According to General Relativity, objects moving in a curved space-time geometry experience the effect of gravity, which causes them to follow a curved path rather than a straight line. This effect is commonly known as “gravitational lensing” and can be observed when light from distant stars and galaxies is bent by massive objects such as black holes or clusters of galaxies.

The theory of General Relativity has been confirmed by numerous experiments and observations, and it remains one of the most successful theories in physics. It has played a crucial role in our understanding of the universe and has led to the development of new technologies such as GPS and gravitational wave detectors.

**Some analogies to day-to-day life:**
**Gravitational Lensing:** Imagine you are looking at a distant star through a curved piece of glass. The glass will bend the light coming from the star, causing it to appear distorted or magnified. Similarly, massive objects such as black holes or galaxy clusters can warp the fabric of space-time, bending light from distant objects around them. This effect is known as gravitational lensing and is a direct consequence of General Relativity.
**GPS and Time Dilation:** GPS satellites rely on precise timekeeping to accurately determine the location of GPS receivers on Earth. However, the satellites are moving at high speeds relative to the surface of the Earth, which causes them to experience time dilation, meaning they experience time slower than objects on the Earth’s surface. If the timekeeping on the satellites were not corrected for this effect, GPS signals would be inaccurate by several kilometers.
**The Rubber Sheet Analogy:** Imagine a heavy ball sitting on a rubber sheet, causing the sheet to sag and stretch around it. Now imagine rolling a smaller ball around the heavy ball. The smaller ball will follow a curved path around the heavy ball, following the curved space-time geometry caused by the massive object. This is similar to how planets and stars move around massive objects like black holes or galaxies in the curved space-time.

**Some Mathematics behind this:**
The mathematical equation of the General Theory of Relativity is known as the Einstein Field Equation, which relates the geometry of spacetime to the distribution of matter and energy within it.

It can be written as:

where Rμν represents the

**Ricci curvature** tensor, R is the scalar curvature, gμν represents the metric tensor, and Tμν is the stress-energy tensor that describes the distribution of matter and energy in spacetime.

This equation shows the interplay between spacetime curvature and matter-energy distribution. It tells us that matter and energy are gravity sources and warp the fabric of spacetime. The curvature of spacetime, in turn, affects the motion of matter and energy, leading to the observed effects of gravity.

The Einstein Field Equation is a cornerstone of modern physics and has led to many important discoveries, including predicting black holes and gravitational waves.

To calculate the gravitational redshift of a light wave near a massive object, we can use the equation:

General Relativity has helped us understand the properties of black holes and the detection of gravitational waves. Black holes are massive objects so dense and have such a strong gravitational pull that nothing, not even light, can escape their grasp. The theory of General Relativity explains that a black hole is created when a massive object collapses in on itself, forming a singularity at the center.

Photo by

Yong Chuan Tan on

Unsplash

Black holes have unique properties, such as an event horizon, which is the point of no return beyond which anything that enters cannot escape the black hole’s gravitational pull. They also have a mass, a spin, and an angular momentum, which can be measured by observing the objects around them.

# Define the function that describes the warped spacetime

def spacetime(x, y):

r = np.sqrt(x**2 + y**2)

return 1 - (2/r)

# Set up the plot

fig = plt.figure(figsize=(8,8))

ax = fig.add_subplot(111, projection='3d')

# Define the range of x and y values

x = np.linspace(-10, 10, 100)

y = np.linspace(-10, 10, 100)

X, Y = np.meshgrid(x, y)

# Generate the surface representing the warped spacetime

Z = spacetime(X, Y)

ax.plot_surface(X, Y, Z, cmap='coolwarm')

# Add labels and titles

ax.set_xlabel('x')

ax.set_ylabel('y')

ax.set_zlabel('Warped spacetime')

ax.set_title('Visualization of warped spacetime around a massive object')

In 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) detected gravitational waves for the first time, providing direct evidence of the existence of black holes. Gravitational waves are ripples in the fabric of space-time caused by the motion of massive objects. When two black holes merge, they emit strong gravitational waves that can be detected by sensitive instruments like LIGO.

