Question about Tong's cosmology lecture notes eqn. 1.19

[haushofer]

TL;DR

Question about Tong's cosmology lecture notes eqn. 1.19 (integral identity used?).

Dear all,

I have a rather basic question about an equation in David Tong's lecture notes on cosmology; see

http://www.damtp.cam.ac.uk/user/tong/cosmo.html

My question is about eqn. 1.19 (page 14), in which the cosmological redshift is derived. It's not about the physics, but about some basic integral identity I apparently don't see: how does he arrive at the conclusion after the arrow in that eqn. 1.19? To reproduce that eqn.,

$$\int_{t_1 + \delta t_1}^{t_0 + \delta t_0} \frac{dt}{a(t)} - \int_{t_1}^{t_0} \frac{dt}{a(t)} = 0 \rightarrow \frac{\delta t_1}{a(t_1)} = \frac{\delta t_0}{a(t_0)} \ \ \ \ (1.19) $$

I do understand that we can write the LHS as

$$\int_{t_0}^{t_0 + \delta t_0} \frac{dt}{a(t)} - \int_{t_1}^{t_1 + \delta t_1} \frac{dt}{a(t)} = 0$$

and I've drawn the integral equation geometrically, but I don't see how Tong arrives at the RHS of eqn. 1.19. Is there some mean value theorem involved? Differentiating the equation also leads me nowhere. I feel a bit silly. Thanks!

 

[martinbn]

I don't get it either. I think it is not mathematically clear/rigorous. Probably he means that the time interval between successive signals is small therefore the integral is equal to the value of the function times the length of the interval. Something like $\int_a^{a+\varepsilon} f(x) dx = f(a)\varepsilon$.

 

[Orodruin]

It is the mean value theorem along with the assumption that $\delta t_i \to 0$. Tong could have been more explicit here.

While perfectly fine, I prefer a treatment in terms of 4-frequency and using invariants.

 

[haushofer]

It is the mean value theorem along with the assumption that $\delta t_i \to 0$. Tong could have been more explicit here.

While perfectly fine, I prefer a treatment in terms of 4-frequency and using invariants.

Thanks! But why $\delta t_i \to 0$? This result of redshift also holds if we increase the time between two signals, right? What happens if the second light signal is send out/absorbed billions of years later such that $\delta t_i = \mathcal{O}(10^9 \ yr)$?

 

[Orodruin]

Thanks! But why $\delta t_i \to 0$? This result of redshift also holds if we increase the time between two signals, right? What happens if the second light signal is send out/absorbed billions of years later such that $\delta t_i = \mathcal{O}(10^9 \ yr)$?

Then you will need the expansion history because you can no longer use the small delay approximation. Redshift is something that by construction deals with the rate of phase increase.

 

[haushofer]

Then you will need the expansion history because you can no longer use the small delay approximation. Redshift is something that by construction deals with the rate of phase increase.

Ok, thanks, I'll have another look and compare it with the usual derivation involving invariants.

 

[Orodruin]

Ok, thanks, I'll have another look and compare it with the usual derivation involving invariants.

I’d say the method presented by Tong is by far the most common one. I’m actually not sure what sources contain the argument based on parallel transport of the 4-frequency, but it should be covered in this Insight I wrote several years back: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

 

[martinbn]

I’d say the method presented by Tong is by far the most common one. I’m actually not sure what sources contain the argument based on parallel transport of the 4-frequency, but it should be covered in this Insight I wrote several years back: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

Wald has it in his book, and Geroch in his lecture notes.