Constancy of Surface Gravity in Gravitational Physics

• Constancy of surface gravity is a fundamental concept defining uniform gravitational acceleration at a horizon, crucial for black hole thermodynamics and stellar structure.
• Empirical techniques, such as asteroseismic calibrations and light curve 'flicker' analyses, enable precise measurements of surface gravity in stars.
• Theoretical frameworks confirm that under stationary and symmetric conditions, energy constraints enforce constant surface gravity, with extensions to dynamic scenarios.

Surface gravity is a fundamental concept in gravitational physics, characterizing the acceleration experienced at the “surface” or horizon of a gravitating body, particularly black holes and stars. The constancy of surface gravity has deep theoretical significance, underpinning the zeroth law of black hole mechanics, governing aspects of stellar structure, entering into thermodynamic analogies, and motivating refined energy inequalities in general relativity and its extensions. While in stationary, highly symmetric spacetimes surface gravity is constant over the horizon, in more dynamical or complex scenarios multiple inequivalent notions arise, necessitating precise definitions and careful consideration of underlying symmetries and contributions from matter, angular momentum, and gravitational radiation.

1. Constancy of Surface Gravity and the Zeroth Law

The zeroth law of black hole mechanics asserts that the surface gravity $\kappa$ is constant over the event horizon of a stationary black hole. In the standard formulation, surface gravity is defined in terms of the inaffinity of the null generators of a Killing horizon: for a Killing vector $\xi^a$ null on the horizon,

$\xi^{b} \nabla_{b} \xi^{a} = \kappa \xi^{a}.$

The constancy follows from the Killing equation and the Einstein field equations, provided the dominant energy condition holds and the horizon is regular. In black hole thermodynamics, this result is essential, as the Hawking temperature $T_H$ is proportional to $\kappa$, establishing thermal equilibrium at the horizon. In higher curvature generalizations such as Lanczos–Lovelock gravity, rigorous proofs show the zeroth law remains valid: the transverse derivatives of surface gravity vanish, provided a smooth general-relativity limit and suitable energy conditions, so $\kappa$ is uniform across the horizon (Ghosh et al., 2020).

For black hole binaries with helical symmetry (e.g., corotating binaries in quasi-equilibrium), each individual horizon possesses its own Killing generator, and the zeroth law applies separately: numerically, the surface gravity is found to be constant over each horizon up to sub-percent deviations, even in strong-field regimes (Tiec et al., 2017). This is an explicit realization that the symmetry assumptions guarantee local equilibrium, a prerequisite for the extension of the laws of black hole mechanics to multi-body and dynamical systems.

2. Dimensionless Formulation and Physical Measurability

A recurring theme is that only dimensionless combinations of constants are physically meaningful, especially in the context of possible time- or space-variation of “constants.” The paper (Moss et al., 2010) emphasizes that variations in Newton’s constant $G$ are only measurable when expressed via dimensionless combinations such as the gravitational fine structure constant,

$\alpha_{\rm g} = \frac{G m_{\rm p}^2}{\hbar c},$

where $m_{\rm p}$ is the proton mass. This focus is crucial because $G$ by itself is a dimensional quantity subject to arbitrary rescalings of units. Only changes in dimensionless parameters such as $\alpha_{\rm g}$ have invariant operational meaning—affecting, for instance, stellar masses, luminosities, and characteristic length scales. Thus, any purported “constancy” of surface gravity must be interpreted in terms of such dimensionless combinations, ensuring that constraints are physically robust and not artifacts of unit conventions.

3. Stationary versus Dynamical Definitions

In stationary, spherically symmetric spacetimes (e.g., Schwarzschild, Kerr), all robust definitions (Killing, Kodama-Hayward, geodesic deviation–based, etc.) of surface gravity converge to a common, constant value on the horizon. However, in dynamical or non-stationary contexts, such as in stellar collapse or black hole evaporation, various definitions can yield different, even time-dependent results.

