Einstein field equations

General relativity

$G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}\,$

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The Einstein field equations (EFE) or Einstein's equations are a set of ten equations in Einstein's theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy.^[1] They were first published in 1915.^[2].

The EFE collectively form a tensor equation and equate the curvature of spacetime (as expressed using the Einstein tensor) with the energy and momentum within the spacetime (as expressed using the stress-energy tensor).

The EFE are used to determine the curvature of spacetime resulting from the presence of mass and energy. That is, they determine the metric tensor of spacetime for a given arrangement of stress-energy in the spacetime. Because of the relationship between the metric tensor and the Einstein tensor, the EFE become a set of coupled, non-linear differential equations when used in this way.

1 Mathematical form
- 1.1 Equivalent formulations
2 Properties
3 The cosmological constant
4 Solutions
5 Vacuum field equations
6 Einstein-Maxwell equations
7 The linearised EFE
8 See also
9 References
10 External links

Mathematical form

The Einstein field equations (EFE) may be written in the form:^[1]

$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$

where $R_{\mu \nu}$ is the Ricci curvature tensor, $R$ the scalar curvature, $g_{\mu \nu}$ the metric tensor, $\Lambda \!$ is the cosmological constant, $G$ is the gravitational constant, $c$ the speed of light, and $T_{\mu \nu}$ the stress-energy tensor.

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. Some authors, including Einstein, have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative

$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R - g_{\mu \nu} \Lambda = -{8 \pi G \over c^4} T_{\mu \nu}.$

Notice that the sign of the (very small) cosmological term would change in both of these versions, if the author is using the +--- metric sign convention rather than the −+++ metric sign convention which we have adopted here (following MTW).

The EFE is a tensor equation relating a set of symmetric 4 x 4 tensors. It is written here using the abstract index notation. Each tensor has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6 in number.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations (if the dimension is clear).

Despite the simple appearance of the equations they are, in fact, quite complicated. Given a specified distribution of matter and energy in the form of a stress-energy tensor, the EFE are understood to be equations for the metric tensor $g_{\mu \nu}$ , as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations.

One can write the EFE in a more compact form by defining the Einstein tensor

$G_{\mu \nu} = R_{\mu \nu} - {1 \over 2}R g_{\mu \nu},$

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

$G_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu}\,$

where the cosmological term has been absorbed into the stress-energy tensor as dark energy.

Using geometrized units where G = c = 1, this can be re-written as

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

The expression on the left represents the curvature of spacetime as determined by the metric and the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how the curvature of spacetime is related to the matter/energy content of the universe.

These equations, together with the geodesic equation, form the core of the mathematical formulation of general relativity.

Equivalent formulations

Einstein's field equations can be rewritten in the following equivalent "trace-reversed" form

$R_{\mu \nu} - g_{\mu \nu} \Lambda = {8 \pi G \over c^4} (T_{\mu \nu} - {1 \over 2}T\,g_{\mu \nu})$

which may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace $g_{\mu\nu}$ in the expression on the right with the Minkowski tensor without significant loss of accuracy).

Properties

Conservation of energy and momentum

General relativity is consistent with the local conservation of energy and momentum. This result arises by using the differential Bianchi identity to obtain

$\nabla_b G^{ab}=G^{ab}{}_{;b}=0$

which is necessary to be consistent with

$\nabla_b T^{ab}= T^{ab}{}_{;b}=0$

which expresses the local conservation of stress-energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.

Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics which is linear in the wavefunction.

The correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant appearing in the EFE is determined by making these two approximations.

