Linear Effective MSH Mass Function

• The linear effective Misner–Sharp–Hernandez mass function is a quasi-local mass model defined by a strictly linear relation with the areal radius, primarily used in quantum dust core models.
• It arises from the spherical specialization of the Hawking–Hayward energy, linking geometric foundations with effective mass perturbations in primordial-black-hole collapse and modified-gravity contexts.
• The linear profile not only helps regularize central singularities and control horizon properties but also guides refined matching techniques between interior quantum models and exterior Schwarzschild solutions.

The linear effective Misner–Sharp–Hernandez mass function is not a universally standardized term in the quasi-local mass literature. Its most explicit current use is in a quantum model of black-hole interiors, where the Misner–Sharp–Hernandez mass inside a dust core is taken to grow linearly with areal radius, $m(r)\simeq 2r/(3G)$, thereby defining an interior effective mass profile hidden by the event horizon (Bambagiotti et al., 1 Sep 2025). More broadly, the expression points to a family of related constructions in spherical symmetry: the Misner–Sharp–Hernandez mass itself as the spherical limit of the Hawking–Hayward quasi-local energy, linearized mass perturbations in primordial-black-hole collapse, and generalized or transformed quasi-local masses in modified-gravity and mapped spacetimes (Prain et al., 2015).

1. Geometric and quasi-local basis

In the modern quasi-local framework, the Misner–Sharp–Hernandez mass is the preferred spherical-symmetry specialization of the Hawking–Hayward energy. For a compact spacelike orientable 2-surface $S$, the Hawking–Hayward quasi-local energy is defined by

$$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$

and in spherical symmetry this reduces to the Misner–Sharp–Hernandez mass (Prain et al., 2015). In that sense, the linear effective Misner–Sharp–Hernandez mass function is not a separate mass notion, but a special spherical realization or reinterpretation of a broader quasi-local energy concept.

A standard geometric expression used in spherical symmetry is

$$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$

with $R$ the areal radius (Hammad, 2016). In a static spherically symmetric gauge,

$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$

this becomes

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$

showing directly how the mass function is encoded in the radial metric coefficient (Faraoni et al., 2020). The same quantity also controls trapping, since apparent or trapping horizons satisfy $1-2M_{\rm MSH}/r=0$.

This geometric basis matters because later “effective” or “linear” constructions do not replace the Misner–Sharp–Hernandez mass so much as alter its source content, perturbative normalization, or transformation properties. The term therefore belongs to a family of quasi-local reparameterizations rather than to a single universally adopted definition.

2. Exact linear profile in quantum dust core models

The exact phrase “linear effective Misner–Sharp–Hernandez mass function” appears explicitly in the black-hole quantum dust core model studied in “Quantum dust cores of black holes and their quasi-normal modes” (Bambagiotti et al., 1 Sep 2025). There the spacetime is written in Schwarzschild-like form,

$$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$

and $m(r)$ is described as “the Misner-Sharp-Hernandez (MSH) fraction of the Arnowitt-Deser-Misner (ADM) mass inside a sphere of radius $S$0.”

The central effective interior profile is obtained from the density

$S$1

which implies

$S$2

This linear law is the defining content of the term in the contemporary literature. It applies to the collapsed dust core in the interior region $S$3, with

$S$4

so that the core is hidden inside the Schwarzschild horizon $S$5 (Bambagiotti et al., 1 Sep 2025).

Several immediate geometric consequences follow. Substituting the linear profile into

$S$6

gives $S$7 in the idealized core, so the interior lapse coefficient is finite and negative rather than Schwarzschild-like singular at $S$8. The model does not claim a fully regular center, because $S$9 still diverges at $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$0, but it does state that the linear profile “weakens the central singularity,” “regularises the singularity,” and “makes the singularity integrable” (Bambagiotti et al., 1 Sep 2025). The core also remains entirely trapped because $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$1 throughout the linear interior.

The simplest version matches continuously to an exterior Schwarzschild region with $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$2 at $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$3, yielding

$$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$4

This matching is only $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$5: $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$6 is continuous, but $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$7 jumps at the core surface. The same paper therefore introduces refinements. One is a slightly parabolic interior profile

$$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$8

meant to account for overlap of neighboring layer wave functions. Another is a $$M_H := \frac{1}{8\pi}\sqrt{\frac{A}{16\pi} \int_S \mu \left( \mathcal{R} + \theta_+ \theta_- - \frac{1}{2}\sigma^+_{ab}\sigma_-^{ab} - 2\omega_a\omega^a \right) },$$9-matched interpolating profile

$$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$0

with continuity conditions imposed on $$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$1, $$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$2, and $$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$3 at both boundaries. These refinements do not abolish the linear profile as the leading effective description; they regulate the transition between the core and the Schwarzschild exterior (Bambagiotti et al., 1 Sep 2025).

