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

Problem 1: proper time and distance in diagonal metrics

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: observable invariants

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: covariant derivative

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: locally inertial frame

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: geodesics

Free falling particles' worldlines in General Relativity are 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: principle of least action

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: symmetries and Killing vectors

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: Killing vectors and selected frames of reference

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: Killing vectors and integrals of motions

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: Riemann tensor

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: Bianchi identity

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: energy-momentum tensor

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: Einstein equations

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$^*$ 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: energy conservation

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?