What is the binding energy of a black hole?

Question

As the particles which constitute a black hole collapse they become tightly bound. I assume this means a lot of energy would be required to liberate a particle from that bound state. Is it a finite amount or not?

Answer 1

Actually, the relativistic formula for the Binding Energy ratio BE of a non-rotating pulsar of mass $M$ and radius $R$, is estimated to be $BE = \frac{E}{M c^{2}} = \frac{1.2}{((2 R c^{2})/G M - 1)}$ (see Wikipedia entry for "Gravitational Binding Energy", where $BE = \frac{0.6 b}{(1-b/2)}$, with $b = \frac{G M}{R c^{2}}$).

For a non-rotating black hole,$R$ may be estimated to be of the order of the Schwarzschild radius $R = \frac{2 G M }{c^{2}}$, and the formula simply gives $E = 0.4 M c^{2}$ or $40 \%$ of its own mass! This is an asymptotic guess from a neutron star nearing black hole collapse (the nonrelativistic approximation for a uniform sphere, would give $30 \%$ of its mass, as $E = 0.6 \frac{G M^{2}}{R}$).

Actually, the binding energy is equal to the energy radiated preceding and during the collapse and is seen eventually as kind of a mass defect (as in the formation of atomic nuclei by nuclear fusion. Helium's mass is less than the total mass of the constituents, 2 protons + 2 neutrons; the mass defect gives the exact amount of energy released). So the collapse process from initially dispersed matter emits the equivalent of the binding energy until the collapse is terminated, or in the case of a black hole, until the moment of the formation of its horizon, as no energy can escape after. This also happens in particular in the case of merging black holes. The merger mass is always lower than the sum of the masses of the initial black holes, as gravitational waves are produced and irradiated during the process.

Yuan K. Ha (The Gravitational Energy of a Black Hole, in 'General Relativity and Gravitation', 35, 2003, pp. 2045-2050) evaluates $BE=0.5$ or 50%. So half of the infalling mass+energy (or the mass seen at the horizon), is seen at an infinite distance (the Bondi mass, I think) not taking into account the radiated energy for this type of non-rotating black hole. See also discussion in ADM mass of a black hole and mass of the associated matter.

Answer 2

As far as I know, no value in the observable physical universe is infinite, but we don't know:

1. If the universe itself is infinite or not

2. We don't know what is inside a black hole, so I don't think there is a definite answer to your question.

Answer 3

Particles may be able to escape a black hole through quantum tunneling, the possibility that a particle confronted with an energy barrier it doesn't have sufficient energy to surmount, nevertheless overcomes by, in effect, tunneling through the barrier. It's a quantum mechanical effect that depends on the particle's probability function extending through the black hole's event horizon, giving it some chance, however slight, of appearing outside the black hole. As the barrier would not be surmounted in a classical sense, there is neither a small nor a large amount of energy that can be measured, to my knowledge.

However, Hawking radiation, which is an emission of particles indirectly caused by the black hole's gravitation, has a theoretical temperature and theoretically can be measured. Some researchers have modeled Hawking radiation as a form of quantum tunneling, with the particle/antiparticle pair forming inside the event horizon, and one of them tunneling out. Another explanation says that the pair pops into existence outside the event horizon because of vacuum fluctuations, and one falls into the black hole with hypothetical negative energy (an exotic form of energy that exists in theory) while the other escapes with positive energy. Both explanations require the black hole to lose some of its mass.

If Hawking radiation is a form of quantum tunneling, then conceivably its temperature might be a measure of the energy required to separate mass from a black hole (or a measure of the separated mass/energy itself). This sort of radiation takes place very slowly, and would be difficult to measure, although this paper claims that an experiment succeeded in quantifying the thermal spectrum of Hawking radiation: http://link.springer.com/article/10.1007%2Fs10701-014-9778-0#page-1

It may be that the only way to extract mass from a non-rotating black hole is through quantum mechanical processes. If you confine yourself to relativistic and classical processes, regardless of how much energy you direct at the black hole, no mass escapes. Rather, such energy augments the non-rotating black hole's mass. So, conventional energy may not do the trick. It requires either quantum tunneling or exotic negative energy.

The Penrose process theorizes that mass/energy in the form of momentum may be extracted from a rotating black hole, as the black hole's rotational energy is located outside the event horizon in an area called the ergosphere. An object entering the ergosphere is dragged by rotating space-time and splits in two, with one part falling through the event horizon, but the other escaping with some of the black hole's rotational momentum. Eventually, this process would drain all the rotational momentum, and the black hole would cease to rotate, though it would remain as a black hole with a Schwarzschild radius. Here is a link to a paper that seems a good explanation of the thermodynamic processes, both classical and quantum, that are involved in the formation and theoretical evaporation of black holes: http://www.physics.umd.edu/grt/taj/776b/lectures.pdf

Answer 4

This is a complicated question that opens up a lot of avenues for further discussion. I'm not sure the question as you posed it has an answer that will be helpful to you, so I'll answer a related question that might satisfy your curiosity - "Under what circumstances is it possible to remove energy from a black hole?"

Remember, everything is energy, so this encompasses your question of "how hard is it to liberate particles form a black hole?" Your original question is difficult to answer because we have no idea what happens to particles once they fall into a black hole. They may very well disappear for good.

In fact, the technical definition of a black hole is a region of spacetime from which nothing can ever escape (sort of - that's not a complete definition, but it's good enough for now). So the naive answer to your question is that the "binding energy" is "infinite," since nothing can ever escape from a black hole.

But that's a technical definition, and it doesn't fit with what you intuitively think of as a black hole. What you think of as a black hole might be more simply described as a curvature singularity -- an area of spacetime where the curvature becomes infinite (this is really only one kind of singularity, but it suffices for now - the point is that if a bunch of mass collapses into one point, as occurs in the formation of what you think of as a "black hole," a curvature singularity is formed).

Luckily, there is a major theorem in general relativity that connects singularities and black holes. Unluckily, this theorem is unproven (but there is some VERY good evidence for believing it to be true). This principle is known as the Cosmic Censorship conjecture. Roughly speaking, the Cosmic Censorship conjecture states that "all singularities are contained within black holes" -- that is, if the curvature of space becomes too strong (as it would when a lot of matter collapses into a tiny space), anything that comes close enough to it can never escape.

If Cosmic Censorship is true, then it is relatively easy to prove the area increase theorem, also known as the second law of black hole mechanics. It states that the area of a black hole's event horizon can only ever increase -- it follows that a black hole can never be destroyed by classical processes. This relates to your original question - you might think of destroying a black hole by pumping in enough energy to "unbind" all of the particles inside, but the area increase theorem says that's impossible.

There are two major ways that things might be recovered from a black hole, though, and I'll describe them briefly now.

The Penrose process allows some energy to be extracted from a rotating black hole. It's possible to shoot in a particle that has "negative energy" and thus get some energy back out of the black hole. Up to ~20% of the total mass-energy of the black hole can be extracted this way. We aren't extracting particles from the black hole, per se, but we are taking some of their energy back. This does not violate cosmic censorship.

In Hawking radiation, all black holes undergo quantum fluctuations near the event horizon that result in a constant steady outward stream of radiation. Over the course of many billions of years, this may cause a black hole to evaporate. Where this stands with regard to cosmic censorship is tricky -- cosmic censorship is a classical theory, and does not take into account quantum effects.

Let me know if there's anything you'd like me to clarify!

Answer 5

OP assumes gravity is a force and has a "binding energy" which is most likely not probably true.As you past the event horizon the space and time coordinates are switched which means that moving in reverse is impossible because you would need to travel backwards in time ...