Gravitational Entanglement Entropy

• Gravitational entanglement entropy is a measure of quantum correlations in dynamical gravity, following an area law modified by a natural cutoff like the Planck length.
• It connects quantum field theory and general relativity by revealing how renormalization and holographic methods underpin the emergence of Einstein’s equations and black hole entropy.
• Challenges such as edge mode contributions, anomaly-induced effects, and splitting ambiguities drive ongoing research in higher curvature corrections and non-equilibrium gravitational dynamics.

Gravitational entanglement entropy is a central concept at the intersection of quantum field theory, general relativity, and quantum information theory. It quantifies the quantum correlations ("entanglement") between spatial subregions of a system where gravity itself is dynamical, and has emerged as a key diagnostic in the paper of black hole thermodynamics, emergent spacetime, and the statistical origin of gravitational entropy. The gravitational setting dramatically alters both the mathematical definition of entanglement entropy and its physical implications, leading to characteristic area law scaling, connections to Einstein’s equations, the emergence of entropy formulas analogous to black hole entropy, and new phenomena such as edge mode contributions, anomaly-induced effects, and holographic dualities.

1. Area Law Scaling and Gravity/Entanglement Connection

The vacuum of quantum fields—when restricted to a region separated by a boundary—exhibits an entanglement entropy that generically diverges in local QFT. The leading divergence scales as the area $A$ of the boundary divided by the square of a short-distance cutoff $\ell_c^2$: $S \sim \frac{A}{\ell_c^2}$ The presence of gravity fundamentally modifies this behavior. If ultraviolet (UV) physics introduces a natural cutoff $\ell_c$ (often associated with the Planck length $L_p$), the entanglement entropy becomes finite and is given, to leading order, by the Bekenstein–Hawking formula: $$S_{\text{BH}} = \frac{A}{4 L_p^2}, \qquad L_p^2 = \hbar G$$ A thermodynamic argument shows that if local causal horizons carry a finite entropy density obeying

$$\delta S = \frac{\delta Q}{T}, \quad T = \frac{\hbar}{2\pi}$$

with $\delta Q$ the boost energy flux, the requirement that this Clausius relation hold for all local horizons forces the dynamics of the geometry to satisfy Einstein’s equation. Specifically, the Newton constant $G$ is set by the entropy density: $\text{Entropy density} = \frac{1}{4\hbar G}$ Thus, gravity emerges as a consequence of the finiteness and universal area scaling of vacuum entanglement entropy, with $G$ inversely proportional to the entropy density (Jacobson, 2012).

2. Renormalization, Effective Action, and Higher Curvature Contributions

Entanglement entropy in QFTs is UV divergent, but in the gravitational context the same divergences appear in the effective action, enabling their cancellation via renormalization of gravitational couplings. Using the Callan–Wilczek formula, the geometrical definition of entanglement entropy for a spatial region can be recast as the variation of the Euclidean effective action with respect to a conical singularity: $$S_{\text{ent}} = -\lim_{\delta\to 0} \left(\frac{\partial}{\partial\delta} + 1\right) W_e(\delta)$$ After renormalization, the leading (area law) term is universal and matches the Bekenstein–Hawking entropy. Subleading corrections in gravitational theories with higher derivative terms (e.g., Ricci and Riemann squared, cubic invariants) match the Wald formula for black hole entropy, and further corrections reflect quantum state dependence and infrared physics (Cooperman et al., 2013).

In theories with cubic curvatures, the so-called “splitting problem” arises: different prescriptions for separating regular and singular curvature contributions yield distinct functionals for the entanglement entropy, especially in K⁴ terms (extrinsic curvature to the fourth power) (Cáceres et al., 2020). For quadratic/Lovelock gravities, both splittings agree, but in generic cubic gravity, they lead to physically different corrections.

Theory Type	Wald Term	Anomaly Terms (Splitting)	Causal Wedge Inclusion
Einstein/Quadratic	Yes	No ambiguity	Satisfied
Cubic (Minimal Split)	Yes	K⁴ terms; prescription 1	Satisfied
Cubic (Non-Minimal)	Yes	K⁴ terms; prescription 2	Satisfied

For highly symmetric regions (e.g., a disk in AdS), extrinsic curvature vanishes, so all prescriptions coincide and entropy corrections reduce to Wald terms; for less symmetric regions (strips), the splitting ambiguities manifest in the entanglement surface’s higher curvature corrections.

