Normal-density materials have internal energy, which is the sum of the average energies associated to each of the degrees of freedom. Degrees of freedom can be described as vibrational, translational, and rotational. If you compress this material in such a way that matter cannot escape, you can detect energy escaping in the form of heat.

Now imagine an extremely high-density material such as a neutron star or black hole. I suspect it's not possible to describe the internal energy of (or anything about) an individual atom inside a black hole, since it's impossible to inspect beyond the event horizon, and since I assume that atoms don't exist in an distinguishable form there. But presumably the atoms had energy when they became part of the thing. How is that energy accounted for?


The internal energy of a black hole is just its mass. You can measure the mass of a black hole by its gravitational effects on outside bodies, and then extrapolate an equivalent energy using $E=mc^{2}$.

An isolated Neutron star also has a well-defined total mass, (the ADM mass), which can be used to define an internal energy. In the case of a neutron star, since it has X numbers of neutrons, and baryon number is conserved in gravitational collapse, you can even divide the total energy by the number of baryons to get a 'total energy per baryon.'

  • $\begingroup$ Really, the tricky part isn't the energy, it's the temperature--and so long as you have enough degrees of freedom to define a number of states for a given energy, you can define $S=k \log \Omega$ and $\frac{1}{T} = \frac{\partial S}{\partial E}$ $\endgroup$ – Jerry Schirmer Mar 4 '11 at 15:12
  • $\begingroup$ Does the angular momentum of a black hole contribute to its internal energy? $\endgroup$ – alfC Dec 17 '13 at 10:49

It is any kind of constituent particles (and not atoms as such) that move about.

  • In "ordinary" gases, liquids, and solids it is the molecules and atoms which move about making up the internal energy.
  • In a plasma that is ions and free electrons.
  • In a degenerate gas (white dwarf material is electron degenerate and neutron star material is neutron degenerate), the bits continue to move, but their choices of energy levels are constrained by Fermi exclusion, resulting in a uniformly filled phase space up to some level called the "Fermi energy" or "Fermi surface". Note the heavy nuclei are degenerate in protons and neutrons and they have internal degrees of freedom which can be excited, so we have first hand experience of this.
  • In a quark--gluon plasma it is, well, quarks and gluons.

And it doesn't make sense to talk about blackholes having "material" at they represent a physical singularity and by definition we don't know the rule there.

An interesting side effect of this is that we have a ready made definition of temperature for these systems in terms of the average energy of the internal modes.

I haven't talked about what happens at the transitions between these regimes because that can be pretty complicated, but one possible case is the existing constituents get torn apart (as in ionizing a gas).

  • $\begingroup$ I didn't start out thinking about black holes. I was really just wondering what happens to internal energy in systems where degrees of freedom are very restricted. I knew adding the term "black hole" to my question would get me in trouble, but I couldn't think of a denser thing, and I felt it would be cumbersome to have to re-qualify material every time I used it. Can you suggest a rephrase? $\endgroup$ – kojiro Mar 4 '11 at 3:25
  • $\begingroup$ @kojiro: "states" or "phases" might work in some places. The interesting thing here is that the dense forms continue to have internal degrees of freedom---albeit mostly simple ones rather then the spiffy vibrational and rotational modes that you get in molecular systems. $\endgroup$ – dmckee Mar 4 '11 at 4:49

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