it that Isaac Newton had the moment of inspiration that would lead to his theory of gravity when, on a warm afternoon, he saw an apple fall from a tree and wondered why it should fall down instead of up. (Some versions of the story have the apple falling and hitting poor Newton on the head.) He named his new hypothesis after the Latin gravitas , for "weight." The tale of Newton's bruised noggin may be apocryphal, but his interest in what makes things move—and especially what makes them fall —was very real. Newton had come home to stay when his university closed down due to a bout of bubonic plague, and he was looking for something to occupy his mind. In 1665, he found it, apple or no apple.

It took Newton, one of the world's greatest mathematical minds and, by all accounts, an irascible jerk, 20 years to articulate his thoughts on gravity to his satisfaction. His biggest problem? The question of whether the Earth's gravitational influence could extend all the way to the Moon.

Two hundred years later, the calculations that took humanity to the Moon were based on Newton's mechanics. But how does gravity work in the first place? What did Newton understand that was such a revolution?

Gravity: The Basics

Can someone just... explain gravity to me?

Let's start with a definition. Gravity, or gravitational attraction, is the tendency of mass to gather toward itself, drifting together even across great distances due to curvature in spacetime. This tendency allows the formation of stars, planets, galaxies, and black holes.

Standing on Earth's surface, the planet's mass creates a gravitational force sufficient to accelerate any object downward (toward the core of the planet, or perpendicular to the planet's surface) at 9.8 m/s²—that is, an additional 9.8 meters per second, each second. Everything experiences the same amount of gravity, but if you take a hammer in one hand and a feather in the other and drop them at the same time, the hammer will hit the ground first. Why? Drag (air resistance) opposes acceleration due to gravity. In a vacuum, such as on the Moon, both objects will hit the ground at the same time.

How does gravity work?

Gravity applies an effective force of mutual attraction to things with inertial mass, including physical matter and photons. The force of gravity is transmitted through spacetime at the speed of light, which creates wavefronts we can detect with special equipment like the LIGO gravitational wave detectors.

In classical (or "Newtonian") mechanics, which describes the motion of macroscopic objects (i.e., things larger than an atom, such as planets), gravity is sometimes called a central force . A central force is directed towards or away from a point called the center of force . Gravity, electrical charge, and magnetism are three examples of central forces. Centers of electromagnetic energy radiating inward or outward are known as poles.

Newton used a mathematical approach to gravity not unlike Coulomb had done with electrostatics, with a field that falls off as the inverse square of the distance between two objects. Looking at it this way, the gravitational force at a point can be expressed as a vector, with magnitude and direction.

How is gravity transmitted?

Christian Huygens, a contemporary of Isaac Newton, discovered that light carries energy. This suggests a force-carrying graviton as an obvious theoretical parallel to a photon. But where photons are the force carrier of the electromagnetic field, relativity frames gravity as an emergent consequence of the way inertial mass warps spacetime. Instead of requiring a force carrier, according to general relativity, what we think of as gravity is more like the idea of "downhill."

Massive or high-energy objects warp the mesh of spacetime, dragging it in toward themselves and creating a gravitational field of influence or "gravity well" from which it can be difficult to escape.

Central Forces

When matter is collected in one place, it forms a center of mass from which its inward-pointing field of gravitational influence extends. The force of attraction between two objects falls off as distance increases from the center of mass.

Objects under the influence of a gravitational field will move toward the field's center of gravity . Sometimes, as with the Sun and Jupiter, their mutual center of gravity or barycenter lies slightly outside one of the bodies; Jupiter is large enough that it drags the Sun in a little circle, centered slightly outside of the Sun's radius, as Jupiter makes each orbit.

On Earth, we have to contend with our own gravity well when we launch rockets and spacecraft; if a rocket isn't powerful enough to escape its gravity well, it will fall back to Earth. As matter becomes more and more dense, that effect becomes more pronounced. Black holes create a gravity well so deep that there's a threshold around black holes called an event horizon, a boundary in space marking the point of no return. Nothing inside the event horizon can escape from a black hole. Indeed, it's thought that the only thing that can ever escape a black hole's gravity is a frisson of virtual particles called Hawking radiation, thrown off every so often when subatomic symmetries align.