The detection of gravitational waves has opened up a new window into the universe, allowing us to observe the most extreme events in the cosmos, such as the collision of black holes and neutron stars. It has also provided us with new insights into the nature of gravity and the behavior of massive objects in space.

**Conclusion:**
General Relativity is a scientific theory describing how massive objects bend spacetime and cause other objects to move in curved paths. This theory has far-reaching consequences that affect our daily lives, from GPS technology to detect gravitational waves. It has helped us understand the properties of black holes and opened a new window into the universe.

Our understanding of General Relativity has led to new technologies and insights that have revolutionized our world, from the development of GPS to the discovery of black holes and the detection of gravitational waves. Its impact on our understanding of the cosmos will continue to shape our view of the universe for years to come.

As we continue to learn about science and its impact on our daily lives, we can appreciate how General Relativity has shaped our world. We can also continue to explore the universe around us and discover new insights that will lead to even more remarkable discoveries and technologies.

**References:**
- A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie” (“The Foundation of the General Theory of Relativity”), Annalen der Physik, vol. 354, pp. 769–822, 1916.
- S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, 1972.
- C. Misner, K. Thorne, and J. Wheeler, Gravitation, W.H. Freeman and Company, 1973.
- R. Wald, General Relativity, University of Chicago Press, 1984.
- L. Smolin, The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next, Houghton Mifflin Harcourt, 2006.
- S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley, 2004.
- A. Lightman, W. Press, R. Price, and S. Teukolsky, Problem Book in Relativity and Gravitation, Princeton University Press, 1975.
- P. Schneider, J. Ehlers, and E.E. Falco, Gravitational Lenses, Springer-Verlag, 1992.
- L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, 2013.
- S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, 1983.

See:

https://www.britannica.com/science/relativity/Curved-space-time-and-geometric-gravitation
Whereas Newton thought that gravity was a

force, Einstein showed that gravity arises from the shape of space-time. While this is difficult to visualize, there is an

analogy that provides some insight—although it is only a guide, not a definitive statement of the theory.

The analogy begins by considering space-time as a rubber sheet that can be deformed. In any region distant from massive cosmic objects such as stars, space-time is uncurved—that is, the rubber sheet is absolutely flat. If one were to probe space-time in that region by sending out a ray of

light or a test body, both the ray and the body would travel in perfectly straight lines, like a child’s marble rolling across the rubber sheet.

curved space-time

The four dimensional space-time continuum itself is distorted in the vicinity of any mass, with the amount of distortion depending on the mass and the distance from the mass. Thus, relativity accounts for Newton's inverse square law of gravity through geometry and thereby does away with the need for any mysterious “action at a distance.” (more)

However, the presence of a massive body curves space-time, as if a bowling ball were placed on the rubber sheet to create a cuplike depression. In the analogy, a marble placed near the depression rolls down the slope toward the bowling ball as if pulled by a force. In addition, if the marble is given a sideways push, it will describe an orbit around the bowling ball, as if a steady pull toward the ball is swinging the marble into a closed path.

In this way, the

curvature of space-time near a

star defines the shortest natural paths, or

geodesics—much as the shortest path between any two points on Earth is not a straight line, which cannot be constructed on that curved surface, but the arc of a

great circle route. In Einstein’s theory, space-time geodesics define the

deflection of light and the orbits of planets. As the American theoretical physicist

John Wheeler put it, matter tells space-time how to curve, and space-time tells matter how to move.

### The mathematics of general relativity

The rubber sheet analogy helps with visualization of space-time, but Einstein himself developed a complete quantitative theory that describes space-time through highly abstract mathematics. General relativity is expressed in a set of interlinked

differential equations that define how the shape of space-time depends on the amount of matter (or, equivalently, energy) in the region. The solution of these so-called

field equations can yield answers to different physical situations, including the behaviour of individual bodies and of the entire

universe.

