Stephen Hawking was born #OTD in 1942. He developed theorems with Penrose that determine when general relativity produces singularities, established classical laws of black hole mechanics, and hypothesized that quantum effects make black holes radiate.
Image: Santi Visalli/Getty
Hawking's work on black holes radiating quantum mechanically led to the black hole information paradox, which is one of the instructive puzzles that guides our attempts to develop a complete theory of quantum gravity.
Black hole radiation and the information paradox is probably Hawking's best-known result. How did it come about?
In 1970 Hawking proposed an “area law” for black holes: the area of the event horizon never gets smaller. This is true in general relativity, a classical theory.
You can get a sense of how this works in the simplest case. The radius of a non-rotating black hole is proportional to its mass. As more material crosses the horizon its radius and area both increase. Since it can’t lose mass via things escaping, the area never decreases.
Hawking’s area law is of course much broader than this simple observation, accommodating the full range of dynamical black holes allowed by general relativity and holding under quite general assumptions.
"In other words, the area of the boundary of a black hole cannot decrease with time, and if two or more black holes merge to form a single black hole, the area of its boundary will be greater than the areas of the boundaries of the original black holes."
— Hawking & Ellis (1973)
This passage from Hawking & Ellis’s “Large Scale Structure of Spacetime” isn't academic. Nowadays when LIGO is running it detects the merger of black holes several times a year, via the gravitational waves they emit.
The black hole mergers detected by LIGO all satisfy Hawking’s area law. In any process involving one or more black holes the horizon area of the final configuration is greater than the initial areas.
Image: LIGO-Virgo/ Northwestern U./ Frank Elavsky & Aaron Geller
In January of 1973, Bardeen, Carter, and Hawking published their “four laws of black hole mechanics.”
There was a zeroth law which, as is often the case with zeroth laws, you need not worry about.
The First Law related the change in a black hole's internal energy to changes in its area, angular momentum, and charge. The Second Law was Hawking’s area theorem. And the Third Law said a black hole can’t reach zero surface gravity through any physical process.
If you’ve studied thermodynamics this might sound familiar.
Hawking’s work in the early 1970s established a really suggestive analogy between the laws of thermodynamics and the laws governing black holes. Bardeen, Carter, and Hawking flesh it out right there in the paper.
But it's just an analogy, right? Thermodynamics is about stuff that is in some sense "hot," and the most basic property of a hot object is that it radiates or emits energy. According to GR black holes don't emit anything!
Around the same time (the paper was submitted in November 1972 and appeared in April 1973) Jacob Bekenstein suggested that perhaps it was more than an analogy.
Maybe black holes really are thermodynamic objects with temperature and entropy!
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.2333
Bekenstein was motivated by discussions with his advisor John Wheeler, who was worried about what would happen to the entropy of a hot cup of tea poured into a black hole.
Would the entropy associated with the tea vanish like everything else? That would seem to contradict the second law of thermodynamics, which would have profound implications for our understanding of physics.
Hawking got right to work, and by the following January he had shown that, indeed, quantum mechanical effects cause black holes to radiate.
As far as Hawking could tell, the spectrum of emissions from a black hole was that of an ideal blackbody.
Since energy and mass are equivalent, the black hole must lose mass as it radiates energy. This causes the area of its event horizon to decrease, a quantum mechanical phenomenon that directly contradicts the classical result Hawking had established just a few years earlier!
Hawking's discovery turned Bekenstein’s proposal into something more concrete.
A black hole has a temperature, and therefore an entropy. This Bekenstein-Hawking entropy is one-quarter of the area of the black hole event horizon, as measured in units of the Planck length squared.
To get a feel for how this works for astrophysical black holes, let's look at the simplest case: Schwarzschild's non-rotating black hole.
In that case the radius of the horizon is proportional to the mass of the black hole.
Here G is Newton's constant and c is the speed of light. A solar-mass black hole has a Schwarzschild radius of around 3 km. But our sun isn't massive enough to form a black hole. A more massive star that became a 20 solar mass black hole would have a radius of ~59 km.
