Black holes as collapsed polymers

We propose that a large Schwarzschild black hole (BH) is a bound state of highly excited, long, closed strings at the Hagedorn temperature. According to our proposal, the interior of the BH consists, on average, of a uniform distribution of matter with low curvature and large quantum fluctuations about the average. This proposal represents a dramatic departure from any conventional state of matter and from the longstanding expectation that the interior of a BH should look like empty space except for a very small, dense core (the singularity). Standard effective field theory in terms of the metric and other quantum fields is incapable of describing such a state in a meaningful way. However, in polymer physics, such states can be described by a mean field theory in terms of the polymer concentration. We therefore propose that the interior of the BH be described in terms of an effective free-energy density which is a function of the string concentration or entropy density; this density being a highly non-perturbative quantity in terms of the metric and other quantum fields. For a macroscopic BH, our proposed free-energy density contains only linear and quadratic terms, in analogy with that of the theory of collapsed polymers. We calculate the coefficient of the linear term under the accepted assumption that the dominant interaction of the strings at large distances is the gravitational interaction and the coefficient of the quadratic term by relying on explicit string calculations to determine the rate of interaction in terms of the string coupling. Using the effective free energy, we find that the size of the bound state is determined dynamically by the string attractive interactions and derive scaling relations for the entropy, energy and size of the bound state. We show that these agree with the scaling relations of the BH; in particular, with the area law for the BH entropy. The fact that the entropy is not extensive is a result of having strong correlations in the interior state, and the specific form of the entropy-area law originates from the inverse scaling of the effective temperature with the bound-state radius. We also find that the energy density of the bound state is equal to its pressure.