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Einstein field equations
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{{Use American English|date=January 2019}}{{Short description|Field equations in general relativity}}{{General relativity sidebar |equations}}The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.JOURNAL, Einstein, Albert, The Foundation of the General Theory of Relativity, Annalen der Physik, 354, 7, 769, 1916,weblink 10.1002/andp.19163540702, PDF, 1916AnP...354..769E,weblink" title="web.archive.org/web/20120206225139weblink">weblink 2012-02-06, First published by Einstein in 1915 as a tensor equation,JOURNAL, Einstein, Albert, Albert Einstein, November 25, 1915, Die Feldgleichungen der Gravitation, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 844â€“847,weblink 2017-08-21, the EFE relate local {{vanchor|spacetime curvature|SPACETIME_CURVATURE}} (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stressâ€“energy tensor).{{sfnp|Misner|Thorne|Wheeler|1973|p=916 [ch. 34]}}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 massâ€“energy and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stressâ€“energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear 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 energyâ€“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.BOOK, Carroll, Sean, Sean M. Carroll, 2004, Spacetime and Geometry â€“ An Introduction to General Relativity, 151â€“159, 0-8053-8732-3, 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.- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
Mathematical form
{{Spacetime|cTopic=Mathematics}}The Einstein field equations (EFE) may be written in the form:BOOK, Einstein's General Theory of Relativity: With Modern Applications in Cosmology, illustrated, Ã˜yvind, GrÃ¸n, Sigbjorn, Hervik, Springer Science & Business Media, 2007, 978-0-387-69200-5, 180,weblink {{Equation box 1|indent =:|equation = R_{mu nu} - tfrac{1}{2}R g_{mu nu} + Lambda g_{mu nu} = frac{8 pi G }{c^4} T_{mu nu}|cellpadding|border|border colour = #50C878|background colour = #ECFCF4}}File:EinsteinLeiden4.jpg|{{largethumb}}|EFE on a wall in LeidenLeidenwhere {{mvar|R{{sub|Î¼Î½}}}} is the Ricci curvature tensor, {{mvar|R}} is the scalar curvature, {{mvar|g{{sub|Î¼Î½}}}} is the metric tensor, {{mvar|Î›}} is the cosmological constant, {{mvar|G}} is Newton's gravitational constant, {{mvar|c}} is the speed of light in vacuum, and {{mvar|T{{sub|Î¼Î½}}}} is the stressâ€“energy tensor.The EFE is a tensor equation relating a set of symmetric 4 Ã— 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 four-dimensional theory, some theorists have explored their consequences in {{mvar|n}} dimensions.BOOK, Stephani, Hans, D., Kramer, M., MacCallum, C., Hoenselaers, E., Herlt, Exact Solutions of Einstein's Field Equations, Cambridge University Press, 2003, 0-521-46136-7, The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when {{mvar|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 stressâ€“energy tensor, the EFE are understood to be equations for the metric tensor {{mvar|g{{sub|Î¼Î½}}}}, 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, hyperbolic-elliptic partial differential equations.Rendall, Alan D. "Theorems on existence and global dynamics for the Einstein equations." Living Reviews in Relativity 8.1 (2005): 6.One can write the EFE in a more compact form by defining the Einstein tensor
G_{mu nu} = R_{mu nu} - tfrac{1}{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} + Lambda g_{mu nu} = frac{8 pi G}{c^4} T_{mu nu}.
In standard units, each term on the left has units of 1/length2. With this choice of Einstein constant as 8Ï€G/c4, then the stress-energy tensor on the right side of the equation must be written with each component in units of energy-density (i.e., energy per volume = pressure).Using geometrized units where {{math|G {{=}} c {{=}} 1}}, this can be rewritten as
G_{mu nu} + Lambda g_{mu nu} = 8 pi T_{mu nu},.
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,BOOK, Weinberg, Steven, Dreams of a Final Theory: the search for the fundamental laws of nature, 1993, Vintage Press, 107, 233, 0-09-922391-0, which dictates how freely-falling matter moves through space-time, 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.{{sfnp|Misner|Thorne|Wheeler|1973|p=501ff}} The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
R_{mu nu}=[S2]times [S3] times {R^alpha}_{mualphanu}
With these definitions Misner, Thorne, and Wheeler classify themselves as {{math|(+ + +)}}, whereas Weinberg (1972){{sfnp|Weinberg|1972}} and Peacock (1994){{sfnp|Peacock|1994}} are {{math|(+ âˆ’ âˆ’)}}, Peebles (1980)BOOK, Peebles, Phillip James Edwin, The Large-scale Structure of the Universe, Princeton University Press, 1980, 0-691-08239-1, and Efstathiou et al. (1990)JOURNAL, Efstathiou, G., W. J., Sutherland, S. J., Maddox, The cosmological constant and cold dark matter, Nature (journal), Nature, 348, 6303, 1990, 705, 10.1038/348705a0, are {{math|(âˆ’ + +)}}, Rindler (1977){{citation needed|date=October 2014}}, Atwater (1974){{citation needed|date=October 2014}}, Collins Martin & Squires (1989)BOOK, Collins, P. D. B., Martin, A. D., Squires, E. J., 1989, Particle Physics and Cosmology, New York, Wiley, 0-471-60088-1, are {{math|(âˆ’ + âˆ’)}}.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} - tfrac{1}{2} R g_{mu nu} - Lambda g_{mu nu} = -frac{8 pi G}{c^4} T_{mu nu}.
