The Definition of Redshift for Photons

[binbagsss]

Hi

Can I just confirm that when we associate a T-L KvF with energy it should be the form ##\dot{t}^2## and not ##\dot{t}##, since I have seen a mix of the notation amongst notes/books, well with just one example using just ##\dot{t}##, I suspect it must be the ## \dot{t}^2 ## , since classically mv^2, ##\omega^2 ## etc, but just to double-check? (for example things like deriving cosmological red-shift, writing a Lagrangian in an energy conservation form, with an 'effective potential' .)

Many thanks

 

[PeterDonis]

Can I just confirm that when we associate a T-L KvF with energy it should be the form ##\dot{t}^2## and not ##\dot{t}##

It's neither. You appear to be confusing energy as measured by a particular observer, which is ##m dt / d\tau## (##m## is the invariant mass) in a local inertial frame in which the observer making the measurement is at rest, with energy associated with a timelike KVF, which is ##g_{\mu \nu} k^\mu P^\nu##, where ##g_{\mu \nu}## is the metric tensor, ##k^\mu## is the timelike KVF, and ##P^\nu## is the 4-momentum of an object moving on a geodesic trajectory. This latter energy is a constant of geodesic motion.

I have seen a mix of the notation amongst notes/books

Can you give some references?

 

[binbagsss]

It's neither. You appear to be confusing energy as measured by a particular observer, which is ##m dt / d\tau## (##m## is the invariant mass) in a local inertial frame in which the observer making the measurement is at rest, with energy associated with a timelike KVF, which is ##g_{\mu \nu} k^\mu P^\nu##, where ##g_{\mu \nu}## is the metric tensor, ##k^\mu## is the timelike KVF, and ##P^\nu## is the 4-momentum of an object moving on a geodesic trajectory. This latter energy is a constant of geodesic motion.
Can you give some references?

Many thanks for your reply.

Probably a stupid question but is ##m## invariant mass of the observer or the body whose energy is being measured?

Ok, so I am looking at a derivation of cosmological red-shift in the FRW space-time (flat so ##k=0##)

The derivation goes as follows:
- consider an observer at ##r=R## and another at ##r=0##
- the observer at ##r=R## emits a light beam at ##t=t_1## and it is observed at ##t=t_2##.
- It emits another lightbeam at ##t=t_1 + \Delta(t_1) ##, how does ## t_2 + \Delta t_2 ## compare?

Then we use a light-beam Lagrangian FRW metric , L=0 , and for simplicitly let the dudes be radial.
Then we have:
##0=dt^2-a(t)^2dr^2##
and the FoC
to get:
##\frac{\Delta t_2}{\Delta t_1}=\frac{a(t_2)}{a(t_1)}##

so the wavelength increases if ##a(t_2) > a(t_1) ##
Now from ##E=dt/d\tau ## I need to compute the proper times . However it is clear that coordinate time and proper time are proportional and the relationship between the two for a particular observer indepdent of any position since ##g_{00}## is just ##1##.

Since I am only interested in the ratio of the energies I also do not need to know ##m##? (unless the answer to the above is it is the mass of the observer, but in which case you would want to assume it is the same anyway). and since coordinate time prop. proper time, the ratio of the energies is proportional to the ratio of coordinate times. HOWEVER, this ofc makes no sense since energy is proportional to frequency and not wavelength? so what have I done wrong? (or at least in flat space it is, but even so you would still expect the negative or positive correlation to hold I suspect?).

Also, apologies probably a stupid question, how is this working in a 'local inertial frame'? because I have specified coordinates rather than working with a tensor expression or?

Lastly here is one resource attached.

Many thanks

 

[PeterDonis]

is ##m## invariant mass of the observer or the body whose energy is being measured?

The body.

I am looking at a derivation of cosmological red-shift in the FRW space-time

FRW spacetime does not have a timelike Killing vector field, so I don't see what relevance this has to this thread.

so the wavelength increases if ##a(t2)>a(t1)##

Yes. More generally, the redshift of light emitted by one comoving object and received by another comoving object in FRW spacetime tells you by what ratio the scale factor changed between emission and reception.

Now from E=dt/dτE=dt/d\tau I need to compute the proper times .