### Cosmological solutions

Einstein immediately understood that the field equations could describe the entire cosmos. In 1917 he modified the original version of his equations by adding what he called the “cosmological term.” This represented a force that acted to make the universe expand, thus counteracting gravity, which tends to make the universe contract. The result was a static universe, in accordance with the best knowledge of the time.

In 1922, however, the Soviet mathematician

Aleksandr Aleksandrovich Friedmannshowed that the field equations predict a

dynamic universe, which can either expand forever or go through cycles of alternating expansion and contraction. Einstein came to agree with this result and abandoned his cosmological term. Later

work, notably pioneering measurements by the American astronomer

Edwin Hubble and the development of the

big-bang model, has confirmed and amplified the concept of an

expanding universe.

In 1916 the German astronomer

Karl Schwarzschild used the field equations to calculate the gravitational effect of a single spherical body such as a star. If the mass is neither very large nor highly concentrated, the resulting calculation will be the same as that given by Newton’s theory of gravity. Thus, Newton’s theory is not incorrect; rather, it

constitutes a valid approximation to

general relativity under certain conditions.

Schwarzschild also described a new effect. If the mass is concentrated in a vanishingly small volume—a

singularity—gravity will become so strong that nothing pulled into the surrounding region can ever leave. Even light cannot escape. In the rubber sheet analogy, it as if a tiny massive object creates a depression so steep that nothing can escape it. In recognition that this severe space-time distortion would be invisible—because it would absorb light and never emit any—it was dubbed a

black hole.

In quantitative terms, Schwarzschild’s result defines a sphere that is centred at the

singularity and whose radius depends on the density of the enclosed mass. Events within the sphere are forever isolated from the remainder of the universe; for this reason, the

Schwarzschild radius is called the

event horizon.

No human technology could compact matter sufficiently to make

black holes, but they occur as final steps in the life cycle of stars. After millions or billions of years, a star uses up all of its hydrogen and other elements that produce

energy through

nuclear fusion. With its nuclear furnace banked, the star no longer maintains an internal pressure to expand, and gravity is left unopposed to pull inward and compress the star. For stars above a certain mass, this gravitational collapse will produce a black hole containing several times the mass of the

Sun. In other cases, the gravitational collapse of huge dust clouds can create supermassive black holes containing millions or billions of solar masses.

black hole in M87

Black hole at the centre of the massive galaxy M87, about 55 million light-years from Earth, as imaged by the Event Horizon Telescope (EHT). The black hole is 6.5 billion times more massive than the Sun. This picture was the first direct visual evidence of a supermassive black hole and its shadow. The ring is brighter on one side because the black hole is rotating, and thus material on the side of the black hole turning toward Earth has its emission boosted by the Doppler effect. The shadow of the black hole is about five and a half times larger than the event horizon, the boundary marking the black hole's limits, where the escape velocity is equal to the speed of light. Created from data collected in 2017, this picture was released in 2019.(more)

Astrophysicists have found many

cosmicobjects that contain such a dense concentration of mass in a small volume. These black holes include one at the centre of the

Milky Way Galaxy (Sagittarius A*) and certain binary stars that emit

X-rays as they orbit each other. One, at the centre of the galaxy M87, has even been directly imaged.

The theory of black holes has led to another predicted entity, a

wormhole. This is a solution of the field equations that resembles a tunnel between two black holes or other points in space-time. Such a tunnel would provide a shortcut between its end points. In analogy, consider an ant walking across a flat sheet of paper from point

*A* to point

*B*. If the paper is curved through the third

dimension, so that

*A* and

*B* overlap, the ant can step directly from one point to the other, thus avoiding a long trek.

The possibility of short-circuiting the enormous distances between stars makes wormholes attractive for

space travel. Because the tunnel links moments in time as well as locations in space, it also has been argued that a wormhole would allow travel into the past. However, wormholes are intrinsically unstable. While exotic stabilization schemes have been proposed, there is as yet no evidence that these can work or indeed that wormholes exist.

