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Redshift in Frequency - Universe

  1. May 20, 2015 #1
    1. The problem statement, all variables and given/known data

    (a) Show the relation between frequency received and emitted
    (b) Find the proper area of sphere
    (c) Find ratio of fluxes

    2007_B5_Q4.png
    2. Relevant equations


    3. The attempt at a solution

    Part (a)
    Metric is ##ds^2 = -c^2dt^2 + a(t)^2 \left( \frac{dr^2}{1-kr^2}+ r^2(d\theta^2 + \sin^2\theta) \right)##. For a light-like geodesic, we have ##ds^2=0##, which means
    [tex]c\frac{1}{a(t)} dt = \frac{1}{\sqrt{1-kr^2}} dr[/tex]
    Since RHS is purely in terms of spatial distance, we have
    [tex]\frac{1}{a(t_1)}\delta t_1 = \frac{1}{a(t_2)}\delta t_2 [/tex]

    Part(b)
    Proper area is:
    [tex]dA = \left( a r d\theta \right)\left( a r \sin\theta d\phi \right)[/tex]
    [tex]A = 4\pi r^2 a^2(t_2)[/tex]

    Part(c)
    Let's first start with emitter at A.
    From part (a), frequency observed is ##\frac{a(t_{1A})}{a(t_{2A})}f_0## where ##t_1## and ##t_2## is time emitted and received.
    Area at reception is ##4\pi r_a^2 a^2(t_{2A})##.
    Flux is then proportional to ##\frac{a(t_{1A})}{a(t_{2A})^3 r_a^2}##. Flux for B is then ##\frac{a(t_{1B})}{a(t_{2B})^3 r_b^2}##.

    Ratio of flux is then:
    [tex]\frac{F_B}{F_A} = \frac{a(t_B)}{a(t_A)} \frac{r_a^2}{r_b^2} \frac{a^3(t_{2A})}{a^3(t_{2B})}[/tex]

    How do I find the time the radiation is received ##t_{2A}## and ##t_{2B}##? Clearly they are different.
     
  2. jcsd
  3. May 22, 2015 #2
  4. Jun 9, 2015 #3
    Solved.
     
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