# General relativity

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General relativity (GR), also known as the General Theory of Relativity, is an extension of special relativity, dealing with curved coordinate systems, accelerating frames of reference, curvilinear motion, and curvature of spacetime itself. It could be said that general relativity is to special relativity as vector calculus is to vector algebra. General relativity is best known for its formulation of gravity as a fictitious force arising from the curvature of spacetime. In fact, "general relativity" and "Einstein's formulation of gravity" are nearly synonymous in many people's minds.

The general theory of relativity was first published by Albert Einstein in 1916.

General relativity, like quantum mechanics, (relativity and quantum mechanics are the two theories comprising "modern physics") has a reputation for being notoriously complicated and difficult to understand. In fact, in the early decades of the 20th century, general relativity had a sort of cult status in this regard. General relativity and quantum mechanics are both advanced college-level and postgraduate level topics. We won't attempt to give a comprehensive explanation of general relativity at the expert level. But we will attempt to give a rough outline, for reasonably advanced students, of the general relativistic formulation of gravity, below. A somewhat simpler, but, we hope, still reasonably literate, introduction for students may be found here.

Modern science does not say that Newtonian (classical) gravity is wrong. It is obviously very very nearly correct. In the weak field approximation, such as one finds in our solar system, the differences between general relativity and Newtonian gravity are minuscule. It takes very sensitive tests to show the difference. The history of those tests is a fascinating subject, and will be covered near the end of this article. But in all tests conducted so far, where there are discrepancies between the predictions of general relativity and Newtonian gravity (or other competing theories for that matter), experimental results have shown general relativity to be a better description.

Outside of the solar system, one can find stronger gravitational fields, and other phenomena, such as quasars and neutron stars, that permit even more definitive tests. General relativity appears to pass those tests as well.

This is not to say, by any means, that general relativity is the ultimate, perfect theory. It has never been unified with modern formulations of quantum mechanics, and it is therefore known to be incorrect at extremely small scales. Just as Newtonian gravity is very nearly correct, and completely correct for its time, general relativity is believed to be very nearly correct, but not completely so. Contemporary speculation on the next step involves extremely esoteric notions such as string theory, gravitons, and "quantum loop gravity".

The theory was inspired by a thought experiment developed by Einstein involving two elevators. The first elevator is stationary on the Earth, while the other is being pulled through space at a constant acceleration of g. Einstein realized that any physical experiment carried out in the elevators would give the same result. This realization is known as the equivalence principle and it states that accelerating frames of reference and gravitational fields are indistinguishable. General relativity is the theory of gravity that incorporates special relativity and the equivalence principle.

Click here for video lectures by Kip Thorne of Caltech on the mathematics of General Relativity.

## Curved space-time

To fully describe the location of an event in our universe, something that occurs at a particular time and place, requires three dimensions of space and one time. In flat spacetime the Cartesian coordinates are often represented in index notation, x=x1, y=x2, z=x3 and Einstein referred to time as a fourth dimension ct=x4, though it is more common today to list the time first as ct=x0. In the early development of relativity it seemed simple to consider time as an imaginary number ict where i is the quadratic root of -1 and c the speed of light. Then the space-time has the following four dimensions: (x,y,z,w=ict), but in modern pedagogy this has been abandoned as it is now understood that the metric tensor is what carries the sign difference in inner product operation yielding a spacetime displacement (line element) between events in flat spacetime of ds2 = dct2 - (dx2 + dy2 + dx2) Calculation of the Riemann tensor for this line element yields all zero elements and so when line elements are frame transformations of this one they also yield a zero Riemann tensor and are said to correspond to flat spacetime. When there is a matter source, the Reimann tensor is not zero and then the line element that yields that Riemann tensor can not be globally transformed into that of the flat spacetime line element of special relativity above, so it is said that spacetime is curved.

