Equations of General Relativity
Contents
Equations of General Relativity
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.
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 $g=\det(g_{\mu\nu})$.
We assume that the coordinate $x^{0}$ is timelike, so $g_{00}>0$, and that $x^{1}$, $x^2$ and $x^3$ is spacelike, so $g_{11},g_{22},g_{33}<0$. The interval along the worldline of a particle at rest ($dx^1 = dx^2 = dx^3 = 0$) takes the form $ds^{2}=g_{00}(dx^{0})^2$. On the other hand, in the reference frame of the particle $ds^{2}$ is the proper time interval: $ds^{2}=c^{2}d\tau^{2}$, so $d\tau=\sqrt{g_{00}}\;dx^{0}/c$. The change of variables $\sqrt{-g_{ii}}dx^{i}=d\tilde{x}^{i}$ (there is no summation over indices here) transforms the metric to locally Lorentz form \[ds^{2}=c^{2}d\tau^{2}-(d\tilde{x}^{1})^{2} -(d\tilde{x}^{2})^{2}-(d\tilde{x}^{3})^{2},\] and it is easy to see that the element of three-dimensional length (line element) is given by \[dl^{2}=-\big[g_{11}(dx^{1})^2 +g_{22}(dx^{2})^{2}+g_{33}(dx^{3})^{2}\big].\] The element of $4$-volume is then \[cd\tau d\tilde{x}^{1}d\tilde{x}^{2}d\tilde{x}^{3} =\sqrt{-g_{00}g_{11}g_{22}g_{33}}\; dx^{0}dx^{1}dx^{2}dx^{3} =\sqrt{-g}\;dx^{0}dx^{1}dx^{2}dx^{3}.\]
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.
