The Temperature of a Black Hole

Dr. Stephen Hawking's gravestone in Westminster Abbey

Dr. Stephen Hawking’s gravestone in Westminster Abbey

In a stylized engraving of a black hole, in Westminster Abbey, London, England – not far from the stones marking the remains of Sir Issac Newton and Charles Darwin – is the signature achievement of the late Stephen Hawking, his formula for the temperature, the radiation level, emerging from a black hole. In the formulation on his gravestone,

T stands for temperature, the level of radiation emerging from the black hole;

h-bar is Planck’s constant which is used to describe the relationship between frequency and energy, about 1.054571800(13)×10−34;

c stands for the speed of light, just under 300,000 meters per second;

Pi is the ratio between the diameter and circumference of a circle (a black hole will always be a perfect sphere);

G is Newton’s constant and specifically the ratio of attraction between two point masses, about 6.674×10−8 cm3⋅g−1⋅s−2;

M is the mass of the black hole; and

k stand for Boltzmann’s constant, which describes the relationship between the average kinetic energy of particles in a gas with the temperature of the gas, about 1.380649×10−23 J/K..

Hawking’s formulation was brilliant and revolutionary. Classic Einsteinian physics says that nothing can escape a black hole, because to do so it would have to travel faster than the speed of light. Hawking showed that radiation could escape, that black holes could have a “temperature.” If you made it all the way through Hawking’s improbable bestseller A Brief History of Time1 you’ve got that figured out in principle, even if you don’t have the abstruse math.

But Hawking’s formulation created the Black Hole Information Paradox. The paradox is even more complex than Hawking radiation, but at the risk of oversimplification, says that information is vanishing from the universe, which quantum physics says shouldn’t happen. A black hole, famously, devours everything around it, including the information contained in what is devoured. The Hawking radiation that emerges from a black hole and that eventually (in several lifetimes of the universe) might cause the black hole to evaporate is free of information. When the black hole evaporated, all the information would be lost.

Hawking died this past March. But this week his final paper was published. And, with a number of co-authors, Hawking, from beyond the grave, takes a stab at resolving the Black Hole Information Paradox. In principle, the information falling in to the black hole encodes the “soft hair,” the radiation, coming out. The paper is beyond abstruse. The abstract states:

A set of infinitesimal Virasoro L ⊗ Virasoro R diffeomorphisms are presented which act non-trivially on the horizon of a generic Kerr black hole with spin J. The covariant phase space formalism provides a formula for the Virasoro charges as surface integrals on the horizon. Integrability and associativity of the charge algebra are shown to require the inclusion of ‘Wald-Zoupas’ counterterms. A counterterm satisfying the known consistency requirement is constructed and yields central charges cL = cR = 12J. Assuming the existence of a quantum Hilbert space on which these charges generate the symmetries, as well as the applicability of the Cardy formula, the central charges reproduce the macroscopic area-entropy law for generic Kerr black holes.

Sure, if you say so.

Translated into grossly over-simplified English, the information isn’t lost because it is encoded in the “soft hair” of Hawking radiation emerging from the black hole.

It’s a hypothesis, mind you. The authors are candid about their assumptions. But it’s a kind of vindication from beyond the grave, too. Hawking was an absolutely extraordinary man, an Issac Newton confined to a wheelchair and handicapped by a dreadful disease, a rock star who had to talk through a computer. But in his lifetime he revolutionized the area where astrophysics meets quantum physics, where the very biggest objects meet the very smallest.

Now, from beyond the grave, he just might have revolutionized it again.


  1. A wag called Hawking’s popular treatment of black holes “the book bought by the most people and read by the fewest.” WC, needless to say, has read it a couple of times now. 

One thought on “The Temperature of a Black Hole

  1. Thank You !! That is one of the most understandable translations/explanations for this mathematically challenged guy of the abstrusely written concept of information evaporation … but wait maybe it does not thanks to the return/release of the “soft hair” of the universe. Now if I can figure out how to take advantage of the mathematical information in that radiation maybe I could finally get math beyond the simple ciphering of business math 🙂

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