Einstein field equations
In general theory of relativity the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime with the distribution of matter within it.[1]
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The equations were first published by Einstein in 1915 in the form of a tensor equation[2] which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).[3]
Similar to the way that electromagnetic fields are determined using charges and currents via Maxwell's equations, the EFE are used to determine the spacetime geometry resulting from the presence of mass–energy and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation.
As well as obeying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation where the gravitational field is weak and velocities are much less than the speed of light.[4]
Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.
Mathematical form
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The Einstein field equations (EFE) may be written in the form:[5][1]
where R_{μν} is the Ricci curvature tensor, R is the scalar curvature, g_{μν} is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and T_{μν} is the stress–energy tensor.
The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.
Although the Einstein field equations were initially formulated in the context of a fourdimensional theory, some theorists have explored their consequences in n dimensions.[6] The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.
Despite the simple appearance of the equations they are actually 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_{μν}, 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 ten coupled, nonlinear, hyperbolicelliptic partial differential equations.[7]
One can write the EFE in a more compact form by defining the Einstein tensor
which is a symmetric secondrank tensor that is a function of the metric. The EFE can then be written as
In standard units, each term on the left has units of 1/length^{2}. With this choice of Einstein constant as 8πG/c^{4}, then the stressenergy tensor on the right side of the equation must be written with each component in units of energydensity (i.e., energy per volume = pressure).
Using geometrized units where G = c = 1, this can be rewritten as
The expression on the left represents the curvature of spacetime as determined by the metric; 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 matter/energy determines the curvature of spacetime.
These equations, together with the geodesic equation,[8] which dictates how freelyfalling matter moves through spacetime, form the core of the mathematical formulation of general relativity.
Sign convention
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler.[9] The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
The third sign above is related to the choice of convention for the Ricci tensor:
With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972)[10] and Peacock (1994)[11] are (+ − −), Peebles (1980)[12] and Efstathiou et al. (1990)[13] are (− + +), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989)[14] are (− + −).
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
The sign of the (very small) cosmological term would change in both these versions, if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.
Equivalent formulations
Taking the trace with respect to the metric of both sides of the EFE one gets
where D is the spacetime dimension. This expression can be rewritten as
If one adds −1/2g_{μν} times this to the EFE, one gets the following equivalent "tracereversed" form
For example, in D = 4 dimensions this reduces to
Reversing the trace again would restore the original EFE. The tracereversed form may be more convenient in some cases (for example, when one is interested in weakfield limit and can replace g_{μν} in the expression on the right with the Minkowski metric without significant loss of accuracy).
The cosmological constant
Einstein modified his original field equations to include a cosmological constant term Λ proportional to the metric
Since Λ is constant, the energy conservation law is unaffected.
The cosmological constant term was originally introduced by Einstein to allow for a universe that is not expanding or contracting. This effort was unsuccessful because:
 the universe described by this theory was unstable, and
 observations by Edwin Hubble confirmed that our universe is expanding.
So, Einstein abandoned Λ, calling it the "biggest blunder [he] ever made".[15]
Despite Einstein's motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. For many years the cosmological constant was almost universally considered to be 0. However, recent improved astronomical techniques have found that a positive value of Λ is needed to explain the accelerating universe.[16][17] However, the cosmological constant is negligible at the scale of a galaxy or smaller.
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:
The resulting vacuum energy density is constant and given by
The existence of a cosmological constant is thus equivalent to the existence of a nonzero vacuum energy. Thus, the terms "cosmological constant" and "vacuum energy" are now used interchangeably in general relativity.
Features
Conservation of energy and momentum
General relativity is consistent with the local conservation of energy and momentum expressed as
 .
Derivation of local energymomentum conservation Contracting the differential Bianchi identity
with g^{αβ} gives, using the fact that the metric tensor is covariantly constant, i.e. g^{αβ}_{;γ} = 0,
The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
which is equivalent to
using the definition of the Ricci tensor.
Next, contract again with the metric
to get
The definitions of the Ricci curvature tensor and the scalar curvature then show that
which can be rewritten as
A final contraction with g^{εδ} gives
which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,
Using the EFE, this immediately gives,
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 weakfield approximation and the slowmotion approximation. In fact, the constant G appearing in the EFE is determined by making these two approximations.
Derivation of Newton's law of gravity Newtonian gravitation can be written as the theory of a scalar field, Φ, which is the gravitational potential in joules per kilogram
where ρ is the mass density. The orbit of a freefalling particle satisfies
In tensor notation, these become
In general relativity, these equations are replaced by the Einstein field equations in the tracereversed form
for some constant, K, and the geodesic equation
To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero
and thus
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
where two factors of dt/dτ have been divided out. This will reduce to its Newtonian counterpart, provided
Our assumptions force α = i and the time (0) derivatives to be zero. So this simplifies to
which is satisfied by letting
Turning to the Einstein equations, we only need the timetime component
the low speed and static field assumptions imply that
So
and thus
From the definition of the Ricci tensor
Our simplifying assumptions make the squares of Γ disappear together with the time derivatives
Combining the above equations together
which reduces to the Newtonian field equation provided
which will occur if
Vacuum field equations
If the energymomentum tensor T_{μν} is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting T_{μν} = 0 in the tracereversed field equations, the vacuum equations can be written as
In the case of nonzero cosmological constant, the equations are
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_{μν} = 0, are referred to as Ricciflat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.
Einstein–Maxwell equations
If the energymomentum tensor T_{μν} is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor
is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant Λ, taken to be zero in conventional relativity theory):
Additionally, the covariant Maxwell equations are also applicable in free space:
where the semicolon represents a covariant derivative, and the brackets denote antisymmetrization. The first equation asserts that the 4divergence of the twoform F is zero, and the second that its exterior derivative is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential A_{α} such that
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.[18] However, there are global solutions of the equation which may lack a globally defined potential.[19]
Solutions
The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are nonlinear, 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 postNewtonian 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.
