Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Einstein field equations

From Wikipedia, the free encyclopedia
Field-equations in general relativity
"Einstein equation" redirects here. For the equation E=mc2, seeMass–energy equivalence.

General relativity
Spacetime curvature schematic

In thegeneral theory of relativity, theEinstein field equations (EFE; also known asEinstein's equations) relate the geometry ofspacetime to the distribution ofmatter within it.[1]

The equations were published byAlbert Einstein in 1915 in the form of atensor equation[2] which related the localspacetimecurvature (expressed by theEinstein tensor) with the local energy,momentum and stress within that spacetime (expressed by thestress–energy tensor).[3]

Analogously to the way thatelectromagnetic fields are related to the distribution ofcharges andcurrents viaMaxwell's equations, the EFE relate thespacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine themetric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinearpartial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. Theinertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using thegeodesic equation.

As well as implying local energy–momentum conservation, the EFE reduce toNewton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than thespeed of light.[4]

Exact solutions for the EFE can only be found under simplifying assumptions such assymmetry. Special classes ofexact solutions are most often studied since they model many gravitational phenomena, such asrotating black holes and theexpanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations fromflat spacetime, leading to thelinearized EFE. These equations are used to study phenomena such asgravitational waves.

Mathematical form

[edit]
Part of a series on
Spacetime
EFE on the wall of the Rijksmuseum Boerhaave inLeiden, Netherlands

The Einstein field equations (EFE) may be written in the form:[5][1]

Gμν+Λgμν=κTμν,{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu },}

whereGμν is the Einstein tensor,gμν is the metric tensor,Tμν is thestress–energy tensor,Λ is thecosmological constant andκ is the Einstein gravitational constant.

The Einstein tensor is defined as

Gμν=Rμν−12Rgμν,{\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu },}

whereRμν is theRicci curvature tensor, andR is thescalar curvature. This is a symmetricdivergenceless second-degree tensor that depends on only the metric tensor and its first and second derivatives.

TheEinstein gravitational constant is defined as[6][7]

κ=8πGc4≈2.07665×10−43N−1,{\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.07665\times 10^{-43}\,{\textrm {N}}^{-1},}

whereG is theNewtonian constant of gravitation andc is thespeed of light invacuum.

The EFE can thus also be written as

Rμν−12Rgμν+Λgμν=κTμν.{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.}

In standard units, each term on the left has quantity dimension ofL−2.

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.

These equations form the core of themathematical formulation ofgeneral relativity. They readily imply thegeodesic equation,[8] which dictates how freely falling test objects move through spacetime.[9]

The EFE is a tensor equation relating a set ofsymmetric 4 × 4 tensors. Each tensor has 10 independent components. The fourBianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with fourgauge-fixingdegrees 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 inn dimensions.[10] The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained whenTμν is everywhere zero) defineEinstein manifolds.

The equations are more complex than they appear. 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 tensorgμν, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-ellipticpartial differential equations.[11]

Sign convention

[edit]

The above form of the EFE is the standard established byMisner, Thorne, and Wheeler (MTW).[12] The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):gμν=[S1]×diag⁡(−1,+1,+1,+1)Rμαβγ=[S2]×(Γαγ,βμ−Γαβ,γμ+ΓσβμΓγασ−ΓσγμΓβασ)Gμν=[S3]×κTμν{\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times \kappa T_{\mu \nu }\end{aligned}}}

The third sign above is related to the choice of convention for the Ricci tensor:Rμν=[S2]×[S3]×Rαμαν{\displaystyle R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }}

With these definitionsMisner, Thorne, and Wheeler classify themselves as(+ + +), whereas Weinberg (1972)[13] is(+ − −), Peebles (1980)[14] and Efstathiou et al. (1990)[15] are(− + +), Rindler (1977),[citation needed] Atwater (1974),[citation needed] Collins Martin & Squires (1989)[16] and Peacock (1999)[17] 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:Rμν−12Rgμν−Λgμν=−κTμν.{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-\kappa T_{\mu \nu }.}

The sign of the cosmological term would change in both these versions if the(+ − − −) metricsign convention is used rather than the MTW(− + + +) metric sign convention adopted here.

