# Introduction to general relativity: Wikis

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# Encyclopedia

High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of space and time (blue lines) due to the Sun's mass.
General relativity
$G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}$
Einstein field equations
Introduction
Mathematical formulation
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General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915. According to general relativity, the observed gravitational attraction between masses results from the warping of space and time by those masses.

Before the advent of general relativity, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. Under Newton's model, gravity was the result of an attractive force between massive objects. Although even Newton was bothered by the unknown nature of that force, the basic framework was extremely successful at describing motion.

However, experiments and observations show that Einstein's description accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment, while others are the subject of ongoing research. For example, although there is indirect evidence for gravitational waves, direct evidence of their existence is still being sought by several teams of scientists in experiments such as the LIGO and GEO 600 projects.

General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.

Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. Nevertheless, a number of open questions remain: the most fundamental is how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.

## From special to general relativity

In September 1905, Albert Einstein published his theory of special relativity, which reconciles Newton's laws of motion with electrodynamics (the interaction between objects with electric charge). Special relativity introduced a new framework for all of physics by proposing new concepts of space and time. Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of gravity, which describes the mutual attraction experienced by bodies due to their mass.

Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity. Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking between 1907 and 1915, from his simple thought experiment involving an observer in free fall to his fully geometric theory of gravity. [1]

### Equivalence principle

A person in a free-falling elevator experiences weightlessness during their fall, and objects either float alongside them or drift at constant speed. Since everything in the elevator is falling together, no gravitational effect can be observed. In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any sufficient source of gravity. Such observers are the privileged ("inertial") observers Einstein described in his theory of special relativity: observers for whom light travels along straight lines at constant speed.[2]

Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his equivalence principle. Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.[3]

### Gravity and acceleration

Ball falling to the floor in an accelerated rocket (left) and on Earth (right)

Just as most effects of gravity can be made to vanish by observing them in free fall, the same effects can be produced by observing objects in an accelerated frame of reference. An observer in a closed room cannot tell which of the following is true:

• Objects are falling to the floor because the room is resting on the surface of the Earth and the objects are being pulled down by gravity.
• Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 m/s2 and is far from any source of gravity. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into his seat.

Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength. This principle allowed Einstein to predict several novel effects of gravity in 1907, as explained in the next section.

An observer in an accelerated reference frame must introduce what physicists call fictitious forces to account for the acceleration experienced by himself and objects around him. One example, the force pressing the driver of an accelerating car into his or her seat, has already been mentioned; another is the force you can feel pulling your arms up and out if you attempt to spin around like a top. Einstein's key insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces.[4] Since fictitious forces are always proportional to the mass of the object on which they act, an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in Newton's law of gravitation.

### Physical consequences

In 1907, Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle.[5]

The gravitational redshift of a light wave as it moves upwards against a gravitational field (caused by the yellow star below).

The first new effect is the gravitational frequency shift of light. Consider two observers aboard an accelerating rocket-ship. Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and unattached objects accelerate in the opposite direction, falling "downward". Assume that one of the observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration causes the light to be red-shifted, as may be calculated from special relativity; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the higher observer to the lower is blue-shifted, that is, shifted towards higher frequencies.[6] Einstein argued that such frequency shifts must be also observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below.

This gravitational frequency shift corresponds to a gravitational time dilation: Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer. Thus, time runs slower for observers who are lower in a gravitational field.

It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame. Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity. It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher.[7] This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described below.

In a similar way, Einstein predicted the gravitational deflection of light: in a gravitational field, light is deflected downward. Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle.[8]

### Tidal effects

Two bodies falling towards the center of the Earth accelerate towards each other as they fall.

The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity. Notably, it does not answer the following simple question: what keeps people on the other side of the world from falling off? When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that fictitious forces are to be expected, provides a suitable explanation. But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.

A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly – but not exactly so. These bodies are not falling in precisely the same direction, but towards a single point in space: namely, the Earth's center of gravity. Consequently, there is a component of each body's motion towards the other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for skydivers on opposite sides of the Earth, the effect is large. Such differences in force are also responsible for the tides in the Earth's oceans, so the term "tidal effect" is used for this phenomenon.

The equivalence between inertia and gravity cannot explain tidal effects – it cannot explain variations in the gravitational field.[9] For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.

### From acceleration to geometry

In exploring the equivalence of gravity and acceleration as well as the role of tidal forces, Einstein discovered several analogies with the geometry of surfaces. An example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speeds) to a rotating reference frame (in which extra terms corresponding to fictitious forces have to be introduced in order to explain particle motion): this is analogous to the transition from a Cartesian coordinate system (in which the coordinate lines are straight lines) to a curved coordinate system (where coordinate lines need not be straight).

A deeper analogy relates tidal forces with a property of surfaces called curvature. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is equivalent to a plane. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.[10]

The elementary objects of geometrypoints, lines, triangles – are traditionally defined in three-dimensional space or on two-dimensional surfaces. In 1907, however, the mathematician Hermann Minkowski introduced a geometric formulation of Einstein's special theory of relativity in which the geometry included not only space, but also time. The basic entity of this new geometry is four-dimensional spacetime. The orbits of moving bodies are lines in spacetime; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.[11]

For surfaces, the generalization from the geometry of a plane – a flat surface – to that of a general curved surface had been described in the early nineteenth century by Carl Friedrich Gauss. This description had in turn been generalized to higher-dimensional spaces in a mathematical formalism introduced by Bernhard Riemann in the 1850s. With the help of Riemannian geometry, Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces.[12]

After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how matter influences spacetime's curvature. Having formulated what are now known as Einstein's equations (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late 1915.[13]

## Geometry and gravitation

Paraphrasing the doyen of American relativity research, John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.[14] What this means is addressed in the following three sections, which explore the motion of so-called test particles, examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime.