### Experimental evidence for general relativity

experimental evidence for general relativity

In 1919 observation of a solar eclipse confirmed Einstein's prediction that light is bent in the presence of mass. This experimental support for his general theory of relativity garnered him instant worldwide acclaim. (more)

Soon after the theory of

general relativity was published in 1915, the English astronomer

Arthur Eddington considered Einstein’s

prediction that

light rays are bent near a massive body, and he realized that it could be verified by carefully comparing

star positions in images of the

Sun taken during a solar

eclipse with images of the same region of

space taken when the Sun was in a different portion of the sky. Verification was delayed by

World War I, but in 1919 an excellent opportunity presented itself with an especially long total

solar eclipse, in the vicinity of the bright

Hyades star cluster, that was visible from northern Brazil to the African coast.

Eddington led one expedition to Príncipe, an island off the African coast, and Andrew Crommelin of the

Royal Greenwich Observatory led a second

expedition to Sobral, Brazil. After carefully comparing photographs from both expeditions with reference photographs of the Hyades, Eddington declared that the starlight had been deflected about 1.75 seconds of arc, as predicted by general relativity. (The same effect produces gravitational lensing, where a massive cosmic object focuses light from another object beyond it to produce a distorted or magnified image. The astronomical discovery of gravitational lenses in 1979 gave additional support for general relativity.)

Further evidence came from the

planet Mercury. In the 19th century, it was found that Mercury does not return to exactly the same spot every time it completes its

ellipticalorbit. Instead, the

ellipse rotates slowly in space, so that on each orbit the

perihelion—the point of closest approach to the Sun—moves to a slightly different angle. Newton’s law of

gravity could not explain this perihelion shift, but general relativity gave the correct orbit.

Another confirmed prediction of general relativity is that time dilates in a gravitational

field, meaning that clocks run slower as they approach the mass that is producing the field. This has been measured directly and also through the

gravitational redshift of light. Time dilation causes light to vibrate at a lower frequency within a gravitational field; thus, the light is shifted toward a longer wavelength—that is, toward the red. Other measurements have

verified the

equivalence principle by showing that inertial and gravitational mass are precisely the same.

The most striking prediction of general relativity is that of

gravitational waves. Electromagnetic waves are caused by accelerated electrical charges and are detected when they put other charges into

motion. Similarly, gravitational waves would be caused by masses in motion and are detected when they initiate motion in other masses. However, gravity is very weak compared with

electromagnetism. Only a huge cosmic event, such as the collision of two stars, can generate detectable gravitational waves. Efforts to sense gravitational waves began in the 1960s, and such waves were first detected in 2015 when

LIGO observed two black holes 1.3 million light-years away spiralling into each other.

### Applications of relativistic ideas

Although relativistic effects are negligible in ordinary life, relativistic ideas appear in a range of areas from fundamental

science to civilian and

military technology.

The relationship

*E* =

*mc*2 is essential in the study of

subatomic particles. It determines the

energy required to create particles or to convert one type into another and the energy released when a particle is

annihilated. For example, two

photons, each of energy

*E*, can collide to form two particles, each with mass

*m* =

*E*/

*c*2. This pair-production process is one step in the early evolution of the

universe, as described in the

big-bang model.

Knowledge of elementary particles comes primarily from

particle accelerators. These machines raise subatomic particles, usually

electrons or

protons, to nearly the

speed of light. When these energetic bullets smash into selected targets, they elucidate how subatomic particles interact and often produce new species of elementary particles.

Particle accelerators could not be properly designed without

special relativity. In the type called an

electron synchrotron, for instance, electrons gain energy as they

traversea huge circular raceway. At barely below the speed of light, their mass is thousands of times larger than their rest mass. As a result, the

magnetic field used to hold the electrons in circular orbits must be thousands of times stronger than if the mass did not change.

### Fission and fusion: bombs and stellar processes

Energy is released in two kinds of nuclear processes. In

nuclear fission a heavy nucleus, such as

uranium, splits into two lighter nuclei; in

nuclear fusion two light nuclei combine into a heavier one. In each process the total final mass is less than the starting mass. The difference appears as energy according to the relation

*E* = Δ

*mc*2, where Δ

*m*is the mass

deficit.

