Gravitational physics uses the word entropy in several specific and distinct senses, but there is no single, universally accepted local definition of gravitational entropy within classical general relativity. In particular settings the concept is well-defined and tightly constrained, while in general spacetimes the idea remains ambiguous and contested.
Black holes and well-defined entropy
The clearest and most robust notion of gravitational entropy applies to black hole horizons. Jacob Bekenstein Hebrew University of Jerusalem identified a proportionality between entropy and horizon area, and Stephen Hawking University of Cambridge showed that black holes radiate with a temperature, giving that area-based quantity bona fide thermodynamic significance in the semiclassical regime. Robert M. Wald University of Chicago extended this by showing how, for diffeomorphism-invariant theories, one can construct an entropy from a Noether-charge associated with the horizon; this provides a precise mathematical prescription for stationary black holes. These results are well-documented, experimentally relevant through astrophysical inferences about compact objects, and central to debates about information loss and quantum gravity.
Entropy beyond horizons: causes and open problems
Outside horizons, proposals diverge. Roger Penrose University of Oxford suggested the Weyl curvature hypothesis, arguing that low initial gravitational entropy corresponds to a small Weyl tensor in the early universe and thereby explains the thermodynamic arrow of time. Ted Jacobson University of Maryland derived Einstein’s equations from local thermodynamic relations, indicating deep links between gravity and entropy, but these derivations are context-dependent and rely on semiclassical assumptions. Attempts to define a local gravitational entropy density using curvature invariants face conceptual and technical problems: general relativity lacks a canonical, local energy density and different curvature scalars capture different physical aspects.
The consequences of these ambiguities are substantial. Without a general definition, cosmological questions about initial conditions, the emergence of structure, and the microscopic origin of gravitational degrees of freedom remain unsettled. Practical implications appear in attempts to quantize gravity and resolve the black hole information paradox: well-defined horizon entropy fits into semiclassical quantum field theory, but a full microscopic account requires a quantum-gravitational framework, as pursued in string theory and loop quantum gravity.
In short, gravitational entropy is well-defined for horizons and in specific thermodynamic derivations but lacks a universally agreed local formulation in classical general relativity. Progress requires input from both mathematical relativity and quantum gravity, and the debate remains tied to foundational questions about time, information, and cosmological initial conditions.