### Probing the gravitational field

Converging geodesics: two lines of longitude (green) that start out in parallel at the equator (red) but converge to meet at the pole

In order to map a body's gravitational influence, it is useful to think about what physicists call probe or test particles: particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of spacetime, this is equivalent to saying that such test particles move along straight world lines in spacetime. In the presence of gravity, however, spacetime is non-Euclidean, or curved. In such a spacetime, straight world lines may not exist. Instead, test particles move along lines called geodesics, which are "as straight as possible".

A simple analogy is the following: In geodesy, the science of measuring Earth's size and shape, a geodesic (from Greek "geo", Earth, and "daiein", to divide) is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a great circle, such as a line of longitude or the equator. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint.

The properties of geodesics differ from those of straight lines. For example, in a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of test particles in free fall are spacetime geodesics, the straightest possible lines in spacetime. But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity. In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case. If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.[15]

Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens. Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A person sitting on a chair is trying to follow a geodesic, that is, to fall freely towards the center of the Earth. But the chair applies an external upwards force preventing the person from falling. In this way, general relativity explains the daily experience of gravity on the surface of the Earth not as the downwards pull of a gravitational force, but as the upwards push of external forces. These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow.[16] For matter objects whose own gravitational influence cannot be neglected, the laws of motion are somewhat more complicated than for test particles, although it remains true that spacetime tells matter how to move.[17]

### Sources of gravity

In Newton's description of gravity, the gravitational force is caused by matter. More precisely, it is caused by a specific property of material objects: their mass. In Einstein's theory and related theories of gravitation, curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.

The equivalence between mass and energy, as expressed by the formula E = mc2, is perhaps the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its temperature or the binding energy of systems such as nuclei or molecules, contribute to that body's mass, and hence act as sources of gravity.[18]

In special relativity, energy is closely connected to momentum. Just as space and time are, in that theory, different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call four-momentum. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal pressure and tension. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the energy-momentum tensor.[19]

### Einstein's equations

Einstein's equations are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of Riemannian geometry, in which the geometric properties of a space (or a spacetime) are described by a quantity called a metric. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).

Distances corresponding to 30 degrees difference in longitude, at different latitudes.

A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic latitude and longitude. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,051 mi). On the other hand, someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,181 mi). Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.[20]

The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the space (or spacetime) is curved at each point. In general relativity, the metric and the Riemann curvature tensor are quantities defined at each point in spacetime. As has already been mentioned, the matter content of the spacetime defines another quantity, the Energy-momentum tensor T, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity G, now called the Einstein tensor, which describes some aspects of the way spacetime is curved. Einstein's equation then states that

$\mathbf{G}=\frac{8\pi G}{c^4}\mathbf{T},$

i.e., up to a constant multiple, the quantity G (which measures curvature) is equated with the quantity T (which measures matter content). The constants involved in this equation reflect the different theories that went into its making: G is the gravitational constant that is already present in Newtonian gravity; c is the speed of light, the key constant in special relativity; and π is one of the basic constants of geometry.

This equation is often referred to in the plural as Einstein's equations, since the quantities G and T are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions.[21] A solution of these equations describes a particular geometry of space and time; for example, the Schwarzschild solution describes the geometry around a spherical, non-rotating mass such as a star or a black hole, whereas the Kerr solution describes a rotating black hole. Still other solutions can describe a gravitational wave or, in the case of the Friedmann-Lemaître-Robertson-Walker solution, an expanding universe. The simplest solution is the uncurved Minkowski spacetime, the spacetime described by special relativity.[22]

## Experimental tests

No scientific theory is apodictically true; each is a model that must be checked by experiment. A theory is falsified if it unambiguously fails even a single experiment. Newton's law of gravity was accepted because it accounted for the motion of planets and moons in the solar system with exquisite accuracy. However, as the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed. These discrepancies were accounted for in the general theory of relativity, but the predictions of that theory must also be checked with experiment. Three experimental tests were devised by Einstein himself and are now known as the classical tests of the theory:

Newtonian (red) vs. Einsteinian orbit (blue) of a single planet orbiting a spherical star.
• Newtonian gravity predicts that the orbit which a single planet traces around a perfectly spherical star should be an ellipse. Einstein's theory predicts a more complicated curve: the planet behaves as if it were travelling around an ellipse, but at the same time, the ellipse as a whole is rotating slowly around the star. In the diagram on the right, the ellipse predicted by Newtonian gravity is shown in red, and part of the orbit predicted by Einstein in blue. For a planet orbiting the Sun, this deviation from Newton's orbits is known as the anomalous perihelion shift. The first measurement of this effect, for the planet Mercury, dates back to 1859. The most accurate results for Mercury and for other planets to date are based on measurements which were undertaken between 1966 and 1990, using radio telescopes.[23] General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth).
• According to general relativity, light does not travel along straight lines when it propagates in a gravitational field. Instead, it is deflected in the presence of massive bodies. In particular, starlight is deflected as it passes near the Sun, leading to apparent shifts of up 1.75 arc seconds in the stars' positions in the night sky (an arc second is equal to 1/3600 of a degree). In the framework of Newtonian gravity, a heuristic argument can be made that leads to light deflection by half that amount. The different predictions can be tested by observing stars that are close to the Sun during a solar eclipse. In this way, a British expedition to Brazil and West Africa in 1919, directed by Arthur Eddington, confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong. Eddington's results were not very accurate; subsequent observations of the deflection of the light of distant quasars by the Sun, which utilize highly accurate techniques of radio astronomy, have confirmed Eddington's results with significantly better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).[24]
• Gravitational redshift was first measured in a laboratory setting in 1959 by Pound and Rebka. It is also seen in astrophysical measurements, notably for light escaping the White Dwarf Sirius B. The related gravitational time dilation effect has been measured by transporting atomic clocks to altitudes of between tens and tens of thousands of kilometers (first by Hafele and Keating in 1971; most accurately to date by Gravity Probe A launched in 1976).[25]

Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in 1916. The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted Einstein to international stardom.[26] These three experimental tests justified adopting general relativity over Newton's theory and, incidentally, over a number of alternatives to general relativity that had been proposed.

