Quantum theory may be incompatible with black holes
One of Stephen Hawking’s most important contributions to physics is the Bekenstein-Hawking formula for the entropy of a black hole.
One of Stephen Hawking’s most important contributions to physics is the Bekenstein-Hawking formula for the entropy of a black hole.
The law is simple: the entropy of a black hole, in the left side, is equal to a constant (Boltzmann’s constant divided by four times the square of the Planck length) and the area of the surface of the black hole’s event horizon, A.
Everything in the universe has entropy, but before Hawking physicists assumed that black holes must some how be an exception because nothing can escape them. Since nothing can escape a black hole, it cannot thermally radiate.
Anything with a temperature must radiate energy. The only thing that would not is something at the theoretical absolute zero. This is a consequence of Planck’s law of black body radiation, the law that governs the emission of radiation from non-reflective, non-translucent, i.e., black, sources. Thus, it seemed as if black holes must all be at absolute zero with no entropy at all. The universe radiates into them and nothing comes out.
That didn’t sit well with Hawking or his contemporary Jacob Bekenstein. The problem with black holes having no entropy and no temperature is that anything that falls into them does. So what happens to it? Thermodynamically, entropy cannot decrease over time. Somehow, that entropy has to get added to the black hole.
Using a bit of quantum field theory, Hawking eventually showed that black holes do have entropy but it is not proportional to their mass, as you might think, but their surface area. More importantly, black holes radiate.
The way they radiate has often been misrepresented both in the media and by scientists. Unfortunately, this misrepresentation can be traced back to Hawking himself who made up a plausible explanation for his radiation, called Hawking radiation, for popular consumption that is nevertheless completely wrong and different from his scientific explanation. He claimed that in the quantum vacuum near the event horizon particle-antiparticle pairs are created all the time. These normally aren’t real but virtual, but sometimes the pairs can “borrow” energy from the black hole to become real. Then, instead of annihilating each other, one particle falls into the black hole and the other escapes.
This isn’t what happens at all of course. If it did, we would see about half matter and half anti-matter coming from black holes. And it would only come from right around the event horizon. Instead we mostly see light and it comes from the area all around the black hole.
What’s really going on is that the intense gravity around the black hole is exciting the electromagnetic field in the vacuum which is normally in its ground state, doing nothing. Those excitations borrow energy from the black hole’s mass to produce light. Thus, what gets created are pairs of photons both of which may escape.
It turns out this happens to the vacuum around all massive bodies, but black holes do it more intensely.
The Bekenstein-Hawking formula, despite being around for 50 years, has only recently received its first evidence.
Despite being a formula based on quantum physics, it also spells trouble for gravity in general because it points to a fundamental flaw in the theory of gravity, general relativity, when you try to quantize it.
Quantization is the process of turning a classical, non-quantum theory into a quantum one, and it is fraught with peril. All known forces and matter have been successfully quantized, except for gravity.
Many quantum field theory textbooks will point to problems “renormalizing” gravity. This is true on the face of it. In order to quantize a theory, you have to renormalize it, which means you have to demonstrate that it behaves well as you go to smaller and smaller scales and higher and higher energies.
The reason other theories are renormalizable is because they become “conformal” field theories at some very high energy.
A conformal field theory is one that doesn’t change with scale. Intuitively, this makes sense. If a theory doesn’t change with scale, then once you hit a particular scale, you know your theory is stable and fixed. You don’t have to care what happens beyond that scale.
Gravity, unfortunately, doesn’t behave like this. At very high energies, it forms black holes. In fact, at a particular scale, called the Planck scale, it forms a particular kind of black hole called a Planck black hole, which has about the mass of a flea. Once you go to smaller scales, energies get higher and the black holes get bigger, so the Planck scale is the theoretical minimum.
Planck scale black holes don’t act like conformal field theories. Not at all. Gravity is about as strong as possible, while other forces become completely free and non-interacting at their maximal energy (called asymptotically free).
Conformal field theories have their own version of the Bekenstein-Hawking formula. It is called the Cardy-Verlinde formula.
The Cardy-Verlinde formula comes from studying a radiation dominated universe of a particular spherical radius R. It is a special case, mainly applicable to one spatial dimension, of a general principle that entropy scales with the Casimir or vacuum energy to a power of the number of spatial dimension divided by one plus that number. So, if you have three dimensions, then entropy scales as vacuum energy to 3/4 power.
It turns out that the Bekenstein-Hawking formula fails this test miserably, entropy scaling instead as energy squared. This is because the radius of the black hole is proportional to its mass, so the area of the event horizon is proportional to the mass squared. And mass is energy.
The generalized Cardy-Verlinde formula is applicable to any conformal field theory, so it matters because it shows that anything that allows black holes cannot be a conformal field theory.
But maybe we don’t need conformal theory. Maybe it is just a crutch that quantum theorists should throw off and good riddance.
The problem with that is that, if gravity isn’t a conformal field theory, then its behavior at small scales is uncontrollable. It cannot be valid at those high energies. This is true of even so-called asymptotically safe gravity. It cannot be valid at both low energy and high energy if it isn’t conformal or nearly so. The form that Einstein’s gravity takes at low energy is simply incompatible with a high energy theory. And the universe as we know it doesn’t appear to be one made entirely of black holes at the smallest scales.
Most physicists think, for this reason, that gravity is an effective low energy theory of some conformal field theory like Weyl gravity, aka conformal gravity, or string theory.
You don’t have to pick a particular high energy theory, however, to find the solution to the Bekenstein-Hawking vs. Cardy-Verlinde disagreement. That is because of the AdS/CFT correspondence, general principle that applies to many theories. This says that a theory in an anti-deSitter space, that is one with a negative cosmological constant, corresponds to a conformal field theory in one less dimension.
The Bekenstein-Hawking formula bears this out. If you compute the entropy of a black hole in an anti-deSitter space, rather than the usual flat space, you get that the entropy scales with the energy in one less than the number of spatial dimensions over the number of spatial dimensions. So, if I have 3 spatial dimensions, entropy scales with energy to the power 2/3. But if I have four spatial dimensions, then entropy scales with energy to the power of 3/4, just like an ordinary conformal field theory. This is a powerful argument that gravity may be a 5-D theory.
This is a pretty amazing result because it effectively resolves the conflict between Bekenstein-Hawking and conformal field theory, as long as you have anti-deSitter space. And while our space at very low energies and long distances may seem the opposite, with a positive cosmological constant, it is possible that at high energies that flips. (We don’t know because we don’t know what the cosmological constant is.)
If there is a theory of gravity out there, even if it isn’t string theory, you can be almost guaranteed that it will rely on the AdS/CFT correspondence. It is the only way that black holes can exist along side quantum theory.
Shomer, Assaf. “A pedagogical explanation for the non-renormalizability of gravity.” arXiv preprint arXiv:0709.3555 (2007).