Derivation

Newtonian gravitation can be written as the theory of a scalar field, $\Phi \!$ , which is the gravitational potential in Joules per kilogram

$\nabla^2 \Phi [\vec{x},t] = 4 \pi G \rho [\vec{x},t]$

where $\rho \!$ is the mass density. The orbit of a free-falling particle satisfies

$\ddot{\vec{x}}[t] = - \nabla \Phi [\vec{x} [t],t] \,.$

In tensor notation, these become

$\Phi_{,i i} = 4 \pi G \rho \,$

$\frac{d^2 x^i}{{d t}^2} = - \Phi_{,i} \,.$

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form

$R_{\mu \nu} = K (T_{\mu \nu} - {1 \over 2} T g_{\mu \nu})$

for some constant, K, and the geodesic equation

$\frac{d^2 x^\alpha}{{d \tau}^2} = - \Gamma^\alpha_{\beta \gamma} \frac{d x^\beta}{d \tau} \frac{d x^\gamma}{d \tau} \,.$

To see how the latter reduce to the former, we assume that the test particle's velocity is approximately zero

$\frac{d x^\beta}{d \tau} \approx (\frac{d t}{d \tau}, 0, 0, 0)$

and thus

$\frac{d}{d t} \left( \frac{d t}{d \tau} \right) \approx 0$

and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives

$\frac{d^2 x^i}{{d t}^2} \approx - \Gamma^i_{0 0}$

where two factors of $\frac{d t}{d \tau}$ have been divided out. This will reduce to its Newtonian counterpart, provided

$\Phi_{,i} \approx \Gamma^i_{0 0} = {1 \over 2} g^{i \alpha} (g_{\alpha 0 , 0} + g_{0 \alpha , 0} - g_{0 0 , \alpha}) \,.$

Our assumptions force α=i and the time (0) derivatives to be zero. So this simplifies to

$2 \Phi_{,i} \approx g^{i j} (- g_{0 0 , j}) \approx - g_{0 0 , i} \,$

which is satisfied by letting

$g_{0 0} \approx - c^2 - 2 \Phi \,.$

Turning to the Einstein equations, we only need the time-time component

$R_{0 0} = K (T_{0 0} - {1 \over 2} T g_{0 0})$

the low speed and static field assumptions imply that

$T_{\mu \nu} \approx \mathrm{diag} (T_{0 0}, 0, 0, 0) \approx \mathrm{diag} (\rho c^4, 0, 0, 0) \,.$

$T = g^{\alpha \beta} T_{\alpha \beta} \approx g^{0 0} T_{0 0} \approx {-1 \over c^2} \rho c^4 = - \rho c^2 \,$

and thus

$K (T_{0 0} - {1 \over 2} T g_{0 0}) \approx K (\rho c^4 - {1 \over 2} (- \rho c^2) (- c^2)) = {1 \over 2} K \rho c^4 \,.$

From the definition of the Ricci tensor

$R_{0 0} = \Gamma^\rho_{0 0 , \rho} - \Gamma^\rho_{\rho 0 , 0} + \Gamma^\rho_{\rho \lambda} \Gamma^\lambda_{0 0} - \Gamma^\rho_{0 \lambda} \Gamma^\lambda_{\rho 0} .$

Our simplifying assumptions make the squares of Γ disappear together with the time derivatives

$R_{0 0} \approx \Gamma^i_{0 0 , i} \,.$

Combining the above equations together

$\Phi_{,i i} \approx \Gamma^i_{0 0 , i} \approx R_{0 0} = K (T_{0 0} - {1 \over 2} T g_{0 0}) \approx {1 \over 2} K \rho c^4 \,$

which reduces to the Newtonian field equation provided

${1 \over 2} K \rho c^4 = 4 \pi G \rho \,$

which will occur if

$K = \frac{8 \pi G}{c^4} \,.$

The cosmological constant

Einstein modified his original field equations to include a cosmological term proportional to the metric

$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu} \,.$

The constant $\Lambda$ is called the cosmological constant. Since $\Lambda$ is constant, the energy conservation law is unaffected.

The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by Hubble a decade later confirmed that our universe is, in fact, not static but expanding. So $\Lambda$ was abandoned, with Einstein calling it the "biggest blunder [he] ever made".^[3] For many years the cosmological constant was almost universally considered to be 0.

Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved astronomical techniques have found that a positive value of $\Lambda$ is needed to explain some observations.^[4]^[5]

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor:

$T_{\mu \nu}^{\mathrm{(vac)}} = - \frac{\Lambda c^4}{8 \pi G} g_{\mu \nu} \,.$

The constant

$\rho_{\mathrm{vac}} = \frac{\Lambda c^2}{8 \pi G}$

is called the vacuum energy. The existence of a cosmological constant is equivalent to the existence of a non-zero vacuum energy. The terms are now used interchangeably in general relativity.