Gravity Probe B with its solar panels folded

Further tests of general relativity include precision measurements of the Shapiro effect or gravitational time delay for light, most recently in 2002 by the Cassini space probe. One set of tests focuses on effects predicted by general relativity for the behavior of gyroscopes travelling through space. One of these effects, geodetic precession, has been tested with Lunar laser ranging experiments (high precision measurements of the orbit of the Moon). Another, which is related to rotating masses, is called frame-dragging. It is due to be tested by the Gravity Probe B satellite experiment launched in 2004, with results expected in late 2008.[27]

By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields. This has become possible thanks to precision observations of binary pulsars. In such a star system, two highly compact neutron stars orbit each other. At least one of them is a pulsar – an astronomical object that emits a tight beam of radiowaves. Similar to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink, these beams strike the Earth at very regular intervals, and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to those predicted by general relativity.[28]

One particular set of observations is related to eminently useful practical applications, namely to satellite navigation systems such as the Global Positioning System that are used both for precise positioning and timekeeping. Such systems rely on two sets of atomic clocks: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, the satellite clocks are either slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of universal coordinated time) are testament to the validity of the relativistic predictions.[29]

A number of other tests have probed the validity of various versions of the equivalence principle; strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself. So far, general relativity has passed all observational tests.[30]

## Astrophysical applications

Models based on general relativity play an important role in astrophysics, and the success of these models is further testament to the theory's validity.

### Gravitational lensing

Einstein cross: four images of the same astronomical object, produced by a gravitational lens

Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a quasar can pass along one side of a massive galaxy and be deflected slightly so as to reach an observer on Earth, while light passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction. As a result, that particular observer will see one astronomical object in two different places in the night sky. This kind of focussing is well-known when it comes to optical lenses, and hence the corresponding gravitational effect is called gravitational lensing.[31]

Observational astronomy uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the mass distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of dark matter, which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos.[32]

### Gravitational waves

Gravitational waves, a direct consequence of Einstein's theory, are distortions of geometry which propagate at the speed of light, and can be thought of as ripples in space-time. They should not be confused with the gravity waves of fluid dynamics, which are a different concept.

Indirectly, the effect of gravitational waves has been detected in observations of specific binary stars. Such pairs of stars orbit each other and, as they do so, gradually lose energy by emitting gravitational waves. For ordinary stars like our sun, this energy loss would be too small to be detectable. However, in 1974, this energy loss was observed in a binary pulsar called PSR1913+16. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as neutron star, for which gravitational wave emission is much stronger than for ordinary stars. Also, a pulsar emits a narrow beam of electromagnetic radiation from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate "clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of space-time in its immediate neighborhood.

The discoverers of PSR1913+16, Russell Hulse and Joseph Taylor, were awarded the Nobel prize in physics in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide the most accurate tests of general relativity.[33]

Currently, one major goal of research in relativity is the direct detection of gravitational waves. To this end, a number of land-based gravitational wave detectors are in operation, and a mission to launch a space-based detector, LISA, is currently under development, with a precursor mission (LISA Pathfinder) due for launch in late 2009. If gravitational waves are detected, they could be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early universe fractions of a second after the Big Bang.[34]

### Black holes

When mass is concentrated into a sufficiently compact region of space, general relativity predicts the formation of a black hole – a region of space with a gravitational attraction so strong that not even light can escape. Certain types of black holes are thought to be the final state in the evolution of massive stars. On the other hand, supermassive black holes with the mass of millions or billions of Suns are assumed to reside in the cores of most galaxies, and they play a key role in current models of how galaxies have formed over the past billions of years.[35]

Black hole-powered jet emanating from the central region of the galaxy M87

Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are quasars and other types of active galactic nuclei. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of jets, in which focused beams of matter are flung away into space at speeds near that of light.[36]

There are several properties that make black holes most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong. Another reason follows from what are called black hole uniqueness theorems: over time, black holes retain only a minimal set of distinguishing features (since different hair styles are a crucial part of what gives different people their different appearances, these theorems have become known as "no hair" theorems). For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass, but with one important difference: in its transition to a spherical shape, the black hole formed by the collapse of a cube will emit gravitational waves.[37]

### Cosmology

Image of radiation emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope WMAP

One of the most important aspects of general relativity is that it can be applied to the universe as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: All current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately homogeneous and isotropic. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's matter content, namely thermodynamics, nuclear- and particle physics. According to these models, our present universe emerged from an extremely dense high-temperature state (the Big Bang) roughly 14 billion years ago, and has been expanding ever since.[38]

Einstein's equations can be generalized by adding a term called the cosmological constant. When this term is present, empty space itself acts as a source of attractive or, unusually, repulsive gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term; prematurely, as we know today: From about 1998 on, a steadily accumulating body of astronomical evidence has shown that the expansion of the universe is accelerating in a way that suggests the presence of a cosmological constant or, equivalently, of a dark energy with specific properties that pervades all of space.[39]

## Modern research: general relativity and beyond

General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.[40]