The proper times of what? The light ray itself doesn't have a proper time; its worldline is null, not timelike. The proper times of the two observers (emitter and receiver) are, as you note, just the FRW coordinate times (since you assumed both observers were comoving, i.e., at rest in FRW coordinates). But that has nothing to do with the energy carried by the light ray.

here is one resource attached

This doesn't help much. What book, what chapter, what page, etc?

 

[binbagsss]

The body.
FRW spacetime does not have a timelike Killing vector field, so I don't see what relevance this has to this thread.
Yes. More generally, the redshift of light emitted by one comoving object and received by another comoving object in FRW spacetime tells you by what ratio the scale factor changed between emission and reception.
The proper times of what? The light ray itself doesn't have a proper time; its worldline is null, not timelike. The proper times of the two observers (emitter and receiver) are, as you note, just the FRW coordinate times (since you assumed both observers were comoving, i.e., at rest in FRW coordinates). But that has nothing to do with the energy carried by the light ray.
This doesn't help much. What book, what chapter, what page, etc?

yeh well previous to your replies I thought it was the time-like vector, but rather it must be the energy as observed so i tried to apply this. apologies for trying to act upon your reply after learning from it my original ideas were wrong?

so as i said and you didnt reply to, I was trying to use the formula given for the energy, and asked about the mass which you didnt reply to. ##E=mdt/d\tau ##

since both observers in the situation described are at rest this applies?no?

yes i meant proper time of the observers.

surely once youve deduced how the wavelength varies between two observers, you have enough information to deduce how the energy does, if I had the right formula? i assume what I thought was the application of ##E=mdt/d\tau ## is miles off track, by your reply, and therefore I ask this.

 

[PeterDonis]

as i said and you didnt reply to, I was trying to use the formula given for the energy

Energy of what? I'm still not clear on what you are trying to calculate.

asked about the mass which you didnt reply to

Sure I did. See the first quote and my first response in post #4.

i assume what I thought was the application of ##E=mdt/d\tau## is miles off track

It depends on what you want the energy of, relative to what. ##E = m dt / d\tau## is the energy of a timelike object with invariant mass ##m##, relative to an observer at rest in the coordinates you are using (assuming that those are the usual FRW coordinates), if the two objects are passing each other. Is that what you're trying to calculate?

 

[Mentz114]

yeh well previous to your replies I thought it was the time-like vector, but rather it must be the energy as observed so i tried to apply this. apologies for trying to act upon your reply after learning from it my original ideas were wrong?

so as i said and you didnt reply to, I was trying to use the formula given for the energy, and asked about the mass which you didnt reply to. ##E=mdt/d\tau ##

since both observers in the situation described are at rest this applies?no?

yes i meant proper time of the observers.

surely once youve deduced how the wavelength varies between two observers, you have enough information to deduce how the energy does, if I had the right formula? i assume what I thought was the application of ##E=mdt/d\tau ## is miles off track, by your reply, and therefore I ask this.

If you assume ##E=\hbar\nu## then a change in ##\nu=c/\lambda## causes a change in ##E##.

Is that what you mean ?

 

[binbagsss]

If you assume ##E=\hbar\nu## then a change in ##\nu=c/\lambda## causes a change in ##E##.

Is that what you mean ?

Yes wasnt entirely sure it held though, in curved space-time

So we can I interpret the difference( in coordinate time) ##\Delta (t_2) , \Delta (t_1) ## as the change in the observed wavelength of the light ray, and in FRW metric proper time = coordinate time, and so, if the relationship you said stated holds I conclude a decrease in energy, as expected, if the space-time expands between the point of emission and point of observation.

Edit : apologies to reword the above can someone confirm that it is the difference in proper time you need to be comparing to deduce the change in emitted / observed wavelength and not coorintaec time ? ( Here the same but in general , many thanks )

 

[binbagsss]

Energy of what? I'm still not clear on what you are trying to calculate.
Sure I did. See the first quote and my first response in post #4.
It depends on what you want the energy of, relative to what. ##E = m dt / d\tau## is the energy of a timelike object with invariant mass ##m##, relative to an observer at rest in the coordinates you are using (assuming that those are the usual FRW coordinates), if the two objects are passing each other. Is that what you're trying to calculate?

