by Josh Kirklin
Awaken, quantum relativist.
Have breakfast, and notice that a black hole has found its way into your laboratory. You measure its mass M, electric charge Q and angular momentum J, double-check the statement of the no hair theorem, and tell yourself that you can learn no more about this particular black hole.
But the quantum mechanic inside of you objects. As a devout believer in unitarity, you are convinced that the black hole must contain a complete description of the matter involved in its formation. So, you think, the no hair theorem must not apply. Finding no logical inconsistency in the mathematical steps involved in its proof, you decide that something must be wrong with its initial assumptions.
One such assumption is that two black holes related by a gauge transformation are physically equivalent. But this cannot be correct, since if we do a gauge transformation whose action does not vanish sufficiently quickly at infinity, there are observable consequences. The quickest way to become convinced of this fact is to find the Poisson brackets appropriate for a description of gravity and electromagnetism, and to use them to compute the actions generated by M, Q and J. The action of each is a gauge transformation that is non-trivial at infinity, and it is a basic fact of Hamiltonian mechanics that if a quantity is observable (M, Q and J certainly are), then so too must be the action that it generates.
About the author: Josh Kirklin is a PhD student in the Relativity and Gravitation group at the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge. He studies black hole thermodynamics and the role of information in quantum gravity.
You deduce that you have at least three new gauge-dependent observable parameters (one for each of M, Q and J) to describe your black hole. In fact, you have infinitely many more, since the dipole, quadrupole and higher order moments of M, Q and J also generate gauge transformations that are non-trivial at infinity. The no hair theorem guarantees that the higher order moments of M, Q and J themselves must vanish, but it does not claim the same for the resultant new gauge-dependent variables.
So you conclude that your stationary black hole is actually described by infinitely many degrees of freedom. These are known as soft hairs, and the quantities which generate their transformations are known as soft charges.
During lunch, you manage to devise a clever experiment which makes use of the gravitational and electromagnetic memory effects to measure the soft hairdo of your black hole. Partially out of respect for the event horizon, but mostly out of fear, you decide to stay far enough away from the black hole that you must treat yourself as an idealised observer at infinity in your calculations. So, carrying out your experiment, you obtain a set of numbers describing how the soft hair appears to an observer at infinity.
But the black hole is an isolated body in spacetime, whose properties should be intrinisic to it. What you really desire is a description of the soft hair that is local to the black hole – one that would reflect what a more courageous observer, who was more willing to closely approach the event horizon, might see. You wonder what the best way would be to deduce such a local description from your observations at infinity, and, absent-mindedly leafing through the latest issue of CQG, you stumble upon a paper that takes you part of the way towards the answer. Following its advice, you are able to write the soft charges in terms of fields close to the black hole.
At dinner, you decide that you would like to justify your scientific credentials by predicting the future of the black hole. Before the discovery of soft hair, you would have only been able to treat the black hole as a single thermodynamical system, supporting an entropy and emitting Hawking radiation. This was necessary because you only had a macroscopic description for your black hole, being ignorant of its microscopic physics. But now you feel that you can improve on this, since you have a candidate for the black hole’s microscopic degrees of freedom – its soft hair. To make some predictions, all you need is a theory that governs the dynamics of the soft hair, and that same paper seems to again provide some assistance. It argues that a softly hairy black hole should actually be treated as an infinity of thermodynamic systems, all in thermal contact with each other. This thermal contact manifests as a heat current on the event horizon.
You use these results to make your predictions, and climb into bed after an exhausting day of hands-on relativity. Drifting off to sleep, you wonder whether the discovery of soft hair will be enough to solve one of the biggest mysteries of black hole physics – the information paradox. To have any hope of this being the case, you ought to be able to use the existence of soft hair to derive the black hole area entropy relation. You have heard rumors that this result is close at hand, and that the paper announcing it will bear Stephen Hawking’s name (making it his last published work).
Until then, you can only dream of tomorrow’s meals and measurements…
Read the full article in Classical and Quantum Gravity: Localisation of soft charges, and thermodynamics of softly hairy black holes
Josh Kirklin, 2018 Class. Quantum Grav. 35 175010
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