How is a Closed String Loop like a Black Hole?

We demonstrate that under plausible assumptions the entropy and temperature associated with the small oscillations on a circular loop of radius R and a black hole of mass M = R/2G are identical. 5/95 † [email protected] ‡ [email protected] According to Hawking [1] and Bekenstein [2], the inverse temperature and entropy of a Schwarzschild black hole of mass M are respectively given by βbh = 1 T = 8πGM h̄ , Sbh = 4πGM h̄ = βM 2 . (1) If we consider Ebh = M as the energy of this system, then the free energy is given by F = Ebh − TbhSbh = M/2. (2) Ever since they were published, these relations have given rise to much speculation in both gravity and quantum field theory. Is the evolution of a quantum state non-unitary in the presence of a black hole? Do ultraviolet divergences of quantum fields modify gravitational singularities? What has surface area got to do with entropy? In this letter we add to the speculation and report a most peculiar coincidence. The formulae relating the mass, entropy and temperature of a black hole including the numerical coefficients, are identical to those of a simple model, namely, a system comprising of small oscillations on a closed string wrapped around what would be the event horizon of a Schwarzschild black hole of the same mass. Although intriguing, this result seems to rely on a few key assumptions, every one of which can be justified on physical grounds. We shall point out these assumptions as we go along. To begin with we consider as our statistical system the small oscillations on a closed circular string. In four dimensions there are two polarizations, φr and φ⊥, one in the radial direction and the other perpendicular to it. Following [3], we write the coordinates of the perturbed world-sheet as x̃ = x + ∑