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.
@mcnees
Thank you! I see a dive into basic Wikipedia info in my very near future; cosmology is good stuff! 👍
@AndyLowry At a minimum, you must me over the Chandrasekhar limit, which is around 1.4-ish solar masses. I don't think we have direct confirmation of any black holes that small, though.