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I noticed a thread on PF in which TOM STOER and others were discussing how to calculate the redshift for an arbitrary metric. I need to talk to Tom if he is still on this list.

The question has arisen in an applied physics field whether the following conformally flat metric will exhibit a Hubble shift or not:

ds^{2}=a(t)^{2}[-dt^{2}+dr^{2}] (cartesian coordinates)

This is a much simpler problem than Tom's "arbitrary" metric, and since this metric is well known (indeed common) in Relativity literature, someone must know the answer to this question off the top of his head.

I presume that there is no Hubble shift for this metric b/c while the scale factor stretches the wavelengths, the scale factor also multiplies dt producing a "cosmic time dilation" which blue shifts the distant light and exactly offsets the wavelength stretching redshift. However, I am unable to produce a simple mathematical demonstration of this.

I have been told that Wald, Weinberg and other texts outline how to calculate the Hubble shift but I find most texts only treat the FLRW metric which is a very special case b/c dt is actually the proper time in FLRW and objects at a fixed comoving coordinate are actually at rest (have a geodesic worldline)). This is not so in the conformally flat metric, so the simple calculation is not applicable.

In the discussion Tom Stoer posted he noted that if the emitting star had 4 momentum u_{e}and the observer at the origin had 4 momentum u_{o}that you can deduce the following:

f_{o}/f_{e}=[itex]\tau[/itex]_{e}/[itex]\tau[/itex]_{o}

for a train of light pulses.

Given that light travels in straight 45 degree lines at constant speed c=1 in the conformal metric, it seems to me that it can't be more than a few lines of math from the above equation to the result that:

f_{o}/f_{e}=1 (i.e. no Hubble shift)

which is what I think is the case for the conformally flat metric.

So, if anyone can simply tell me, or indicate how I can prove this, we would be very grateful, b/c it turns out that this metric is of fundamental importance in a field only remotely related to Relativity, and the question of whether a Hubble shift exists or does not exist for this metric turns out to be an immediate proof or disproof of the theory being investigated.

best wishes, tardy

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# Is there a redshift in a conformally flat metric space?

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