Fission is used in atomic bombs and in reactors that produce power for civilian and military applications. The fusion of hydrogen into helium is the energy source in stars and provides the power of a

hydrogen bomb. Efforts are now under way to develop controllable hydrogen fusion as a clean, abundant power source.

Learn about the significance of gravitational waves in science and in everyday life.(more)

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The

global positioning system (GPS) depends on relativistic principles. A GPS receiver determines its location on Earth’s surface by processing radio signals from four or more satellites. The distance to each satellite is calculated as the product of the speed of light and the time lag between transmission and reception of the signal. However, Earth’s gravitational field and the motion of the satellites cause time-dilation effects, and Earth’s rotation also has relativistic

implications. Hence, GPS technology includes relativistic corrections that enable positions to be calculated to within several centimetres.

Cosmology, the study of the structure and origin of the universe, is intimately connected with gravity, which determines the macroscopic behaviour of all matter. General relativity has played a role in cosmology since the early calculations of Einstein and Friedmann. Since then, the theory has provided a framework for accommodating observational results, such as Hubble’s discovery of the

expanding universe in 1929, as well as the

big-bang model, which is the generally accepted explanation of the origin of the universe.

The latest solutions of Einstein’s field equations depend on specific

parameters that characterize the fate and shape of the universe. One is

Hubble’s constant, which defines how rapidly the universe is expanding; the other is the density of matter in the universe, which determines the strength of gravity. Below a certain critical density, gravity would be weak enough that the universe would expand forever, so that space would be unlimited. Above that value, gravity would be strong enough to make the universe shrink back to its original minute size after a finite period of expansion, a process called the “big crunch.” In this case, space would be limited or bounded like the surface of a sphere. Current efforts in observational cosmology focus on measuring the most accurate possible values of Hubble’s constant and of critical density.

### Relativity, quantum theory, and unified theories

Cosmic behaviour on the biggest scale is described by general relativity. Behaviour on the subatomic scale is described by

quantum mechanics, which began with the

work of the German physicist

Max Planck in 1900 and treats energy and other physical quantities in discrete units called

quanta. A central goal of

physics has been to combine relativity theory and

quantum theory into an overarching “theory of everything” describing all physical phenomena. Quantum theory explains electromagnetism and the strong and weak forces, but a quantum description of the remaining

fundamental force of gravity has not been achieved.

After Einstein developed relativity, he unsuccessfully sought a so-called

unified field theory with a

space-time geometry that would

encompass all the fundamental forces. Other theorists have attempted to merge general relativity with quantum theory, but the two approaches treat forces in fundamentally different ways. In quantum theory, forces arise from the interchange of certain elementary particles, not from the shape of space-time. Furthermore, quantum effects are thought to cause a serious distortion of space-time at an extremely small scale called the Planck length, which is much smaller than the size of elementary particles. This suggests that quantum gravity cannot be understood without treating space-time at unheard-of scales.

Although the connection between general relativity and quantum mechanics remains

elusive, some progress has been made toward a fully unified theory. In the 1960s, the

electroweak theory provided partial unification, showing a common basis for electromagnetism and the

weak force within quantum theory. Recent research suggests that

superstring theory, in which elementary particles are represented not as mathematical points but as extremely small strings vibrating in 10 or more dimensions, shows promise for supporting complete unification, including gravitation. However, until confirmed by experimental results, superstring theory will remain an untested

hypothesis.

### Reactions in general culture

The impact of relativity has not been limited to

science. Special relativity arrived on the scene at the beginning of the 20th century, and

general relativity became widely known after World War I—eras when a new sensibility of “modernism” was becoming defined in art and literature. In addition, the confirmation of general relativity provided by the

solar eclipse of 1919 received wide publicity. Einstein’s 1921

Nobel Prize for Physics (awarded for his

work on the photon nature of light), as well as the popular perception that relativity was so complex that few could grasp it, quickly turned Einstein and his theories into cultural icons.

