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

  1. Apr 27, 2015 #1
    1. The problem statement, all variables and given/known data

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

    [tex] \dot r = \frac{\alpha}{a(t)^2} [/tex]
    [tex]c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)} [/tex]

    2011_B5_Q2.png


    2. Relevant equations


    3. The attempt at a solution

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

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

    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. jcsd
  3. May 2, 2015 #2
    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?
     
  4. May 2, 2015 #3
    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.
     
  5. May 4, 2015 #4
    How do I get the final expression?
     
  6. May 7, 2015 #5
    bumpp
     
  7. May 10, 2015 #6
    bumpp anyone?
     
  8. May 14, 2015 #7
    bumping on light-like, limits.
     
  9. May 16, 2015 #8
  10. May 18, 2015 #9
  11. May 22, 2015 #10
  12. May 23, 2015 #11
    bumpp on last part
     
  13. May 25, 2015 #12
  14. May 31, 2015 #13
  15. Jun 1, 2015 #14
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