Editing Einstein field equations (section)

== Features ==

=== Conservation of energy and momentum ===
General relativity is consistent with the local conservation of energy and momentum expressed as
<math display="block">\nabla_\beta T^{\alpha\beta} = {T^{\alpha\beta}}_{;\beta} = 0.</math>

{{math proof|title=Derivation of local energy–momentum conservation|proof=
[[Tensor contraction|Contracting]] the [[Riemann curvature tensor#Symmetries and identities|differential Bianchi identity]]
<math display="block">R_{\alpha\beta[\gamma\delta;\varepsilon]} = 0</math>
with {{mvar|g{{sup|αβ}}}} gives, using the fact that the metric tensor is covariantly constant, i.e. {{math|1=''g''{{sup|''αβ''}}{{sub|;''γ''}} = 0}},
<math display="block">{R^\gamma}_{\beta\gamma\delta;\varepsilon} + {R^\gamma}_{\beta\varepsilon\gamma;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0</math>

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
<math display="block">{R^\gamma}_{\beta\gamma\delta;\varepsilon} - {R^\gamma}_{\beta\gamma\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0</math>
which is equivalent to
<math display="block">R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0</math>
using the definition of the [[Ricci tensor]].

Next, contract again with the metric
<math display="block">g^{\beta\delta}\left(R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma}\right) = 0</math>
to get
<math display="block">{R^\delta}_{\delta;\varepsilon} - {R^\delta}_{\varepsilon;\delta} + {R^{\gamma\delta}}_{\delta\varepsilon;\gamma} = 0 .</math>

The definitions of the Ricci curvature tensor and the scalar curvature then show that
<math display="block">R_{;\varepsilon} - 2{R^\gamma}_{\varepsilon;\gamma} = 0 ,</math>
which can be rewritten as
<math display="block">\left({R^\gamma}_{\varepsilon} - \tfrac{1}{2}{g^\gamma}_{\varepsilon}R\right)_{;\gamma} = 0 .</math>

A final contraction with {{math|''g''{{sup|''εδ''}}}} gives
<math display="block">\left(R^{\gamma\delta} - \tfrac{1}{2}g^{\gamma\delta}R\right)_{;\gamma} = 0 ,</math>
which by the symmetry of the bracketed term and the definition of the [[Einstein tensor]], gives, after relabelling the indices,
<math display="block"> {G^{\alpha\beta}}_{;\beta} = 0 .</math>

Using the EFE, this immediately gives,
<math display="block">\nabla_\beta T^{\alpha\beta} = {T^{\alpha\beta}}_{;\beta} = 0</math>
}}

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 field|electric]] and [[magnetic field]]s, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is the [[Schrödinger equation]] of [[quantum mechanics]], which is linear in the [[wavefunction]].

=== Correspondence principle ===
The EFE reduce to [[Newton's law of gravity]] by using both the [[weak-field approximation]] and the [[Classical limit|low-velocity approximation]]. The constant {{mvar|G}} appearing in the EFE is determined by making these two approximations.

{{math proof |title=Derivation of Newton's law of gravity | proof=
Newtonian gravitation can be written as the theory of a scalar field, <math>\Phi</math>, which is the gravitational potential in joules per kilogram of the gravitational field <math>g=-\nabla\Phi</math>, see [[Gauss's law for gravity#Poisson's equation and gravitational potential|Gauss's law for gravity]]
<math display="block">\nabla^2 \Phi \left(\vec{x},t\right) = 4 \pi G \rho \left(\vec{x},t\right)</math>
where {{mvar|ρ}} is the mass density. The orbit of a [[free-fall]]ing particle satisfies
<math display="block">\ddot{\vec{x}}(t) = \vec{g} = - \nabla \Phi \left(\vec{x} (t),t\right) \,.</math>

In tensor notation, these become
<math display="block">\begin{align}
\Phi_{,i i} &= 4 \pi G \rho \\
\frac{d^2 x^i}{d t^2} &= - \Phi_{,i} \,.
\end{align}</math>

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
<math display="block">R_{\mu \nu} = K \left(T_{\mu \nu} - \tfrac{1}{2} T g_{\mu \nu}\right)</math>
for some constant, {{mvar|K}}, and the [[geodesic equation]]
<math display="block">\frac{d^2 x^\alpha}{d \tau^2} = - \Gamma^\alpha_{\beta \gamma} \frac{d x^\beta}{d \tau} \frac{d x^\gamma}{d \tau} \,.</math>

To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero
<math display="block">\frac{d x^\beta}{d \tau} \approx \left(\frac{dt}{d \tau}, 0, 0, 0\right) </math>
and thus
<math display="block">\frac{d}{d t} \left( \frac{dt}{d \tau} \right) \approx 0 </math>
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
<math display="block">\frac{d^2 x^i}{d t^2} \approx - \Gamma^i_{0 0} </math>
where two factors of {{math|{{sfrac|''dt''|''dτ''}}}} have been divided out. This will reduce to its Newtonian counterpart, provided
<math display="block">\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) \,.</math>

Our assumptions force {{math|1=''α'' = ''i''}} and the time (0) derivatives to be zero. So this simplifies to
<math display="block">2 \Phi_{,i} \approx g^{i j} \left(- g_{0 0 , j}\right) \approx - g_{0 0 , i} \,</math>
which is satisfied by letting
<math display="block">g_{0 0} \approx - c^2 - 2 \Phi \,.</math>

Turning to the Einstein equations, we only need the time-time component
<math display="block">R_{0 0} = K \left(T_{0 0} - \tfrac{1}{2} T g_{0 0}\right)</math>
the low speed and static field assumptions imply that
<math display="block">T_{\mu \nu} \approx \operatorname{diag} \left(T_{0 0}, 0, 0, 0\right) \approx \operatorname{diag} \left(\rho c^4, 0, 0, 0\right) \,.</math>

So
<math display="block">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 \,</math>
and thus
<math display="block">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^2\right) \left(- c^2\right)\right) = \tfrac{1}{2} K \rho c^4 \,.</math>

From the definition of the Ricci tensor
<math display="block">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}.</math>

Our simplifying assumptions make the squares of {{mvar|Γ}} disappear together with the time derivatives
<math display="block">R_{0 0} \approx \Gamma^i_{0 0 , i} \,.</math>

Combining the above equations together
<math display="block">\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 </math>
which reduces to the Newtonian field equation provided
<math display="block">\tfrac{1}{2} K \rho c^4 = 4 \pi G \rho ,</math>
which will occur if
<math display="block">K = \frac{8 \pi G}{c^4} \,.</math>
}}