Equations of General Relativity

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I have thought seriously about this question,
and have come to the conclusion that
what I have to say cannot reasonably be conveyed
without a certain amount of mathematical notation
and the exploration of genuine mathematical concepts.

Roger Penrose
The Road to Reality

Equations of General Relativity

Problem 1.

Consider a spacetime with diagonal metric \[ds^2=g_{00}(dx^{0})^2 + g_{11}(dx^{1})^2 +g_{22}(dx^{2})^{2}+g_{33}(dx^{3})^{2}.\] Find the explicit expressions for the intervals of proper time and spatial length, and for the 4-volume. Show that the invariant $4$-volume is given by \[\sqrt{-g}\;d^{4}x\equiv \sqrt{-g}\;dx^{0}dx^{1}dx^{2}dx^{3},\] where $g=\det(g_{\mu\nu})$.


Problem 2.

Let there be an observer with $4$-velocity $u^{\mu}$. Show that the energy of a photon with $4$-wave vector $k^\mu$ that he registers is $u^{\mu}k_{\mu}$, and the energy of a massive particle with $4$-momentum $p^{\mu}$ is $u^{\mu}p_{\mu}$.


Problem 3.

The covariant derivative (or connection) $\nabla_{\mu}$ is a tensorial generalization of partial derivative of a vector field $A^{\mu}(x)$ in the curved space-time. It's action on vectors is defined as \begin{equation}\label{nabla} \nabla_{\mu}A^{\nu}=\partial_{\mu}A^{\nu}+ {\Gamma^{\nu}}_{\lambda\mu}A^{\lambda};\qquad \nabla_{\mu}A_{\nu}=\partial_{\mu}A_{\nu}- {\Gamma^{\rho}}_{\nu\mu}A_{\rho}, \end{equation} where matrices ${\Gamma^{i}}_{jk}$ are called the connection coefficients, so that $\nabla_{\mu}A^{\nu}$ and $\nabla_{\mu}A_{\nu}$ are tensors. The connection used in GR is symmetric in lower indices (${\Gamma^{\lambda}}_{\mu\nu}= {\Gamma^{\lambda}}_{\nu\mu}$) and compatible with the metric $\nabla_{\lambda}g_{\mu\nu}=0$. It is called the Levi-Civita's connection, and the corresponding coefficients ${\Gamma^{\lambda}}_{\mu\nu}$ the Christoffel symbols. The action on tensors is defined through linearity and Leibniz rule. Express the Christoffel symbols through the metric tensor.


Problem 4.

Derive the transformation rule for matrices ${\Gamma^{\lambda}}_{\mu\nu}$ under coordinate transformations. Show that for any given point of spacetime there is a coordinate frame, in which ${\Gamma^{\lambda}}_{\mu\nu}$ are equal to zero in this point. It is called a locally inertial, or locally geodesic frame.


Problem 5.

Free falling particles' worldlines in General Relativity are \textit{geodesics} of the spacetime--- i.e the curves $x^{\mu}(\lambda)$ with tangent vector $u^{\mu}=dx^{\mu}/d\lambda$, such that covariant derivative of the latter along the curve equals to zero: \[u^{\mu}\nabla_{\mu}u^{\nu}=0.\]

In a (pseudo-)Euclidean space the geodesics are straight lines. Obtain the general equation of geodesics in terms of the connection coefficients. Show that the quantity $u^{\mu}u_{\mu}$ is conserved along the geodesic.


Problem 6.

Consider the action for a massive particle of the form \[S_{AB}=-mc\int_{A}^{B} ds,\quad\text{where}\quad ds=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}\] and derive the geodesic equation from the principle of least action. Find the canonical $4$-momentum of a massive particle and the energy of a photon.


Problem 7.

A Killing vector field, or just Killing vector, is a vector field $K^{\mu}(x)$, such that infinitesimal coordinate transformation $x\to x'+\varepsilon K $ (where $\varepsilon\to 0$) leaves the metric invariant in the sense* \[g_{\mu\nu}(x)=g'_{\mu\nu}(x).\] A Killing vector defines a one-parametric symmetry group of the metric tensor, called isometry.