The ideas of relativity were widely applied—and misapplied—soon after their advent. Some thinkers interpreted the theory as meaning simply that all things are relative, and they employed this concept in arenas distant from

physics. The Spanish humanist philosopher and essayist

José Ortega y Gasset, for instance, wrote in

*The Modern Theme* (1923),

The theory of Einstein is a marvelous proof of the harmonious multiplicity of all possible points of view. If the idea is extended to

morals and

aesthetics, we shall come to experience

history and life in a new way.

The revolutionary aspect of Einstein’s thought was also seized upon, as by the American art critic Thomas

Craven, who in 1921 compared the break between classical and modern art to the break between Newtonian and Einsteinian ideas about

spaceand time.

Some saw specific relations between relativity and art arising from the idea of a four-dimensional

space-time continuum. In the 19th century, developments in geometry led to popular interest in a fourth spatial dimension, imagined as somehow lying at right angles to all three of the ordinary dimensions of length, width, and height. Edwin Abbott’s

*Flatland*(1884) was the first popular presentation of these ideas. Other works of fantasy that followed spoke of the fourth dimension as an arena apart from ordinary existence.

Einstein’s four-dimensional

universe, with three spatial dimensions and one of time, is conceptually different from four spatial dimensions. But the two kinds of four-dimensional world became

conflated in interpreting the new art of the 20th century. Early

Cubist works by

Pablo Picasso that simultaneously portrayed all sides of their subjects became connected with the idea of higher dimensions in space, which some writers attempted to relate to relativity. In 1949, for example, the art historian Paul LaPorte wrote that “the new pictorial

idiomcreated by [C]ubism is most satisfactorily explained by applying to it the concept of the space-time continuum.” Einstein specifically rejected this view, saying, “This new artistic ‘language’ has nothing in common with the Theory of Relativity.” Nevertheless, some artists explicitly explored Einstein’s ideas. In the new

Soviet Union of the 1920s, for example, the poet and illustrator

Vladimir Mayakovsky, a founder of the artistic

movement called Russian

Futurism, or

Suprematism, hired an expert to explain relativity to him.

The widespread general interest in relativity was reflected in the number of books written to elucidate the subject for nonexperts. Einstein’s popular

exposition of special and general relativity appeared almost immediately, in 1916, and his article on space-time appeared in the 13th edition of

*Encyclopædia Britannica* in 1926. Other scientists, such as the Russian mathematician

Aleksandr Friedmann and the British astronomer

Arthur Eddington, wrote popular books on the subjects in the 1920s. Such books continued to appear decades later.

When relativity was first announced, the public was typically awestruck by its complexity, a justified response to the intricate mathematics of general relativity. But the abstract, nonvisceral nature of the theory also generated reactions against its apparent violation of common sense. These reactions included a political undertone; in some quarters, it was considered undemocratic to present or support a theory that could not be immediately understood by the common person.

In contemporary usage, general

culture has accepted the ideas of relativity—the impossibility of faster-than-light travel,

*E* =

*mc*2,

time dilation and the

twin paradox, the

expanding universe, and black holes and wormholes—to the point where they are immediately recognized in the media and provide plot devices for works of

science fiction. Some of these ideas have gained meaning beyond their strictly scientific ones; in the business world, for instance, “black hole” can mean an unrecoverable financial drain.

In 1925 the British philosopher

Bertrand Russell, in his

*ABC of Relativity*, suggested that Einstein’s

work would lead to new philosophical concepts. Relativity has indeed had a great effect on philosophy,

illuminating some issues that go back to the ancient Greeks. The idea of the

ether,

invoked in the late 19th century to carry

light waves, harks back to

Aristotle. He divided the world into earth, air, fire, and water, with the

ether (aether) as the fifth element representing the pure

celestial sphere. The

Michelson-Morley experiment and relativity eliminated the last vestiges of this idea.

