Equation of motion in curved spacetime

[Markus Kahn]

Homework Statement

Consider the spacetime metric given by
$$d s^{2}=-r^{2}\left(1-\frac{1}{r^{3}}\right) d t^{2}+\frac{d r^{2}}{r^{2}\left(1-\frac{1}{r^{3}}\right)}+r^{2}\left(d x^{2}+d y^{2}\right)$$.

1) Identify three quantities which are constant along geodesics, corresponding to the Killing vectors ##\partial_k## for ##k\in\{t,x,y\}##.
2) Show that the geodesic equations can be reduced to an equation of the form
$$\dot r ^2 + V(r)=K^2,$$
for some potential function ##V(r)##. Fix ##K## in terms of the constants of motion from 1).

optional: 3) Sketch the potential ##V(r)## fro timelike and null-geodesics.

Relevant Equations

All given above.

1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities
$$Q^t := -r^2 (1-r^{-3})\dot q^0,\quad \text{and}\quad Q^k := r^2\dot q^k \quad \text{for}\quad k \in \{x,y\}.$$

2) We first construct a Lagragian
$$L=\frac{1}{2}g_{\mu\nu} \dot q^\mu \dot q^\nu = -\frac{1}{2}r^2\left(1-\frac{1}{r^3}\right)\dot t^2 + \frac{1}{2r^2(1-\frac{1}{r^3})}\dot r^2 +\frac{1}{2}r^2(\dot x^2 +\dot y^2). $$
We then find
$$\frac{\partial L}{\partial q^k} = 0 \quad \text{and}\quad \frac{\partial L }{\partial \dot q^k} = Q^k\quad\text{for}\quad k\in\{t,x,y\}$$
and
$$\frac{\partial L}{\partial r} = -\frac{1}{2}\left(\frac{1}{r^2}+2r\right)\dot t^2 - \frac{1}{2}\frac{2r^3+1}{(r^3-1)^2}\dot r^2 + r(\dot x^2 + \dot y^2)\quad \text{and}\quad \frac{\partial L }{\partial \dot r} = \frac{1}{2}\left(1-\frac{1}{r^3}\dot r\right).$$
Now, since ##Q^k## is conserved in time, three of the four geodesic equations are automatically satisfied:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot q^k}-\frac{\partial L}{\partial q^k} =0 \quad\Longleftrightarrow \quad \frac{d Q^k}{dt}= 0 \quad \text{for}\quad k\in\{t,x,y\}.$$
The only non-trivial eom is the one for ##r##. The issue is, I can't get it into the desired form... What I get is:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot r}-\frac{\partial L}{\partial r} =0 \quad\Longleftrightarrow \quad \ddot r - \frac{2r^3+1}{2r(r^3-1)}\dot r^2 -\frac{(Q^x)^2 + (Q^y)^2}{r}\left(1-\frac{1}{r^3}\right) -\frac{1}{2}\frac{2 r^3 + 1}{r - r^4}(Q^t)^2 =0$$
I tried to incorporate the different ##Q^k##, but I just don't see how this is supposed to reduce to the desired result...

3) Here I'm just curious how exactly the fact that the particle is spacelike, timelike or lightlike could influence how ##V(r)## looks like.. I mean, where exactly could that information enter?

 

[Nate810]

A:For the second part of your question, it may be easier to solve using the Hamiltonian formulation. The canonical momentum conjugate to $r$ is given by\begin{equation}p_r = \frac{\partial L}{\partial \dot{r}} = \frac{1}{2}\left(1 - \frac{1}{r^3}\right) \dot{r}.\end{equation}The Hamiltonian is then given by\begin{equation}H = p_r \dot{r} - L = \frac{1}{4}\left(1 - \frac{1}{r^3}\right)\dot{r}^2 + \frac{1}{2}r^2 \left(\dot{x}^2 + \dot{y}^2\right) - \frac{1}{2}r^2 \left(1 - \frac{1}{r^3}\right)\dot{t}^2.\end{equation}Using $Q^k$ and $H$, you can rewrite the equation of motion as\begin{equation}\ddot{r} = -\frac{1}{2}\left(\frac{1}{r^2} + \frac{2}{r}\right)\dot{t}^2 - \frac{1}{2}\frac{2r^3 + 1}{(r^3 - 1)^2}\dot{r}^2 + \frac{1}{r}\left(1 - \frac{1}{r^3}\right)(Q^x)^2 + \frac{1}{r}\left(1 - \frac{1}{r^3}\right)(Q^y)^2 + \frac{1}{2}\frac{2r^3 + 1}{r^3 - 1}(Q^t)^2 - \frac{\partial H}{\partial r}.\end{equation}Now consider the constraint that the particle must be either spacelike, timelike or lightlike. This is equivalent to the constraint that the Hamiltonian must vanish, i.e.\begin{equation}H(q^k,p_k) = 0.\end{equ