For spherically symmetric collapsing stars (Sadeghi et al., 12 Feb 2024), the Killing definition remains valid only outside the (static) star. Inside the star—describable by a Friedmann–Robertson–Walker metric in Painlevé–Gullstrand coordinates—definitions such as Hayward’s (based on the Kodama vector), Nielsen–Visser’s (via locally defined mass functions), and Fodor’s (using affinely normalized null vectors) diverge. Crucially, at the moment when the star’s surface meets the horizon, Fodor’s approach allows tuning the normalization so that the interior and exterior surface gravity match, providing a bridge between static and dynamic sectors. In these dynamical scenarios, the extremality condition (e.g., vanishing expansion derivative $n^a\nabla_a\theta_\ell=0$) does not necessarily result in $\kappa=0$ for all definitions.

A similar situation arises for generalized notions of surface gravity based on tidal acceleration (Dahal, 2023): by integrating the tidal acceleration (related to Riemann tensor components) from the horizon outward, one obtains definitions that coincide with traditional Killing values in static spacetimes, while in slowly evolving settings (e.g., Vaidya spacetime) the dynamical corrections are subleading, preserving approximate constancy of surface gravity on the horizon.

4. Relationship to Energy Inequalities and Quasi-local Surfaces

Refined geometric inequalities for quasi-local surfaces—such as loosely trapped surfaces (LTS) and attractive gravity probe surfaces (AGPS)—encode the relationship between surface area, mass, angular momentum, and gravitational wave content, and directly relate to the constancy or variability of an effective “surface gravity” (Lee et al., 2021). The conditions under which the inequalities reduce to Penrose-like bounds or minimize additional correction terms are those where the effective gravitational pull (often measured by the mean curvature $k$) is spatially uniform, precisely mirroring the situation on a stationary black hole's horizon.

When angular momentum, gravitational waves, and matter contributions are negligible, the surfaces exhibit near-constant effective surface gravity, and the inequalities attain their tightest forms. Deviations from constancy are reflected in additional terms—quadratic in angular momentum or involving local gravitational wave energy densities—that relax the maximal allowed area for a given mass. Thus, the refined inequalities serve as sensitive probes of the departure from constancy in the gravitational field, both theoretically and in numerical relativity studies.

5. Observational and Empirical Aspects

For astrophysical objects, particularly stars, the direct measurement of surface gravity is complex. Photometric and spectroscopic techniques are limited in precision, but recent large-scale analyses leveraging high-cadence light curves—e.g., “flicker” metrics encoding RMS brightness variations on short time scales—enable precise empirical determination of surface gravity (Bastien et al., 2013). There exists a tight correlation, calibrated against asteroseismic standards, permitting the inference of surface gravity within $<25\%$ uncertainty for Sun-like stars. Notably, in solar data, this approach demonstrates that surface gravity remains constant across solar cycles, validating the theoretical expectation embedded in the constancy of surface gravity—at least for relatively inactive stars and specific timescales.

6. Theoretical Generalizations and Thermodynamic Implications

In modified gravity theories, particularly those with higher curvature terms (such as Gauss–Bonnet or more general Lovelock terms), the constancy of surface gravity remains intact on stationary Killing horizons provided the field equations have a smooth general-relativity limit and satisfy appropriate energy conditions (Ghosh et al., 2020). The proof techniques involve projecting derivatives of surface gravity onto the transverse horizon directions and showing that the resulting equations, constrained by the geometric structure and energy conditions, force vanishing of these derivatives. As a consequence, the Hawking temperature remains uniform across the horizon, maintaining the thermodynamical analogy embodied by the zeroth law.

Further, on dynamical trapping horizons (which may not coincide with event horizons in general spacetimes), the surface gravity can be defined in several non-equivalent ways, leading to differing contributions in the first law of horizon thermodynamics (Sadeghi et al., 12 Feb 2024). Thus, operational constancy and thermodynamic interpretation are tightly linked but dependent on the chosen geometric and physical framework.

7. Summary and Significance

The constancy of surface gravity represents a unifying principle across gravitational physics, from black hole thermodynamics to stellar structure and numerical relativity. Its precise operational meaning relies on symmetry, stationarity, and the careful use of dimensionless quantities. With the advent of new measurement techniques and theoretical generalizations, the paper of the extent and limitations of this constancy remains a critical area for both foundational theory and empirical application, as reflected in the broad swath of contemporary research spanning relativistic thermodynamics, black hole dynamics, and astrophysical phenomenology.