Equivalent formulations

[edit]

Taking thetrace with respect to the metric of both sides of the EFE one getsR−D2R+DΛ=κT,{\displaystyle R-{\frac {D}{2}}R+D\Lambda =\kappa T,}whereD is the spacetime dimension. Solving forR and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:Rμν−2D−2Λgμν=κ(Tμν−1D−2Tgμν).{\displaystyle R_{\mu \nu }-{\frac {2}{D-2}}\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).}

InD = 4 dimensions this reduces toRμν−Λgμν=κ(Tμν−12Tgμν).{\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }=\kappa \left(T_{\mu \nu }-{\frac {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 replacegμν in the expression on the right with theMinkowski metric without significant loss of accuracy).

Cosmological constant

[edit]
Main article:Cosmological constant

In the Einstein field equationsGμν+Λgμν=κTμν,{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }\,,}the term containing the cosmological constantΛ was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for auniverse that is not expanding or contracting. This effort was unsuccessful because:

  • any desired steady state solution described by this equation is unstable, and
  • observations byEdwin Hubble showed that our universe isexpanding.

Einstein then abandonedΛ, remarking toGeorge Gamow "that the introduction of the cosmological term was the biggest blunder of his life".[18]

The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recentastronomical observations have shown anaccelerating expansion of the universe, and to explain this a positive value ofΛ is needed.[19][20] The effect of 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 and incorporated as part of the stress–energy tensor:Tμν(vac)=−Λκgμν.{\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{\kappa }}g_{\mu \nu }\,.}

This tensor describes avacuum state with anenergy densityρvac and isotropic pressurepvac that are fixed constants and given byρvac=−pvac=Λκ,{\displaystyle \rho _{\mathrm {vac} }=-p_{\mathrm {vac} }={\frac {\Lambda }{\kappa }},}where it is assumed thatΛ has SI unit m−2 andκ is defined as above.

The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.

Features

[edit]

Conservation of energy and momentum

[edit]

General relativity is consistent with the local conservation of energy and momentum expressed as∇βTαβ=Tαβ;β=0.{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.}

Derivation of local energy–momentum conservation

Contracting thedifferential Bianchi identityRαβ[γδ;ε]=0{\displaystyle R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=0}withgαβ gives, using the fact that the metric tensor is covariantly constant, i.e.gαβ;γ = 0,Rγβγδ;ε+Rγβεγ;δ+Rγβδε;γ=0{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:Rγβγδ;ε−Rγβγε;δ+Rγβδε;γ=0{\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}which is equivalent toRβδ;ε−Rβε;δ+Rγβδε;γ=0{\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}using the definition of theRicci tensor.

Next, contract again with the metricgβδ(Rβδ;ε−Rβε;δ+Rγβδε;γ)=0{\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0}to getRδδ;ε−Rδε;δ+Rγδδε;γ=0.{\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0.}

The definitions of the Ricci curvature tensor and the scalar curvature then show thatR;ε−2Rγε;γ=0,{\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0,}which can be rewritten as(Rγε−12gγεR);γ=0.{\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0.}

A final contraction withgεδ gives(Rγδ−12gγδR);γ=0,{\displaystyle \left(R^{\gamma \delta }-{\tfrac {1}{2}}g^{\gamma \delta }R\right)_{;\gamma }=0,}which by the symmetry of the bracketed term and the definition of theEinstein tensor, gives, after relabelling the indices,Gαβ;β=0.{\displaystyle {G^{\alpha \beta }}_{;\beta }=0.}

Using the EFE, this immediately gives,∇βTαβ=Tαβ;β=0{\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=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

[edit]

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example,Maxwell's equations ofelectromagnetism are linear in theelectric andmagnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is theSchrödinger equation ofquantum mechanics, which is linear in thewavefunction.