The same work shows that observable consequences are controlled mainly by the quantum nature of the core surface rather than by the deep interior linearity alone. If the exterior is forced to be exactly Schwarzschild, the quasi-normal mode spectrum is exactly Schwarzschild. If the outermost-layer wave function leaks into the exterior, both linear and parabolic mass functions produce small deviations, with the parabolic case closer to Schwarzschild than the linear one (Bambagiotti et al., 1 Sep 2025). This establishes the linear effective Misner–Sharp–Hernandez mass function as an interior effective description whose phenomenological imprint is surface-sensitive.

3. Linearized mass variables in primordial-black-hole collapse

In primordial-black-hole collapse, the exact phrase is usually absent, but closely related linearized Misner–Sharp mass variables are standard. In the Misner–Sharp–Hernandez formalism for spherical collapse,

$$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$4

the exact Misner–Sharp mass is

$$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$5

with $$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$6 and $$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$7 (Bloomfield et al., 2015). For cosmological applications, the mass is normalized by the FRW background through

$$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$8

The variable

$$m(t,r)=\frac{R}{2G}\left(1-g^{\mu\nu}\partial_\mu R\,\partial_\nu R\right),$$9

is the background-subtracted cosmological mass perturbation and is the principal linear analogue of an effective Misner–Sharp–Hernandez mass function in the primordial-black-hole literature (Bloomfield et al., 2015).

This perturbative mass variable has a direct density relation,

$R$0

and at horizon crossing it becomes the usual fractional mass excess,

$R$1

Its linear growing mode satisfies

$R$2

The same analysis also introduces a conserved linear combination,

$R$3

which is constant in the growing mode. These are not called linear effective Misner–Sharp–Hernandez mass functions, but they are the nearest exact equivalents in linear PBH perturbation theory (Bloomfield et al., 2015).

A more recent reformulation for type-II curvature fluctuations preserves the same geometric content while changing how the mass is computed numerically. There the Misner–Sharp mass is defined by

$R$4

and, geometrically,

$R$5

In the long-wavelength expansion one writes

$R$6

so $R$7 is the explicit linearized mass perturbation. The same paper uses the compaction function

$R$8

and the trapping relation $R$9 as the practical mass-based diagnostics of PBH formation (Escrivà, 8 Apr 2025). This suggests that in PBH theory the most useful “linear effective” mass objects are not exact linear-in-$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$0 profiles, but normalized or perturbative mass excess variables tied to the quasi-local Misner–Sharp geometry.

4. Effective and linearly corrected generalizations

Several modified-gravity and spacetime-mapping constructions produce effective Misner–Sharp-type masses with explicitly linear pieces, although the exact phrase is usually not adopted.

In generalized Rastall theory, for static spherical metrics with $ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$1, the generalized Misner–Sharp energy is

$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$2

At the horizon this becomes

$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$3

The paper is explicit that the mass is not generically linear in $ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$4; rather, it contains a linear-in-$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$5 core term plus a generalized Rastall correction. It therefore supports an “effective linear” interpretation only in the restricted sense that the leading horizon contribution is linear when the correction is negligible or vanishing (Moradpour et al., 2019).

Under conformal mappings, a related distinction appears between geometric and hydrodynamic mass notions. In the hydrodynamic collapse formulation, the transformed mass is

$ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$6

This formula is linear in the original mass $ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$7, linear in the matter term $ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$8, and linear in the conformal-gradient factor $ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega_{(2)}^2,$9, although it remains nonlocal because of the radial integral. The same paper argues that this is the closest object to a linearly corrected effective Misner–Sharp mass under conformal mapping (Hammad, 2016).

A complementary study of spacetime mappings shows that exact linearity survives under Kerr–Schild transformations but not generically under conformal rescalings. For a Kerr–Schild map,

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$0

the transformed Misner–Sharp–Hernandez mass is

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$1

an exact additive linear correction. By contrast, the conformally transformed mass satisfies

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$2

which is linear in the seed mass but not a fully linear law in the conformal factor (Faraoni et al., 2014).

