Fuzzballs and black hole thermodynamics

TL;DR

The paper addresses whether fuzzball microstate replacements for the traditional horizon can preserve black hole thermodynamics. By drawing an analogy to modular invariance and comparing Euclidean path-integral and Hamiltonian decompositions, the authors show that if fuzzball entropy $S_{fuzzball}$ matches the traditional $S_{bh}$, then the temperature and emission rates also align, and the standard entropy is recovered via a degeneracy of cap configurations: ${\\cal N}_{fuzzball}=e^{4\\pi G M^2}$, yielding $S_{fuzzball}=A/(4G)$. This leads to a coherent picture where fuzzball microstates reproduce Hawking radiation while eliminating the horizon, provided two key assumptions hold: the same Euclidean saddle contributes, and fuzzballs form a complete microstate set. The work thus provides a pathway to resolve the information paradox within the fuzzball framework and clarifies the conditions under which traditional black hole thermodynamics can be maintained.

Abstract

The fuzzball construction resolves the black hole information paradox by making spacetime end just before the horizon is reached. But if there is no traditional horizon, then do we lose the elegant relations of black hole thermodynamics? Using an argument similar to modular invariance, we argue that the answer is no; the completeness of fuzzball states implies that the generic fuzzball indeed reproduces the thermal properties attributed to the traditional hole.

Figures (6)

• Figure 1: (a) In the traditional hole, compact directions appear as a tensor product with the noncompact ones (the tensor product is denoted by the cross). (b) In a fuzzball microstate the compact directions 'cap-off' before the horizon is reached; different choices of 'caps' at the various angular positions yield the entropy $S_{fuzzballls}$.
• Figure 2: (a) The $r, \tau$ space is a 'cigar' with the topology of a disc. (b) The bold line is the spatial slice on which a state of energy $E=M$ lives. Euclidean time evolution is in the $\tau$ direction. A small hole is cut around the point $r=2M$ where slices of different $\tau$ meet.
• Figure 3: A fuzzball microstate (defined on slice depicted by the bold line) 'caps-off' before reaching the horizon; the cap is denoted by the black dot. Thus we do not need to cut out a hole around $r=2M$, but do have to take into account the degeneracy factor ${\cal N}_{fuzzball}$ arising from different possible caps.
• Figure 4: (a) A slicing of the $r, \tau$ space that is regular at the center of the disc $r=2M$. (b) The slice through $r=2M$ will be a cylinder, in the toy example where we assume there is one compact dimension.
• Figure 5: The field theory of the metric variables $g$ on the cylinder of fig.\ref{['c3']}(b). The spatial slice has been approximated by a lattice of points on which the variables $g_i$ live; the gradient term $\nabla g$ is represented by springs joining the lattice points.