Conservation of energy and momentum General relativity is consistent with the local conservation of energy and momentum expressed as \nabla_\beta T^{\alpha\beta} = {T^{\alpha\beta}}_{;\beta} = 0. {{math proof|title=Derivation of local energy–momentum conservation|proof=
Contracting the
differential Bianchi identity R_{\alpha\beta[\gamma\delta;\varepsilon]} = 0 with gives, using the fact that the metric tensor is covariantly constant, i.e. , {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: {R^\gamma}_{\beta\gamma\delta;\varepsilon} - {R^\gamma}_{\beta\gamma\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0 which is equivalent to R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma} = 0 using the definition of the
Ricci tensor. Next, contract again with the metric g^{\beta\delta}\left(R_{\beta\delta;\varepsilon} - R_{\beta\varepsilon;\delta} + {R^\gamma}_{\beta\delta\varepsilon;\gamma}\right) = 0 to get {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 that R_{;\varepsilon} - 2{R^\gamma}_{\varepsilon;\gamma} = 0 , which can be rewritten as \left({R^\gamma}_{\varepsilon} - \tfrac{1}{2}{g^\gamma}_{\varepsilon}R\right)_{;\gamma} = 0 . A final contraction with gives \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 the
Einstein tensor, gives, after relabelling the indices, {G^{\alpha\beta}}_{;\beta} = 0 . Using the EFE, this immediately gives, \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 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 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
low-velocity approximation. The constant 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, \Phi, which is the gravitational potential in joules per kilogram of the gravitational field g=-\nabla\Phi, see
Gauss's law for gravity \nabla^2 \Phi \left(\vec{x},t\right) = 4 \pi G \rho \left(\vec{x},t\right) where is the mass density. The orbit of a
free-falling particle satisfies \ddot{\vec{x}}(t) = \vec{g} = - \nabla \Phi \left(\vec{x} (t),t\right) \,. 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, , 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{dt}{d \tau}, 0, 0, 0\right) and thus \frac{d}{d t} \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 gives \frac{d^2 x^i}{d t^2} \approx - \Gamma^i_{0 0} where two factors of 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 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 \operatorname{diag} \left(T_{0 0}, 0, 0, 0\right) \approx \operatorname{diag} \left(\rho c^4, 0, 0, 0\right) \,. 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^2\right) \left(- c^2\right)\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 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 ==