What is a Gravitational Path Integral? {\it or} Gravitational Path Integrals as Fluctuating Gravito-Hydrodynamics

Abstract: We show how Gravitational Path Integral formulae for various quantities that have been computed in the literature, follow from a few coarse grained hydrodynamic assumptions about the relations between space-time geometry, entropy, and fluctuations of the modular Hamiltonian of causal diamonds. These remarks have implications for the way we think about such path integrals in relation to a more fundamental model of quantum gravity, and to questions about which space-time topologies are actually summed over in real models.

• The paper introduces a hydrodynamic reinterpretation of gravitational path integrals as statistical functionals over fluctuating spacetime variables.
• It derives Einstein’s equations from coarse-grained entropy and modular Hamiltonian fluctuations, linking thermodynamics with gravitational dynamics.
• The study challenges universal topology sums in quantum gravity, restricting nontrivial contributions to specific AdS/CFT contexts with random matrix statistics.

Gravitational Path Integrals as Fluctuating Gravito-Hydrodynamics

Introduction

This work articulates a comprehensive reinterpretation of the gravitational path integral (GPI), challenging standard quantum mechanical perspectives by proposing a hydrodynamic and statistical interpretation grounded in coarse-grained properties of space-time, entropy, and the fluctuations of the modular Hamiltonian associated with causal diamonds. The arguments synthesize and extend results from the thermodynamic derivation of Einstein’s equations, modular Hamiltonian fluctuations, and the statistical mechanics of horizon degrees of freedom, with the conceptual foundation provided by Jacobson’s identification of Einstein's equations as an equation of state.

The Covariant Entropy Principle and Entropic Gravity

Central to this approach is the Covariant Entropy Principle (CEP), a strong form of the covariant entropy bound, which posits that the modular Hamiltonian expectation for any causal diamond satisfies $\langle K \rangle = \frac{A_{\diamond}}{4G_N}$, where $A_{\diamond}$ is the maximal area of the diamond boundary and $G_N$ is Newton’s constant. Remarkably, this relation is asserted to hold even for "empty" diamonds, i.e., those in vacuum states, leading to the claim that spacetime geometry encodes a fundamental entropic content, independent of excitation.

This entropic perspective unifies horizon entropy and causal diamond entanglement entropy, and further implies that local energy density and excitations are interpreted as entropy deficits—constraints on “holographic q-bits” on the diamond’s boundary. The analysis connects this structure to the Holographic Space-Time (HST) models and relates localized energy to the freezing of degrees of freedom within a coarse-grained bulk.

Statistical Fluctuations and Hydrodynamics of Causal Diamonds

The formulation asserts that, much like in condensed matter systems, gravity at low energies emerges as a hydrodynamic description, with Einstein’s equations recovering as a coarse-grained limit of statistical entropy flow. The hydrodynamic variables are taken to be the metric $h_{ab}$ on the maximal area surface and the conjugate momenta $\pi^{ab}$ (extrinsic curvature), with flow defined Hamiltonianly via the Einstein equations.

Utilizing results from Carlip and Solodukhin, and connections to boundary conformal field theory (CFT), the note postulates that the fluctuating entropy current $S_{vv}$ on “stretched horizons” satisfies large central charge CFT statistics, with two-point functions fixed by the area.

Figure 1: Nested causal diamonds with Planck-separated tips, illustrating the region of the stretched horizon where hydrodynamic entropy fluctuations are supported.

This structure, where nested causal diamonds are separated by a stretched horizon, underpins the identification of the hydrodynamic regime with a white-noise-driven stochastic process at the Planck scale. The independence of fluctuations at successive Planck-separated intervals further enforces the analogy with fluctuating hydrodynamics and allows a functional integral description over these hydrodynamic variables.

Gravitational Path Integral: Functional and Topological Structure

In this hydrodynamic paradigm, the gravitational path integral no longer reflects a sum over quantum histories in the traditional sense but computes the statistical properties (e.g., spectral form factors, entropy) of fluctuating hydrodynamic fields defined by nested causal diamonds along a given geodesic. The non-locality in the transverse directions is a direct consequence of the stretched horizon CFT, and the white-noise-driven functional integral manifests as a statistical path integral that mirrors elements of the collective field formalism.

The action consists of a kinetic term over the fields $h_{ab}$ and $\pi^{ab}$ and a coupling to the stochastic entropy current $S_{vv}$. Integrating over the entropy fluctuations gives rise to effective non-local interactions, reminiscent of wormhole contributions but rooted explicitly in hydrodynamical fluctuations, not in summing over arbitrary topologies. The Lorentzian double-cone geometries, as in the SYK context, are identified as the relevant semi-classical configurations for black hole spectral form factors, but in general, Euclidean wormholes are not universally justified within this entropic hydrodynamics framework.

Implications for the Sum over Topologies

A strong claim is advanced: neither physical nor mathematical considerations support a sum over all topologies in the gravitational path integral beyond two-dimensional wormhole sectors. The path integral in this formalism is only justified as a statistical sum over the fluctuating configurations of hydrodynamic variables on nested causal diamonds, with the inclusion of higher-genus wormholes relevant only in AdS/CFT contexts that admit a random matrix interpretation in the regime of stable, finite-entropy black holes. For non-negative cosmological constant backgrounds (e.g., Minkowski or de Sitter), no observable detector can access the regime where higher topology effects are relevant, restricting measurable phenomena to the linear “ramp” in spectral form factors.

AdS vs. Non-Negative Curvature: Global Structure and Entropy

The appendix contrasts AdS with non-negative curvature spacetimes regarding the global structure of causal diamonds. In AdS, the area of causal diamonds diverges at finite proper time, and the empty diamond state seen by a bulk observer is pure by the correspondence with boundary CFT ground states. In contrast, in flat and de Sitter space the empty diamond state is maximally mixed at large scales. This distinction is captured by tensor network models of AdS/CFT and reflects fundamentally different features of entanglement, thermalization, and hydrodynamic excitation spectra in these backgrounds.

Conclusion

This work reframes gravitational path integrals as statistical functionals encoding the fluctuating hydrodynamics of Einstein gravity, rooted in three postulates connecting entropy, modular Hamiltonian fluctuations, and the hydrodynamic description of maximal surfaces. The implications are twofold:

• The gravitational path integral should be interpreted as a coarse-grained, background-dependent, hydrodynamic/statistical tool, not as a direct quantization of Einstein gravity.
• There is no theoretical support for a universal sum over topologies in the GPI; nontrivial topology enters only in specific situations admitting random matrix statistics, as seen in certain AdS/CFT contexts.

Practically, this viewpoint narrows the expected scope of GPI-based quantum gravity predictions to coarse-grained, time-averaged observables within regimes accessible to thermodynamic detectors or CFT correlators. Theoretically, it motivates a program of further developing gravito-hydrodynamic models, especially the identification and quantization of the relevant degrees of freedom on holographic screens, and refining the connection to boundary CFT data in AdS.

Future directions include clarifying the consistency constraints across overlapping causal diamonds for a full Hilbert bundle description, exploring corrections beyond leading hydrodynamic order, and rigorously deriving when (and whether) random matrix statistics should be expected in semi-classical gravitating systems outside of AdS/CFT.

This summary is based on "What is a Gravitational Path Integral? {\it or} Gravitational Path Integrals as Fluctuating Gravito-Hydrodynamics" (2601.10834).