Misner-Sharp Mass Function

• Misner-Sharp mass function is a quasi-local definition of gravitational energy that encodes the energy within spherical boundaries via geometric invariants.
• It recasts the Einstein field equations into mass evolution laws, establishing a direct link with thermodynamic identities and applications in black hole and cosmological contexts.
• Generalizations in modified gravity theories introduce additional masslike functions that are crucial for ensuring energy conservation and verifying horizon thermodynamics.

The Misner-Sharp mass function is a quasi-local definition of energy in spherically symmetric spacetimes, fundamental for connecting geometry, thermodynamics, and gravitational dynamics. It originally encapsulates the internal energy within a sphere of areal radius in Einstein gravity, but its generalization enables robust formulations of energy in modified gravity theories, cosmology, dynamical collapse, and black hole thermodynamics. The formalism allows the gravitational field equations to be recast into compact mass evolution laws, supporting integration techniques, consistency checks, and thermodynamic identities such as the first law.

1. Mathematical Definition and Geometric Context

In a general spherically symmetric spacetime with the line element

$$ds^2 = -A^2(r,t) dt^2 + B^2(r,t) dr^2 + Y^2(r,t) d\Omega^2,$$

the Misner-Sharp (MS) mass $M$ is defined through the invariants of the geometry as

$$M = \frac{Y}{2G}\left(1 - g^{ab}\partial_a Y \partial_b Y\right)$$

where $Y(r, t)$ is the area radius and $G$ is Newton's gravitational constant. This function appears naturally when expressing the Einstein equations as: $$M_{,a} = 4\pi Y^2 (T^b_a - \delta^b_a T) \partial_b Y,$$ which asserts that changes in the quasi-local energy are sourced only by the energy-momentum crossing the sphere $Y$. At the apparent horizon (AH), defined by

$Y_A = \frac{1}{\sqrt{H^2 + k/a^2}},$

$M$ yields the total energy inside the cosmological horizon.

2. Unified First Law and Thermodynamic Connections

The MS mass function supports formulation of the unified first law: $dE = A \Psi_a dx^a + W dV$ where $E$ is the enclosed energy, $A$ is the surface area, $\Psi_a$ is the energy-supply vector, $W$ is the work density, and $V$ is volume. In the presence of an apparent horizon, the Friedmann equations project directly onto the first law: $dE = T_A dS_A$ with horizon temperature

$T_A = \frac{1}{2\pi Y_A},$

and entropy

$S_A = \frac{\pi Y_A^2}{G}$

(see (0704.0793)). This formalism shows that the local cosmological dynamical equations encode a thermodynamic identity at the horizon, suggesting a profound geometric-thermodynamic correspondence.

3. Masslike Functions in Modified Gravity Theories

In extensions such as scalar-tensor, Lovelock, $f(R)$, massive gravity, and $f(R,\mathcal{G})$ theories, the standard MS mass is modified by additional degrees of freedom:

• Scalar-Tensor Theories: The masslike function is generalized (e.g., in Brans–Dicke theory) as

$$M = \frac{\phi\, Y}{2}\Bigl(1 + g^{ab}\partial_a Y\,\partial_b \phi\Bigr),$$

where $\phi$ is the scalar field (0704.0793).

• $f(R)$ Gravity: The generalized MS mass reads

$$M = f'(R)\frac{Y}{2}\left(1 - g^{ab}\partial_a Y \partial_b Y\right),$$

with horizon entropy $S_A = \frac{A f'(R)}{4G}$ (Zhang et al., 2014).

• Massive Gravity: The MS mass function in the dRGT model contains terms proportional to the graviton mass squared and reference metric parameters, preserving equilibrium thermodynamics (Hu et al., 2015, Hu et al., 2016).
• $f(R,\mathcal{G})$ Gravity: Includes additional contributions involving the Gauss–Bonnet invariant $\mathcal{G}$ and its derivatives, significantly altering both static and dynamic expressions for $M$ (Akbarieh et al., 4 Jun 2025).

In all cases, the equilibrium form $dE = T_A dS_A$ at the AH is recovered only when the masslike function is properly chosen to account for extra gravitational fields; otherwise, nonequilibrium corrections are necessary.

4. Applications: Cosmology, Black Holes, and Collapse

Cosmology

In Friedmann–Robertson–Walker (FRW) backgrounds, the MS mass provides the quasi-local energy inside the causal horizon: $E_{MS}(R_g) = \frac{R_g}{2G},$ whose conservation selects de Sitter solutions and relates global curvature energy to matter energy. This conservation law naturally accommodates the cosmological constant and captures gravitational time dilation effects that render open de Sitter cosmologies spatially flat, reconciling with $\Lambda$CDM (Telkamp, 2017, Chu et al., 19 Feb 2025).

Black Hole Thermodynamics

For black hole solutions (Schwarzschild, Schwarzschild–de Sitter, generalized Vaidya), the MS mass appears in the derivation of the Smarr formula and accounts for both local and global aspects of gravitational energy. It supports thermodynamic identities involving the horizon areas, surface gravities, and (in SdS) the negative pressure from cosmological constant (Bhattacharya et al., 2013). In massive gravity, the MS mass is essential for establishing the equilibrium and Clausius relation at the horizon (Hu et al., 2015, Hu et al., 2016).

Gravitational Collapse

During spherical collapse (e.g., scalar field collapse), the MS mass tracks local energy, diverging near the singularity (“mass inflation”) due to scalar backreaction, even though it does not represent global ADM/Bondi mass (Guo et al., 2015). Its behavior signals the local intensity of dynamics rather than the total black hole mass observable at infinity.

Primordial Black Hole (PBH) Formation

The MS mass formalism underpins modern simulations of PBH formation: in numerical codes, it appears in dynamical evolution equations and in horizon identification. Handling non-monotonic type-II fluctuations requires auxiliary variables (trace of extrinsic curvature) to regularize computation, retaining correct threshold and scaling law physics for PBH production (Escrivà, 8 Apr 2025).

5. Generalizations and Alternative Currents

Almost-Killing Currents: In dynamical or non-symmetric spacetimes lacking exact Killing symmetries, the MS prescription is generalized via almost-Killing vectors, yielding mass functions satisfying elliptic–hyperbolic systems, providing more realistic energy quantification in time-dependent or perturbed backgrounds (Ruiz et al., 2013).

Alternative Mass Definitions: The geometric and hydrodynamic (material) definitions of the MS mass behave differently under conformal transformations:

• The geometric definition inherits an induced energy density and transforms anomalously under conformal mappings.
• The hydrodynamic definition recovers the expected $m \mapsto m/\Omega$ scaling for physical mass under conformal rescaling, restoring physical consistency, especially in collapse scenarios and scalar–tensor gravity (Hammad, 2016).

6. Pathologies, Asymptotics, and Quasilocal Mass

Mass anomalies (negative, divergent, or zero MS/Hernandez mass) signal unphysical geometric structures, arising when asymptotic flatness fails in quantum-corrected black holes and other models. The proper asymptotic scaling, notably the Newtonian $1/r$ monopole term, is essential for physical isolation; its failure is reliably diagnosed by the MS mass behavior (Faraoni et al., 2020).

7. Thermodynamic Phase Structure in Extended Gravity

In quasi-topological cosmology, the MS energy equals $\rho V$ inside the apparent horizon, and the unified first law naturally yields thermodynamic variables such as pressure, temperature, and volume. This supports the derivation of an equation of state, P–V criticality, and phase transitions akin to Van der Waals fluids, with conventional mean-field exponents ($\alpha = 0, \beta = 1/2, \gamma = 1, \delta = 3$), providing a correspondence between microscopic gravity theory and macroscopic thermodynamics (Chu et al., 19 Feb 2025).

8. Summary and Physical Implications

The Misner-Sharp mass function and its generalizations serve as a bridge between gravitational field equations and thermodynamic laws, underpin both equilibrium and nonequilibrium thermodynamics at horizons, and enable computation of exact solutions, numerical simulations, and consistency checks in diverse gravitational theories. Its proper use and modification are critical for defining energy localization, interpreting cosmological dynamics, and diagnosing geometric validity in black hole and collapse scenarios. The MS mass, either alone or through adapted masslike functions, is indispensable in any attempt to unify gravitational dynamics with thermodynamic principles in general relativity and its extensions.