Apologies re the mass don't know how i missed it ( too focused on my yummy porridge that I spilt everywhere )

Ok thanks, no I am a timeline observer trying to measure the energy of a light ray. Is there a similar formula ?

 

[binbagsss]

It depends on what you want the energy of, relative to what. ##E = m dt / d\tau## is the energy of a timelike object with invariant mass ##m##, relative to an observer at rest in the coordinates you are using (assuming that those are the usual FRW coordinates), if the two objects are passing each other. Is that what you're trying to calculate?

Also in Sean Carroll's notes under eq 8.65 ( 97 unrevised version ) it says the observed frequency of a photon by a comoving observer is ## w= - U^u V_u ## where ##U^u ## is the four velocity of a comoving observer i.e. ##(1,0,0,0 ) ## and ##V_u ##is the four velocity of the photon - can I confirm that in the scenario described in my 3rd post to derive cosmological red shift fgalaxies hat we do not know the four velocity of the photon and so can not proceed by use of this equation, however if we did, we could.

 

[PeterDonis]

I am a timeline observer trying to measure the energy of a light ray. Is there a similar formula ?

There is certainly a formula: it's the one you quoted from Carroll, which actually ends up reducing to the one @Mentz114 gave you. But that formula does not involve a mass ##m## anywhere; it can't, because the photon is massless, and, as I said before, the mass of the timelike observer, which appears in the formula for the observer's own energy, has nothing whatever to do with the photon's energy.

can I confirm that in the scenario described in my 3rd post to derive cosmological red shift fgalaxies hat we do not know the four velocity of the photon

Yes, we do. You have assumed that the photon is traveling radially, and we know it moves on a null worldline; that is sufficient to express both the time and radial components of its 4-velocity in terms of the scale factor ##a## in the FRW metric, and a parameter, which I'll call ##k##. Then, since the observer measures the frequency of the photon, you can apply Carroll's formula to determine ##k##. 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.

 

[pervect]

Hi

Can I just confirm that when we associate a T-L KvF with energy it should be the form ##\dot{t}^2## and not ##\dot{t}##, since I have seen a mix of the notation amongst notes/books, well with just one example using just ##\dot{t}##, I suspect it must be the ## \dot{t}^2 ## , since classically mv^2, ##\omega^2 ## etc, but just to double-check? (for example things like deriving cosmological red-shift, writing a Lagrangian in an energy conservation form, with an 'effective potential' .)

Many thanks

1) Write down the line element
2) Write down the killing vector field
3) The dot product of the tangent vector of a geodesic and a killing vector field is a conserved quantity - see most GR textbooks. The energy-momentum 4 vector of a point particle is just the tangent vector - i.e. the four velocity of the particle - multiplied by the mass of the particle. Therefore the dot product of the energy-momentum 4-vector of a test particle following a geodesic, and the Killing vector field is a conserved quantity. This quantity is the conserved energy you're looking for.

 

[binbagsss]

1) Write down the line element
2) Write down the killing vector field
3) The dot product of the tangent vector of a geodesic and a killing vector field is a conserved quantity - see most GR textbooks. The energy-momentum 4 vector of a point particle is just the tangent vector - i.e. the four velocity of the particle - multiplied by the mass of the particle. Therefore the dot product of the energy-momentum 4-vector of a test particle following a geodesic, and the Killing vector field is a conserved quantity. This quantity is the conserved energy you're looking for.

Pervect. Thanks.

 

[binbagsss]

1) Write down the line element
2) Write down the killing vector field
3) The dot product of the tangent vector of a geodesic and a killing vector field is a conserved quantity - see most GR textbooks. The energy-momentum 4 vector of a point particle is just the tangent vector - i.e. the four velocity of the particle - multiplied by the mass of the particle. Therefore the dot product of the energy-momentum 4-vector of a test particle following a geodesic, and the Killing vector field is a conserved quantity. This quantity is the conserved energy you're looking for.

And I am banging on about a photon, who doesn't have mass..