In contrast to all other modern theories of fundamental interactions, general relativity is a classical theory: it does not include the effects of quantum physics. The quest for a quantum version of general relativity addresses one of the most fundamental open questions in physics. While there are promising candidates for such a theory of quantum gravity, notably string theory and loop quantum gravity, there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called singularity theorems which predict that such singularities must exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the beginning of the universe.[41]

Other attempts to modify general relativity have been made in the context of cosmology. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. There have been several controversial proposals to obviate the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion, for example modified Newtonian dynamics.[42]

It is possible that another reason to modify Einstein's theory can be found much closer to home, in the shape of what is called the Pioneer anomaly, after the Pioneer 10 and Pioneer 11 space probes. Taking into account all known effects, gravitational or otherwise, it is possible to make very specific predictions for these probes' trajectories. Yet observations show ever-so-slight divergences between these predictions and the actual positions. The possibility of new physics has not been ruled out, despite thorough attempts to find more conventional explanations.[43]

Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations,[44] ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run,[45] and the race for the first direct detection of gravitational waves continues apace.[46] More than ninety years after the theory was first published, research is more active than ever.[47]

## Notes

1. ^ This development is traced e.g. in Renn 2005, p. 110ff., in chapters 9 through 15 of Pais 1982, and in Janssen 2005. A precis of Newtonian gravity can be found in Schutz 2003, chapters 2–4. It is impossible to say whether the problem of Newtonian gravity crossed Einstein's mind before 1907, but by his own admission, his first serious attempts to reconcile that theory with special relativity date to that year, cf. Pais 1982, p. 178.
2. ^ This is described in detail in chapter 2 of Wheeler 1990.
3. ^ While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and Einstein's original concept, cf. Norton 1985.
4. ^ E. g. Janssen 2005, p. 64f. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by Ernst Mach, Einstein also explored centrifugal forces and their gravitational analogue, cf. Stachel 1989.
5. ^ More specifically, Einstein's calculations, which are described in chapter 11b of Pais 1982, use the equivalence principle, the equivalence of gravity and inertial forces, and the results of special relativity for the propagation of light and for accelerated observers (the latter by considering, at each moment, the instantaneous inertial frame of reference associated with such an accelerated observer).
6. ^ This effect can be derived directly within special relativity, either by looking at the equivalent situation of two observers in an accelerated rocket-ship or by looking at a falling elevator; in both situations, the frequency shift has an equivalent description as a Doppler shift between certain inertial frames. For simple derivations of this, see Harrison 2002.
7. ^ See chapter 12 of Mermin 2005.
8. ^ Cf. Ehlers & Rindler 1997; for a non-technical presentation, see Pössel 2007.
9. ^ These and other tidal effects are described in Wheeler 1990, pp. 83–91.
10. ^ Tides and their geometric interpretation are explained in chapter 5 of Wheeler 1990. This part of the historical development is traced in Pais 1982, section 12b.
11. ^ For elementary presentations of the concept of spacetime, see the first section in chapter 2 of Thorne 1994, and Greene 2004, p. 47–61. More complete treatments on a fairly elementary level can be found e.g. in Mermin 2005 and in Wheeler 1990, chapters 8 and 9.
12. ^ See Wheeler 1990, chapters 8 and 9 for vivid illustrations of curved spacetime.
13. ^ Einstein's struggle to find the correct field equations is traced in chapters 13–15 of Pais 1982.
14. ^ E.g. p. xi in Wheeler 1990.
15. ^ A thorough, yet accessible account of basic differential geometry and its application in general relativity can be found in Geroch 1978.
16. ^ See chapter 10 of Wheeler 1990.
17. ^ In fact, when starting from the complete theory, Einstein's equation can be used to derive these more complicated laws of motion for matter as a consequence of geometry; however, deriving from this the motion of idealized test particles is a highly non-trivial task, cf. Poisson 2004.
18. ^ A simple explanation of mass-energy-equivalence can be found in sections 3.8 and 3.9 of Giulini 2005.
19. ^ See chapter 6 of Wheeler 1990.
20. ^ For a more detailed definition of the metric, but one that is more informal than a textbook presentation, see chapter 14.4 of Penrose 2004.
21. ^ The geometrical meaning of Einstein's equations is explored in chapters 7 and 8 of Wheeler 1990; cf. box 2.6 in Thorne 1994. An introduction using only very simple mathematics is given in chapter 19 of Schutz 2003.
22. ^ The most important solutions are listed in every textbook on general relativity; for a (technical) summary of our current understanding, see Friedrich 2005.
23. ^ More precisely, these are VLBI measurements of planetary positions; see chapter 5 of Will 1993 and section 3.5 of Will 2006.
24. ^ For the historical measurements, see Hartl 2005, Kennefick 2005, and Kennefick 2007; Soldner's original derivation in the framework of Newton's theory is Soldner 1804. For the most precise measurements to date, see Bertotti 2005.
25. ^ See Kennefick 2005 and chapter 3 of Will 1993. For the Sirius B measurements, see Trimble & Barstow 2007.
26. ^ Pais 1982, Mercury on pp. 253–254, Einstein's rise to fame in sections 16b and 16c.
27. ^ For the Cassini measurements of the Shapiro effect, see Bertotti 2005. For more information about Gravity Probe B, see the Gravity Probe B website, retrieved 2007-06-13
28. ^ Kramer 2004.
29. ^ An accessible account of relativistic effects in the global positioning system can be found in Ashby 2002; details are given in Ashby 2003.
30. ^ An accessible introduction to tests of general relativity is Will 1993; a more technical, up-to-date account is Will 2006.
31. ^ The geometry of such situations is explored in chapter 23 of Schutz 2003.
32. ^ Introductions to gravitational lensing and its applications can be found on the webpages Newbury 1997 and Lochner 2007.
33. ^ Schutz 2003, pp. 317–321; Bartusiak 2000, pp. 70–86.
34. ^ The ongoing search for gravitational waves is described vividly in Bartusiak 2000 and in Blair & McNamara 1997.
35. ^ For an overview of the history of black hole physics from its beginnings in the early twentieth century to modern times, see the very readable account by Thorne 1994. For an up-to-date account of the role of black holes in structure formation, see Springel et al. 2005; a brief summary can be found in the related article Gnedin 2005.
36. ^ See chapter 8 of Sparke & Gallagher 2007 and Disney 1998. A treatment that is more thorough, yet involves only comparatively little mathematics can be found in Robson 1996.
37. ^ An elementary introduction to the black hole uniqueness theorems can be found in Chrusciel 2006 and in Thorne 1994, pp. 272–286.
38. ^ Detailed information can be found in Ned Wright's Cosmology Tutorial and FAQ, Wright 2007; a very readable introduction is Hogan 1999. Using undergraduate mathematics but avoiding the advanced mathematical tools of general relativity, Berry 1989 provides a more thorough presentation.
39. ^ Einstein's original paper is Einstein 1917; good descriptions of more modern developments can be found in Cowen 2001 and Caldwell 2004.
40. ^ Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.
41. ^ With a focus on string theory, the search for quantum gravity is described in Greene 1999; for an account from the point of view of loop quantum gravity, see Smolin 2001.
42. ^ For dark matter, see Milgrom 2002; for dark energy, Caldwell 2004.
43. ^ See Nieto 2006.
44. ^ See Friedrich 2005.
45. ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002.
46. ^ See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO 600 and LIGO.
47. ^ A good starting point for a snapshot of present-day research in relativity is the electronic review journal Living Reviews in Relativity.

