Dr. Gyula Bene Department for Theoretical Physics, Loránd Eötvös University Pázmány Péter sétány 1/A, 1117 Budapest 8. week
Conservation laws
In curved spacetime the energy-momentum tensor satisfies the divergence equation
\[\begin{align}
T^{k}_{i;k}=0
\end{align}\]
which may be written as
\[\begin{align}
T^{k}_{i;k}=\frac{1}{\sqrt{-g}}\frac{\partial \left(T^{k}_{i}\sqrt{-g}\right)}{\partial x^k}-
\frac{1}{2}\frac{\partial g_{kl}}{\partial x^i}T^{kl}=0\;.
\end{align}\]
It does not imply a conservation law.
Reason: the combined energy and momentum of gravitational field and matter is conserved.
Determining the conserved four-momentum
Based on Einstein's equation we are looking for such an expression
that goes
over to the energy-momentum tensor of matter ( \(T^{ik}\)) when switching off
gravity,
whose volume integral is a conserved quantity, and
which differs from \(T^{ik}\) by an expression that contains at most
first derivatives of the metric.
We do the calculation in a locally geodetic (inertial) frame, so first
derivatives of the metric and Christoffel's symbols vanish at a given point.
Thus we have there
\[\begin{align}
T^{ik}_{;k}=\frac{\partial T^{ik}}{\partial x^k}=0
\end{align}\]
We cast \(T^{ik}\) with the help of Einstein's equation to the form
\[\begin{align}
T^{ik}=\frac{\partial \eta^{ikl}}{\partial x^l}\;.
\end{align}\]
where
\[\begin{align}
\eta^{ikl}=-\eta^{ilk}
\end{align}\]
Hence the continuity equation is automatically satisfied.
To this end we start from Einstein's equations:
\[\begin{align}
T^{ik}=\frac{c^4}{8\pi k}\left({\mathcal R}_{ik}-\frac{1}{2}g_{ik}{\mathcal R}\right)
\end{align}\]
In a locally geodetic system we have
\[\begin{align}
{\mathcal R}^{ik}
=
\frac{1}{2}\;g^{im}g^{kp}g^{ln}\left\{\frac{\partial^2 g_{lp}}{\partial x^m \partial x^n}
+\frac{\partial^2 g_{mn}}{\partial x^l \partial x^p}
-\frac{\partial^2 g_{ln}}{\partial x^m \partial x^p}
-\frac{\partial^2 g_{mp}}{\partial x^l \partial x^n}
\right\}
\end{align}\]
We get:
\[\begin{align}
T^{ik}=\frac{\partial}{\partial x^l}\left\{\frac{1}{(-g)}\right.&\underbrace{\frac{c^4}{16\pi k}\frac{\partial}{\partial x^m}
\left[(-g)\left(g^{ik}g^{lm}-g^{il}g^{km}\right)\right]}&\left.\phantom{\frac{1}{(-g)}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\right\}\\
&b^{ikl}&
\end{align}\]
(\(\eta^{ikl}=\frac{1}{(-g)}b^{ikl}\))
Still in locally geodetic system:
\[\begin{align}
T^{ik}=\frac{1}{(-g)}\frac{\partial b^{ikl}}{\partial x^l}
\end{align}\]
or
\[\begin{align}
(-g)T^{ik}=\frac{\partial b^{ikl}}{\partial x^l}
\end{align}\]
We return to curved coordinates. Then
\[\begin{align}
(-g)\left(T^{ik}+t^{ik}\right)=\frac{\partial b^{ikl}}{\partial x^l}
\end{align}\]
where
\(t^{ik}\) depends at most on the first derivatives of the metric. Explicitely
we have
\[\begin{align}
t^{ik}&=\frac{c^4}{16\pi k}\left\{\left(g^{il}g^{km}-g^{ik}g^{lm}\right)\left(2\Gamma^n_{lm}\Gamma^p_{np}-\Gamma^n_{lp}\Gamma^p_{mn}-\Gamma^n_{ln}\Gamma^p_{mp}\right)\right. \\
&+\left.g^{il}g^{mn}\left(\Gamma^k_{lp}\Gamma^p_{mn}+\Gamma^k_{mn}\Gamma^p_{lp}-\Gamma^k_{np}\Gamma^p_{lm}
-\Gamma^k_{lm}\Gamma^p_{np}\right)\right. \\
&+\left.g^{kl}g^{mn}\left(\Gamma^i_{lp}\Gamma^p_{mn}+\Gamma^i_{mn}\Gamma^p_{lp}-\Gamma^i_{np}\Gamma^p_{lm}
-\Gamma^i_{lm}\Gamma^p_{np}\right)\right. \\
&+\left.g^{lm}g^{np}\left(\Gamma^i_{ln}\Gamma^k_{mp}-\Gamma^i_{lm}\Gamma^k_{np}\right)
\right\} \phantom{pszeud}
\end{align}\]
\(t^{ik}\) is the pseudo-tensor of energy-momentum of gravity.
Since
\[\begin{align}
b^{ikl}=-b^{ilk}\;,
\end{align}\]
it is identically true, that
\[\begin{align}
\frac{\partial}{\partial x^k}(-g)\left(T^{ik}+t^{ik}\right)=0
\end{align}\]
Therefore
\[\begin{align}
P^i=\frac{1}{c}\int (-g)\left(T^{ik}+t^{ik}\right)dS_k
\end{align}\]
is a conserved quantity. This is the combined four momentum of mattar and
gravity. (Not a four vector!)
Since
\((-g)\left(T^{ik}+t^{ik}\right)\)
is symmetric, there exist a conserved four-angular momentum, too:
\[\begin{align}
J^{ik}=\int \left(x^idP^k-x^kdP^i\right)=\frac{1}{c}\int \left[x^i\left(T^{kl}+t^{kl}\right)
-x^k\left(T^{il}+t^{il}\right)\right](-g)dS_l
\end{align}\]
Conservation of center of mass corresponds to conservation of
\(J^{0\alpha}\):
\[\begin{align}
x^0\int \left(T^{\alpha 0}+t^{\alpha 0}\right)(-g)dV-\int x^\alpha\left(T^{00}+t^{00}\right)(-g)dV=const.
\end{align}\]
Otherwise:
\[\begin{align}
X^\alpha=const.'+\frac{P^\alpha}{P^0}x^0
\end{align}\]
where
\[\begin{align}
X^\alpha=\frac{\int x^\alpha\left(T^{00}+t^{00}\right)(-g)dV}{\int \left(T^{00}+t^{00}\right)(-g)dV}
\end{align}\]
Momentum and angular momentum expressed as surface integrals:
\[\begin{align}
P^i=\frac{1}{c}\int \frac{\partial b^{i0l}}{\partial x^l}dV=\frac{1}{c}\int \frac{\partial b^{i0\alpha}}{\partial x^\alpha}dV=
\frac{1}{c}\oint b^{i0\alpha}df_{\alpha}
\end{align}\]
Similarly we get
\[\begin{align}
J^{ik}=\frac{1}{c}\oint \left(x^ib^{k0\alpha}-x^kb^{i0\alpha}+\lambda^{i0\alpha k}\right)df_{\alpha}
\end{align}\]
bene@arpad.elte.hu