3. Holographic Entanglement and the Emergence of Geometry

The Ryu–Takayanagi (RT) proposal and its generalizations provide a geometric (holographic) prescription for entanglement entropy in theories with gravitational duals: $S_A = \frac{\text{Area}(\widetilde{A})}{4 G_N}$ where $\widetilde{A}$ is an extremal (minimal) co-dimension two surface homologous to the boundary region $A$ in the bulk AdS geometry. In the presence of higher-curvature corrections, Wald and anomaly-like contributions are included via functional derivatives of the action (Cooperman et al., 2013, Cáceres et al., 2020).

Perturbative analyses in holographic CFTs demonstrate that the first law-like relation $\delta S = \delta E$ (where $\delta E$ is the hyperbolic energy) for ball-shaped regions, together with the RT formula and holographic dictionary for the stress tensor, is so constraining that the bulk geometry must satisfy Einstein’s equation linearized about AdS (Lashkari et al., 2013). This provides an explicit mechanism for the emergence of gravitational dynamics from quantum entanglement properties in a CFT, reinforcing the narrative of spacetime geometry as emergent from entanglement.

More generally, in setups with entangled gravitational subsystems or “islands,” the path integral over gravitational saddles can produce nontrivial wormhole geometries (swap wormholes, tidal island formulas), which compute entropy on new joined Cauchy slices interpolating between boundaries, reflecting ER=EPR-type connectedness (Balasubramanian et al., 2021).

4. Edge Modes, Gauge Invariance, and the Algebraic Structure

The definition of entanglement entropy in gauge theories (including gravity) is subtle as the Hilbert space does not naturally tensor factorize due to constraints (like the Gauss law). It is necessary to introduce an extended Hilbert space where additional “edge modes” or “centre degrees of freedom” live on the entangling surface or its boundary. The gravitational path integral formalism must then sum over all embeddings of a model region into the spatial slice, and integrating over diffeomorphisms induces a sum over these edge modes (Balasubramanian et al., 2023).

This process naturally leads to a “generalized entropy” formula: $$S(\rho_a) = \min_{\phi(a) \supset A} \left[ \frac{\text{Area}(\partial\phi(a))}{4 G_N} + S_\text{out}(\phi(a)) \right]$$ where the minimization is over all outward “deformations” of the fixed region $A$ and $S_\text{out}$ denotes the bulk entanglement contribution outside $A$. In free graviton and higher-spin field theories, the universal (logarithmic) part of the Rényi and entanglement entropies can be calculated from the sum over physical superselection sectors labelled by normal components of the curvature, matching the calculation via partition functions on hyperbolic cylinders (e.g., using Harish–Chandra characters) (David et al., 2022).

5. Anomalous Contributions: Gravitational Anomalies and Chiral Entropy

In the presence of gravitational or mixed gauge-gravitational anomalies (e.g., in CFT₂ with $c_L \neq c_R$ or chiral higher-dimensional theories), entanglement entropy acquires anomalous contributions that depend on the choice of reference frame or regulator. Under infinitesimal local boosts, the regulated entropy varies as: $$\delta S = \frac{c_R - c_L}{12} (\delta\chi_1 + \delta\chi_2)$$ where $\delta\chi$ are the local hyperbolic angles at entangling surface points (Iqbal et al., 2015, He et al., 2023). Holographically, such anomalies arise from gravitational Chern–Simons terms in the bulk; the RT minimal surface generalizes to a ribbon whose twist measures the anomalous contribution, and the corresponding extremal problem yields Mathisson–Papapetrou–Dixon equations for spinning particles in AdS₃ (Castro et al., 2014). These contributions are essential for matching CFT entanglement entropy with the black hole entropy in topologically massive gravity and reproduce the parity-odd (boost/momentum) dependence.

