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String Theory and the Thermodynamics of a Black Hole

When Stephen Hawking described the Hawking radiation emitted by a black hole, he had to use his physical and mathematical intuition, because quantum physics and general relativity aren’t reconciled. One of the major successes of string theory is in offering a complete description of (some) black holes.

Hawking radiation takes place when radiation is emitted from a black hole, causing it to lose mass. Eventually, the black hole evaporates into nothing (or almost nothing).

Stephen Hawking’s incomplete argument

Hawking’s paper on the way a black hole radiates heat (also called thermodynamics) begins a line of reasoning that doesn’t quite work all the way through to the end. In the middle of the proof there’s a disconnect, because no theory of quantum gravity exists that would allow the first half of his reasoning (based on general relativity) to connect with the second half of his reasoning (based on quantum mechanics).

The reason for the disconnect is that performing a detailed thermodynamics analysis of a black hole involves examining all the possible quantum states of the black hole. But black holes are described with general relativity, which treats them as smooth — not quantum — objects. Without a theory of quantum gravity, there seems to be no way to analyze the specific thermodynamic nature of a black hole.

In Hawking’s paper, this connection was made by means of his intuition, but not in the sense that most of us probably think of intuition. The intuitive leap he took was in proposing precise mathematical formulas, called greybody factors, even though he couldn’t absolutely prove where they came from.

Most physicists agree that Hawking’s interpretation makes sense, but a theory of quantum gravity would show whether a more precise process could take the place of his intuitive step.

String theory may complete the argument

Work by Andrew Strominger and Cumrun Vafa on the thermodynamics of black holes is seen by many string theorists as the most powerful evidence in support of string theory. By studying a problem that is mathematically equivalent to black holes — a dual problem — they precisely calculated the black hole’s thermodynamic properties in a way that matched Hawking’s analysis.

Sometimes, instead of simplifying a problem directly, you can create a dual problem, which is essentially identical to the one you’re trying to solve but is much simpler to handle. Strominger and Vafa used this tactic in 1996 to calculate the entropy in a black hole.

In their case, they found that the dual problem of a black hole described a collection of 1-branes and 5-branes. These “brane constructions” are objects that can be defined in terms of quantum mechanics. They found that the results matched precisely with the result Hawking anticipated 20 years earlier.

Now, before you get too excited, the Strominger and Vafa results only work for certain very specific types of black holes, called extremal black holes. These extremal black holes have the maximum amount of electric or magnetic charge that is allowed without making the black hole unstable. An extremal black hole has the odd property of possessing entropy but no heat or temperature. (Entropy is a measure of disorder, often related to heat energy, within a physical system.)

At the same time Strominger and Vafa were performing their calculations, Princeton student Juan Maldacena was tackling the same problem (along with thesis advisor Curt Callan). Within a few weeks of Strominger and Vafa, they had confirmed the results and extended the analysis to black holes that are almost extremal. Again, the relationship holds up quite well between these brane constructions and black holes, and analyzing the brane constructions yields the results Hawking anticipated for black holes. Further work has expanded this work to even more generalized cases of black holes.

To get this analysis to work, gravity has to be turned down to zero, which certainly seems strange in the case of a black hole that is, quite literally, defined by gravity. Turning off the gravity is needed to simplify the equations and obtain the relationship. String theorists conjecture that by ramping up the gravity again you’d end up with a black hole, but string theory skeptics point out that without gravity you really don’t have a black hole.

Still, even a skeptic can’t help but think that there must be some sort of relationship between the brane constructions and the black holes because they both follow the Hawking thermodynamics analysis created 20 years earlier. What’s even more amazing is that string theory wasn’t designed to solve this specific problem, yet it did. The fact that the result falls out of the analysis is impressive, to say the least.

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