Why redshift of light occurs between galaxies only?

Question

Edwin Hubble discovery of redshift of distant galaxies make me wonder why is this phenomenon only occurs when object is further away, is it possible that redshift happens in short distance too such as a millimetre, nanometer etc? Is it more prominent the larger the distance or the flatter the spacetime curvature?

Answer 1

I guess in principle, there may be an effect, though things can be tricky. Let me expand on how I understand this works:

Relativistic frequency shifts can be understood as a kinematic effect: Initial energy (which corresponds to frequency up to a factor of $h$) depends on the relative orientation of photon 4-momentum at time of emission and the 4-velocity of the emitter, and final energy depends on the relative orientation of photon 4-momentum at time of absorption and the 4-velocity of the absorber.

If spacetime is flat, there exist inertial frames where photon momentum is constant, so the shift will be completely determined by the relative velocity of emitter and absorber. This is the special-relativistic Doppler effect.

In curved spacetimes, there is no distance parallelism, so we can't just compare the velocities of emitter and absorber, and the momentum the photon will end up with will depend on the path taken as momentum is parallel transported along the trajectory.

In addition to this abstract analysis, in many cases we can also give heuristic explanations in terms of specific physical effects: For example, in gravitational fields photons will lose (or gain) energy climbing up (or falling down) a gravity well. In case of static spacetimes such as Schwarzschild spacetime, we can also explain this in terms of time dilation (observers at fixed Schwarzschild coordinates will measure different frequencies because their clocks tick differently).

We can make a similar argument for cosmological redshift: If we use conformal time and comoving distance as our coordinates, radial null geodesics will be manifestly straight lines, ie the time interval between two consecutive wave fronts measured by emitters and absorbers comoving with the Hubble flow will be the same. We then just have to convert the conformal time coordinate to the observers' eigentimes.

Note that generally, cosmological redshift is directly attributed to spatial expansion instead, ie as light waves getting stretched. This is a perfectly fine alternative point of view, as the scale factor enters the definition of conformal time in the right manner.

Now, on to your actual question: If we want to figure out the effect of spatial expansion on locally measured frequency shifts, we have to consider spacetime geometry along the photon trajectory. One possible model to consider is Schwarschild-de Sitter spacetime, a spherically symmetric mass in a universe domiated by dark energy. At the relevant scales, the effect of the cosmological constant will be overwhelmed by the gravitational effect, so even if we could in principle attribute part of the shifting to that, in practice, it can be safely ignored.

Things get more complicated in models that are somewhat more realistic, such as Swiss cheese cosmologies. Here, the cosmological constant won't have a direct effect on the metric within the holes, but I imagine may still affect things indirectly because the hole metric has to be matched to the FLRW metric at its edge via appropriate boundary conditions. I don't know enough about these models to say anything definitive here.

So in summary, I would answer that I would expect spatial expansion to indirectly affect frequency shifts at small distances in principle, but it only becomes relevant at large distances because the effect on local spacetime geometry is minor.