6. Dynamical Settings: Collapse, Quenches, and Measurement

During time-dependent processes such as gravitational collapse (as in Oppenheimer–Snyder spacetimes), the entanglement entropy of quantum fields evolves nontrivially. In covariant regularizations suitable for curved, dynamical backgrounds, entropy generation tracks Hawking radiation, exhibits nontrivial scaling and time dependence, and can depart from the area law especially near horizons (Bianchi et al., 2014, Belfiglio et al., 20 Feb 2025). For example, the entropy of regions deep inside collapsing dust can initially decrease, then increase sharply near horizon or singularity formation, and deviate from $S \propto A$ as rapid geometry and field configuration changes “pollute” the area scaling.

For local quenches (excitations) by fields of integer spin $s$, the time evolution of entanglement entropy as the excitation crosses into a spatial region is governed by universal polynomials of degree $2s+1$ in $r = y_0/t$, where $y_0$ is the distance from the entangling surface. For spin-2 (linearized graviton) quenches, one probes curvature densities (via the Kretschmann scalar), and entropy growth is fully determined by the representation under preserved $SO(2)_T \times SO(2)_L$ symmetries (David et al., 2022).

In the context of gravitational wave detections, where direct graviton detection is impractical, measurement-induced bipartite entanglement entropy between spatially separated detectors provides an experimentally meaningful, normalized signature of the quantum nature of gravitational radiation. The entanglement entropy, scaling as a few percent of the mean graviton number, is a robust indicator of non-classicality that avoids the limitations of single-point detection strategies (Jones et al., 23 Nov 2024).

7. Conceptual Implications and Future Directions

Gravitational entanglement entropy acts as a unifying diagnostic across several domains:

• It links quantum field degrees of freedom to the emergence of spacetime dynamics and geometry (gravity as emergent from entanglement).
• It provides the statistical origin of black hole entropy, with the universal area law arising via intrinsic regularization from gravity itself and edge mode counting.
• It signals deep ties between quantum information (e.g., quantum extremal surfaces, mutual information) and the global causal structure of spacetimes, with holography providing precise dualities (e.g., terrestrial holography for finite regions, ER=EPR bridges for entangled universes).
• Anomalies and edge mode ambiguities point toward a nuanced, local dependence of entropy on frame, background fields, and algebraic choices, inviting further investigation into the microscopic substrate of quantum gravity.

While robustly explaining the universality of the area law in gravitational entropy, current frameworks face limitations—especially in capturing corrections beyond Einstein gravity, dealing with ambiguity in region splits linked to gauge invariance, and quantifying entropy during strongly dynamical or non-equilibrium processes. Continued advances in mathematical formalism, holographic dual models, and (potentially) experimental correlates in gravitational-wave detectors are active areas of research.

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Key Equations Referenced Across the Literature

• Entanglement entropy scaling: $S \sim \frac{A}{\ell_c^2}$
• Bekenstein–Hawking entropy: $S_{\text{BH}} = \frac{A}{4 L_p^2}$
• Clausius relation: $\delta S = \delta Q/T$
• Wald formula (higher curvature): $$S = -2\pi \int \frac{\delta \mathcal{L}}{\delta R_{\mu\nu\rho\sigma}} \epsilon_{\mu\nu} \epsilon_{\rho\sigma} \sqrt{\gamma} \, d^{D-2}x$$
• Holographic RT formula: $S_A = \frac{\text{Area}(\widetilde{A})}{4 G_N}$
• Anomaly contribution (2d CFT): $$\delta S = \frac{c_R - c_L}{12} (\delta\chi_1 + \delta\chi_2)$$

References to arXiv preprints:

• (Jacobson, 2012, Cooperman et al., 2013, Lashkari et al., 2013, Neiman, 2013, Karch et al., 2014, Castro et al., 2014, Bianchi et al., 2014, Iqbal et al., 2015, Ma, 2016, Speranza, 2018, Sahakian, 2019, Cáceres et al., 2020, Balasubramanian et al., 2021, David et al., 2022, David et al., 2022, He et al., 2023, Balasubramanian et al., 2023, Jones et al., 23 Nov 2024, Belfiglio et al., 20 Feb 2025).