Correspondence principle

[edit]

The EFE reduce toNewton's law of gravity by using both theweak-field approximation and thelow-velocity approximation. The constantG 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{\displaystyle \Phi }, which is the gravitational potential in joules per kilogram of the gravitational fieldg=−∇Φ{\displaystyle g=-\nabla \Phi }, seeGauss's law for gravity∇2Φ(x→,t)=4πGρ(x→,t){\displaystyle \nabla ^{2}\Phi \left({\vec {x}},t\right)=4\pi G\rho \left({\vec {x}},t\right)}whereρ is the mass density. The orbit of afree-falling particle satisfiesx→¨(t)=g→=−∇Φ(x→(t),t).{\displaystyle {\ddot {\vec {x}}}(t)={\vec {g}}=-\nabla \Phi \left({\vec {x}}(t),t\right)\,.}

In tensor notation, these becomeΦ,ii=4πGρd2xidt2=−Φ,i.{\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}}

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed formRμν=K(Tμν−12Tgμν){\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)}for some constant,K, and thegeodesic equationd2xαdτ2=−Γβγαdxβdτdxγdτ.{\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.}

To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zerodxβdτ≈(dtdτ,0,0,0){\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)}and thusddt(dtdτ)≈0{\displaystyle {\frac {d}{dt}}\left({\frac {dt}{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 givesd2xidt2≈−Γ00i{\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}}where two factors of⁠dt/dτ⁠ have been divided out. This will reduce to its Newtonian counterpart, providedΦ,i≈Γ00i=12giα(gα0,0+g0α,0−g00,α).{\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {1}{2}}g^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right)\,.}

Our assumptions forceα =i and the time (0) derivatives to be zero. So this simplifies to2Φ,i≈gij(−g00,j)≈−g00,i{\displaystyle 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,}which is satisfied by lettingg00≈−c2−2Φ.{\displaystyle g_{00}\approx -c^{2}-2\Phi \,.}

Turning to the Einstein equations, we only need the time-time componentR00=K(T00−12Tg00){\displaystyle R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)}the low speed and static field assumptions imply thatTμν≈diag⁡(T00,0,0,0)≈diag⁡(ρc4,0,0,0).{\displaystyle T_{\mu \nu }\approx \operatorname {diag} \left(T_{00},0,0,0\right)\approx \operatorname {diag} \left(\rho c^{4},0,0,0\right)\,.}

SoT=gαβTαβ≈g00T00≈−1c2ρc4=−ρc2{\displaystyle T=g^{\alpha \beta }T_{\alpha \beta }\approx g^{00}T_{00}\approx -{\frac {1}{c^{2}}}\rho c^{4}=-\rho c^{2}\,}and thusK(T00−12Tg00)≈K(ρc4−12(−ρc2)(−c2))=12Kρc4.{\displaystyle K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.}

From the definition of the Ricci tensorR00=Γ00,ρρ−Γρ0,0ρ+ΓρλρΓ00λ−Γ0λρΓρ0λ.{\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.}

Our simplifying assumptions make the squares ofΓ disappear together with the time derivativesR00≈Γ00,ii.{\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.}

Combining the above equations togetherΦ,ii≈Γ00,ii≈R00=K(T00−12Tg00)≈12Kρc4{\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}}which reduces to the Newtonian field equation provided12Kρc4=4πGρ,{\displaystyle {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho ,}which will occur ifK=8πGc4.{\displaystyle K={\frac {8\pi G}{c^{4}}}\,.}

Vacuum field equations

[edit]
A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).

If the energy–momentum tensorTμν is zero in the region under consideration, then the field equations are also referred to as thevacuum field equations. By settingTμν = 0 in thetrace-reversed field equations, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written asRμν=0.{\displaystyle R_{\mu \nu }=0\,.}

In the case of nonzero cosmological constant, the equations areRμν=ΛD2−1gμν.{\displaystyle R_{\mu \nu }={\frac {\Lambda }{{\frac {D}{2}}-1}}g_{\mu \nu }\,.}

The solutions to the vacuum field equations are calledvacuum solutions. FlatMinkowski space is the simplest example of a vacuum solution. Nontrivial examples include theSchwarzschild solution and theKerr solution.

