Dr. Gyula Bene Department for Theoretical Physics, Loránd Eötvös University Pázmány Péter sétány 1/A, 1117 Budapest 7. week
Einstein's equations
Combined action of matter and gravity. Derivation of Einstein's
equations. Properties of Einstein's
equations. Second time derivatives only of spacelike metric tensor
components. Constraints.
Initial value problem. Independent physical quantities.
Variational principle:
\[\begin{align}
\delta (S_g+S_m)=0
\end{align}\]
Variation is done with respect to metric tensor components.
\[\begin{align}
\delta S_g\propto \delta \int {\mathcal R}\sqrt{-g}d\Omega &= \delta \int g^{ik}{\mathcal R}_{ik}\sqrt{-g}d\Omega \\&
=\int \left({\mathcal R}_{ik}\sqrt{-g} \delta g^{ik}+ g^{ik}{\mathcal R}_{ik}\delta \sqrt{-g}+g^{ik}\sqrt{-g}\delta {\mathcal R}_{ik}\right)d\Omega
\end{align}\]
\[\begin{align}
\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}\;g_{ik}\;\delta g^{ik}
\end{align}\]
\[\begin{align}
\delta \int R\sqrt{-g}d\Omega = \int \left({\mathcal R}_{ik}-\frac{1}{2}g_{ik}R\right)\delta g^{ik}d\Omega
+\int g^{ik}\delta {\mathcal R}_{ik}\;\sqrt{-g}d\Omega
\end{align}\]
Second term vanishes:
\(\delta \Gamma^k_{il}A_k dx^l\)
is a vector, since it is difference of vectors at the same point. Hence
\(\delta \Gamma^k_{il}\)
is a tensor.
We do calculation in a locally geodetic (inertial) system, thus first derivatives
of the metric tensor vanish.
\[\begin{align}
g^{ik}\delta {\mathcal R}_{ik}=g^{ik}\left(\frac{\partial }{\partial x^l}\delta \Gamma^l_{ik}
-\frac{\partial }{\partial x^k}\delta \Gamma^l_{il}
\right)=\frac{\partial w^l}{\partial x^l}
\end{align}\]
where
\[\begin{align}
w^l=g^{ik}\delta \Gamma^l_{ik}-g^{il}\delta \Gamma^k_{ik}
\end{align}\]
Therefore, in the general case
\[\begin{align}
g^{ik}\delta {\mathcal R}_{ik}=\frac{1}{\sqrt{-g}}\frac{\partial\left(\sqrt{-g}\; w^l\right)}{\partial x^l}
\end{align}\]
Gauss's theorem proves the statement. QED.
So
\[\begin{align}
\delta S_g=-\frac{c^3}{16\pi k}\int \left({\mathcal R}_{ik}-\frac{1}{2}g_{ik}{\mathcal R}\right)\delta g^{ik}d\Omega
\end{align}\]
Variation of matter action integral (cf. derivation of energy-momentum tensor):
\[\begin{align}
\delta S_m=\frac{1}{2c}\int T_{ik}\delta g^{ik}\;\sqrt{-g}\;d\Omega
\end{align}\]
Complete variation:
\[\begin{align}
-\frac{c^3}{16\pi k}\int \left({\mathcal R}_{ik}-\frac{1}{2}g_{ik}{\mathcal R}-\frac{8\pi k}{c^4} T_{ik}\right)\delta g^{ik}d\Omega
\end{align}\]
This implies Einstein's equations:
\[\begin{align}
{\mathcal R}_{ik}-\frac{1}{2}g_{ik}{\mathcal R}=\frac{8\pi k}{c^4} T_{ik}
\end{align}\]
Otherwise:
\[\begin{align}
{\mathcal R}^k_{i}-\frac{1}{2}\delta^k_{i}{\mathcal R}=\frac{8\pi k}{c^4} T^k_{i}\phantom{ein_v}
\end{align}\]
Contracting indices:
\[\begin{align}
{\mathcal R}=-\frac{8\pi k}{c^4} T
\end{align}\]
This implies
\[\begin{align}
{\mathcal R}_{ik}=\frac{8\pi k}{c^4} \left(T_{ik}-\frac{1}{2}g_{ik}T\right)
\end{align}\]
Einstein's equations are nonlinear, second order partial differential
equations. They contain the second time derivatives of the spacelike metric tensor
components only, and these derivatives appear in mixed components (one covariant and one contravariant index)
only in spacelike (type \(\dots^\alpha_\beta\)) equations. The other four
equations contain only first derivatives, so these are constraints.
Initial values of matter and gravity cannot be given independently.
There are eight independent initial conditions, four of them characterizes
matter (e.g. components of four velocity) and four of them characterizes
gravity (two independent polarization of gravitational radiation, amplitudes
and first time derivatives).
Equations of motion of matter is implied by Einstein's equations.
A unique solution also requires equation of state of matter, that is not
contained in Einstein's equations.
Consistency with divergence equation
Vanishing of covariant four-divergence of energy-momentum tensor was a
defining property. We show that this property is consistent with Einstein's equations:
\[\begin{align}
{\mathcal R}^k_{i;k}-\frac{1}{2}{\mathcal R}_{,i}=\frac{8\pi k}{c^4} T^k_{i;k}\phantom{ein_v2}
\end{align}\]
The left hand side vanishes due to Bianchi's identity. QED.
bene@arpad.elte.hu