Show that a Killing vector obeys the equation \[\nabla_{\mu}K_{\nu}+\nabla_{\nu}K_{\mu}=0,\] called the Killing equation.


*That is, let $g_{\mu\nu}(x)$ be the components of the metric at some point $A$ in the original frame. Then $g'_{\mu\nu}(x)$ are the components of the metric in the new frame, taken at point $A'$, which has the same coordinates in the new frame as $A$ had in the old frame.


Problem 8.

Suppose there is a coordinate frame, in which the metric does not depend on one of the coordinates $x^{k}$. Show that in this case the vectors $\partial_{k}$ constitute the Killing vector field, and that the inverse is also true: if there is a Killing vector, we can construct such a coordinate frame.


Problem 9.

Prove that if $K^{\mu}$ is a Killing vector, the quantity $K^{\mu}u_{\mu}$ is conserved along a geodesic with tangent vector $u^{\mu}$.


Problem 10.

The Riemann curvature tensor ${R^{i}}_{klm}$ can be defined through the so-called Ricci identity, written for arbitrary $4$-vector $A^i$: \[\nabla_{m}\nabla_{l}A^{i}-\nabla_{l}\nabla_{m}A^{i} ={R^{i}}_{klm}A^{k}.\] Express ${R^{i}}_{klm}$ in terms of the Christoffel symbols. Show that the Ricci tensor \[R_{km}={R^{l}}_{klm}\] is symmetric.


Problem 11.

Prove the differential Bianchi identity for the curvature tensor: \begin{equation}\label{BianchiId} \nabla_{i}{R^{j}}_{klm}+\nabla_{l}{R^{j}}_{kmi} +\nabla_{m}{R^{j}}_{kil}=0, \end{equation} and show that \[\nabla_{i}R^{i}_{j}=\tfrac{1}{2}\partial_{j}R.\]


Problem 12.

The energy-momentum tensor in General Relativity is defined through the variational derivative of the action for matter \[S_{m}[g^{\mu\nu},\psi] =\frac{1}{c}\int d^{4}x\sqrt{-g} \;L_{m}(g^{\mu\nu},\psi)\] with respect to metric $g^{\mu\nu}$: \[\delta_{g}S=\tfrac{1}{2}\int d^{4}x\sqrt{-g}\; \delta g^{\mu\nu}T_{\mu\nu}.\] Here $L_{m}$ is the Lagrange function for the matter fields $\psi$. Show that for the cases of a massless scalar field and electromagnetic field the above definition reduces to the usual one.


Problem 13.

The full action consists of the action for matter, discussed in the previous problem, and the action for the gravitational field $S_{g}$: \[S=S_{g}+S_{m};\qquad\text{where}\quad S_{g}=-\frac{c^{3}}{16\pi G}\int d^{4}x\sqrt{-g}\;R,\] $R=R^{i}_{i}=g^{ik}R_{ik}$ is scalar curvature, $G$ is the gravitational constant. Starting from the variational principle, derive the Einstein-Hilbert equations\footnote{Also called Einstein field equations, or just Einstein equation (either in singular or plural); the action $S_{g}$ is referred to as Hilbert or Einstein-Hilbert action.} for the gravitational field \begin{equation}\label{EinsteinHilbetEq} R_{\mu\nu}-\tfrac{1}{2}Rg_{\mu\nu} =\frac{8\pi G}{c^4}T_{\mu\nu}. \end{equation}


Problem 14.

Show that the Einstein's equation can be presented in the following form \[R_{\mu\nu} = \frac{8\pi G}{c^4} \left( T_{\mu\nu}-\tfrac{1}{2}T g_{\mu\nu}\right), \quad\text{where}\quad T=T^{\mu}_{\mu}.\] Show that it leads to the energy-momentum conservation law for matter \[\nabla_{\mu}T^{\mu\nu}=0.\] Does it mean the energy and momentum of matter are actually conserved in general?