## References

Additional resources, including more advanced material, can be found in General relativity resources.

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

 Educational level: this is a secondary education resource.
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A great many attempts have been made to explain the basic concepts of general relativity to non-experts. These can range from museum exhibits that roll a ball around on a curved surface, to treatments that are mathematically quite daunting. This article will attempt to explain it at the level of undergraduate, or ambitious high-school, mathematics and physics.

The general relativity formulation of gravity states that gravity arises from the curvature of spacetime, and, in analogy with the classical notion that massive objects create a gravitational field attracting other objects, matter causes spacetime to curve. In the words of physicist John Wheeler, "Space tells matter how to move, matter tells space how to curve."[1]

## Coordinate Systems and Spacetime Diagrams

Figure 1—A flat and stationary coordinate system for spacetime

Figure 1 shows a "spacetime diagram", with my house, my neighbor's house, and my neighbor walking from his house to mine.

This is the same kind of diagram that is used in explanations of special relativity. The "spacetime" is sometimes called "Minkowski space". Spacetime is actually four-dimensional, but we can only show two dimensions, so we leave out y and z. The single x spatial coordinate is good enough for our purposes, so the diagram has x going from left to right, and t (time) going upward. For the purposes of this explanation, don't worry about the considerations of special relativity such as the speed of light, the Lorentz transform, or light cones. None of that is important just now.

The diagram shows the calibration, in space (that is, x) and time. These measurements are made with respect to my (stationary) frame of reference. My house is at x=0, and my neighbor's house is at x=5 (all units are arbitrary). My neighbor walks at 1 unit per minute.

The diagram shows some "events"—my house, now; my house, 5 minutes from now; and my neighbor's house now and 5 minutes from now. The green diagonal line depicts my neighbor walking from his house to mine, arriving 5 minutes from now. That line is called his world line. The red line going straight up in my house is my own world line (I'm sitting at home.)

Figure 2—A flat and uniformly moving coordinate system for spacetime

A car is driving down the street, from left to right. Figure 2 shows the same four events and two world lines, but with different calibration—the car's own coordinate system. The car is driving at 1 unit per minute, but in the opposite direction. The event of my neighbor's arrival at my house is now at x=-5. It's way behind the car, though the car was directly adjacent to my house at t=0. The car passes my neighbor (that is, my neighbor is observed to be at x=0 in this coordinate system) at t=2.5.

Because the car's frame of reference is in motion, the calibration lines in figure 2 are not perpendicular. The formerly vertical lines are now slanted. But there is something very important to notice about both this coordinate system and the previous one: They are both flat. The flatness comes from the fact that the calibration lines are straight and parallel. The boxes created by the lines are parallelograms. But note that the lines don't have to be perpendicular, and the boxes don't have to be rectangles. Straight parallel lines and parallelograms are all that is required.

These two flat coordinate systems have a very important physical property: Neither I, sitting at home, nor a passenger in the car, experiences any "fictitious forces" (this concept is described below.) That is, people in the car don't feel any recoil from acceleration, or centrifugal force, or Coriolis force. These frames of reference are said to be inertial. This leads to an important principle of geometrical physics:

• Inertial frames of reference have flat coordinate systems. Flat coordinate systems lead to an absence of fictitious forces.

Figure 3—A curved, accelerating coordinate system for spacetime

Now consider figure 3. The coordinate system is once again that of the car, but the car is accelerating, starting at a standstill in front of my house at t=0. Its world line is curved. Once again, it crosses paths with my neighbor. This case is very different from the other two. The calibration lines are curved, and the boxes that they create are not parallelograms. This coordinate system is curved. Another thing to notice is that people in the car will feel a fictitious force—a "recoil" force agains the back of the seat. This frame of reference is not inertial.

• Accelerating frames of reference have curved coordinate systems. Curved coordinate systems lead to fictitious forces.

