A Relativistic Redshift and understanding it's approximation

[Arman777]

I was reading an article, and I saw this expression.

$$
1+z=\frac{(g_{\mu\nu}k^{\mu}u^{\nu})_e}{(g_{\mu\nu}k^{\mu}u^{\nu})_o}
$$

Where $e$ represents the emitter frame, $o$ the observer frame, $g_{\mu\nu}$ is the metric, $k^{\mu}$ is the photon four-momentum and $u^{\nu}$ is the four-velocity of the source or observer.

Has anyone seen this expression before? I want to understand how we can obtain $$1+z=\frac{a_o}{a_e}$$ from this expression and understand the metric potentials etc. Any reference would be appreciated. Thanks.

 

[Orodruin]

Where e represents the emitter frame, o the observer frame,

No, they represent the emitter and observer, respectively. The expressions themselves are invariant. However, they equal the frequencies measured by emitter/observer.

Has anyone seen this expression before?

g(k,u) is by definition the frequency of wave vector k as measured by an observer with 4-velocity u. The expression follows directly from that and the definition of the redshift parameter z.

I want to understand how we can obtain 1+z=aoae from this expression and understand the metric potentials etc. Any reference would be appreciated. Thanks.

This follows directly from making the computation for comoving observers in a Robertson-Walker spacetime.

 

[Orodruin]

Addendum: The computation is found in this PF Insight: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/

 

[Arman777]

No, they represent the emitter and observer, respectively

hmm. That's what is says in the original article...not my fault. But you are also right.

This follows directly from making the computation for comoving observers in a Robertson-Walker spacetime.

In other types of metric (LTB, Bianchi) the $1+z=a_o/a_e$ will not hold then right ?

 

[Orodruin]

In other types of metric (LTB, Bianchi) the 1+z=ao/ae will not hold then right ?

The scale factors are particular for the RW spacetimes in standard coordinates.