Landauer entropy of spacetime

Abstract: Based on Landauer's principle, we provide a geometrical definition for the entropy of a given static, spherically symmetric spacetime. Considering a congruence of geodesics across a surface, one defines the entropy of a congruence as the surface integral of the entropy of the constituent geodesics. Under certain mild assumptions, we establish a second law for the entropy function thus defined (Landauer entropy), and relate it to Bekenstein-Hawking entropy.

• The paper introduces a novel gravitational entropy defined by Landauer's principle, linking entropy increase to the spacetime metric.
• It demonstrates that the proposed entropy function is monotonic along outgoing radial geodesics in static, spherically symmetric spacetimes.
• It establishes a direct proportionality between Landauer entropy and Bekenstein-Hawking entropy, providing new insights into gravitational thermodynamics.

Landauer Entropy as a Geometric Construct for Spacetime

Introduction

The interplay between gravity, thermodynamics, and information theory has long motivated efforts to elucidate the microscopic and macroscopic meaning of gravitational entropy. While the Bekenstein-Hawking prescription connects black hole entropy to horizon area, various approaches seek to generalize or fundament the notion of entropy for arbitrary spacetimes. "Landauer entropy of spacetime" (2605.22172) presents a novel definition of gravitational entropy rooted in Landauer’s principle, leveraging purely geometric and macroscopic arguments, and explores its properties across static, spherically symmetric spacetimes. This essay provides a technical overview of the paper’s formulation, results, implications, and theoretical context.

Landauer’s Principle and Entropic Gravity

Landauer’s principle postulates that erasing a single bit of information at temperature $T$ mandates an energy cost of at least $k_B T \ln 2$, producing a corresponding minimal entropy increase in the environment. This principle offers a direct bridge between logical irreversibility and physical irreversibility through thermodynamic entropy, and has been previously related to black hole thermodynamics.

Extending Landauer’s reasoning to gravitational contexts, the authors note that in a weak gravitational field, the local Tolman relation modifies the effective temperature, implying that the entropy contribution from information processing acquires a dependence on the spacetime metric. This observation enables the proposal of an entropy function for static, spherically symmetric spacetimes, constructed from the lapse function $f(r)$ appearing in the standard static metric form:

$$ds^2 = -f^2(r)\,dt^2 + \frac{dr^2}{f^2(r)} + r^2 d\Omega_2^2$$

Landauer entropy for a radial geodesic is then defined as $S(r) = f(r)$, with appropriate sign choices to ensure monotonicity along outgoing trajectories.

General Definition and Properties of Landauer Entropy

A central contribution is the construction of a Landauer entropy for congruences of radial geodesics through a surface $\mathbb{S}$:

$${\cal S}[\mathbb{S}] = \frac{1}{4G\hbar}\int_{\mathbb{S}} dA\, f(r)$$

For spheres of radius $r$, this reduces to ${\cal S}(r) = (\pi r^2/G\hbar) f(r)$, formally paralleling the Bekenstein-Hawking entropy $S_{BH} = A/(4G\hbar)$ while differing by the multiplicative factor $k_B T \ln 2$0 derived directly from the metric.

The authors rigorously demonstrate that, under mild conditions (i.e., non-vanishing derivative $k_B T \ln 2$1), the Landauer entropy is monotonic along outward, radial geodesics—ensuring a second law–type behavior. In cases where $k_B T \ln 2$2 changes sign, the definition is adjusted piecewise (including a sign flip) to maintain monotonicity. This general method produces entropy functions that always align with the required thermodynamic arrow of time along geodesic flows.

Application to Prototypical Spacetimes

The general formulation is applied to several key spacetime backgrounds:

• Spherical Rindler: Near static horizons, the entropy function scales as $k_B T \ln 2$3, manifestly monotonic for $k_B T \ln 2$4.
• Schwarzschild: $k_B T \ln 2$5 increases monotonically for $k_B T \ln 2$6.
• de Sitter: $k_B T \ln 2$7 (with sign flip), also monotonic within the causal patch.
• Schwarzschild–de Sitter: Piecewise sign assignments enforce monotonicity between cosmological and event horizons.
• AdS and Schwarzschild–AdS: Both admit monotonic entropy functions with $k_B T \ln 2$8 (plus Schwarzschild correction).

A key result is that the Landauer congruence entropy for a sphere is always proportional to the corresponding Bekenstein-Hawking entropy by $k_B T \ln 2$9, which is constant on the surface. This proportionality, rooted in geometry, provides an alternative to the microstate-centric statistical mechanical derivations.

Theoretical Implications

The construction connects the thermodynamic arrow of time, entropy increase, and gravitational field geometry via an information-theoretic principle, absent any specific microscopic model for spacetime degrees of freedom. This suggests that macroscopic constraints (here, monotonicity from the second law) may suffice to define meaningful geometric entropy functions for large classes of spacetimes. Notably, the Landauer entropy is not covariant under general diffeomorphisms, reflecting its reliance on static slicing—consistent with the observer dependence of entropy in frameworks like Unruh and Rindler physics.

The claim that Landauer entropy is always monotonic along outgoing radial geodesics, for wide classes of spherically symmetric spacetimes, is a strong assertion supported by explicit case-by-case construction. Moreover, the explicit proportionality to Bekenstein-Hawking entropy (which itself lacks a universally accepted microscopic foundation) offers a new pathway toward a generalized geometric entropy for arbitrary horizons and possibly other spacetime regions.

Future Directions

This macroscopic construction opens several lines of inquiry:

• Extension beyond Spherical Symmetry: The analysis leverages spherical symmetry for tractability. Generalization to spacetimes with less symmetry, dynamical metrics, or inclusion of matter fields may require new techniques or additional principles.
• Covariant Formulation: The lack of diffeomorphism invariance points to the need for a more covariant or observer-independent entropy notion, paralleling the broader efforts in defining gravitational entropy.
• Quantum Gravity and Information Bounds: The explicit dependence on Landauer’s principle, which is rooted in quantum statistical mechanics, may have implications for emergent gravity scenarios and holographic approaches.
• Relation to Black Hole Information Paradox: Since Landauer entropy ties information processing to geometric constructs, it could provide new tools for analyzing information recovery and loss in gravitational collapse or evaporation.

Conclusion

"Landauer entropy of spacetime" (2605.22172) provides a geometric, macroscopic, and information-theoretic definition of spacetime entropy for static, spherically symmetric backgrounds, grounded in Landauer's principle and enforcing a monotonic second law behavior. The resulting Landauer entropy is directly related to the Bekenstein-Hawking area law by a proportionality constant dependent on the metric. This approach bypasses assumptions about microscopic degrees of freedom and offers a robust operational definition of gravitational entropy, with direct implications for thermodynamics in curved spacetimes and the ongoing effort to connect information theory with gravity.