Energy equipartition and minimal radius in entropic gravity

In this article, we investigate the assumption of equipartition of energy in arguments for the entropic nature of gravity. It has already been pointed out by other authors that equipartition is not valid for low temperatures. Here we additionally point out that it is similarly not valid for systems with bounded energy. Many explanations for black hole entropy suggest that the microscopic systems responsible have a finite dimensional state space, and thus finite maximum energy. Assuming this to be the case leads to drastic corrections to Newton’s law for high gravitational fields, and, in particular, to a singularity in acceleration at finite radius away from a point mass. This is suggestive of the physics at the Schwarzschild radius. We show, however, that the location of the singularity scales differently.

Figure 1

Energy versus temperature for the system described in Sec. 3 (in units of the energy spacing $E_0$, with $N=10$ energy levels).

Figure 2

Energy versus temperature in units of some fixed energy $E$, for $E_0=E$, $N=10$ (upper solid curve) $E_0=E/5$, $N=50$ (dashed curve) $E_0=E/10$, $N=100$ (dot-dashed curve) and for (17), with $E_{\max }^{\prime }=10E_0$.

Figure 3

Acceleration versus radius in SI units, with $M=M_{\mathrm{Earth}}$ and $E_{\max }$ such that $R_{\mathrm{at}}=R_{\mathrm{S}}$: General relativity (solid curve), nonrelativistic entropic gravity (dashed curve), Newton’s law (dot-dashed curve).