Figure 4—Cartesian (flat) coordinates on a flat space
Figure 5—Polar (curved) coordinates on a flat space

The notion of flat and curved coordinate systems is actually quite familiar from analytic geometry. Figures 4 and 5 show two common coordinate systems (or "ways of drawing the calibration lines on graph paper".) Figure 4 shows Cartesian coordinates, which are flat, and figure 5 shows polar coordinates, which are curved.

 More Advanced Mathematical Treatment While we can often visualize curved and flat coordinate systems in terms of pictures, the mathematically correct way is with the "metric", or "metric tensor". This is a formula for measuring the distance between two points in space. (Actually, it only measures the distance between two points that are infinitesimally close to each other.) The measurement is in terms of the two points' coordinates—different coordinate systems mean different coordinates for any given point, so the distance formula will be different. In ordinary Cartesian coordinates, the Pythagorean theorem applies, so the distance is given by: $s^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2\,$ Or, since we are only interested in the distance between points that are infinitesimally close, we write it in terms of "differentials": $ds^2 = dx^2 + dy^2\,$ For polar coordinates, the formula is: $ds^2 = dr^2 + r^2 d\theta^2\,$ These expressions are sometimes known as "quadratic forms" or as "symmetric second-rank covariant tensors". These metrics are often written as 2x2 (or whatever the dimensionality is) matrices. For Cartesian coordinates the matrix is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$ For polar coordinates it is $\begin{bmatrix} 1 & 0 \\ 0 & r^2 \\ \end{bmatrix}$ For the spacetime coordinate system of Figure 1, it could be $\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$, depending on our choice of physical calibration factors, including the scale factor between space and time. In true special relativity we would choose these very carefully, using the Lorentz metric, and with the speed of light as the scale factor. But we promised not to get into that yet. The important point is that all the coefficients are constants. For the "oblique" coordinate system of Figure 2, it is $\begin{bmatrix} 1 & 1 \\ 1 & 2 \\ \end{bmatrix}$, or $ds^2 = dx^2 + 2\ dx\ dt + 2\ dt^2\,$ Once again, all the coefficients are constants. For the coordinate system of the accelerating car, in Figure 3, it is $\begin{bmatrix} 1 & t/2 \\ t/2 & 1 + t^2/4 \\ \end{bmatrix}$, or $ds^2 = dx^2 + t\ dx\ dt + (1 + t^2/4)\ dt^2\,$ The noteworthy fact here is that the coefficients depend on time. They are not constants. The general principle is that: A coordinate system is flat if its metric coefficients are constant.

## Geodesics and Fictitious Forces

A geodesic is a line that is "objectively straight". What we mean by that is that a line's straightness transcends different coordinate systems. Different coordinate systems may have different equations for a line, but whether a line is a geodesic is independent of that.

We have drawn the figures above (and those below) so that geodesics are straight lines on your computer screen. The various colored world lines of Figures 1, 2, and 3 are all geodesics except for the blue line of Figure 3, for the accelerating car.

In a flat coordinate system, the equation of a geodesic is a linear equation. For example, the equation of the green line in figure 2 is x = 5 − t.

The equation of the blue line in Figure 3 is x = 0, but that doesn't make it a geodesic, because the coordinate system is not flat.

In a curved coordinate system, the formula for a geodesic is much more complicated. It will be explained below that, in the presence of gravitational fields, the space itself is curved. This means that all coordinate systems are curved, which makes the calculation of geodesics very complicated. This is one of the reasons that general relativity is so difficult.

Everyone except the passengers in the accelerating car (blue line, Figure 3), are undergoing inertial motion, and feel no fictitious forces:

Observers following geodesics in spacetime are undergoing inertial motion, and feel no fictitious forces.

An observer following a geodesic is said to be in "free fall". Planets are in free fall around the Sun. People falling down elevator shafts are also in free fall. (We consider the various observers in Figures 1 and 2 to be in free fall, because we are ignoring gravity for now. They feel no fictitious forces other than gravity.)

How about observers whose world lines are not geodesics? They are being accelerated by some external force, which is what makes their world line deviate from straightness. If they are "at rest", or moving uniformly, in their own coordinate system, the force making their world line curve may not seem real, but their recoil against that force will seem real. That recoil is a fictitious force.

For example, passengers in the car of Figure 3 feel a recoil force against the back of the car seat. People in various rotating amusement park rides feel a "centrifugal force", another fictitious force, seeming to push them outward. An observer standing on the ground can see that the centrifugal force is fictitious—what is really happening is that the ride mechanism is pushing them into circular motion. There is another fictitious force associated with rotation: the Coriolis force.

## Gravity as a Fictitious Force

Finally, we come to gravity itself.

Gravity is a fictitious force, just like all the others. It arises from particles not following geodesics.
Figure 6—Spacetime diagram of a person falling down an elevator shaft

Figure 6 shows a spacetime diagram of the coordinate system of a person falling down an elevator shaft. His world line is shown in red. The left-to-right axis, which was the "x" axis in the earlier figures, is now the "z" axis, that is, height. The coordinate system travels down the shaft with him, and he thinks he is at rest in it. He does not feel any fictitious forces, including gravity. The coordinate system is flat and inertial.

Figure 7—The same person, in the coordinate system of "stationary" people

Figure 7 shows the same person in the coordinate system of "stationary" people standing on the top floor of the building. The world line of the falling person is still red, and that of the stationary people is blue. The stationary coordinate system is curved; the stationary people are not following geodesics. The floor is pushing up against them, making them deviate from geodesics. Their recoil against this is the fictitious force that they call "gravity". The curved coordinate lines are the lines of "stationary" constant height. Think of them as being like the floors of the building. Note that the unfortunate victim is passing downward through them.
The aspect of gravity that indicates that it is a fictitious force is that it shares an important property with the standard fictitious forces: it is exactly proportional to the object's mass. For the standard fictitious forces, it is easy to see why this is so—they are actually the result of accelerations. By Newton's law F = ma, the force is proportional to the mass for a given acceleration.

