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Why do photons add mass to a black hole?

When photons are taken irreversibly into a black hole does the mass of the BH increase?

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Related: – Qmechanic Aug 20 '12 at 21:24
possible duplicate of If photons have no mass, how can they have momentum? – genneth Aug 20 '12 at 22:00
up vote 1 down vote accepted

This is really just an expansion of Graham's answer.

It's a commonly made mistake that gravity, and therefore a black hole, is caused by matter. In fact the spacetime curvature is related to a quantity called the stress-energy tensor. This is usually represented by a matrix with ten independant values in it (it's a 4x4 matrix but it's symmetric so six of the elements in it are duplicated).

Only one of the elements in the matrix, $T_{00}$, depends directly on the mass, and actually that element gives the energy density, where mass is counted as energy using Einstein's equation $e = mc^2$.

So photons affect spacetime curvature because they contribute to the energy density even though they have no mass. Actually photons contribute to other elements of the matrix as well because they have a non-zero momentum and this too affects the spacetime curvature.

Re your question:

When photons are taken irreversibly into a black hole does the mass of the BH increase?

Yes, the mass of the black hole will increase by the photon energy divided by $c^2$.

Re your comment to Graham's question, yes, provided you add more energy than the black hole is radiating you will maintain or increase the black hole. You could add the energy using lots of low energy photons or a few high energy photons. It's the total energy added that matters.

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The first part of your answer is a gem, but I was quite taken aback by the statement that you could measure a different $\Delta M$ from the gravitational field created and the inertia. Wasn't this equivalency the entire point of GR? I'm sitting here thinking "no way that's what he meant". My understanding would be completely wrong about everything if that's true. – Alan Rominger Aug 21 '12 at 14:02
@alanse: I didn't say you could measure a different inertia from gravitational mass, I said I wasn't sure if you would. Having said that, you're quite right of course because even though photons are massless they have a momentum. I was just suffering a temporary brain fade. Thanks for the injection of sanity :-) – John Rennie Aug 21 '12 at 14:27
I don't even understand how you can be unsure, if a photon hits the event horizon it will increase the mass of the BH. I don't understand where there is room for skepticism. Photons have no rest mass but they can impart mass onto other things. If this were about a particle of ordinary matter, a fully absorbed photon would impart mass through increased KE plus heat energy. In the case of a BH taken to be stationary, photon hitting at CM, there is no KE change, and 100% of the photon's energy adds to the BH mass. I'm baffled b/c you're the expert of the 2 of us! – Alan Rominger Aug 21 '12 at 14:36
You're quite correct of course. Put it down to old age and impending senility. – John Rennie Aug 21 '12 at 14:39
To expand on my "soft" understanding the photon does not hit the eh just the backup of material created by the eh. Does the same increase occur if the photon crosses the eh "un-interrupted" not yet quite far enough to prove GR mute when would the momentum effect the BH as soon as it passes the eh or is there a certain point were the energy becomes part of the BH. – Argus Aug 21 '12 at 19:10

Although photons are rest massless and the Einstein equation E=mc^2 isn't appropriate here they still have massenergy corresponding to E=plancks constant multiplied by the photon frequency. Black holes do have a temperature, very low for astronomical BHs, so all photon energies above those of the cosmic background radiation temperature at 2.725 Kelvin, incident upon the BH will slowly increase its mass.

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Is this added mass arbitrary compared to the Hawking radiation or can the singularity be maintained indefinitely by photons? – Argus Aug 20 '12 at 21:50
@Argus I would think the answer to your question is "yes, the mass added to the BH is arbitrary relative to the Hawking radiation". The in-falling matter is a result of what goes on in our universe, on our side of the horizon. Hawking radiation is determined only by the BH parameters. If it is "fed" sufficiently, it needn't ever decrease in mass. – Alan Rominger Aug 23 '12 at 17:24

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