The temperature of a Schwarzschild black hole is inversely proportional to its radius and hence inversely proportional to its mass.
Various factors of fundamental constants, along with the large mass, mean that the temperature of an astrophysical black hole is phenomenally small.
A 20 solar mass black hole, comparable to some of the black holes involved in mergers detected by LIGO, would have a temperature of around 3 x 10⁻⁹ K.
That's three billionths of a Kelvin above absolute zero for a Schwarzschild black hole twenty times as massive as our Sun.
The results for a rotating black hole (as far as we can tell, all astrophysical black holes have some amount of rotation) look a bit more complicated, but aren't quantitatively that different.
What does this mean for the entropy?
The area of the Schwarzschild horizon is 4π times its radius squared. And the Planck length is astonishingly small – about 1.6 x 10⁻³⁴ meters. So the entropy of a 20 solar mass black hole is about 4 x 10⁷⁹ k_B.
This is many orders of magnitude larger than a self-gravitating ball of hot gas with comparable mass.
Black holes are far and away the most entropic objects in the Universe. Most of the entropy in the Universe is in the form of black holes.
Anyway, before we went on this numerical detour we were talking about radiation.
As you can see from the calculations above, astrophysical black holes are far colder than their surroundings.
The Cosmic Microwave Background is an ambient bath of photons with a temperature of about 2.7 K.
Black holes have a temperature, so they can radiate. But since astrophysical black holes are colder than their surroundings, they are out of thermal equilibrium and therefore absorb more energy than they emit.
But the fact that black holes can *in principle* radiate energy raises a profound question.
In Hawking’s calculation the radiation emitted by a black hole seemed to be perfectly thermal. This means that the radiation does not contains any information about the particular quantum mechanical state of the matter that formed the black hole.
Why is that a problem?
Well, the CMB is cooling down as the Universe expands. Its temperature will eventually (in the far, far future) drop below the temperatures of astrophysical black holes.
Once a black hole is hotter than the surrounding environment it will radiate more than it absorbs and begin to lose mass.
As the black hole loses energy it gets smaller, of course. But it also gets *hotter*. And hotter objects emit more radiation.
This suggests a runaway process in which the black hole could completely “evaporate.” But if the black hole evaporates, and the radiation is thermal, does that mean we lost all the information about the matter that preceded it?
This would be at odds with our understanding of quantum mechanics, which demands in a very precise sense that certain types of information can’t be lost.
This “black hole information paradox” has been one of the deep questions guiding work on quantum gravity over the last 50 years or so.
There are many possible resolutions to this paradox.
Maybe the evaporation process changes when the black holes are small and hot enough for other quantum effects to become important.
Maybe the original calculation missed something, and the information is carried away by the radiation, subtly encoded in delicate spacetime structures of which Hawking was not aware.
We don't know for sure!
Studying this paradox has led to many insights about the nature of quantum gravity.
Progress comes from sharp questions that show the gaps in our understanding. This is one of those. I think of this work as Hawking's greatest scientific achievement not because he came up with an answer, but because he led us to a deep and instructive question.
If you are curious about the Black Hole Information Paradox, take a look at @gmusser's 2020 piece in @QuantaMagazine explaining recent progress.
We may finally be close to resolving it.
https://www.quantamagazine.org/the-most-famous-paradox-in-physics-nears-its-end-20201029
@mcnees @gmusser @QuantaMagazine
Thanks very much for the link. I've pulled up the article and will have a look at it shortly. I'd seen at least one discussion of the paradox on Sabine Hossenfelder's site some time back, but didn't read it at the time because I was still trying to get a basic handle on particle physics at the time. 🤔
@mcnees
And thanks again! I should be able to track it down.
@mcnees
And one of the great things about her site is the occasional original song performance, complete with stage gestures and everything. When I first discovered her site-- I think because I'd just read "Lost in Math"-- I surely did not expect the musical numbers. Really delightful!
@AndyLowry You should read Sabine's explanation of "where" Hawking radiation originates. It is, in my opinion, a fantastic piece of science writing.