The sign of the (very small) cosmological term would change in both these versions, if the {{math|(+ âˆ’ âˆ’ âˆ’)}} metric sign convention is used rather than the MTW {{math|(âˆ’ + + +)}} metric sign convention adopted here.Equivalent formulations
Taking the trace with respect to the metric of both sides of the EFE one gets
R - frac{D}{2} R + D Lambda = frac{8 pi G}{c^4} T ,
where {{mvar|D}} is the spacetime dimension. This expression can be rewritten as
-R + frac{D Lambda}{frac{D}{2} -1} = frac{8 pi G}{c^4} frac{T}{frac{D}{2}-1} ,.
If one adds {{math|âˆ’{{sfrac|1|2}}g{{sub|Î¼Î½}}}} times this to the EFE, one gets the following equivalent "trace-reversed" form
R_{mu nu} - frac{ Lambda g_{mu nu}}{frac{D}{2}-1} = frac{8 pi G}{c^4} left(T_{mu nu} - frac{1}{D-2}Tg_{mu nu}right) .,
For example, in {{math|D {{=}} 4}} dimensions this reduces to
R_{mu nu} - Lambda g_{mu nu} = frac{8 pi G}{c^4} left(T_{mu nu} - tfrac{1}{2}T,g_{mu nu}right) .,
Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{mvar|g{{sub|Î¼Î½}}}} 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 {{mvar|Î›}} proportional to the metric
R_{mu nu} - tfrac{1}{2} R , g_{mu nu} + Lambda g_{mu nu} = frac{8 pi G}{c^4} T_{mu nu} ,.
Since {{mvar|Î›}} 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.
, Turner, Michael S.
, May 2001, Making Sense of the New Cosmology
, Int. J. Mod. Phys. A
, 17, S1
, 180â€“196
, 10.1142/S0217751X02013113
, astro-ph/0202008, 2002IJMPA..17S.180T, 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 stressâ€“energy tensor:
, May 2001, Making Sense of the New Cosmology
, Int. J. Mod. Phys. A
, 17, S1
, 180â€“196
, 10.1142/S0217751X02013113
, astro-ph/0202008, 2002IJMPA..17S.180T, However, the cosmological constant is negligible at the scale of a galaxy or smaller.
T_{mu nu}^mathrm{(vac)} = - frac{Lambda c^4}{8 pi G} g_{mu nu} ,.
The resulting vacuum energy density is constant and given by
rho_mathrm{vac} = frac{Lambda c^2}{8 pi G}
The existence of a cosmological constant is thus equivalent to the existence of a non-zero 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
nabla_beta T^{alphabeta} = {T^{alphabeta}}_{;beta} = 0.
{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of local energy-momentum conservation|Contracting the differential Bianchi identity
R_{alphabeta[gammadelta;varepsilon]} = 0
with {{mvar|g{{sup|Î±Î²}}}} gives, using the fact that the metric tensor is covariantly constant, i.e. {{math|g{{sup|Î±Î²}}{{sub|;Î³}} {{=}} 0}},
{R^gamma}_{betagammadelta;varepsilon} + {R^gamma}_{betavarepsilongamma;delta} + {R^gamma}_{betadeltavarepsilon;gamma} = , 0
The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
{R^gamma}_{betagammadelta;varepsilon} - {R^gamma}_{betagammavarepsilon;delta} + {R^gamma}_{betadeltavarepsilon;gamma} = 0
which is equivalent to
R_{betadelta;varepsilon} - R_{betavarepsilon;delta} + {R^gamma}_{betadeltavarepsilon;gamma} = 0
using the definition of the Ricci tensor.Next, contract again with the metric
g^{betadelta}left(R_{betadelta;varepsilon} - R_{betavarepsilon;delta} + {R^gamma}_{betadeltavarepsilon;gamma}right) = 0
to get
{R^delta}_{delta;varepsilon} - {R^delta}_{varepsilon;delta} + {R^{gammadelta}}_{deltavarepsilon;gamma} = 0
The definitions of the Ricci curvature tensor and the scalar curvature then show that
R_{;varepsilon} - 2{R^gamma}_{varepsilon;gamma} = 0
which can be rewritten as
left({R^gamma}_{varepsilon} - tfrac{1}{2}{g^gamma}_{varepsilon}Rright)_{;gamma} = 0
A final contraction with {{mvar|g{{sup|ÎµÎ´}}}} gives
left(R^{gammadelta} - tfrac{1}{2}g^{gammadelta}Rright)_{;gamma} = 0
which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,
{G^{alphabeta}}_{;beta} = 0
Using the EFE, this immediately gives,
nabla_beta T^{alphabeta} = {T^{alphabeta}}_{;beta} = 0
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 SchrÃ¶dinger'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 {{mvar|G}} appearing in the EFE is determined by making these two approximations.
