Static Patch Holography in Gravitational Thermodynamics

• Static Patch Holography is a framework that defines gravitational dynamics through thermodynamic behavior on holographic screens in static spacetimes.
• It leverages equipotential surfaces to compute key observables like energy, temperature, and entropy, supporting emergent gravity models.
• The approach extends to diverse static spacetimes, including de Sitter, wormholes, and multi-black-hole systems, unifying gravitational and thermodynamic insights.

Static patch holography describes the paradigm in gravitational and quantum field theory wherein the causal region (the "static patch") accessible to an inertial observer in a static or stationary spacetime—most notably, de Sitter space—is treated as the fundamental domain for holographic duality and emergent thermodynamic behavior. Unlike global spacetime holography as in AdS/CFT, this approach focuses on codimension-one surfaces ("holographic screens") within the patch which encode gravitational, thermodynamic, and quantum information, often motivated by entropic or thermodynamic analogies. Key theoretical developments include the recasting of gravity as an entropic force, the identification of horizon thermodynamics, and the emergence of static-patch localized dual quantum systems with rich symmetry (conformal, supersymmetric, and algebraic) structure.

1. Entropic Force, Holographic Screens, and Verlinde's Principle

In Verlinde's entropic gravity framework, gravity emerges from changes in entropy of the information content on equipotential surfaces, termed "holographic screens" (Konoplya, 2010). For a static spacetime, these screens coincide with loci of constant generalized potential

$$\varphi = \frac{1}{2}\log(-g^{\alpha\beta}\xi_\alpha\xi_\beta),$$

where $\xi_\alpha$ is a timelike Killing vector. The entropic force relation reads

$F\Delta x = T\Delta S,$

with $\Delta S = 2\pi k_B$ and displacement scale $\Delta x = \hbar/(mc)$. Thermodynamic quantities on a screen are directly computed: the acceleration is

$a^\alpha = -g^{\alpha\beta}\nabla_\beta\varphi,$

and the Unruh-Verlinde temperature is

$$T = \frac{\hbar}{2\pi}e^{\varphi}n^\alpha\nabla_\alpha\varphi,$$

where $n^\alpha$ is the unit normal to the surface. These constructs provide the foundation for static patch holography—gravitational and thermodynamic observables are encoded on the equipotential screen.

2. Geometrical and Thermodynamic Character of Holographic Screens

Equipotential surfaces are not merely geometric objects; in static patch holography they manifest the full thermodynamic content of the spacetime. The energy associated to a screen is given by

$$E = \frac{1}{4\pi}\oint e^{\varphi}\nabla\varphi\cdot dA,$$

where the integral extends over the screen. This thermodynamic assignment generalizes naturally to all stationary, static, or otherwise symmetrical solutions (spherical, axial, multi-center configurations), supporting the concept that gravitational observables (energy, temperature, acceleration) are localized and computable entirely from screen data. For traversable wormhole metrics, the Unruh temperature and acceleration vanish at the throat, indicating thermodynamically neutral surfaces; in multi-black-hole solutions (Majumdar–Papapetrou, dilatonic), temperature minima appear between horizon sites, directly manifesting the "balance" points of gravitational information.

3. Unruh-Verlinde Temperature and Thermodynamic Holography

The Unruh-Verlinde temperature, extending the horizon temperature concept to arbitrary static patches, takes the general form, for spherically symmetric metrics,

$$T(r) = \frac{\hbar}{4\pi}\frac{A'(r)}{\sqrt{A(r)B(r)}},$$

where $ds^2 = -A(r)dt^2 + B(r)dr^2 + \cdots$. In Morris-Thorne wormhole geometries,

$$T(r) = \frac{\hbar}{2\pi}\Phi'(r)e^{\Phi(r)}\sqrt{1-b(r)/r},$$

vanishes at $r = b_0$ (the throat), and verified to coincide with the standard Hawking temperature for black hole event horizons (e.g., $T_H = 1/(8\pi M)$ for Schwarzschild solutions). In more general spacetimes, e.g., Ernst–Schwarzschild black holes immersed in magnetic fields, the temperature profile adapts to deformations in the equipotential surfaces yielded by extra fields—leading to angular dependence and low-temperature zones.

4. Static Patch Solutions: Spherically Symmetric Systems, Wormholes, Black Holes

The formalism of static patch holography encompasses several classes of static spacetimes:

• Generic Spherical Solutions: All thermodynamic and geometric quantities defined above extend to arbitrary $A(r),B(r)$ metrics.
• Traversable Wormholes: For metrics $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/(1-b(r)/r) + r^2d\Omega^2$$, with shape and redshift functions controlling the screen geometry, symmetry, and thermodynamics.
• Magnetized Black Holes: Ernst–Schwarzschild solutions, where conformal symmetry is broken and equipotential screens become distorted; yet, established temperature matching at horizons persists.
• Multi-Black-Hole Equilibrium: For metrics $ds^2 = -U^{-2}dt^2 + U^2(dx^i dx^i)$ with harmonic $U$, equipotential surfaces again encode full gravitational and thermodynamic content, and isotherms reveal "balanced zones" in direct analogy to energy surfaces of harmonic superposition.

5. Thermodynamic Interpretation and Informational Emergence

A significant development is the interpretation that gravitational interaction is a statistical effect—a manifestation of entropy gradients encoded on lower-dimensional screens. The equipotential ("holographic") surface thermodynamics—including the equipartition formula for energy, temperature profiles, and entropy assignments—demonstrate that the macroscopic gravitational field is in fact an emergent thermodynamic phase, underpinned by the holographic storage and coarse-graining of information. For wormholes, the zero temperature at the throat signals maximal coarse-graining (i.e., quantum erased information), paralleling the maximum entropy horizon structure at black holes. In multi-black-hole systems, temperature minima are correlated with balanced entropic flows.

6. Physical Implications and Generalizations

The results collectively reinforce the static patch holography paradigm: gravitational degrees of freedom are localized and quantifiable on equipotential screens, with thermodynamic, informational, and geometric character fully interlinked. The approach generalizes to arbitrary static spacetimes (including wormholes and multi-center black hole configurations), supports the correspondence between local acceleration and Unruh temperature, and provides a rigorous basis for interpreting gravity as emergent from the entropy and energy content of holographic surfaces.

The structure is robust under additional field content—magnetic, dilatonic, etc.—with the core holographic thermodynamic assignment persisting. The correspondence between geometry (metric data), thermodynamics (temperature, energy, entropy), and information content (coarse-grained screen data) is exact and foundational in static patch holography.

7. Summary Table: Key Quantities in Static Patch Holography

Quantity	Formula	Physical Interpretation
Equipotential	$$\varphi = \frac{1}{2}\log(-g^{\alpha\beta}\xi_\alpha\xi_\beta)$$	Defines holographic screen locality
Unruh-Verlinde Temp.	$$T = \frac{\hbar}{2\pi}e^{\varphi}n^\alpha\nabla_\alpha\varphi$$	Local screen temperature via acceleration
Energy (screen)	$$E = \frac{1}{4\pi}\oint e^\varphi\nabla\varphi\cdot dA$$	Equipartition on screen

These relations specify, for any static spacetime, the complete thermodynamic assignment to the holographic screen and define the operational meaning of static patch holography.

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Static patch holography thus offers a precise and thermodynamically consistent encoding of gravitational dynamics, applicable to a wide host of static or stationary space-times. The mathematical framework developed here provides tools for further extension to quantum, semiclassical, or emergent gravitational regimes, always grounded in the informational content of the holographic screen.