 

[binbagsss]

There is certainly a formula: it's the one you quoted from Carroll, which actually ends up reducing to the one @Mentz114 gave you. But that formula does not involve a mass ##m## anywhere; it can't, because the photon is massless, and, as I said before, the mass of the timelike observer, which appears in the formula for the observer's own energy, has nothing whatever to do with the photon's energy.
Yes, we do. You have assumed that the photon is traveling radially, and we know it moves on a null worldline; that is sufficient to express both the time and radial components of its 4-velocity in terms of the scale factor ##a## in the FRW metric, and a parameter, which I'll call ##k##. Then, since the observer measures the frequency of the photon, you can apply Carroll's formula to determine ##k##. 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.

By parameter are you referring to the parameter chosen for the tangent vector ?

Nobody answered should it be the proper times or coordinate times you should compare to deduce the change in wavelength . ( I know they are equal in this case as we've said hundred times but in general ). Thanks

 

[pervect]

The dot-product of the energy-momentum four-vector of a particle <<moving along a geodesic>> and a killing vector field is still a conserved quantity even for a massless particle. The quantity is conserved because it's constant along the worldline of the particle.

Consider a Minkowskii space-time. For some observer O, with coordinates (t,x,y,z), there is a killing vector field ##\partial / \partial t##. I am making use of the notion that vector fields are just partial derivative operators to write this down the vector field as a partial derivative operator, I hope that it is clear. It's an important point for what follows.

For some observer O', moving relative to O, with coordinates (t', x', y', z'), there is a different killing vector field ##\partial / \partial t'##. One will get a conserved quantity along the worldline of either a massive or a masless <<geodesic-following>> particle in O and O' using either the Killing vector field ##\partial / \partial t## or ##\partial / \partial t'##, but the value of this conserved quantity will depend on which timelike Killing vector field one chooses. One choice is more natural in O, the other choice is more natural in O', each will give a correct answer.

Consider a photon in O, and the same photon in O' for instance. There's a natural choice of the timelike KVF in O that gives a conserved quantity in O, and there's an equally natural choice of the timelike KVF in O' that gives rise to a different quantity which is also conserved. But they are not the same quantity, because the choice of the timelike KVF was different.

 

[PeterDonis]

By parameter are you referring to the parameter chosen for the tangent vector ?

For the photon's 4-velocity (or more properly 4-momentum), which is the tangent vector to its worldline, yes. If you do the math it will make what I said a lot clearer.

 

[binbagsss]

The dot-product of the energy-momentum four-vector of a particle <<moving along a geodesic>> and a killing vector field is still a conserved quantity even for a massless particle. The quantity is conserved because it's constant along the worldline of the particle.

Consider a Minkowskii space-time. For some observer O, with coordinates (t,x,y,z), there is a killing vector field ##\partial / \partial t##. I am making use of the notion that vector fields are just partial derivative operators to write this down the vector field as a partial derivative operator, I hope that it is clear. It's an important point for what follows.

For some observer O', moving relative to O, with coordinates (t', x', y', z'), there is a different killing vector field ##\partial / \partial t'##. One will get a conserved quantity along the worldline of either a massive or a masless <<geodesic-following>> particle in O and O' using either the Killing vector field ##\partial / \partial t## or ##\partial / \partial t'##, but the value of this conserved quantity will depend on which timelike Killing vector field one chooses. One choice is more natural in O, the other choice is more natural in O', each will give a correct answer.

Consider a photon in O, and the same photon in O' for instance. There's a natural choice of the timelike KVF in O that gives a conserved quantity in O, and there's an equally natural choice of the timelike KVF in O' that gives rise to a different quantity which is also conserved. But they are not the same quantity, because the choice of the timelike KVF was different.

mmm ok i believe i followed that ok. however, i apologise in advance for any wrong terminology etc, but from my course, we tend not to work in such a regime where the same type of observer, i.e. both timelike, both null have different kvfs , rather the starting point is a Lagrangian in some coordinattes , and set this equal to 0 or 1 depending on whether null or time-like observers (signature (+,-,-,-,) say, and in these cooridnates (t,x,y,z) that the space-time is specified in , we would say there is a kvf ##\partial_t## and then, and I assume what follows is a equivalent way to proceed as 'the natural choice of ' the KVF, proceed by specificying a observer via it's value of L=0,1 and then relating proper-time and coordinate time, and comparing the proper times.

and this is what I am still asking, it must b be the proper times that we are 'actually' comparing to deduce the realtive red-shift in the FRW case also?