## Riemann coordinates

Understanding of general relativity, like restricted relativity, will be easier by first looking at the calculus of curved spacelike coordinates called Riemannian tensor calculus and first at the case of a surface described by two dimensions (x,y) instead of four. Generalizing the Riemann tensor and the calculus of curved space to four dimensional spacetime is referred to as pseudo-Riemannian tensor calculus. Cartesian coordinates are the most common reference system. The Earth, being spherical, is not a flat space and the Pythagorean theorem is valid only locally. The cartesian frame changes its orientation from place to place but the law of gravity is the same in Paris or in Valparaiso. The Riemann coordinates are local cartesian coordinates. They are such that the Pythagorean theorem is valid even on a curved surface. It is not necessary to know the transformation from curved coordinates to use them. They are not always suitable; for example, it is necessary to compute the Riemann tensor in Gauss (e.g. spherical) coordinates in order to obtain the Schwarzschild metric.

## The metric

The metric of a euclidean space represents, the Pythagorean theorem. With the $\left(-+++\right)$ sign convention reduced to the Euclidean three dimensions of space the theorem is

 $\left. ds ^2 = dx^2 + dy^2 + dz^2 \right.$

The line element for a two dimensional surface of coordinates x and y which is curved is according to Gauss:

 $\left. ds ^2= g_{xx} dx^2 + 2 g_{xy} dx dy + g_{yy} dy^2\right.$

where the gij are the coefficients of the metric. Every curved surface may be approximated, locally, by the osculating paraboloid, becoming the tangent plane z=0 when the principal curvatures kx and ky cancel:

 $z= \frac{1}{2}\left(k_{x} x^2 + k_{y} y^2\right)$

Indeed, in the frame used, the axes Ox and Oy are in the tangent plane z=0, the origin of the coordinates, x=0, y=0 being at the contact point. The Gauss curvature is, by definition, the product of the principal curvatures:

 $K=k_{x}k_{y}= \frac{\partial ^2 z}{\partial x^2} \frac{\partial ^2 z}{\partial y^2}$

In order to be in Riemann coordinates, it remains to orient the axes Ox and Oy in such a manner that the metric be diagonal (the computation is given in [1]):

 $ds^2= dx^2 + \left[1 - K\left( x^2 + y^2\right)\right] dy^2$

where K= kxky is the Gaussian curvature. In this expression, we have gxx=1, gxy=0 and

 $g_{yy} =1 - K\left( x^2 + y^2\right)$

It is not necessary to determine the principal directions to work with the Riemann coordinates since the laws of physics are invariant under a frame change. It is also not necessary to change the scales of the coordinate axes to get a metric with coefficients equal to one. It only assumed that it is always possible to change the coordinates in such a way that the Pythagorean theorem is verified locally, at the contact point, taken as the origin of the coordinates. In Riemann coordinates, all the paraboloids, including the sphere, locally, have the same metric, provided that they have the same Gaussian curvature.

The line element applied to a real particle of any sort describes the world line or path of the particle through spacetime and its length integrated along the path is the proper time for the particle. As such it is proper to use the +--- sign convention for the metric of spacetime so that ds along the path of a real particle is itself real representing the proper time differential dct. With the +--- sign convention the rectilinear coordinate line element of special relativity becomes

 $ds^2 = dct^2 - dx^2 - dy^2 - dz^2$

Just as the introduction of the metric tensor $g_{ij}$ allowed Gauss to describe paths along curved surfaces, the introduction of it in general relativity allows for the description of curved spacetimes

 $ds^2 = g_{\mu \nu}dx^{\mu}dx^{\nu}$

where we here use the Einstein summation convention where a high and low repeated index is summed over from 0 to 3. This set of elements $g_{\mu \nu}$ is known as the covariant "metric tensor".

## Riemann tensor

Gauss found a formula of the curvature K of a surface with a computation, complicated in Gaussian coordinates but much simpler in Riemannian coordinates where the curvature and the Riemann tensor are equal (in two dimensions) [1]:

 $R_{xyxy}= -\frac{1}{2}\left(\frac{\partial ^2 g_{xx}}{\partial y^2} + \frac{\partial ^2 g_{yy}}{\partial x^2}\right)$

Let us check that the Riemann tensor is equal to the total Gauss curvature:

 $R_{xyxy}= -\frac{1}{2}\left(\frac{\partial ^2 g_{xx}}{\partial y^2} + \frac{\partial ^2 g_{yy}}{\partial x^2}\right)= 0 -\frac{1}{2}(-2K)=K$

We have also, by partial derivation of the coefficients of the metric:

 $\frac{\partial ^2 g_{xx}}{\partial x^2} + \frac{\partial ^2 g_{xx}}{\partial y^2}= 0$

The same for gyy

 $\frac{\partial ^2 g_{yy}}{\partial x^2} + \frac{\partial ^2 g_{yy}}{\partial y^2}= -4K$

We have obtained a Laplace equation and a Poisson equation.

