Undergrad Gravitational Redshift: Derivation from Static Metric

[redtree]

I am trying to find a derivation of gravitational redshift from a static metric that does not depend on the equivalence principle and is not a heuristic Newtonian derivation. Any suggestions?

 

[Ibix]

Solve for a null geodesic. Start one at $(r,t_1)$ and the other at $(r,t_1+\Delta t)$. Determine the time between the events where the two geodesics cross some radius $R$?

 

[PeterDonis]

Determine the time

More precisely, the proper time along a worldline of constant radius $R$ (the radius of reception), as compared with the proper time along a worldline of constant radius $r$ (the radius of emission).

Another method would be to compute the energy at infinity of a null geodesic, and then show how that relates to its energy as measured by static observers at $r$ and $R$.

 

[redtree]

Given that time and frequency are Fourier conjugates, how can changes in proper time relate directly to changes in frequency (as measured by redshift)? Or am I imposing a quantum perspective on a non-quantum theory?

 

[Nugatory]

Given that time and frequency are Fourier conjugates, how can changes in proper time relate directly to changes in frequency (as measured by redshift)? Or am I imposing a quantum perspective on a non-quantum theory?

You are.

To get the time dilation, all you need is the proper time between emission of successive peaks at the source and the proper time between the arrival of these peaks at the destination. The ratio between the two is the time dilation factor. You could do this calculation with two flashes of light emitted an hour apart at the source if you wanted.