Relativity also changed the meaning of

geometry as it was developed in

Euclid’s

*Elements* (

*c.* 300 BCE). Euclid’s system relied on the axiom “a straight line is the shortest distance between two points,” among others that seemed self-evidently true. Straight lines also played a special role in Euclid’s

*Optics* as the paths followed by light rays. To philosophers such as the German

Immanuel Kant, Euclid’s straight-line axiom represented a deep level of truth. But

general relativity makes it possible scientifically to examine

space like any other physical quantity—that is, to investigate Euclid’s

premises. It is now known that

space-time is curved near stars; no straight lines exist there, and light follows curved geodesics. Like Newton’s law of

gravity, Euclid’s geometry correctly describes reality under certain conditions, but its axioms are not absolutely fundamental and universal, for the cosmos includes

non-Euclidean geometries as well.

See:

https://www.britannica.com/science/relativity/Philosophical-considerations
Spacetime does not tear. Its fabric just above Earth’s surface experiences the same lateral contractility as it does just below the surface. Not so with the curvature in the two-dimensional domain defined by time and by direction perpendicular to Earth’s surface. In that one plane, curvature within Earth is contractile but suddenly jumps just above Earth’s surface to the opposite character. Hence the tide-producing character of spacetime curvature outside Earth. A point twice as far from Earth’s center lies on an imaginary Earth-centered sphere that encompasses eight times the volume. There the tide-producing curvature experiences eight times the dilution and has one eighth the strength Despite this rapid dilution of tide-producing power with distance, it has strength enough at Moon, 60 Earth radii away from Earth’s center, to deform Moon from sphere to ellipsoid, 1738.35 kilometers in radius along the Earth-Moon direction, 1738.15kilometers in radius for each of the other two perpendicular directions.

Easy as it is to regard Earth as running the whole show, Moon too has its part. Like an infant standing on the trampoline some distance from its mother, it imposes its own small curvature on top of the curvature evoked by Earth. That additional curvature, contractile in Moon’s interior, has tide-driving character outside. Were the Earth an ideal sphere covered by an ideal ocean of uniform depth, then Moon would draw that ocean’s surface 35.6 centimeters higher than the average in two domains, one directly facing Moon, one directly opposite to it - simultaneously lowering those waters 17.8 centimeters below the average on the circle of points midway between the two. (These low figures show how important are funneling and resonant sloshing in determining heights of actual ocean tides on Earth.)

The local contractile curvature of spacetime at Moon’s location added up along Moon’s path yields the appearance of long-range gravitation, similar to that illustrated in Figure 9-6. Box 2-1 tells a little of the many influences that have to be taken into account in any fuller treatment of the tides.

Spacetime controls momenergy Obedience to the eyes of a corps of bookkeepers? No, Einstein taught us. The enforcing agency does not lie far away. It’s close at hand. It’s the geometry of spacetime, right where the crash takes place. Not only does spacetime grip isolated mass, telling it how to move. In addition, in a crash it sees to it that the participants neither gain nor lose momenergy. But there is more! Spacetime, in so acting, cannot maintain the perfection assumed in textbooks of old. To every action there is a corresponding reaction.

*Spacetime acts on momenergy, telling it bow to move; momenergy reacts back on spacetime, telling it bow to curve*. This "handshake" between momenergy and spacetime is the origin of momenergy conservation - and the source of spacetime curvature that leads to gravitation.

See:

https://phys.libretexts.org/Bookshe...n/9.06:_Gravitation_as_Curvature_of_Spacetime
See:

https://www.semanticscholar.org/pap...tein/795e91a5e2779177fb4eb6f6689e1158fde2d19c
The geometry used to describe motion in any local free-float frame is the flat-spacetime geometry of Lorentz (special relativity). Relative to such a local freefloat frame, every nearby electrically neutral test particle moves in a straight line with constant velocity. Slightly more remote particles are detected as slowly changing their velocities, or the directions of their worldlines in spacetime. These changes are described as tidal effects of gravitation. They are understood as originating in the local curvature of spacetime. Gravitation shows itself not at all in the motion of one test particle but only in the change of separation of two or more nearby test particles. "Rather than have one global frame with gravitational forces we have many local frames without gravitational forces." However, these local dimension changes add up to an effect on the global spacetime structure that one interprets as "gravitation" in its everyday manifestations.

Hartmann352