In full four dimensional spacetime the entire expression for the Riemann tensor in terms of the Christoffel symbols is

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

## Einstein equations in vacuum

Einstein's hypothesis is that the curvature of space-time is zero in the vacuum which is thus a flat space. This is true in two dimensions where the Gaussian curvature is zero. In higher dimensions, only the Ricci tensor is zero according to the Einstein equation. In matter, the Ricci tensor is different from zero. We shall not consider this case, here, but it should be considered to describe the universe which contains matter. The Einstein equations are, in the vacuum:

 $\left.R_{ik}= 0\right.$

Rik is a complicated function of the various componants of the Riemann tensorRijkl and of the metric gik. The Ricci tensor, like the Riemann tensor depends only on the coefficients of the metric. The Christoffel symbols are then unnecessary intermediaries. In two dimensions, the Ricci tensor has two components each proportional to the single component of the Riemann tensor. Therefore there is only one Einstein equation in two dimensions:

 $\left.R_{xyxy}= 0\right.$

In two dimensions and in Riemann coordinates, the Riemann tensor is equal to the Gaussian curvature K, which is zero in the vacuum. Then the coefficients of the metric have to satisfy the Laplace equation Δgxx=0 and Δgyy=0. But, in two dimensions, the Laplace equation diverges unless the coefficients of the metric are constants, corresponding to a pseudo-euclidean space.

In full four dimensional spacetime General relativity/Einstein equations

 $G^{\mu \nu}-\lambda g^{\mu \nu}=kT^{\mu \nu}$

in vacuum and for a zero cosmological constant are

 $G^{\mu \nu}=0$
 $R^{\mu \nu} - \frac{1}{2}g^{\mu \nu}R=0$

Contracting this with the metric tensor yeilds

 $R - \frac{1}{2}g^{\mu \nu}g_{\mu \nu}R=0$
 $R - \frac{1}{2}4R=0$

Yeilding a zero Ricci-scalar $R$:

 $R=0$

Inserting above then, Einstein's field equations in vacuum and a zero cosmological constant reduce to a statement that the Ricci tensor $R^{\mu \nu}$ is zero:

 $R^{\mu \nu}=0$

## Gravitational waves

Replacing y by ict in the Laplace equation, one obtains the d'Alembert equation of the plane gravitational waves for two of the coefficients of the metric:

 $\frac{\partial ^2 g_{xx}}{\partial x^2} - \frac{1}{c^2}\frac{\partial ^2 g_{xx}}{\partial t^2} = 0$
 $\frac{\partial ^2 g_{tt}}{\partial x^2} - \frac{1}{c^2}\frac{\partial ^2 g_{tt}}{\partial t^2} = 0$

The Brinkmann solution which is the exact solution to Einstein's field equations for gravitational and electromagnetic plane polorized waves traveling in the + x direction is

 $ds^2 = dct^2 - \left(dx^2 + dy^2 + dz^{2}\right) + h\left(x-ct,y,z\right)\left(dx - dct\right)^2$

and though Khan, Penrose and Szekeres have written an exact solution for colliding plane waves, exact solutions to the field equations for astronomical sources for which the waves would not be plane waves are not known. As such the theoretical modeling of gravitational wave emissions are based on weak field approximations of Einstein's field equations. Measurements on the orbital decay of binary pulsars agree with such weak field approximation predictions on the amount of energy carried away by gravitational waves, but otherwise gravitational waves have not yet been directly detected.