I guess this is a stupid question also, but the Schwazschild metric has a time-like KVF in our notes we didn't derive gravitational red-shift this way but rather the method I describe above,( with the absense of a KVF, )/ in my OP. Can you derive Schwarzschild red-shift using the time-like KVF?

 

[PeterDonis]

we tend not to work in such a regime where the same type of observer, i.e. both timelike, both null have different kvfs

This property is peculiar to Minkowski spacetime (and, IIRC, to a very few other spacetimes with a high degree of symmetry, such as de Sitter). In a general curved spacetime there will not be multiple timelike KVFs at a given event; at most there will be one (as in Schwarzschild spacetime, for example).

Also, even in the Minkowski case, there is no such thing as a "null observer", and obviously a timelike KVF cannot be null.

 

[binbagsss]

sure half my questions are missed.
i am still asking
1)
and this is what I am still asking, it must b be the proper times that we are 'actually' comparing to deduce the realtive red-shift in the FRW case also?

and also, may I ask

2)I guess this is a stupid question also, but the Schwazschild metric has a time-like KVF in our notes we didn't derive gravitational red-shift this way but rather the method I describe above,( with the absense of a KVF, )/ in my OP. Can you derive Schwarzschild red-shift using the time-like KVF?

 

[PeterDonis]

it must b be the proper times that we are 'actually' comparing to deduce the realtive red-shift in the FRW case also?

No. The redshift is a direct observable. As I've already said, it doesn't tell you anything about proper times by itself; it tells you about the ratio by which the scale factor changed between emission and reception. In order to convert that into elapsed proper time (for a comoving observer--if we assume that both the emitter and the receiver are comoving observers then this would be the proper time according to either of their clocks that it took the light to travel), you need to know the expansion history, i.e., the exact function ##a(t)## that gives the scale factor as a function of time (##t## is FRW coordinate time, but by construction that is the same as proper time for comoving observers).

If you know the expansion history, i.e., the function ##a(t)##, and you know the comoving space coordinates of the emitter and receiver, then you can calculate the redshift of a light ray that goes from one to the other, yes--the calculation is the one you did in post #3, the ratio ##\Delta t_2 / \Delta t_1## is the ratio of wavelengths, which is the same as ##1 + z##, where ##z## is the redshift. Note that nowhere in that calculation did you have to know the elapsed proper time between emission and reception; you can calculate that, but you don't need to to calculate the redshift.

Can you derive Schwarzschild red-shift using the time-like KVF?

I'm not sure what you mean by "derive red-shift using the time-like KVF".

 

[binbagsss]

No. The redshift is a direct observable. As I've already said, it doesn't tell you anything about proper times by itself; it tells you about the ratio by which the scale factor changed between emission and reception. In order to convert that into elapsed proper time (for a comoving observer--if we assume that both the emitter and the receiver are comoving observers then this would be the proper time according to either of their clocks that it took the light to travel), you need to know the expansion history, i.e., the exact function ##a(t)## that gives the scale factor as a function of time (##t## is FRW coordinate time, but by construction that is the same as proper time for comoving observers).

.

Ok thanks
I am confused as to why in the derivation of gravitational red shift with the schwarzschild metric we want the difference in proper times but not here ?

 

[stevendaryl]

I am confused as to why in the derivation of gravitational red shift with the schwarzschild metric we want the difference in proper times but not here ?

Well, you can use red shifts in Schwarzschild, as well. If you have some process (such as the decay of an excited state of a hydrogen atom) that produces light with a particular frequency, then the Schwarzschild metric will allow you to compute what the redshifted or blueshifted frequency of that light will be when it is observed at another location. So you don't have to compare proper times.

In the case of cosmological models, I actually don't know what astronomers use as the base light frequency in order to figure out how much the light is redshifted. Either they know ahead of time what the frequency should be, or they compare light from similar origins coming to us via different paths?

 

[stevendaryl]

I'm not sure what you mean by "derive red-shift using the time-like KVF".