Solutions

Main article: Solutions of the Einstein field equations

The solutions of the Einstein field equations are metrics of spacetime. The solutions are hence often called 'metrics'. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are numerous cases where the field equations have been solved completely, and those are called exact solutions.^[6]

The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe.

Vacuum field equations

If the energy-momentum tensor $T_{\mu \nu}$ is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting $T_{\mu \nu} = 0$ in the full field equations, the vacuum equations can be written as

$R_{\mu \nu} = {1 \over 2} R \, g_{\mu \nu} \,.$

Taking the trace of this (contracting with $g^{\mu \nu}$ ) and using the fact that $g^{\mu \nu} g_{\mu \nu} = 4$ , we get

$R = {1 \over 2} R \, 4 = 2 R \,$

and thus

$R = 0 \,.$

Substituting back, we get an equivalent form of the vacuum field equations

$R_{\mu \nu} = 0 \,.$

In the case of nonzero cosmological constant, the equations are

$R_{\mu \nu} = {1 \over 2}R g_{\mu \nu} - \Lambda g_{\mu \nu}$

which gives

$R = 4 \Lambda \,$

yielding the equivalent form

$R_{\mu \nu} = \Lambda g_{\mu \nu} \,.$

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, $R_{\mu \nu}=0$ , are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

Einstein-Maxwell equations

If the energy-momentum tensor $T_{\mu \nu}$ is that of an electromagnetic field, i.e. if the electromagnetic stress-energy tensor

$T_{ab} = \, -\frac{1}{\mu_0} ( F_{a}{}^{s} F_{sb} + {1 \over 4} F_{st} F^{st} g_{ab} )$

is used, then the Einstein field equations are called the Einstein-Maxwell equations:

$R_{ab} - {1 \over 2}R g_{ab} = \frac{8 \pi G}{c^4 \mu_0} (\, F_{a}{}^{s} F_{sb} + {1 \over 4} F_{st} F^{st} g_{ab}) \$

The linearised EFE

Main articles: Linearised Einstein field equations, Linearized gravity

The nonlinearity of the EFE makes finding exact solutions quite difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric. This linearisation procedure can be used to discuss the phenomena of gravitational radiation.

References

See General relativity resources.

↑ ^1.0 ^1.1 Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik. http://www.alberteinstein.info/gallery/gtext3.html.
↑ Einstein, Albert (November 25, 1915). "Die Feldgleichungun der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb. Retrieved on 2006-09-12.
↑ Gamow, George (April 28, 1970). My World Line : An Informal Autobiography. Viking Adult. ISBN-10: 0670503762. http://www.jb.man.ac.uk/~jpl/cosmo/blunder.html. Retrieved on 2007-03-14.
↑ Wahl, Nicolle (2005-11-22). "Was Einstein's 'biggest blunder' a stellar success?". Retrieved on 2007-03-14.
↑ Turner, Michael S. (May, 2001). "A Spacetime Odyssey". Int.J.Mod.Phys. A17S1: 180–196.
↑ Stephani, Hans; D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt (2003). Exact Solutions of Einstein's Field Equations. Cambridge University Press. ISBN 0-521-46136-7.

Aczel, Amir D., 1999. God's Equation: Einstein, Relativity, and the Expanding Universe. Delta Science. A popular account.
Charles Misner, Kip Thorne, and John Wheeler, 1973. Gravitation. W H Freeman.

External links

Caltech Tutorial on Relativity — A simple introduction to Einstein's Field Equations.
The Meaning of Einstein's Equation — An explanation of Einstein's field equation, its derivation, and some of its consequences
Video Lecture on Einstein's Field Equations by MIT Physics Professor Edmund Bertschinger.

Einstein field equations

Contents

Mathematical form

Equivalent formulations

Properties

Conservation of energy and momentum

Nonlinearity

The correspondence principle

The cosmological constant

Solutions

Vacuum field equations

Einstein-Maxwell equations

The linearised EFE

See also

References

External links