General Relativity

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

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

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

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

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

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

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

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

The Equivalence Principle

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

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

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

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

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

Early Predictions and Tests

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

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

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

Modern Tests and Research

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

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

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

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

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

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

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

What is a metric?

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

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

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

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

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

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

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

The Minkowski Metric

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

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

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

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

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

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

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

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

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

The Invariant Interval

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

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

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

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

Parallel Transport of Vectors

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

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

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

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

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

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

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

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

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

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

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

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

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

Covariant Derivatives, the Christoffel Connection, and the Geodesic Equation

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

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

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

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

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

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

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

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

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

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

Poisson's Equation and the Weak-Field Limit

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

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

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

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

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

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

The Stress-Energy Tensor

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

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

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

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

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

Curvature and Einstein's Equations

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

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

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

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

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

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

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

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

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

Einstein's Equation in Vacuum

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

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

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

The Schwarzschild Metric and Black Holes

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

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

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

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

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

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

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

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

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

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