{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation of Newton's law of gravity|Newtonian gravitation can be written as the theory of a scalar field, {{mvar|Î¦}}, which is the gravitational potential in joules per kilogram
nabla^2 Phi left[vec{x},tright] = 4 pi G rho left[vec{x},tright]
where {{mvar|Ï}} is the mass density. The orbit of a free-falling particle satisfies
ddot{vec{x}}[t] = - nabla Phi left[vec{x} [t],tright] ,.
In tensor notation, these become
begin{align}
Phi_{,i i} &= 4 pi G rho frac{d^2 x^i}{d t^2} &= - Phi_{,i} ,.end{align}In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
R_{mu nu} = K left(T_{mu nu} - tfrac{1}{2} T g_{mu nu}right)
for some constant, {{mvar|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 reduces to the former, we assume that the test particle's velocity is approximately zero
frac{d x^beta}{d tau} approx left(frac{d t}{d tau}, 0, 0, 0right)
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 {{math|{{sfrac|dt|dÏ„}}}} have been divided out. This will reduce to its Newtonian counterpart, provided
Phi_{,i} approx Gamma^i_{0 0} = tfrac{1}{2} g^{i alpha} left(g_{alpha 0 , 0} + g_{0 alpha , 0} - g_{0 0 , alpha}right) ,.
Our assumptions force {{math|Î± {{=}} i}} and the time (0) derivatives to be zero. So this simplifies to
2 Phi_{,i} approx g^{i j} left(- g_{0 0 , j}right) 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 left(T_{0 0} - tfrac{1}{2} T g_{0 0}right)
the low speed and static field assumptions imply that
T_{mu nu} approx mathrm{diag} left(T_{0 0}, 0, 0, 0right) approx mathrm{diag} left(rho c^4, 0, 0, 0right) ,.
So
T = g^{alpha beta} T_{alpha beta} approx g^{0 0} T_{0 0} approx -frac{1}{c^2} rho c^4 = - rho c^2 ,
and thus
K left(T_{0 0} - tfrac{1}{2} T g_{0 0}right) approx K left(rho c^4 - tfrac{1}{2} left(- rho c^2right) left(- c^2right)right) = tfrac{1}{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 {{mvar|Î“}} 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 left(T_{0 0} - tfrac{1}{2} T g_{0 0}right) approx tfrac{1}{2} K rho c^4
which reduces to the Newtonian field equation provided
tfrac{1}{2} K rho c^4 = 4 pi G rho ,
which will occur if
K = frac{8 pi G}{c^4} ,.
Vacuum field equations
missing image!
- Swiss-Commemorative-Coin-1979b-CHF-5-obverse.png -
A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).
If the energy-momentum tensor {{mvar|T{{sub|Î¼Î½}}}} is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting {{math|T{{sub|Î¼Î½}} {{=}} 0}} in the trace-reversed field equations, the vacuum equations can be written as
- Swiss-Commemorative-Coin-1979b-CHF-5-obverse.png -
A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).
R_{mu nu} = 0 ,.
In the case of nonzero cosmological constant, the equations are
R_{mu nu} = frac{Lambda}{frac{D}{2} -1} 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, {{math|R{{sub|Î¼Î½}} {{=}} 0}}, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.Einsteinâ€“Maxwell equations
{{see also|Maxwell's equations in curved spacetime}}If the energy-momentum tensor {{mvar|T{{sub|Î¼Î½}}}} is that of an electromagnetic field in free space, i.e. if the electromagnetic stressâ€“energy tensor
T^{alpha beta} = , -frac{1}{mu_0} left( {F^alpha}^psi {F_psi}^beta + tfrac{1}{4} g^{alpha beta} F_{psitau} F^{psitau}right)
is used, then the Einstein field equations are called the Einsteinâ€“Maxwell equations (with cosmological constant {{mvar|Î›}}, taken to be zero in conventional relativity theory):
R^{alpha beta} - tfrac{1}{2}R g^{alpha beta} + Lambda g^{alpha beta} = frac{8 pi G}{c^4 mu_0} left( {F^alpha}^psi {F_psi}^beta + tfrac{1}{4} g^{alpha beta} F_{psitau} F^{psitau}right).
