Homework Help: Light-like Geodesic - What are the limits of integration?

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1. Apr 27, 2015

unscientific

1. The problem statement, all variables and given/known data

Consider the following geodesic of a massless particle where $\alpha$ is a constant:

$$\dot r = \frac{\alpha}{a(t)^2}$$
$$c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)}$$

2. Relevant equations

3. The attempt at a solution

Part (a)
$$c \frac{dt}{d\lambda} = \frac{\alpha}{a}$$
$$a dt = \frac{\alpha}{c} d\lambda$$
$$\frac{1}{H} a = \frac{\alpha}{c} \lambda + \epsilon^{'}$$
$$a = \frac{H}{c} \left( \alpha \lambda + \epsilon \right)$$

Similarly,
$$r = \frac{c^2}{H^2} \left[ -\frac{1}{\alpha \lambda + \epsilon} + \delta \right]$$

Part(b)
I'm confused as to what the limits of integration are. I'm not sure if this is right:

At $t = t_0$, $a(t_0) = 1 = \frac{H}{c} \left(\alpha \lambda + \epsilon \right)$.

At $r = r_e$ what happens to $\delta$?

2. May 2, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 2, 2015

unscientific

I have tried all I can on this question, I think there might be something I'm not seeing? Would appreciate if anyone could tell me what that is.

4. May 4, 2015

unscientific

How do I get the final expression?

5. May 7, 2015

unscientific

bumpp

6. May 10, 2015

unscientific

bumpp anyone?

7. May 14, 2015

unscientific

bumping on light-like, limits.

8. May 16, 2015

unscientific

bumpp

9. May 18, 2015

unscientific

bumpp

10. May 22, 2015

unscientific

bumpp

11. May 23, 2015

unscientific

bumpp on last part

12. May 25, 2015

unscientific

bumpp

13. May 31, 2015

unscientific

bumpp

14. Jun 1, 2015

bump