Manifolds with a vanishingRicci tensor,Rμν = 0, are referred to asRicci-flat manifolds and manifolds with a Ricci tensor proportional to the metric asEinstein manifolds.

Einstein–Maxwell equations

[edit]
See also:Maxwell's equations in curved spacetime

If the energy–momentum tensorTμν is that of anelectromagnetic field infree space, i.e. if theelectromagnetic stress–energy tensorTαβ=−1μ0(FαψFψβ+14gαβFψτFψτ){\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right)}is used, then the Einstein field equations are called theEinstein–Maxwell equations (withcosmological constantΛ, taken to be zero in conventional relativity theory):Gαβ+Λgαβ=κμ0(FαψFψβ+14gαβFψτFψτ).{\displaystyle G^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {\kappa }{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).}

Additionally, thecovariant Maxwell equations are also applicable in free space:[21]Fαβ;β=0F[αβ;γ]=13(Fαβ;γ+Fβγ;α+Fγα;β)=13(Fαβ,γ+Fβγ,α+Fγα,β)=0,{\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{[\alpha \beta ;\gamma ]}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0,\end{aligned}}}where the semicolon represents acovariant derivative, and the brackets denoteanti-symmetrization. The first equation asserts that the 4-divergence of the2-formF is zero, and the second that itsexterior derivative is zero. From the latter, it follows by thePoincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potentialAα such that[21]Fαβ=Aα;β−Aβ;α=Aα,β−Aβ,α{\displaystyle F_{\alpha \beta }=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.[22] However, there are global solutions of the equation that may lack a globally defined potential.[23]

Solutions

[edit]
Main article:Solutions of the Einstein field equations

The solutions of the Einstein field equations aremetrics 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 aspost-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.[10]

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

One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.[24] In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,[25] self-similar solutions to the Einstein field equations are fixed points of the resultingdynamical system. New solutions have been discovered using these methods by LeBlanc[26] and Kohli and Haslam.[27]

Linearized EFE

[edit]
Main article:Linearized gravity

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, thegravitational field is very weak and thespacetime 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 theMinkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena ofgravitational radiation.

Polynomial form

[edit]

Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be writtendet(g)=124εαβγδεκλμνgακgβλgγμgδν{\displaystyle \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\alpha \kappa }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}using theLevi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:gακ=16εαβγδεκλμνgβλgγμgδνdet(g).{\displaystyle g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\kappa \lambda \mu \nu }g_{\beta \lambda }g_{\gamma \mu }g_{\delta \nu }}{\det(g)}}\,.}

Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power ofdet(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. TheEinstein–Hilbert action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.[28]

See also

[edit]

