How is Frequency Redshift Related to Sphere's Proper Area and Flux Ratio?

[unscientific]

Homework Statement

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

Homework Equations

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
$$c\frac{1}{a(t)} dt = \frac{1}{\sqrt{1-kr^2}} dr$$
Since RHS is purely in terms of spatial distance, we have
$$\frac{1}{a(t_1)}\delta t_1 = \frac{1}{a(t_2)}\delta t_2$$

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

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:
$$\frac{F_B}{F_A} = \frac{a(t_B)}{a(t_A)} \frac{r_a^2}{r_b^2} \frac{a^3(t_{2A})}{a^3(t_{2B})}$$

How do I find the time the radiation is received $t_{2A}$ and $t_{2B}$? Clearly they are different.

 

[unscientific]

bumpp

 

[unscientific]

Solved.