# Quiz: mass of photons, E=mc²

1. Feb 12, 2007

### lalbatros

Dear all,

Let us consider a closed waveguide, or a closed fiberoptic or more simply a perfectly conducting box.
Assume there are no wall losses or dissipation in this system.

Let us consider 2 versions of this system:

System 1) the box is empty
System 2) the box contains a huge amount of electromagnetic energy (resonant modes)

Can we say that the mass of these two systems are the same, since photons have no mass?
Can we say that the gravity produced by these two systems is the same?

Back to the basics !

Michel

2. Feb 12, 2007

### masudr

Gravitation is produced by the stress-energy tensor. The stress-energy tensor of System 2 is clearly different from that of System 1 by the extra terms in the stress-energy tensor. The gravity is certainly not the same.

3. Feb 12, 2007

### lalbatros

masudr,

I agree with this point of view.
However, would that not be a way to "weight" the photon, or at least to whole system 2?
And conversly, why do we assume that the mass of the metal box is something fundamentally different from the "mass of the photons"?

Michel

another question: can this be analysed within SR or is some part of GR needed?

4. Feb 12, 2007

### masudr

Not really: the quantity weight is the gravitational force an object feels in some specified gravitational field. Note, these terms are all related to Newton's gravitation, which we know is incorrect. If the nature of the system is such that the Newtonian limit applies, then both systems can be treated as identical.

We haven't really done that: both objects contribute to the stress energy tensor, but each in their own way.

Einstein originally derived $E=mc^2$ by considering the momentum of a photon moving in a box.

By the way, the energy of the boxes in systems 1 & 2 certainly aren't the same. Depending on which definition of mass you use, you will give different answers for whether or not the mass of the box is the same.

5. Feb 12, 2007

### cesiumfrog

I don't think so. If you use relativistic mass, the total is simply that of container plus that of the photons. If you use rest mass, it should be well known that the total in general is not just the sum of the component's rest masses. Either way, the result is the same (because the system as a whole is at rest) and should match up with inertial mass, with weight (by equivalence), and.. (as noted) gravitational charge.

6. Feb 12, 2007

### masudr

You have assumed we are using the centre of momentum frame.

7. Feb 12, 2007

### cesiumfrog

In what way?

8. Feb 13, 2007

### pervect

Staff Emeritus
The mass (by which I mean invariant mass) of the systems #1 and #2 is not the same.

I've worked this out in some detail in http://en.wikipedia.org/wiki/Mass_i...simple_examples_of_mass_in_general_relativity

using a "photon gas" aka "null dust" model for incoherent electromagnetic radiation.

Consider a frame of reference where the momentum of the system is zero. When you add electromagnetic energy E to the system, you increase the total energy of the system, but do not (at least in the example I analyzed) add any momentum to the system. Therefore the invariant mass of the system (in geometric units, E^2 - p^2) increases.

The problem is that you are assuming that the mass of the system is the sum of the masses of the components. This is not correct (at least not for invariant mass).

The energy of the system is the sum of the energies of the components (this is because in a Minkowskian space-time you find the total energy by an integral of components of the stress energy tensor).

The momentum of the system is the sum of the momenta of the components because it is also an intergal of components of the stress-energy tensor.

But the mass is computes via the formula E^2-p^2. While E and p are simple intergals of components of the stress-energy tensor, the mass is not. You basically find the mass by computing E and p, and then use the above formula.

All of this was for Minkowskian space-times. If you want to figure out mass in arbitrary space-times, things get considerably more complex - in fact, mass may not even be defined at all. The Komar mass I mention and use in the above wikipedia article is one of the easier to deal with concepts of mass in full GR, but it does not apply to all systems, only stationary systems. Fortunatlely, a container containing disordered electromagnetic radiation is stationary (actually it's usually even better than that, usually its static).

Last edited: Feb 13, 2007