Entropic force in matrix

TL;DR

This work embeds Verlinde's entropic gravity idea into matrix theory by modeling $N$ D0-branes in a constant RR $4$-form background and treating a fuzzy sphere $X^i=fJ_i$ as a holographic screen with radius $R^2=f^2 j(j+1)$. Quantum fluctuations around this screen are analyzed via off-diagonal modes, yielding bosonic and fermionic mass spectra and mass matrices $\mathcal{M}_B^2$ and $\mathcal{M}_F$, which feed into a finite-temperature partition function. The resulting free energy $\mathcal{F}=\mathcal{F}^{(0)}+\mathcal{F}^{(1)}dx+\cdots$ produces an entropic force $F=\beta\frac{\partial \mathcal{F}^{(1)}}{\partial \beta}$, with two regimes: (i) large $R$ where $S^{(0)}=\frac{21\zeta(3)k_B N^2 \lambda^2}{\beta^2 R^2}$ and $F= -\frac{21\zeta(3) N^2 \lambda^2}{\beta^3 R^3}$, not matching Verlinde's $1/R^2$ law, and (ii) small $R$ with $f\to0$ and $N$ large, where $S^{(0)}\propto N^2$ and $F\propto -1/(\beta R)\propto -1/R^2$, indicating entropic gravity in the RR-free regime. This work clarifies the conditions under which gravity can emerge as an entropic force in a matrix-theory context and highlights the role of the holographic screen choice.

Abstract

We consider the entropic force in matrix theory. We find the gravity in bulk can be emergent from the entropic force.

Figures (1)

• Figure 1: It is a fuzzy sphere with a radius $R$ and a D0 brane at $x^3=R+\Delta x$.