Invariant mass of a photon changes - from Wiki

[Imager]

I'm reading the Wiki article below to say the invariant mass of photons in an expanding volume of space will decrease. I thought invariant mass of a photon was always zero and the energy of photon changed due to the expansion of space. So where did I go wrong?

Quote from Wiki

General relativity
In general relativity, the total invariant mass of photons in an expanding volume of space will decrease, due to the red shift of such an expansion (see Mass in general relativity). The conservation of both mass and energy therefore depends on various corrections made to energy in the theory, due to the changing gravitational potential energy of such systems.

http://en.wikipedia.org/wiki/Conservation_of_mass#General_relativity

 

[Vanadium 50]

Wikipedia never says "Invariant mass of a photon changes". Never. It is talking about the collective mass of an ensemble of photons.

 

[Nugatory]

You didn't go wrong at all. That wikipedia section is quite confusing, as the "invariant mass" that it's describing is not the sum of the rest masses of the photons; it's a quantity associated with the total energy of the system under consideration. Take a look at the accompanying wikipedia article on "invariant mass", but be sure to read the Talk page for that article as well as the one that you found... Wikipedia talk pages can tell you a lot about how much you can trust an article.

 

[wabbit]

This formulation in wikipedia is confusing indeed. What is invariant about that quantity ?

 

[Imager]

Nugatory, Double Thanks you!

One for clarify, and another for the Talk Page button, very cool! I never noticed it. (I really need to wear those reading glasses).

read the Talk page for that article

 

[Orodruin]

This formulation in wikipedia is confusing indeed. What is invariant about that quantity ?

It is the 4-momentum squared of a collection of photons. The Higgs peak in the 2$\gamma$ channel was found by binning in the invariant mass of photon pairs.

 

[wabbit]

It is the 4-momentum squared of a collection of photons.

Could you clarify what this means, I'm not getting it, esp. the "square" of 4 momentum - is that not zero for a photon ?

 

[Orodruin]

Could you clarify what this means, I'm not getting it, esp. the "square" of 4 momentum - is that not zero for a photon ?

For one photon, yes. For several photons, no. In any given system it is the square of the energy minus the square of the momentum. Thus, if you have two photons, each of energy E, traveling in opposite directions, the invariant mass square of the system is $4E^2$, since the total momentum is zero.

 

[wabbit]

Thanks, wasn't parsing that right.

 

[PAllen]

In addition to what others have said, I'll add another critique of that wiki presentation.

Locally, it is (pretty much) unambiguous to talk about invariant mass of a 'box of photons' because spacetime can be considered locally flat. However, globally, in GR, you are proposing to add distant vectors and take the norm of the result. This is, in a word, nonsense. You cannot add distant vectors in curved spacetime. You can parallel transport them together, then add them, but then the result depends almost entirely on how you bring them together. Thus, a correct statement is that invariant mass cannot be defined at all for a large ensemble of particles in GR.

 

[wabbit]

I was wondering about that too. From a given observer viewpoint, there is (usually) a unique geodesic linking an event of the observer's worldine to one on a photon's worldline, so that defines an unambiguous way to sum all those photon's momentums and square that sum, at least for a chosen simultaneity - but does that define an invariant quantity ? I don't see a reason that it should, but I don't see in what way it varies either.

 

[PAllen]

I was wondering about that too. From a given observer viewpoint, there is (usually) a unique geodesic linking an event of the observer's worldine to one on a photon's worldline, so that defines an unambiguous way to sum all those photon's momentums and square that sum, at least for a chosen simultaneity - but does that define an invariant quantity ? I don't see a reason that it should, but I don't see in what way it varies either.

I don't see where observers are involved or help the wiki error. They are summing vectors over a large volume of a spatial slice. This is a nonsense operation in GR.

 

[wabbit]

Agreed but if the operation I described gave a result independent of the observer (and simultaneity) then that definition (suitably completed) would make sense. Not saying it does, but I still wonder in which way the result changes. Would a different choice of simultaneity wreck things ? Would it make sense to define that quantity relative to a comoving class of observers ?

 

[PAllen]

Agreed but if the operation I described gave a result independent of the observer (and simultaneity) then that definition (suitably completed) would make sense. Not saying it does, but I still wonder in which way the result changes. Would a different choice of simultaneity wreck things ? Would it make sense to define that quantity relative to a comoving class of observers ?

I guess I don't understand your suggestion. How are you proposing to use observers sum 4-momenta over large volume of space? If you are thinking of congruence of observers, you can make the result come out literally any value you want with the appropriate congruence. You can even construct a congruence where the sum of measured photon energy increases with cosmologic time, without bound (I have in mind a rather bizarre congruence that achieves this).

If you imagine a large system placed in isolation in asymptotically flat spacetime, there is an unambiguous way to assign a 4-momentum to the system in GR. But that doesn't help you with a cosmologic solution. It is perfectly adequate as a high precision approximation for treating a galaxy in isolation, but if you are asking about cosmologic volumes and time scales it doesn't help you at all.