One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.[20] In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,[21] selfsimilar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc [22] and Kohli and Haslam.[23]
The linearized EFE
The nonlinearity of the EFE makes finding exact solutions 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, with terms that are quadratic in or higher powers of the deviation being ignored. This linearization procedure can be used to investigate the phenomena of gravitational radiation.
Polynomial form
One might think that EFE are nonpolynomial since they contain the inverse of the metric tensor. However, the equations can be arranged so that they contain only the metric tensor and not its inverse. First, the determinant of the metric in 4 dimensions can be written:
using the LeviCivita symbol; and the inverse of the metric in 4 dimensions can be written as:
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by det(g) until there are none left in the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.[24]
See also
Notes
 Einstein, Albert (1916). "The Foundation of the General Theory of Relativity". Annalen der Physik. 354 (7): 769. Bibcode:1916AnP...354..769E. doi:10.1002/andp.19163540702. Archived from the original (PDF) on 20120206.
 Einstein, Albert (November 25, 1915). "Die Feldgleichungen der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. Retrieved 20170821.
 Misner, Thorne & Wheeler (1973), p. 916 [ch. 34].
 Carroll, Sean (2004). Spacetime and Geometry – An Introduction to General Relativity. pp. 151–159. ISBN 0805387323.
 Grøn, Øyvind; Hervik, Sigbjorn (2007). Einstein's General Theory of Relativity: With Modern Applications in Cosmology (illustrated ed.). Springer Science & Business Media. p. 180. ISBN 9780387692005.
 Stephani, Hans; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003). Exact Solutions of Einstein's Field Equations. Cambridge University Press. ISBN 0521461367.
 Alan D. Rendall,“Theorems on Existence and Global Dynamics for the Einstein Equations”,Living Rev. Relativity,8, (2005), 6. [Online Article]: cited [20191210],http://www.livingreviews.org/lrr20056
 Weinberg, Steven (1993). Dreams of a Final Theory: the search for the fundamental laws of nature. Vintage Press. pp. 107, 233. ISBN 0099223910.
 Misner, Thorne & Wheeler (1973), p. 501ff.
 Weinberg (1972).
 Peacock (1994).
 Peebles, Phillip James Edwin (1980). The Largescale Structure of the Universe. Princeton University Press. ISBN 0691082391.
 Efstathiou, G.; Sutherland, W. J.; Maddox, S. J. (1990). "The cosmological constant and cold dark matter". Nature. 348 (6303): 705. doi:10.1038/348705a0.
 Collins, P. D. B.; Martin, A. D.; Squires, E. J. (1989). Particle Physics and Cosmology. New York: Wiley. ISBN 0471600881.
 Gamow, George (April 28, 1970). My World Line : An Informal Autobiography. Viking Adult. ISBN 0670503762. Retrieved 20070314.
 Wahl, Nicolle (20051122). "Was Einstein's 'biggest blunder' a stellar success?". News@UofT. University of Toronto. Archived from the original on 20070307.
 Turner, Michael S. (May 2001). "Making Sense of the New Cosmology". Int. J. Mod. Phys. A. 17 (S1): 180–196. arXiv:astroph/0202008. Bibcode:2002IJMPA..17S.180T. doi:10.1142/S0217751X02013113.
 Brown, Harvey (2005). Physical Relativity. Oxford University Press. p. 164. ISBN 9780199275830.
 Trautman, Andrzej (1977). "Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings". International Journal of Theoretical Physics. 16 (9): 561–565. Bibcode:1977IJTP...16..561T. doi:10.1007/BF01811088..
 Ellis, G. F. R.; MacCallum, M. (1969). "A class of homogeneous cosmological models". Comm. Math. Phys. 12 (2): 108–141. Bibcode:1969CMaPh..12..108E. doi:10.1007/BF01645908.
 Hsu, L.; Wainwright, J (1986). "Selfsimilar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions". Class. Quantum Grav. 3: 1105–1124. Bibcode:1986CQGra...3.1105H. doi:10.1088/02649381/3/6/011.
 LeBlanc, V. G. (1997). "Asymptotic states of magnetic Bianchi I cosmologies". Class. Quantum Grav. 14: 2281. Bibcode:1997CQGra..14.2281L. doi:10.1088/02649381/14/8/025.
 Kohli, Ikjyot Singh; Haslam, Michael C. (2013). "Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model". Phys. Rev. D. 88: 063518. arXiv:1304.8042. Bibcode:2013PhRvD..88f3518K. doi:10.1103/physrevd.88.063518.
 Katanaev, M. O. (July 7, 2005). "Polynomial form of the Hilbert–Einstein action". Gen. Rel. Grav. 38: 1233–1240. arXiv:grqc/0507026. Bibcode:2006GReGr..38.1233K. doi:10.1007/s1071400603105.
References
See General relativity resources.
 Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 9780716703440.
 Weinberg, Steven (1972). Gravitation and Cosmology. John Wiley & Sons. ISBN 0471925675.
 Peacock, John A. (1994). Cosmological Physics. Cambridge University Press. ISBN 9780521410724.
External links
Wikibooks has a book on the topic of: General Relativity 
Wikiversity has learning resources about General Relativity 
 Hazewinkel, Michiel, ed. (2001) [1994], "Einstein equations", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 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.
 Arch and scaffold: How Einstein found his field equations Physics Today November 2015, History of the Development of the Field Equations
 The Einstein field equation on the wall of the Museum Boerhaave in downtown Leiden