This connection is the basis for Einstein's equivalence principle—one can't tell the difference between gravity and a traditional fictitious force. The thought experiment making this plausible was the "elevator experiment". A person in a closed elevator can't tell whether it is stationary in a building on Earth, or is out in space being dragged, at 9.8 meters per second squared, by a cable attached to a rocket.

For ordinary fictitious forces, we can readily explain things by transforming away from the curved coordinate system to some flat coordinate system. For example, we can "transform away" the centrifugal force of a rotating amusement park ride by viewing the action from a stationary observer on the ground.

For gravity, the equivalent transformation is to a coordinate system that is in free fall, such as that of the person falling down an elevator shaft. This leads to the following not-very-satisfactory explanation of the Earth's gravity: The gravity observed on the Earth's surface arises because the Earth is flat, and there are rocket engines on the underside, accelerating everything upward at 9.8 meters per second squared. This would explain everything, at least locally, but it has one fatal flaw: The Earth is round, with gravity going inward everywhere. So the surface of the Earth would have to be accelerating outward in all directions. This would require that the Earth continually get bigger, which it does not do.

## Curvature of Spacetime

The problem with the not-very-satisfactory explanation of gravity given above was that, while we can transform the curved coordinate system of a person standing on the ground to the coordinate system of a person falling down an elevator shaft locally, we can't do it everywhere. People falling down elevator shafts on opposite sides of the Earth will not have consistent coordinate systems. The problem is that spacetime itself is curved.

Up until now, we have been dealing with curved and flat coordinate systems on spaces (the technical term is "manifold") that are themselves flat. We put coordinate systems on them as though we were drawing lines on a flat piece of graph paper. Now we have to deal with curved manifolds.

The curvature of a 2-dimensional manifold embedded in 3-dimensional space can be visualized by looking at it. It's just what you think it means. A typical curved 2-dimensional manifold is the surface of a sphere. We can place a coordinate system on it that is nearly flat over a tiny region. For example, ordinary latitude/longitude coordinates will work near the equator. But this coordinate system doesn't work everywhere. As one moves away from the equator, it measures distances incorrectly, and at the North and South poles it turns into polar coordinates, which are clearly not flat. (One could use a different coordinate system that looks flat at the North pole, but it wouldn't be flat at the equator.) This coordinate system that is flat at the equator is analogous to the coordinate system of the person falling down the elevator shaft—it can't be extended around the world.

Whether a manifold is curved or not is determined by the Riemann tensor, also called the curvature tensor. This is a mathematical object that has an "objective" or "intrinsic" existence, independent of coordinate system. (Tensors are defined to have this property. Doing this correctly is what makes tensor calculus so difficult.) All of the spaces we have discussed so far, even those with curved coordinate systems, are flat. In fact, they are 2-dimensional Euclidean spaces.

Riemann's tensor on the 2-dimensional surface of a sphere is nonzero.

 More Advanced Mathematical Treatment Riemann's tensor can be calculated, by a some long formulas that we won't go into here, from the various partial derivatives of the metric coefficients. If those coefficients are constant (the coordinate system is flat), all of the derivatives are zero, and Riemann's tensor is zero. Conversely, if the manifold is curved (Riemann's tensor is nonzero), there are no flat coordinate systems. For the curved coordinate system of Figure 3, the derivatives are not zero, but they all cancel in the calculation of Riemann's tensor. For the surface of a sphere, Riemann's tensor can be shown to be nonzero, so there are no flat coordinate systems. The number of nontrivial components of Riemann's tensor is $\frac{N^2\ (N^2-1)}{12}$ For a 2-dimensional surface, this is 1, so there is just one number determining the curvature. This is called the Gaussian curvature, usually denoted by K. For 4-dimensional spacetime, there are 20 components.
Figure 8—Examples of negative, zero, and positive Gaussian curvature

For the case of a 2-dimensional surface, the curvature is determined by a single number K, the Gaussian curvature. Figure 8 shows some surfaces with negative, zero, and positive Gaussian curvature.

The figure on the left, with negative Gaussian curvature, is often referred to as being (locally) "saddle-shaped". The figure in the middle has zero curvature. Only "intrinsic" curvature counts; rolling up a sheet of paper does not.

The fact that we can't make a flat coordinate system globally around a non-microscopic region of the Earth's surface tells us that in the vicinity of the Earth, spacetime itself is curved.

It is important to note that the curvature of 4-dimensional spacetime is vastly more complicated than the Gaussian curvature of a surface. It has 20 components instead of 1. Attempts to visualize it as a (perhaps saddle-shaped) surface are not correct; they merely suggest the meaning of curvature. The actual analysis of the curvature, and how it gives rise to gravity, requires careful analysis of tensor calculus.

This point needs to be emphasized: Visualizations of gravitational curvature in terms of an ordinary curved surface, with a ball rolling across it and being deflected, such as one sees in science museum exhibits, do not do justice to the actual curvature of general relativity. They simulate a classical gravity well, such as exists in the Newtonian formulation of gravity.

The curvature of spacetime is manifested in the metric tensor, which has 10 non-trivial parameters, from which one derives the curvature itself in Riemann's tensor (20 parameters), from which one further derives Ricci's tensor and Einstein's tensor (10 parameters each.)

