The entropy of black holes: a primer

TL;DR

The article surveys how black-hole entropy and temperature arise from classical and quantum considerations, culminating in a microscopic accounting for certain extremal black holes via D-branes and the Cardy formula. It introduces the string–black-hole correspondence, arguing that massive string states transition into black holes around a coupling $g$ where $g^2 M \sim M_s$, with entropies matching at the crossover; this picture is clarified through self-gravitating-string microcanonical analyses. While significant progress explains microstates for BPS holes and supports unitary evolution in those regimes, a full microscopic description for non-BPS Schwarzschild black holes and the interior structure remains incomplete, leaving open questions about the fate of information and the end-point of evaporation. Overall, the work highlights a coherent, dimension-dependent bridge between string theory and black-hole thermodynamics, advancing our understanding of quantum gravity while outlining key unresolved issues.

Abstract

After recalling the definition of black holes, and reviewing their energetics and their classical thermodynamics, one expounds the conjecture of Bekenstein, attributing an entropy to black holes, and the calculation by Hawking of the semi-classical radiation spectrum of a black hole, involving a thermal (Planckian) factor. One then discusses the attempts to interpret the black-hole entropy as the logarithm of the number of quantum micro-states of a macroscopic black hole, with particular emphasis on results obtained within string theory. After mentioning the (technically cleaner, but conceptually more intricate) case of supersymmetric (BPS) black holes and the corresponding counting of the degeneracy of Dirichlet-brane systems, one discusses in some detail the ``correspondence'' between massive string states and non-supersymmetric Schwarzschild black holes.

Figures (4)

• Figure 1: Spacetime representation of the formation of a black hole by the collapse of a star. The horizon is the spacetime history of a bubble of light ( i.e. a null hypersurface) which stabilizes itself under the strong pull of relativistic gravity
• Figure 2: Splitting of an initial negative-frequency mode straddling the horizon into a mode falling into the black hole, and an outgoing mode which, after being partially reflected back into the black hole by the potential barrier representing gravitational and centrifugal effects, ends up as positive-frequency Hawking radiation at infinity. The antiparticle mode falling into the black hole can be interpreted as a particle travelling backwards in time, from the singularity down to the horizon [25] (hence the downwards orientation of the arrow).
• Figure 3: Evolution of the mass-energy, and of the physical structure, of an initially point-like configuration of $D$-branes as the string coupling $g$ increases.
• Figure 4: Evolution of the energy levels of massive string states as the string coupling $g$ increases. String states are expected to transform into black-hole states on the correspondence line $g^2 M \sim M_s$. Vertical line: Corresponding transformation of a black hole, loosing mass at fixed $g$ under Hawking radiation, into an initially compact string state, which inflates before decaying into massless radiation.