I think I know what he means. If there is a time-like KVF, there is a corresponding conserved energy-like quantity for geodesics. So he's asking whether gravitational red shift can be derived using that "energy" in the way that people heuristically derive red shift by calculating how much energy a photon loses when it goes from the bottom of a mountain to the top. I was always uncomfortable with the energy-based derivation because I didn't understand why it worked, or under what circumstances it worked.

The derivation I've seen goes something like this:

You have a particle of mass ##m## and energy (rest mass energy plus kinetic energy) ##E## at the bottom of the mountain. When it gets to the top, it has energy (approximately) ##E - mgh \approx E (1-\frac{gh}{c^2})## (since ##E \approx mc^2##). This result is independent of ##m##, so we assume that it continues to hold even in the limit ##m \rightarrow 0## (which doesn't really make sense, because ##E \approx mc^2## is only valid for massive particles.) So we conclude that if a photon has energy ##E## at the bottom of the mountain, it will have energy ##E (1-\frac{gh}{c^2})## at the top of the mountain. Now, since we know that a photon's energy is related to its frequency through ##E = h\nu##, this implies that ##\nu_{top} = \nu_{bottom} (1-\frac{gh}{c^2})##

I don't know how (or if it's possible) to make something like this derivation rigorous.

 

[binbagsss]

at light will be when it is observed at another location. So you don't have to compare proper times.

Ok I would have thought it would have to be proper time because say you have a light ray emitted only in the z direction - two observes align in the z axis, and the metric is such that ##g_{00}## depends on ##z##. Then without converting to proper time, surely you are not taking into consideration relative positions of the observers completely , as ##g_00## will make you plug in the positions of ##z## when converting. Thanks .

 

[PeterDonis]

I am confused as to why in the derivation of gravitational red shift with the schwarzschild metric we want the difference in proper times

You're going to have to give a specific reference to such a derivation. You might be confused by the fact that the ratio of proper times in Schwarzschild spacetime is also the gravitational redshift factor; but the fact that they happen to be equal in this particular spacetime does not mean the second is "derived" from the first.

 

[PeterDonis]

The derivation I've seen goes something like this

The argument I remember from Einstein was actually somewhat different; it went like this:

Suppose we take a mass ##m## at height ##h## and let it fall to ground level (height zero). At ground level, we convert the mass into photons, capturing all of its rest and kinetic energy, and shine the photons back upward to height ##h##, where we convert them back into a mass at rest. The final result must be a mass ##m## or energy conservation will be violated; but the total energy that was converted to photons at height zero was ##m c^2 + m g h##, where ##m g h## is the potential energy that was converted to kinetic energy as the mass fell. So the photons must redshift by a factor ##1 + gh / c^2## in going from height zero to height ##h##.

 

[PeterDonis]

I actually don't know what astronomers use as the base light frequency in order to figure out how much the light is redshifted

They are looking at spectral lines with known frequencies in the lab. One of the difficulties is measuring spectral lines in spectra that are already very faint.

 

[stevendaryl]

The argument I remember from Einstein was actually somewhat different; it went like this:

Suppose we take a mass ##m## at height ##h## and let it fall to ground level (height zero). At ground level, we convert the mass into photons, capturing all of its rest and kinetic energy, and shine the photons back upward to height ##h##, where we convert them back into a mass at rest. The final result must be a mass ##m## or energy conservation will be violated; but the total energy that was converted to photons at height zero was ##m c^2 + m g h##, where ##m g h## is the potential energy that was converted to kinetic energy as the mass fell. So the photons must redshift by a factor ##1 + gh / c^2## in going from height zero to height ##h##.

That's better, because you're not applying nonrelativistic physics to photons.

 

[binbagsss]

You're going to have to give a specific reference to such a derivation. You might be confused by the fact that the ratio of proper times in Schwarzschild spacetime is also the gravitational redshift factor; but the fact that they happen to be equal in this particular spacetime does not mean the second is "derived" from the first.

ok will do ( on phone ATM not laptop ) but this goes back to my initial question , I am really asking what is the definition of red-shift. Crudely, inverse proper time / coordinate time has the same units as frequency. So frequency inversely proportional to wavelength , I would want to invert the ratio etc to get a ratio of wavelengths, red- shift is defined as the difference in time between two light rays as observed by two different observers. What do you mean 'happen to be equal ' what is the definition ?

 

[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!