Notes

[edit]
  1. ^abEinstein, 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 fromthe original(PDF) on 2012-02-06.
  2. ^Einstein, Albert (November 25, 1915)."Die Feldgleichungen der Gravitation".Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin:844–847. Retrieved2017-08-21.
  3. ^Misner, Thorne & Wheeler (1973), p. 916 [ch. 34].
  4. ^Carroll, Sean (2004).Spacetime and Geometry – An Introduction to General Relativity. Addison Wesley. pp. 151–159.ISBN 0-8053-8732-3.
  5. ^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 978-0-387-69200-5.
  6. ^With the choice of the Einstein gravitational constant as given here,κ = 8πG/c4, 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, equivalently pressure). In Einstein's original publication, the choice isκ = 8πG/c2, in which case the stress–energy tensor components have units of mass density.
  7. ^Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975).Introduction to general relativity (2nd ed.). New York: McGraw-Hill.ISBN 0-07-000423-4.OCLC 1046135.
  8. ^Weinberg, Steven (1993).Dreams of a Final Theory: the search for the fundamental laws of nature. Vintage Press. pp. 107, 233.ISBN 0-09-922391-0.
  9. ^Ehlers, Jurgen; Geroch, Robert (2003-09-16),Equation of Motion of Small Bodies in Relativity,arXiv:gr-qc/0309074, arXiv:gr-qc/0309074
  10. ^abStephani, Hans; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003).Exact Solutions of Einstein's Field Equations.Cambridge University Press.ISBN 0-521-46136-7.
  11. ^Rendall, Alan D. (2005)."Theorems on Existence and Global Dynamics for the Einstein Equations".Living Rev. Relativ.8 (1) 6. Article number: 6.arXiv:gr-qc/0505133.Bibcode:2005LRR.....8....6R.doi:10.12942/lrr-2005-6.PMC 5256071.PMID 28179868.
  12. ^Misner, Thorne & Wheeler (1973), p. 501ff.
  13. ^Weinberg (1972).
  14. ^Peebles, Phillip James Edwin (1980).The Large-scale Structure of the Universe. Princeton University Press.ISBN 0-691-08239-1.
  15. ^Efstathiou, G.; Sutherland, W. J.; Maddox, S. J. (1990). "The cosmological constant and cold dark matter".Nature.348 (6303): 705.Bibcode:1990Natur.348..705E.doi:10.1038/348705a0.S2CID 12988317.
  16. ^Collins, P. D. B.; Martin, A. D.; Squires, E. J. (1989).Particle Physics and Cosmology. New York: Wiley.ISBN 0-471-60088-1.
  17. ^Peacock (1999).
  18. ^Gamow, George (April 28, 1970).My World Line: An Informal Autobiography.Viking Adult.ISBN 0-670-50376-2. Retrieved2007-03-14.
  19. ^Wahl, Nicolle (2005-11-22)."Was Einstein's 'biggest blunder' a stellar success?".News@UofT. University of Toronto. Archived fromthe original on 2007-03-07.
  20. ^Turner, Michael S. (May 2001). "Making Sense of the New Cosmology".Int. J. Mod. Phys. A.17 (S1):180–196.arXiv:astro-ph/0202008.Bibcode:2002IJMPA..17S.180T.doi:10.1142/S0217751X02013113.S2CID 16669258.
  21. ^abThe full expression here with partial derivatives, denoted by a comma, is fully covariant, despite the individual terms not being covariant.
  22. ^Brown, Harvey (2005).Physical Relativity. Oxford University Press. p. 164.ISBN 978-0-19-927583-0.
  23. ^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.S2CID 123364248..
  24. ^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.S2CID 122577276.
  25. ^Hsu, L.; Wainwright, J (1986). "Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions".Class. Quantum Grav.3 (6):1105–1124.Bibcode:1986CQGra...3.1105H.doi:10.1088/0264-9381/3/6/011.S2CID 250907312.
  26. ^LeBlanc, V. G. (1997). "Asymptotic states of magnetic Bianchi I cosmologies".Class. Quantum Grav.14 (8): 2281.Bibcode:1997CQGra..14.2281L.doi:10.1088/0264-9381/14/8/025.S2CID 250876974.
  27. ^Kohli, Ikjyot Singh; Haslam, Michael C. (2013). "Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model".Phys. Rev. D.88 (6) 063518.arXiv:1304.8042.Bibcode:2013PhRvD..88f3518K.doi:10.1103/physrevd.88.063518.S2CID 119178273.
  28. ^Katanaev, M. O. (2006). "Polynomial form of the Hilbert–Einstein action".Gen. Rel. Grav.38 (8):1233–1240.arXiv:gr-qc/0507026.Bibcode:2006GReGr..38.1233K.doi:10.1007/s10714-006-0310-5.S2CID 6263993.

References

[edit]

SeeGeneral relativity resources.

External links

[edit]
Wikibooks has a book on the topic of:General Relativity
Wikiversity has learning resources about General Relativity

External images

[edit]
Physics
Works
In popular
culture
Prizes
Books about
Einstein
Family
Related
Special
relativity
Background
Fundamental
concepts
Formulation
Phenomena
Spacetime
General
relativity
Background
Fundamental
concepts
Formulation
Phenomena
Advanced
theories
Solutions
Scientists
International
National
Other
Retrieved from "https://en.wikipedia.org/w/index.php?title=Einstein_field_equations&oldid=1320600487"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp