Hayward Mass Function in Regular Black Holes

• Hayward Mass Function is a covariantly defined measure of quasilocal gravitational energy in regular black hole spacetimes.
• It interpolates between a classical Schwarzschild profile at large radii and a regular de Sitter-like core at small radii to avoid curvature singularities.
• The function is critical for analyzing dynamical horizons, mass inflation under perturbations, and extends to scalar–tensor and f(R) gravity models.

The Hayward Mass Function is a central construct in the study of regular (non-singular) black hole spacetimes and quasilocal gravitational energy. It provides a prescription both for avoiding curvature singularities within black holes and for precisely formulating the concept of energy contained within a given region, especially within the frameworks of general relativity and extended gravity theories. The Hayward mass finds crucial application not only in the analysis of dynamical horizons and black hole thermodynamics, but also in the generalization of energy concepts to scalar–tensor and $f(R)$ gravity models.

1. Quasilocal Mass: Hawking–Hayward Formalism and Its Spherical Specialization

The Hawking–Hayward quasilocal mass, sometimes referred to simply as the Hayward mass, is a covariantly defined quantity associated to a closed, spacelike 2-surface $S$ within a spacetime $(M, g_{ab})$. Its rigorous definition is

$$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$

where $A$ is the area of $S$, $\mu$ the induced volume element, $\mathcal{R}$ the Ricci scalar of $S$, $\theta_{(\pm)}$ the expansions of outgoing/ingoing null congruences, $S$0 their shear tensors, and $S$1 the anholonomicity (twist) vector. This formulation robustly incorporates both matter and gravitational field contributions within $S$2, is gauge-invariant under spatial slicing, and underpins the definition of dynamical horizon mass (Prain et al., 2015).

When restricted to spherically symmetric situations, the Hawking–Hayward mass reduces exactly to the Misner–Sharp–Hernandez (MSH) mass,

$S$3

where $S$4 is the areal radius. This equivalence is central to analytical and numerical relativity.

2. The Hayward Regular Black Hole and Its Mass Function

The Hayward regular black hole is a static, spherically symmetric, non-singular solution whose metric in ingoing Eddington–Finkelstein coordinates takes the form

$S$5

with

$S$6

where $S$7 is the ADM mass at infinity, and $S$8 is a small positive parameter controlling the scale of the regular core (Iofa, 2022). As $S$9, $(M, g_{ab})$0, ensuring avoidance of the $(M, g_{ab})$1 curvature singularity. At $(M, g_{ab})$2, $(M, g_{ab})$3, and the classical Schwarzschild regime is recovered.

This particular mass function is constructed to interpolate smoothly between the usual Schwarzschild profile at large $(M, g_{ab})$4 and a regular, de Sitter-like core at small $(M, g_{ab})$5, making it a prototypical example of a regular black hole solution.

3. Hayward Mass Function under Dynamical Perturbations: Mass Inflation and Inner Horizon Analysis

In the context of dynamical perturbations (such as scalar tails or null fluxes), the behavior of the Hayward mass function near inner (Cauchy) horizons is critical for understanding the stability of regular black holes. A central result derives from the analysis of Ori’s mass inflation scenario: when both ingoing (Price-law) tails and outgoing null shells are present, the mass function on the inner region $(M, g_{ab})$6 obeys a quadratic ODE

$(M, g_{ab})$7

with $(M, g_{ab})$8 and explicit constants $(M, g_{ab})$9. Unlike the Reissner–Nordström case, where mass inflation leads to $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$0, in the Hayward black hole the solution approaches a finite (often negative) asymptote $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$1 at late times, indicating the absence of Ori-type mass inflation in this setting. The quadratic structure of the ODE, directly stemming from the nonlinear core modification in $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$2, regularizes the mass function (Iofa, 2022).

However, the physical interpretation of this finiteness is nuanced; $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$3 itself is not a directly measurable invariant, and blueshift instabilities for energy fluxes at the Cauchy horizon persist, signaling ongoing loss of predictability inside the regular black hole.

4. Generalization to Scalar–Tensor Gravity: The Modified Quasilocal Mass

In scalar–tensor gravities (e.g., Brans–Dicke theory and $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$4 models), the gravitational "constant” becomes a field $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$5, requiring a modification of the original (general-relativistic) mass definition. The generalization of the Hawking–Hayward mass in the Jordan frame (for a round sphere of radius $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$6 in vacuum) is given by

$$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$7

where $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$8 is the Misner–Sharp–Hernandez mass, $$M_H(S) = \frac{1}{16\pi} \sqrt{\frac{A}{16\pi}} \int_S \mu \left[ \mathcal{R} + \theta_{(+)} \theta_{(-)} - \frac{1}{2} \sigma^{(+)}_{ab} \sigma_{(-)}^{ab} - 2 \omega_a \omega^a \right]$$9 the scalar–tensor coupling function, and $A$0 the radial normal (Faraoni et al., 2019). Each term encodes distinct physical contributions: rescaled GR mass, local scalar kinetic energy, and boundary variations of $A$1.

For constant $A$2, this reduces exactly to the standard Hawking–Hayward (Misner–Sharp) mass, corroborating its role as a consistent extension. Furthermore, in asymptotically flat, stationary solutions, the quasilocal mass at infinity $A$3 matches the mass monopole derived from independent multipole expansions of the metric and scalar field, reinforcing the definition's physical validity.

5. Behavior under Conformal Transformations

The transformation properties of the Hawking–Hayward (and thus Hayward) mass under conformal rescalings $A$4 are crucial in alternate gravity theories and frame changes. The mass transforms as

$A$5

with all geometric quantities appropriately mapped (Prain et al., 2015). For constant $A$6, the transformation reduces to a simple scaling. However, for spatially varying $A$7, additional terms appear corresponding to the energy contribution of the conformal factor itself—key for $A$8 and scalar-tensor models where frame transformations are operationally significant.

6. Physical Significance and Limitations

The Hayward mass function and its generalizations provide a robust, covariant measure of energy content in a spacetime region, incorporating both matter and pure gravitational contributions, and admit consistent extensions to dynamical, non-spherical, and scalar–tensor settings (Prain et al., 2015, Faraoni et al., 2019). The regularizing role of the Hayward mass profile in black hole interiors crucially alters the nonlinear evolution under perturbations, moderating but not wholly eliminating blueshift instabilities at Cauchy horizons (Iofa, 2022).

A key limitation is that some mass function parameters, particularly inside regularized black holes, lack invariant measurement and can be sensitive to coordinate choices or matching prescriptions (e.g., in Ori- or DTR-type analyses). While the self-regularizing core prohibits unbounded growth of the internal mass function, the Cauchy horizon retains the feature of divergent fluxes for infalling observers, preserving a locus of instability.

7. Comparative Table: Key Aspects of the Hayward Mass Function

Context	Core Formula for Mass Function	Notable Feature
Hayward regular black hole	$A$9	Nonsingular core, $S$0 as $S$1
Hawking–Hayward (quasilocal)	See Section 1 above	Covariant, geometric, includes matter & gravity
Scalar–tensor generalization	See Section 4 above	Scalar kinetic and boundary terms included

This framework continues to inform the deeper understanding of gravitational energy, black hole interior dynamics, and the landscape of alternative theories of gravity.