## The Schwartzschild Solution

Under general relativity, the metric tensor describing the curved spacetime in the vicinity of the Earth (or the Sun, or any spherically symmetric gravitating body) is the celebrated Schwartzschild solution. Exact closed-form solutions to the gravity equations are notoriously difficult to obtain, and the Schwartzschild solution is one of very few such solutions. (The Kerr solution, which takes the body's rotation into account, is another.) In the Schwartzschild solution, the strength of an object's gravity is determined by a parameter, with the dimensions of length, called the Schwartzschild radius. When the equations are solved, the instantaneous downward acceleration of a particle initially at rest (for example, the instant someone jumps into the elevator shaft) is

$\frac{d^2 r}{dt^2} = - \frac{c^2\,a\ (r - a)}{2\,r^3}$

where a is the Schwartzschild radius. Classically, the acceleration would be

$\frac{d^2 r}{dt^2} = - \frac{G M}{r^2}$

where G is Newton's constant of gravitation, and M is the mass of the gravitating body.

Setting these equal, under the assumption that $a\,$ is very small in comparison to $r\,$, we get

$a = \frac{2\,G M}{c^2}$

For the Earth, a is about 1 centimeter. For the Sun it is about 3 kilometers.

But, as will be seen below, this equation only applies in the absence of gravitating matter, that is, outside of the Earth. The Earth would only have a Schwartzschild radius if its mass were all concentrated inside a sphere of 1 centimeter. If that were the case, it would be a black hole, with an event horizon at the Schwartzschild radius.

 More Advanced Mathematical Treatment The metric tensor, under plain special relativity with no curvature, in Cartesian coordinates, is: $\begin{bmatrix} -c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$ In polar coordinates, it is: $\begin{bmatrix} -c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2\ \sin^2\,\theta \end{bmatrix}$ The curved Schwartzschild metric tensor, in polar coordinates, is: $\begin{bmatrix} -\frac{c^2\ (r - a)}{r} & 0 & 0 & 0 \\ 0 & \frac{r}{(r - a)} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2\ \sin^2\,\theta \end{bmatrix}$ Since the Schwartzschild solution is curved, it has Riemann's tensor not equal to zero. But Einstein's tensor is zero: $G_{\mu\nu} = 0\,$ The gravitational field, being very nearly an inverse square field under Newtonian mechanics, is quite similar to the classical electrostatic field. That field is described by Maxwell's equations, specifically the equation $\nabla \cdot E = \rho / \epsilon$, where $\rho\,$ is the density of electric charge. In the absence of charge, we have $\nabla \cdot E = 0$, giving an inverse-square electric field. The quantity $\nabla \cdot E$ in electrostatics is roughly analogous to $G_{\mu\nu}\,$ in general relativity. Just as the inverse-square electric field is the solution to $\nabla \cdot E = 0$, the (very nearly) inverse square Schwartzschild gravitational field is the solution to $G_{\mu\nu} = 0\,$.

The Schwartzschild solution is just one simple case (gravity in the vacuum surrounding a spherically symmetric massive object) of general relativity, but it provides many of the observed phenomena:

Depiction of the Shapiro effect delaying signals (green wave) from the Cassini space probe, as they pass through the gravity well (blue lines) of the Sun.
• bending of light (both around the Sun during an eclipse and around distant galaxies with gravitational lensing)
• the anomalous precession of the perihelion of Mercury
• gravitational redshift
• gravitational time dilation
• gravitational delay (Shapiro effect) of spacecraft signals past planets

## Nonzero Gravitating Mass

We have seen how the curvature of spacetime tells matter how to move. Now we will examine how matter tells space how to curve.

The "gravitational field", which we might denote by a vector $V\,$, arises from the curvature of spacetime. Two tensor fields—the Ricci tensor $R_{\mu\nu}\,$ and the closely related Einstein tensor $G_{\mu\nu}\,$ govern this. The divergence of $V\,$ is just the 00 component of Ricci's tensor:

$\nabla \cdot V = R_{00}$

Just as the statement that an electric field is spherically symmetric and has zero divergence leads to an inverse square law, the statement that a coordinate system is spherically symmetric and has $R_{\mu\nu} = 0\,$ leads to an inverse square law (in the classical approximation) for the "gravitational field". (The Schwartzschild solution was just the exact solution to $R_{\mu\nu} = 0\,$.)

Now the actual law for electrostatics is Coulomb's law:

$E = \frac{Q}{4 \pi \epsilon r^2}$

where $Q\,$ is the charge and $\epsilon\,$ is the physical constant governing the force. This is equivalent to Maxwell's first equation:

$\nabla \cdot E = \frac{\rho}{\epsilon}$

where $\rho\,$ is the charge density.

Similarly, the equation for classical gravity is:

$V = \frac{G M}{r^2}$

where $M\,$ is the gravitating object's mass and $G\,$ is Newton's constant of gravitation. This is equivalent to:

$\nabla \cdot V = 4 \pi G \rho$

where $\rho\,$ is the matter density.

Therefore we have:

$R_{00} = 4 \pi G \rho\,$

(We had $R_{00} = 0\,$ before because we were calculating the field in a vacuum.)

The actual tensor of interest is Einstein's tensor $G_{\mu\nu}\,$, which has $G_{00} = 2 R_{00}\,$ in this simple case, so:

$G_{00} = 8 \pi G \rho\,$

Specifically:

$G_{\mu\nu} = \begin{bmatrix} 8 \pi G \rho & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

The other components relate to the motion of the gravitating matter—when the matter moves, things become even more complicated.

The stress-energy-momentum tensor $T_{\mu\nu}\,$, describes the configuration of matter and other things (e.g. electrical energy) that create gravity. For motionless matter, we have:

$T_{\mu\nu} = \begin{bmatrix} \rho & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

So the equation of gravity, Einstein's equation, is:

$G_{\mu\nu} = 8 \pi G\ T_{\mu\nu}\,$

## References

1. Misner, Thorne & Wheeler. Gravitation. (1973)