GR: Solving Geodesic Equation

Date:

26 Sep 2019

Derivation

Step 1

\[\begin{split}\begin{align*} & \text{I. Earlier we had}\\ & ds^2 = -(1-{r_s\over r}) dt^2+(1-{r_s\over r})^{-1} dr^2 +r^2 d\theta^2+ r^2 \sin^2 \theta d\phi^2\\ & \boxed{ {1\over 2}( -{\partial g_{ab}\over \partial x^{(c)}}{dx^{(a)} \over d\tau}{dx^{(b)} \over d\tau} ) = {d\over d\tau}[ -g_{bc} {d{x}^{(b)} \over d\tau}] \;\;\;\;\text{Geodesic Equation}}\\ & \text{II. Consider different cases of }x^{(c)}\\ & (1)\; x^{(0)}=t\Rightarrow 0={d\over d\tau}[ -g_{b0} {d{x}^{(b)} \over d\tau}] \\ & \;\;\;\; \to {d\over d\tau}[ -g_{00} {d{x}^{(0)} \over d\tau}] =0 \Rightarrow (1-{r_s\over r}) {dt\over d\tau}=\text{const}\equiv e \\ & (2)\; x^{(3)}=\phi \Rightarrow 0={d\over d\tau}[ -g_{b3} {d{x}^{(b)} \over d\tau}] \\ & \;\;\;\; \to {d\over d\tau}[ -g_{33} {d{x}^{(3)} \over d\tau}] =0 \Rightarrow r^2 \sin^2 \theta {d\phi \over d\tau}=\text{const}\equiv l \\ & (3)\; x^{(2)}=\theta \Rightarrow {1\over 2}( -{\partial g_{ab}\over \partial x^{(2)}}{dx^{(a)} \over d\tau}{dx^{(b)} \over d\tau} ) = {d\over d\tau}[ -g_{b2} {d{x}^{(b)} \over d\tau}] \\ & \;\;\;\; \to {1\over 2} {\partial g_{33}\over \partial \theta}({d\phi \over d\tau})^2 = {d\over d\tau}[ g_{22} {d\theta \over d\tau}] \\ & \;\;\;\; \to r^2\sin\theta\cos\theta({d\phi \over d\tau})^2 = r^2 {d^2\theta \over d\tau^2} + {d\theta \over d\tau} 2r {dr\over d\tau}\\ & \;\;\;\; \to \theta ={\pi \over 2} \text{ (Plane Orbit)} \text{ is a solution}\\ & (4)\; \begin{cases} x^{(1)}=r \\ \theta ={\pi \over 2} \end{cases} \Rightarrow {1\over 2} {\partial g_{00}\over \partial r} ({dt \over d\tau})^2 + {1\over 2} {\partial g_{11}\over \partial r} ({dr \over d\tau})^2 + {1\over 2} {\partial g_{33}\over \partial r} ({d\phi \over d\tau})^2 = {d\over d\tau}[ g_{11} {dr \over d\tau}] \\ & \;\;\;\; \to {1\over 2} (-{r_s\over r^2}) ({dt \over d\tau})^2 + {1\over 2} (1-{r_s\over r})^{-2}{r_s\over r^2} ({dr \over d\tau})^2 + r ({d\phi \over d\tau})^2 = (1-{r_s\over r})^{-1}{d^2r \over d\tau^2} \\ & \text{III. Applications}\\ & (1)\; \text{[ II.(4)+Circular Orbit ]} \\ & \;\;\;\; \text{Circular} \Rightarrow {dr \over d\tau}={d^2r \over d\tau^2}=0 \Rightarrow {1\over 2} (-{r_s\over r^2}) ({dt \over d\tau})^2 + r ({d\phi \over d\tau})^2 = 0 \\ & \;\;\;\; \to {1\over 2} (-{r_s\over r^2}) ({dt \over d\tau})^2 + r ({d\phi \over dt}{dt \over d\tau})^2 = 0 \\ & \;\;\;\; \text{1. Any observer: }\Omega \equiv {d\phi \over d\tau} \\ & \;\;\;\; \text{2. Observer at infinity: } \Omega = {d\phi \over dt} \Rightarrow \Omega^2 ={1\over 2} {r_s\over r^3} \underset{r\gg r_s}{=} {M\over r^3} \end{align*}\end{split}\]