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Black Holes
The most unexpected place thermodynamics turned up is general relativity, and whether it genuinely turned up there or merely appeared to is still argued.
The correspondence
In the early 1970s it was noticed that black holes obey relations formally identical to the laws of thermodynamics. Bardeen, Carter and Hawking set out four laws of black hole mechanics in 1973: surface gravity is constant over the horizon of a stationary black hole, mirroring the zeroth law; changes in mass relate to changes in area, angular momentum and charge, mirroring the first; horizon area never decreases, mirroring the second; and surface gravity cannot be reduced to zero in finite time, mirroring the third.
The authors presented this as an analogy and were initially sceptical that it was more. Jacob Bekenstein had already proposed that the area is an entropy, up to a constant — motivated by the thought that a black hole would otherwise offer a way to destroy entropy by dropping it in, violating the second law. What converted the sceptics was Hawking’s own result: black holes radiate, with a temperature proportional to surface gravity, which supplies the missing element the analogy needed. With a real temperature, the correspondence has entropy equal to a quarter of the horizon area in Planck units.
Literal or analogical
Whether this is entropy or looks like entropy remains open, with articulate positions on both sides.
John Dougherty and Craig Callender argue the analogy is weaker than commonly supposed. Black hole thermodynamics, on their account, corresponds not to thermodynamics but to a caricature of it; it is unclear which systems the theory is even about; and it presupposes a contested epistemic conception of entropy. They press specific mismatches: temperature is intensive and independent of system size, whereas surface gravity is not; and entropy ordinarily scales with volume rather than area.
Robert Wald and David Wallace defend the literal reading, Wallace in a two-part treatment addressing phenomenological thermodynamics and statistical mechanics in turn, arguing that the objections either misidentify what thermodynamics requires or apply equally to ordinary systems. Erik Curiel, and Carina Prunkl with Christopher Timpson, have contributed further to the exchange, which continues in the current literature.
The microstate count
The correspondence gave the horizon an entropy before anyone had a count of states behind it — as in thermodynamics itself, where the macroscopic relations came half a century before Boltzmann’s count. The constructive account has been sought since. The count has been supplied for particular cases. Andrew Strominger and Cumrun Vafa derived the area law in 1996 for a class of five-dimensional extremal black holes in string theory, by counting the degeneracy of bound states of the theory’s solitons; the result reproduces a quarter of the horizon area exactly. Loop quantum gravity has a count of its own: Ashtekar, Baez, Corichi and Krasnov showed in 1998 that the entropy of a large non-rotating black hole is proportional to its horizon area, with agreement to the Bekenstein–Hawking coefficient depending on the choice of the Immirzi parameter. Both are counts within a candidate theory of quantum gravity, for restricted classes of black hole, and neither is a count of the microstates of an astrophysical black hole in a theory known to describe it. What they establish is that the constructive account can be built at all.
The information paradox
If black holes radiate and eventually evaporate, what becomes of the information about what fell in? Unitary quantum evolution says it cannot be destroyed; the radiation appears thermal and featureless. The tension is Hawking’s paradox, and the standard diagnostic is the Page curve: if evolution is unitary, the entanglement entropy of the radiation should rise and then fall, rather than rising indefinitely as the semiclassical calculation gives.
Work from 2019 onward — Penington; Almheiri, Engelhardt, Marolf and Maxfield; and the replica-wormhole calculations of Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini, and of Penington, Shenker, Stanford and Yang — reproduces the Page curve through new gravitational saddle points that dominate after the midpoint of evaporation. This is a genuine advance and is widely taken as strong evidence for unitarity.
What has been computed is an entropy; no state-recovery process has been exhibited, so the mechanism by which information escapes is not shown. The results are largely obtained in two-dimensional and holographic settings, and whether they extend to four-dimensional astrophysical black holes is open. There is also an argument that the island formula gives the wrong radiation entropy where replica symmetry breaking is significant. The accurate summary is that the entropy is now calculated correctly and the paradox is not thereby agreed to be resolved.
See also: Thermodynamics · What entropy is · Hawking