Is There a Universal Redshift Formula for Arbitrary Spacetime Metrics?

[George Jones]

There is no need to look at problem 5.4 (a specific case), because the general method (not just for isotropic observers, and not just for cosmological models) is given in the second paragraph of section 5.3. This paragraph leads quickly and directly to the equation that Tom quoted.

 

[WannabeNewton]

Do you mean where he says "Thus, we can always find the observed frequency by calculating the null geodesic determined by the initial value..."?

 

[George Jones]

Do you mean where he says "Thus, we can always find the observed frequency by calculating the null geodesic determined by the initial value..."?

Yes, except that I get the reciprocal of what Tom wrote. Applying 5.3.1 twice, once at P and once at Q gives

<br /> 1+z = \frac{\lambda_Q}{\lambda_P} = \frac{\omega_P}{\lambda_Q} = \frac{g \left(k , u \right)_P}{g \left(k , u \right)_Q}<br />

There is a change of notation based on the coordinate representation of the lightlike k.

Assume that s is an affine parametrization of the lightlike worldline and that x^\mu is a coordinate system. Loosely,

k= \frac{d}{ds} = \frac{dx^\mu}{ds}\frac{\partial}{\partial x^\mu}

Denote the derivative with respect to s by dot.