On Entropic Gravity from BFSS Matrix Theory

Abstract

We study Matrix theory at strong coupling in a setting describing two static objects a fixed distance apart, using numerical techniques. We reproduce the exact general relativistic force law between the two objects as an entropic force in Matrix theory. This involves employing an operator algebra that represents an external observer measuring the relative positions and momenta of the two objects. We identify the location of the horizons of the objects from this entropic force and are led to a modification of the Schwarzschild spacetime inside the horizon. We find evidence that the inside of a black hole is instead to be described by AdS space. The conclusions constitute numerical validation of Verlinde's entropic gravity proposition and the fuzzball paradigm.

Figures (10)

• Figure 1: Some of the low eigenvalues of the Hamiltonian (\ref{['eq:Heff']}). The lowest line is the ground state. We also show conservative error bars estimated from the convergence of the numerics as a function of a UV cutoff. The shaded region, $g_X\sim 1$, corresponds to the weak-strong coupling transition, and to the $r\sim \ell_P$ point.
• Figure 2: The entropic force as a function of $g_X\sim r/\ell_P$ for spacetime dimension $d=3+1$. At small but still super-Planckian distances, $f_{ent}\sim -r$; at larger distances, $f_{ent}$ is the gravitational force from general relativity, with a horizon at the minimum point of the force profile. In this plot, we show the case $t=5$, but the fit was generated across many values of $t$. The dashed curve is the result from general relativity fitted to the data.
• Figure 4: (a) The low eigenvalues of the Hamiltonian (\ref{['eq:Heff']}) on a log-log scale for $d=3+1$. After a regime of eigenvalue crossing, eigenvalue repulsion sets in at higher couplings. (b) Application of the Gauss constraint to eigenvectors. The horizontal scale is logarithmic. In this case, all shown eigenvectors are physical states, lying below the numerical cutoff.
• Figure 5: The onset of fast scrambling in Matrix theory at large coupling for $d=3+1$.
• Figure 6: The entropic force for $d=3+1$ plotted against $g_X/t$ showing the consistency of the $v=g_X/t$ scaling at large $g_X$, and the breakdown of this parameterization inside the horizon at small $g_X/t$. Error bars shown do not include systematic effects from the matrix UV cutoff and the truncation of the partition function that become important at and below $g_X/t\sim 2$. The systematics in this lower $g_X/t$ region increase the errors to as much as 30% near the minimum of the force.