The Definition of Redshift for Photons

[martinbn]

@binbagsss : It is hard to guess what exactly you are asking, as you can tell from all the posts above. But you can find in Wald's book the derivation of the redshift for the Schwarzschild solution using the time-like Killing field. You can also find the trick how to do that in the FRWL space-time although there is no time-like Killing field.

 

[PeterDonis]

I am really asking what is the definition of red-shift.

The shift in frequency (or wavelength) of spectral lines observed in light from distant objects, as compared with the frequency (or wavelength) of those same lines in a lab.

red- shift is defined as the difference in time between two light rays as observed by two different observers.

Is it? Look at the definition I just gave above. You might be confusing how redshift is defined with how it is calculated in a particular model (more precisely, how a prediction of what redshift would be observed in a particular observation, is calculated in a particular model).

 

[binbagsss]

The shift in frequency (or wavelength) of spectral lines observed in light from distant objects, as compared with the frequency (or wavelength) of those same lines in a lab.
Is it? Look at the definition I just gave above. You might be confusing how redshift is defined with how it is calculated in a particular model (more precisely, how a prediction of what redshift would be observed in a particular observation, is calculated in a particular model).

The shift in frequency (or wavelength) of spectral lines observed in light from distant objects, as compared with the frequency (or wavelength) of those same lines in a lab.
Is it? Look at the definition I just gave above. You might be confusing how redshift is defined with how it is calculated in a particular model (more precisely, how a prediction of what redshift would be observed in a particular observation, is calculated in a particular model).

Ok, so and how does the method for computing the predictions differ for instance, in the frw model and schwarzschild metric - how do you decide which method for a particular model is going to give you the prediction of that observation

 

[binbagsss]

@binbagsss : It is hard to guess what exactly you are asking, as you can tell from all the posts above. But you can find in Wald's book the derivation of the redshift for the Schwarzschild solution using the time-like Killing field. You can also find the trick how to do that in the FRWL space-time although there is no time-like Killing field.

Yes I have seen this in Carroll via a killing vector tensor, instead

 

[PeterDonis]

how does the method for computing the predictions differ for instance, in the frw model and schwarzschild metric

It doesn't have to. You can use the same method for both. But the method that works for both does not involve the proper time of either the emitter or the receiver. It's the method you used in your calculation earlier in the thread, which I discussed in post #21; you can do that same kind of calculation in Schwarzschild spacetime, but of course the specific expression for the metric is different, so you will get a different final expression for the redshift--the one you are used to seeing for Schwarzschild spacetime. Try it!

 

[binbagsss]

. The redshift of the photon, as you will see if you do this analysis, then turns out to be, as I said before, the ratio of the scale factor at reception to the scale factor at emission--more precisely, this ratio is equal to $1 + z$, where $z$ is the redshift.

In previous posts I was trying to establish the definition of red shift, (well the predicted redshift not the lab spectral line observed measurement) and there was no such direct response, other than that of observation. Well if we determine $k$ by the method you described, and $k$ is such that the photon is null, then that is something we are comparing - you used the term red-shift twice above. Somehow it is plausible to immediately state that the 'red shift if the photon' is given by comparing these frequencies, but then you use redshift to refer to $z$ - the only mysterious definition I was pointed to. The important parameter was what I was after, so it's $k$ ? It's so obvious that you can compare frequencies to give the red shift if the photon that you don't even explain this, yet all my questions where pointed toward $z$ ...

 

[PeterDonis]

In previous posts I was trying to establish the definition of red shift, (well the predicted redshift not the lab spectral line observed measurement) and there was no such direct response, other than that of observation.

Sure there was; I gave you the definition of redshift in post #32.

if we determine $k$ by the method you described, and $k$ is such that the photon is null

I don't know what you mean here; $k$ itself doesn't tell anything about whether the photon's worldline is timelike or null; you have to already know that the photon's worldline is null in order to determine $k$ by the method I described.

I've already said once that if you do the actual math, it will make all this a lot clearer. Have you done the actual math?

Somehow it is plausible to immediately state that the 'red shift if the photon' is given by comparing these frequencies, but then you use redshift to refer to z

I don't understand what your issue is. If you want the precise mathematical definition, here it is: we observe a particular spectral line in the lab to have a wavelength $\lambda_{\text{lab}}$. We observe the same spectral line in light from a distant object to have a wavelength $\lambda_{\text{obs}}$. Then

$$
1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{lab}}}
$$

If you want it in terms of frequencies instead, then we have the lab frequency $\nu_{\text{lab}}$ and the frequency observed from the distant object $\nu_{\text{obs}}$, and then

$$
1 + z = \frac{\nu_{\text{lab}}}{\nu_{\text{obs}}}
$$

You can find these definitions in any textbook or in many places online; most of them will call $\nu_{\text{lab}}$ or $\lambda_{\text{lab}}$ the "emitted" frequency or wavelength, because we assume that the frequency/wavelength we observe in the lab will be the same as the frequency/wavelength emitted by the distant object (because the same substances should emit the same frequencies/wavelengths anywhere, since the physical laws governing emission are the same).

Is this what you were looking for? If not, what are you looking for?