"mass" of electromagnetic waves

[JustinLevy]

Consider the stress energy tensor T^{\mu\nu} of a particle. Integrating the trace T over space gives the invariant mass m of the particle. For electromagnetic fields, T=0 which fits with the quantum picture where a photon has zero invariant mass.

However consider two photons each of energy E travelling towards each other. The invariant mass of this two photon system is just 2E/c^2. But in the classical picture T=0 always. That's strange.

1] So is it possible that in semiclassical gravity, a gas of photons would gravitate very differently than the classical GR case of a ball of electromagnetic energy?

2] In just classic GR, if there was a ball of electromagnetic energy such that the stress energy tensor was spherically symmetric, then the solution outside the ball must be the Swartzschild solution which is parameterized by a mass M. What mass should I use here? Or is M=0 (outside the "gas" of electromagnetic waves, the mass is registered as zero)?

[Bill_K]

Consider the stress energy tensor T^{\mu\nu} of a particle. Integrating the trace T over space gives the invariant mass m of the particle. For electromagnetic fields, T=0 which fits with the quantum picture where a photon has zero invariant mass.

Sorry, that's already incorrect. The trace T does not have that interpretation. A particle is described by an energy-momentum 4-vector Pμ, which is gotten by integrating Tμ0. The invariant mass of the particle is the length of Pμ. This is the mass that would go into the Schwarzschild solution.

[atyy]

Try Eq 17 of http://arxiv.org/abs/gr-qc/9909014 .

One of the things involved in the tracelessness is the (-,+,+,+) of the metric, but I'm not sure about specifics beyond that.

[JustinLevy]

Sorry, that's already incorrect. The trace T does not have that interpretation.
Of course it does.

Consider a particle at rest. The elements of the stress energy tensor will all be zero except the T^{00} energy density component. So if you take the trace you just get the energy density, and if you integrate over space you will then get the energy of the particle. So T=E=m for a particle (ignoring factors of c) in the rest frame. While E is not a Lorentz scalar, T is. So spatial integral of T will always be the invariant mass of the particle.

A particle is described by an energy-momentum 4-vector Pμ, which is gotten by integrating Tμ0. The invariant mass of the particle is the length of Pμ. This is the mass that would go into the Schwarzschild solution.
The reason I started with T is because
1) I want to discuss electromagnetic waves and therefore the stress energy tensor is the appropriate tool, as opposed to the four momentum (which admittedly is more convenient for particles)
2) the field equations of GR couple spacetime to the stress energy tensor, so it seems the appropriate place to start from



Try Eq 17 of http://arxiv.org/abs/gr-qc/9909014 .

One of the things involved in the tracelessness is the (-,+,+,+) of the metric, but I'm not sure about specifics beyond that.
Hmm.. it looks like he is using the linear limit to argue what the contributions are. But while I agree the gravity from the electromagnetic waves can be small, they can't be "non-relativistic slow" which I thought was required for using sources in the linear limit.

Am I misunderstanding something?

[atyy]

Hmm.. it looks like he is using the linear limit to argue what the contributions are. But while I agree the gravity from the electromagnetic waves can be small, they can't be "non-relativistic slow" which I thought was required for using sources in the linear limit.

At that point it looks like he has taken weak field g=Minkowski+h, but he hasn't taken slow speeds yet, which usually involve taking something like coordinate time and proper time to be the same.

Edit: I see, in Eq 16 he does use the slow speed limit for a particle, but I believe not in Eq 17 for EM waves.

[JustinLevy]

At that point it looks like he has taken weak field g=Minkowski+h, but he hasn't taken slow speeds yet, which usually involve taking something like coordinate time and proper time to be the same.

Edit: I see, in Eq 16 he does use the slow speed limit for a particle, but I believe not in Eq 17 for EM waves.
Okay, thanks for helping clarify that.

I also found this, which essentially says light contributes a "mass" from the energy density, plus and equal part due to the pressure. So in the Schwarzschild solution I should use M = 2 E/c^2
http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html

Amazingly it also notes that there have actually been experimental tests of the gravitational effect of electric fields. I had no idea this was actually confirmed (well, for static fields anyway... but still amazing).

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EDIT: Another related thread http://www.physicsforums.com/showthread.php?t=442266

[atyy]

I haven't seen the calculation for the case of Schwarzschild matched to a ball of containing photons, but I believe the Schwarzschild mass parameter is typically not the true energy of the interior, because although the integral is done over the energy, but the volume element is not one of proper volume.

http://arxiv.org/abs/gr-qc/0510041

[bcrowell]

I haven't seen the calculation for the case of Schwarzschild matched to a ball of containing photons, [...]

http://www.physicsforums.com/showpost.php?p=2956775&postcount=15

[atyy]

http://www.physicsforums.com/showpost.php?p=2956775&postcount=15

Yes, that's the same as what the Ehler's et al reference gives, and I agree it's very reasonable. The part I haven't seen fleshed out is how this is seen as a solution of the Einstein-Maxwell equations.

[bcrowell]

Yes, that's the same as what the Ehler's et al reference gives, and I agree it's very reasonable. The part I haven't seen fleshed out is how this is seen as a solution of the Einstein-Maxwell equations.
Hmm...well, outside the sphere, it's a solution of the vacuum field equations (the Schwarzschild solution). Inside the sphere, it's just a photon gas on a background of approximately flat spacetime. I guess there may be a hidden assumption in pervect's outline that the fields are relatively weak, so we don't really need to solve the Einstein-Maxwell equations. Is this the kind of thing you had in mind?

[atyy]

Hmm...well, outside the sphere, it's a solution of the vacuum field equations (the Schwarzschild solution). Inside the sphere, it's just a photon gas on a background of approximately flat spacetime. I guess there may be a hidden assumption in pervect's outline that the fields are relatively weak, so we don't really need to solve the Einstein-Maxwell equations. Is this the kind of thing you had in mind?

The Ehlers reference actually shows the weak field assumption isn't needed (which is why it was published it PRD).

The part I'm stuck is why can we assume that a photon gas exists as a solution of the Einstein-Maxwell equations? Or why can we use the perfect fluid stress-energy tensor which goes with the perfect fluid equation of state, rather than the electromagnetic stree-energy tensor which goes with Maxwell's equations?