Additionally, the covariant Maxwell equations are also applicable in free space:
begin{align}
{F^{alphabeta}}_{;beta} &= 0 F_{[alphabeta;gamma]}&=tfrac{1}{3}left(F_{alphabeta;gamma} + F_{betagamma;alpha}+F_{gammaalpha;beta}right)=tfrac{1}{3}left(F_{alphabeta,gamma} + F_{betagamma,alpha}+F_{gammaalpha,beta}right)= 0.end{align}where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the two-form {{mvar|F}} is zero, and the second that its exterior derivative is zero. From the latter, it follows by the PoincarÃ© lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential {{mvar|AÎ±}} such that
F_{alphabeta} = A_{alpha;beta} - A_{beta;alpha} = A_{alpha,beta} - A_{beta,alpha}
in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived.BOOK, Brown, Harvey,weblink Physical Relativity, 164, Oxford University Press, 2005, 978-0-19-927583-0, However, there are global solutions of the equation which may lack a globally defined potential.JOURNAL, Trautman, Andrzej, Andrzej Trautman, Solutions of the Maxwell and Yangâ€“Mills equations associated with Hopf fibrings, 1977, International Journal of Theoretical Physics, 16, 9, 561â€“565, 10.1007/BF01811088, 1977IJTP...16..561T, .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 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.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.JOURNAL, Ellis, G. F. R., MacCallum, M., A class of homogeneous cosmological models, Comm. Math. Phys., 12, 2, 1969, 108â€“141, 1969CMaPh..12..108E, 10.1007/BF01645908, In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,JOURNAL, Hsu, L., Wainwright, J, Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions, Class. Quantum Grav., 3, 1986, 1105â€“1124, 10.1088/0264-9381/3/6/011, 1986CQGra...3.1105H, self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc JOURNAL, LeBlanc, V. G., Asymptotic states of magnetic Bianchi I cosmologies, 1997, Class. Quantum Grav., 14, 2281, 10.1088/0264-9381/14/8/025, 1997CQGra..14.2281L, and Kohli and Haslam.JOURNAL, Kohli, Ikjyot Singh, Haslam, Michael C., Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model, Phys. Rev. D, 88, 063518, 2013,weblink 10.1103/physrevd.88.063518, 1304.8042, 2013PhRvD..88f3518K,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 non-polynomial 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:
det(g) = tfrac{1}{24} varepsilon^{alphabetagammadelta} varepsilon^{kappalambdamunu} g_{alphakappa} g_{betalambda} g_{gammamu} g_{deltanu},
using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:
g^{alphakappa} = frac{tfrac{1}{6} varepsilon^{alphabetagammadelta} varepsilon^{kappalambdamunu} g_{betalambda} g_{gammamu} g_{deltanu} }{ det(g)},.
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by {{math|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.JOURNAL, Katanaev, M. O., July 7, 2005, Polynomial form of the Hilbertâ€“Einstein action, Gen. Rel. Grav., 38, 1233â€“1240, gr-qc/0507026, 10.1007/s10714-006-0310-5, 2006GReGr..38.1233K, See also
{{Div col|colwidth=25em}}- Einsteinâ€“Hilbert action
- Equivalence principle
- Exact solutions in general relativity
- General relativity resources
- History of general relativity
- Hamiltonâ€“Jacobiâ€“Einstein equation
- Mathematics of general relativity
- Numerical relativity
- Ricci calculus
Notes
{{reflist|30em}}References
See General relativity resources.- BOOK, Charles Misner, Charles W., Misner, Kip S. Thorne, Kip S., Thorne, John Archibald Wheeler, John Archibald, Wheeler, Gravitation (book), Gravitation, San Francisco, W. H. Freeman, 1973, 978-0-7167-0344-0, harv,
- BOOK, Weinberg, Steven, 1972, Gravitation and Cosmology, 0-471-92567-5, John Wiley & Sons, Steven Weinberg, harv,
- BOOK, Peacock, John A., 1994, Cosmological Physics, 978-0521410724, Cambridge University Press, John A. Peacock, harv,
External links
{{wikiversity | General Relativity}}- {{springer|title=Einstein equations|id=p/e035210}}
- weblink" title="web.archive.org/web/20110621005940weblink">Caltech Tutorial on Relativity â€” A simple introduction to Einstein's Field Equations.
- The Meaning of Einstein's Equation â€” 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
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