## Einstein and Newton

The two-dimensional Laplace equation may be extrapolated in higher spaces with small curvature. In three dimensions, spherical symmetry and time independent metric, the Einstein equations reduce to the radial laplacian:

 $\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dg_{rr}}{dr}\right)= 0$

and for $g_{tt}$. Its solution is the Coulomb potential in 1/r:

 $ds^2= g_{tt} dct^2 + g_{rr} dr^2= \left(A'+ \frac{B'}{r}\right) dct^2 - \left(A+ \frac{B}{r}\right) dr^2$

The correspondence principle with special relativity will give us the integration constants A and A'. For r=∞, we have:

 $ds^2= A' dct^2 - A dr^2$

It should be the Minkowski metric:

 $ds^2=dct^2 - dr^2$

Identifying these two metrics, we get A=A'=1. Yeilding

 $g_{tt} = \left(1+ \frac{B'}{r}\right)$

The geodesic equation is

 $\frac{d^{2}x^{\lambda}}{d\tau ^{2}} + \Gamma ^{\lambda} _{\mu \nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau}=0$

At low speeds the only contributing terms are

 $\frac{d^{2}x^{\lambda}}{d\tau ^{2}} + c^2 \Gamma ^{\lambda} _{00} \left(\frac{dt}{d\tau}\right)^2 =0$

And in the Newtonian limit $dt=d\tau$ so the equation of motion reduces to

 $\frac{d^{2}x^{\lambda}}{d\tau ^{2}} + c^2 \Gamma ^{\lambda} _{00} =0$

Writting the Christoffel symbol in terms of the metric

 $\frac{d^{2}x^{\lambda}}{d\tau ^{2}} + \frac{c^2}{2}g^{\lambda \rho}\left(g_{0\rho},_{0} + g_{\rho 0},_{0} - g_{00},_{\rho}\right) =0$

This metric is time independent

 $\frac{d^{2}x^{\lambda}}{d\tau ^{2}} - \frac{c^2}{2}g^{\lambda \rho}\left(g_{00},_{\rho}\right) =0$

And diagonal so the only contributing $g^{\lambda \rho}$ is for $\rho =\lambda$. Looking At radial motion for simplicity:

 $\frac{d^{2}r}{d\tau ^{2}} - \frac{c^2}{2}g^{rr}\left(g_{00},_{r}\right) =0$

In the weak field $g^{rr} \approx -1$

 $\frac{d^{2}r}{d\tau ^{2}} + \frac{c^2}{2}\left(g_{00},_{r}\right) =0$

which in terms of the metric element found above is

 $\frac{d^2 r}{d\tau ^2 } + \frac{c^2 }{2} \left(\frac{\partial}{\partial r}\left(1+\frac{B'}{r} \right) \right) = 0$

for this to correspond to Newtonian dynamics then $\frac{c^2 }{2}\frac{B'}{r}$ must therefor be the Newtonian gravitational potential $\Phi = -\frac{GM}{r}$ resulting in

 $B' = - \frac{2GM}{c^2}$

where G is the gravitation constant, M the mass of the attracting star.

According to Einstein, the determinant (or its trace for low gravitation) of the metric should be equal to one. This can be shown by solving the four-dimensional Einstein equations for a static and spherically symmetric gravitational field. Therefore we may write B=-B' and obtain an approximation of the Schwarzschid metric:

 $ds^2= \left(1- \frac{2GM}{rc^2}\right) dct^2 - \left(1+ \frac{2GM}{rc^2}\right) dr^2 - r^2 d\theta ^2 - r^2 sin ^2 \theta d\phi ^2$

This metric gives a light deviation by the sun twice as predicted by the newtonian theory or by the first Einstein theory of 1911 where time is dilated by gravitation. In his 1916 theory, gravitation dilates time and contracts space. The exact Schwarzschild metric is

 $ds^2= \left(1- \frac{2GM}{rc^2}\right) dct^2 - \frac{dr^2 }{ \left(1- \frac{2GM}{rc^2}\right)} - r^2 d\theta ^2 - r^2 sin ^2 \theta d\phi ^2$

The Schwarzschild metric is an exact vacuum solution to General relativity/Einstein equations.

## References

1. Bernard Schaeffer, Relativités et quanta clarifiés, Publibook, 2007