# Light-like Geodesic - What are the limits of integration?

Tags:
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