Sacred Geometry

Astronomers in ancient Greece noticed that the planets sometimes seem to move in retrograde across the sky, backward with respect to their normal orbits. This offended some astronomers' sense of cosmos , the orderliness of the universe. In a universe perfectly ordered by the hands of their gods, there was little room for irrational numbers or eccentric orbits.

In their attempt to reconcile their geocentric models with their empirical observations, they proposed the idea of epicycles: complex orbits that were neither circular nor elliptical, with planets dancing around the Earth in paths that look like geometric lace.

Geocentrism reigned unchallenged for more than two thousand years. Despite the repeated proposal of a heliocentric solar system over the millennia by scholars as respected as Leonardo da Vinci, heliocentrism wasn't taken seriously until the medieval era. However, the scientific consensus began to change in the 1500s. Nicolaus Copernicus developed a heliocentric model, backing up his argument with astronomical observations—and predictions that would confirm his model as correct or invalidate it. Galileo Galilei, using the newly invented refracting telescope, made and published observations showing that the planet Venus went through phases just like the Moon, and that Jupiter was orbited by its own moons.

Music of the Spheres

Then, Johannes Kepler put forth a solution to the problem of retrograde planets that would have satisfied even the strictest Pythagorean. Even with their orbits taking the "imperfect" shape of an ellipse, Kepler showed, a planet swept out an equal geometric area of its orbit over the same length of time, no matter where in its elliptical orbit it might be, nor how eccentric that ellipse. Kepler was a big believer in musica universalis , the music of the spheres; the idea that an inaudible mathematical harmony existed between the orbits of the planets was central to his Mysterium Cosmographicum .

In 1687, Isaac Newton published his opus, Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy , but many affectionately call it just Principia for short), which combined his laws of motion with a new mathematical analysis—calculus!—that could replicate Kepler's empirical observations of the planets and their moons. In Principia , Newton proposed a law of universal gravitation that now bears his name. Newton's law of universal gravitation holds that any two bodies, no matter how far they may be separated in space, are attracted by a force proportional to their mass and inversely proportional to the square of the distance between them.

'Spooky Action at a Distance'

Yet the question remained: How could one planet affect another at such a great distance? Newton considered action at a distance to be, in his own words, "so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it." Familiar as he was with electrostatics, Newton's theory of gravitation didn't require what he viewed as an exotic, unnecessary transmission mechanism when the inverse-square law modeled gravitational attraction entirely well enough.

Newton was by no means in the scientific minority on the topic of action at a distance. Albert Einstein operated under some assumptions of aether theory when developing his theory of relativity. Einstein would eventually dismiss the notion of quantum entanglement between two particles as spukhafte Fernwirkungen (translated as "spooky action at a distance"). Likewise, Newton and others of his day believed there must be a transmission medium, such as the luminiferous aether, through which electromagnetic or gravitational forces could exert a force on bodies separated in space.

Quantum Gravity

What Einstein knew that Newton didn't is that the universe is permeated not by an aetheric substance made of molecules of some kind but by an invisible warp and weft of field lines along which forces such as gravity are transmitted. No aether is required to produce the effects described in Maxwell's laws of electromagnetism or to produce gravity as we understand it. Today, it looks like the graviton will go the way of the aether.

On the deepest level, our cosmos is governed by four fundamental forces or fundamental interactions: electromagnetism and gravity, whose reach appears unlimited, and the weak and strong nuclear forces, which constrain themselves to the smallest scale, the inner workings of the atom. We call them fundamental forces because when we try to answer how spacetime works under the hood, these four forces appear not to be reducible to simpler interactions.

The human understanding of gravity still has some problems. Chief among them is the difficulty of applying current models to the subatomic scale or to extremely high-energy environments such as black holes or the very early universe.

As technology advances, scientists' understanding of what the laws of physics will permit stays in a state of flux. Once fusion, which must surmount the Coulomb barrier, was an exotic fiction; now, it's an engineering problem. It may be the same with gravity and other phenomena on the quantum scale. Until then, we can all enjoy the view from where we stand—on the shoulders of giants.