Cosmological Redshift: Calculating Wavelengths

[fisicist]

Hi all!

I've got a question about the cosmological redshift. We're given the metric
ds^2 = c^2\,dt^2 - a(t)^2 \left[ dr^2 + r^2\,d\theta^2 + r^2\sin^2 \theta\,d\varphi^2 \right]
Now light moves on null geodesics, so c^2\,dt^2 - a(t)^2\,dr^2 for radially moving light. For a GR exercise, we are asked to calculate the redshift. In order to do so, we are given the clue that \lambda = c / f. Why is that true? Why don't we use c/a instead of c? By using c, I get, indeed, the correct result, whereas by using c/a I would get that the observed wavelength and the emitted wavelength are equal.

Thank you!

 

[PeterDonis]

Why don't we use c/a instead of c?

Because it gives the wrong answer.

Perhaps a better way of putting is, why would you want to use c/a instead of c?

 

[fisicist]

Because it gives the wrong answer.

Perhaps a better way of putting is, why would you want to use c/a instead of c?

Because light moves with c/a, right (if ds²=c²dt²-a(t)²dr²=0, then |dr/dt|=c/a(t))? And isn't the general relation that the speed equals to the frequency times the wavelength?

 

[PeterDonis]

Because light moves with c/a

That's a coordinate speed, not a physical speed. No observer will ever measure light rays moving at c/a; all observers will measure them moving at c. So measurements of frequency and wavelength will always multiply to c.

 

[fisicist]

That's a coordinate speed, not a physical speed. No observer will ever measure light rays moving at c/a; all observers will measure them moving at c. So measurements of frequency and wavelength will always multiply to c.

What do you mean by 'physical speed'? To what are wavelength and frequency related if not to coordinate time and length?

 

[PeterDonis]

What do you mean by 'physical speed'?

A speed that's actually measured.

To what are wavelength and frequency related if not to coordinate time and length?

To actually measured physical time and length. Coordinate times and lengths are not the same as actually measured physical times and lengths. Failure to recognize this is one of the elementary mistakes people make when looking at relativistic models.

 

[wabbit]

@fisicist, what is the coordinate velocity of a comoving galaxy receding from us at c, in the FRW coordinates ?

 

[fisicist]

@wabbit: Sorry, but I don't know the answer (actually, I hardly understand the question). I am yet quite new to GR (the problem was posed in the third week of a GR lecture to introduce the concept of metric and geodesics).

@both: Okay, I have an idea. In order to determine what you call 'physical speed' I would have to introduce local inertial coordinates (the coordinates of the observer), right? And because ds²=0 is independent of the coordinate chart, light moves at v=c as ds²=c²dt²-dr² in the local inertial coordinates, right?

 

[wabbit]

@fisicist, yes you know the answer. What is the space coordinate of that comoving galaxy ?

Oh sorry I see you say you don't understand the question - do you know what comoving means here ? And do you know what those FRW coordinates mean ?

 

[fisicist]

@fisicist, yes you know the answer. What is the space coordinate of that comoving galaxy ?

Sorry, but I'm not a native speaker. What does 'comoving' mean (I've looked it up in two dictionaries, one of which is Merriam Webster's, and couldn't find it)? What does "recede with c" mean? I understand it lingually, but not physically: A speed must be related to a coordinate system; So, if the speed c is related to our inertial coordinates, I would suppose, after all, that because of the invariance of the line element the coordinate speed is c/a(t)? Forgive me, but I am not even used to local inertial coordinates (I've heard that they exist, but we haven't gone so far yet).

 

[wabbit]

Ah, not easy then - "comoving" in this context means "moving together with the expansion" - well, more precisely it means static with respect to the FRW coordinates. "Recede" just means "moving away".

But then please just forget my question, I assumed these terms and notions would be familiar since you are working on FRW, and my question was just meant as a hint, something illustrating the difference between coordinate velocity and physical velocity.

As it is, I don't think this is helping so I'll recede away quietly:)

And yes, c/a(t) is the coordinate speed of light in FRW coordinates as you state.

 

[PeterDonis]

In order to determine what you call 'physical speed' I would have to introduce local inertial coordinates (the coordinates of the observer), right? And because ds²=0 is independent of the coordinate chart, light moves at v=c as ds²=c²dt²-dr² in the local inertial coordinates, right?

This is one way of understanding what physical speed is and why the physical speed of light is always $c$, yes. But you don't have to introduce local inertial coordinates to determine what physical speed is; you just measure it. You can measure speed without introducing coordinates; all you need is a single standard ruler and a single standard clock in your local neighborhood. (Local inertial coordinates would require a whole system of rulers and clocks.) The measured speed of light will always be $c$ when you measure it with a standard ruler and clock in your local neighborhood; that's a physical statement that is independent of coordinates.

 

[fisicist]

This is one way of understanding what physical speed is and why the physical speed of light is always $c$, yes. But you don't have to introduce local inertial coordinates to determine what physical speed is; you just measure it. You can measure speed without introducing coordinates; all you need is a single standard ruler and a single standard clock in your local neighborhood. (Local inertial coordinates would require a whole system of rulers and clocks.) The measured speed of light will always be $c$ when you measure it with a standard ruler and clock in your local neighborhood; that's a physical statement that is independent of coordinates.

Okay, because not every 1-form is exact and therefore every coordinate chart defines a moving tetrad, but there are more general moving tetrads.

Then, both of you, thank you very much for your patient help! I believe that this discussion is going to help me in the future. :)