Tensor Calculus General Theory of Relativity

[Sissy]

Hello

I have huge problems with the following exercise. Please give me some hints. No complete Solutions but a little bit help.


Find the differential equations of the paths of test particles in the space-time of which the metric ist

$$\mathrm{d}s^2 = e^{2kx} \left[- \left( \mathrm{d}x^2 + \mathrm{d}y^2 +\mathrm{d}z^2 \right) + \mathrm{d}t^2 \right]$$,

where $$k$$ is a constant. If

$$v^2 = \left( \dfrac{\mathrm{d} x }{\mathrm{d} t } \right)^2 + \left( \dfrac{\mathrm{d}y }{\mathrm{d}t } \right)^2 + \left( \dfrac{ \mathrm{d} z }{\mathrm{d} t } \right)^2$$

and if $$v=V$$ when $$x=0$$, show that

$$1-v^2 = \left( 1-V^2 \right) e^{2kx}$$.


Now I have no idea how to start. I do not want a solution. I will calculate it on my own but I need some assistance.

greetings

 

[phyzguy]

Do you know how to calculate the equation for geodesics given a metric? Test particles will follow geodesics.

 

[Sissy]

In the lecture we had this equation:

$$\dfrac{\mathrm{d}^2 x^i }{ \mathrm{d} s^2 } + \Gamma ^i _{kl} \dfrac{ \mathrm{d} x^k}{ \mathrm{d} s} ~ \dfrac{\mathrm{d} x^l}{ \mathrm{d} s} = 0$$

But how to use this in my problem?

greetings

 

[phyzguy]

First calculate the $$\Gamma^i_{jk}$$ terms. Do you know how to do this? Then you have a set of differential equations for the 4-velocity components.

 

[Sissy]

We introduced $$\Gamma ^m_{kl}$$ as

$$\Gamma ^m_{kl} = g^{im} \Gamma_{ikl}$$

with

$$\Gamma_{ikl} = \dfrac{1}{2} \left( \dfrac{\partial g_{ik}}{\partial x^l} + \dfrac{\partial g_{li}}{\partial x^k} + \dfrac{\partial g_{kl}}{ \partial x^i} \right)$$

and called $$\Gamma_{ikl}$$ Christoffel symbols of first kind and $$\Gamma ^m_{kl}$$ Christoffel symbols of second kind.

I think $$g$$ is the metric tensor coming from the Riemann-metric but how to calculate this $$g$$? This is something I even did not understand it in the lecture.

Also from lecture we know

$$\mathrm{d}s^2 = g_{lm} \mathrm{d}x^l \mathrm{d}x^m$$

But I don't know how to work with this.

You said that I should calculate this christoffel symbols of second kind but what I have to differentiate in the exercise and why? I only have this metric and velocity square?

thanks for help

 

[phyzguy]

You have it right there. You've told me:
$$\mathrm{d}s^2 = g_{lm} \mathrm{d}x^l \mathrm{d}x^m$$

and:

$$\mathrm{d}s^2 = e^{2kx} \left[- \left( \mathrm{d}x^2 + \mathrm{d}y^2 +\mathrm{d}z^2 \right) + \mathrm{d}t^2 \right]$$

So, remembering that

$$dx^i = (dt, dx, dy, dz)$$

Can you tell me what $$g_{ij}$$ is? If not, you need to go back and review whatever textbook or reference materials you are using.