Why Emergence of Gravity in Matrix Theories is Entropic (2507.21996v1)

Abstract: Matrix theories exhibit the phenomenon of spacetime emergence in certain regimes of their dynamics. In this work, I posit that the key to this emergence is a hierarchy between two timescales -- very slow modes that an observer measures plus a chaotic cloud of very fast modes; this then leads to a natural operator algebra for measurements for the slow modes and an associated density matrix for the full system. I show that this density matrix implies the same gravitational potential energy that has been identified previously in the literature; furthermore, I show that the corresponding emergent gravitational force is nothing but the entropic force associated with the entropy of the fast modes of the system. I also demonstrate that the posited regime corresponds to distances being super-Planckian in the dual eleven dimensional supergravity.

• The paper demonstrates that a dynamical separation of slow and fast matrix modes leads to an entropic derivation of gravitational dynamics.
• The paper employs SU(2) Matrix theory techniques with block matrix structure and coherent state analysis to model measurement processes and thermal effects.
• The paper's results bridge quantum matrix entanglement and spacetime emergence, offering a new perspective on nonperturbative quantum gravity.

Entropic Emergence of Gravity in Matrix Theories

Introduction and Motivation

This work addresses the mechanism by which gravity and spacetime emerge in Matrix theories, specifically focusing on the entropic origin of the emergent gravitational force. The central thesis is that a hierarchy between two dynamical timescales—slow modes (diagonal matrix blocks) and fast, chaotic modes (off-diagonal blocks)—naturally leads to an operator algebra and density matrix structure that encodes gravitational dynamics as an entropic force. The analysis is carried out in the context of the bosonic sector of SU(2) Matrix theory, with implications for both finite and large $N$.

Matrix Theory Structure and Timescale Hierarchy

Matrix theories, such as the BFSS model, encode degrees of freedom in $N \times N$ matrices with U(N) gauge symmetry. The author considers a block structure for the matrices, where diagonal blocks ($\bm{D}_1$, $\bm{D}_2$) represent slow modes and off-diagonal blocks ($\bm{R}$) represent fast modes. The regime of interest is characterized by a clear separation of timescales: $\tau_D \gg \tau_R$, where $\tau_D$ is associated with the slow, observable dynamics and $\tau_R$ with the rapid, chaotic evolution of the off-diagonal modes.

This separation is not in conflict with gauge invariance, as the physical operator algebra is constructed to respect the SU(2) symmetry by identifying coherent states up to color rotations. The slow modes correspond to D0-brane positions and momenta, while the fast modes are interpreted as high-frequency excitations mediating interactions.

Operator Algebra, Coherent States, and Measurement

The measurement process is formalized using coherent states parameterized by complex variables $\alpha^i$, encoding the mean position and momentum of the slow modes. The operator algebra is constructed such that physical states are equivalence classes under SU(2) rotations, ensuring gauge invariance. The density matrix relevant for physical measurements is then a tensor product of a pure state for the slow modes and a mixed, thermal state for the fast modes:

$$\bm{\rho} = |\alpha\rangle\langle\alpha| \otimes \bm{R}_\alpha$$

where $\bm{R}_\alpha$ is a thermal density matrix for the fast sector at temperature $T$, with the slow sector acting as an external parameter.

Physical Regime and Emergence of Spacetime

The timescale hierarchy is shown to correspond, via dualities, to the regime where the separation between D0-branes is much larger than the eleven-dimensional Planck length ($r \gg \ell_P$). This is precisely the regime where emergent spacetime and gravity are expected to arise in Matrix theory. The effective couplings governing the dynamics are

$$G_{eff}^2 = g^2 \tau_D^3 = (g X_0^3)^2, \quad g_{eff}^2 = g^2 \tau_R^3 = (g X_0^3)^{-1}$$

with $g X_0^3 \gg 1$ indicating strong coupling for the slow sector and weak, but chaotic, dynamics for the fast sector.

Entropic Force and Gravitational Dynamics

The main technical result is the derivation of the force on the slow modes due to the fast sector. By tracing over the fast modes and employing standard techniques from quantum optics, the author shows that the force is given by

$\frac{d\langle \bm{P}^i\rangle}{dt} = T \nabla_i S$

where $S$ is the von Neumann entropy of the fast sector. This matches the entropic force paradigm, as advocated by Verlinde, and is shown to reproduce the gravitational force law previously derived in Matrix theory via other methods (e.g., background field gauge, one-loop effective potential).

The entropy change is entirely due to the fast sector, as the coherent state for the slow sector is a state of maximal entropy and does not contribute to entropy variation under small displacements. The temperature $T$ of the fast sector is related to the scrambling time, and the analysis suggests that the measurement process (i.e., the resolution of the observer) plays a nontrivial role in the emergence of gravity.

Implications, Limitations, and Future Directions

This approach contrasts with recent proposals that focus on target space entanglement; instead, it emphasizes entanglement between matrix sub-blocks, guided by the timescale separation. The framework is manifestly gauge-invariant and provides a direct transcription of gravitational data into matrix entanglement structure.

The analysis is currently limited to the bosonic sector and $N=2$, but the extension to large $N$ is expected to preserve the essential mechanism, with additional structure arising from self-gravitational effects within matrix sub-blocks. The full dynamics, especially at strong coupling, will require numerical methods for quantitative predictions, particularly for the computation of the microcanonical ensemble and the determination of the temperature-entropy relation.

The work also raises questions about the equivalence principle in Matrix theory, given its background-dependent formulation, and suggests that a deeper understanding may require the construction of analogues of Fermi normal coordinates in matrix phase space.

Conclusion

This paper provides a detailed and technically robust account of how gravity emerges as an entropic force in Matrix theories, rooted in a hierarchy of dynamical timescales and the associated structure of the density matrix. The identification of the gravitational force with an entropic force of the fast, thermalized matrix modes is shown to be consistent with previous results and offers a new perspective on the emergence of spacetime in nonperturbative quantum gravity. The approach is generalizable and sets the stage for future numerical and analytical investigations, including the incorporation of supersymmetry and the exploration of large $N$ dynamics.