- #1

Markus Kahn

- 112

- 14

- 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?

$$Q^t := -r^2 (1-r^{-3})\dot q^0,\quad \text{and}\quad Q^k := r^2\dot q^k \quad \text{for}\quad k \in \{x,y\}.$$

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?