Modified-gravity theories also generate effective Misner–Sharp masses with explicit linear-in-radius terms. In four-dimensional massive gravity, the generalized mass derived from the unified first law is

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$3

The bracket contains an explicit term linear in $M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$4, $M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$5, so the massive-gravity correction is literally linear in the areal radius at the level of the quasi-local energy functional (Hu et al., 2015). In $M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$6 gravity, by contrast, the generalized Misner–Sharp energy is interpreted as an effective enclosed energy containing Ricci-sector, Gauss–Bonnet, derivative, and possible integral contributions; the literature there supports an effective quasi-local mass, but not a named linearized one (Akbarieh et al., 4 Jun 2025).

5. Asymptotic constraints and physical admissibility

A central limitation on any proposed linear effective Misner–Sharp–Hernandez mass function is asymptotic flatness. For static spherical metrics in the Abreu–Visser form,

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$7

the function $M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$8 is precisely the Misner–Sharp–Hernandez mass, and asymptotic flatness requires

$M_{\rm MSH}=\frac{r}{2}\left(1-\frac{1}{B}\right),$9

A strictly linear profile $1-2M_{\rm MSH}/r=0$0 would imply

$1-2M_{\rm MSH}/r=0$1

so unless $1-2M_{\rm MSH}/r=0$2 it describes a $1-2M_{\rm MSH}/r=0$3 matter distribution extending outward rather than an asymptotically flat vacuum exterior. The same analysis treats quasilocal masses that become negative, vanish, or diverge at infinity as anomalous and physically pathological for isolated systems (Faraoni et al., 2020).

The sign of the Misner–Sharp–Hernandez mass also has direct coordinate and dynamical consequences. In Painlevé–Gullstrand form, the radial infall/outflow velocity is

$1-2M_{\rm MSH}/r=0$4

If $1-2M_{\rm MSH}/r=0$5, this becomes imaginary and the Painlevé–Gullstrand construction fails. The paper identifies anti-de Sitter space and the inner region of Reissner–Nordström as examples of such breakdown, interpreting them in terms of repulsive gravity preventing the free-fall congruence underlying the slicing (Faraoni et al., 2020). Thus an effective mass profile is not merely bookkeeping; its sign controls whether standard horizon-penetrating coordinates exist.

A further caution comes from Schwarzschild–de Sitter thermodynamics. There, a local mass function

$1-2M_{\rm MSH}/r=0$6

is introduced and shown to be a simple linear transform of the standard Misner–Sharp mass,

$1-2M_{\rm MSH}/r=0$7

This is an effective, thermodynamically useful quasi-local energy, but it is not identical to the canonical Misner–Sharp–Hernandez mass (Bhattacharya et al., 2013). The example shows that “effective mass” may denote a linearly shifted quasi-local energy tailored to a particular physical problem rather than the geometric MSH mass itself.

6. Terminological status and related usages

The literature therefore supports several distinct but related meanings of the phrase. The following summary organizes the main usages already present in the arXiv record.

Context	Representative quantity	Use of phrase
Quantum dust core	$1-2M_{\rm MSH}/r=0$8	Explicit (Bambagiotti et al., 1 Sep 2025)
PBH collapse	$1-2M_{\rm MSH}/r=0$9; $$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$0	Nearest linearized analogue (Bloomfield et al., 2015, Escrivà, 8 Apr 2025)
Generalized Rastall	$$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$1 correction	Phrase absent; linear leading term (Moradpour et al., 2019)
Spacetime mappings	$$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$2, $$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$3	Effective transformed masses (Hammad, 2016, Faraoni et al., 2014)
Modified gravity	$$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$4 with $$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$5, or generalized $$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$6 in $$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$7	Generalized rather than named linear mass (Hu et al., 2015, Akbarieh et al., 4 Jun 2025)

This suggests that the expression is not yet standardized. In current practice, it refers most precisely to the quantum-dust-core profile $$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$8, while elsewhere it functions as a shorthand for one of three nearby ideas: a linear-in-radius contribution inside a broader effective mass, a linearized mass perturbation around FRW or collapse backgrounds, or a linearly corrected transformed/generalized quasi-local mass.

Related usages reinforce this conclusion. In the effective-fluid description of the dark universe, the standard Misner–Sharp mass remains tied to the energy density,

$$ds^2 = -\left(1-\frac{2\,G\, m(r)}{r} \right)dt^2 + \left(1-\frac{2\,G\, m(r)}{r} \right)^{-1}dr^2 +r^2\,d\Omega^2,$$9

while the dark-force contribution can nevertheless be rewritten as an effective enclosed mass

$m(r)$0

so the linear growth arises from pressure rather than from an additional Misner–Sharp density term (Cadoni et al., 2017). Likewise, the general Hawking mass decomposition shows that only the electric part of the Weyl tensor contributes to the pure gravitational part of the quasilocal mass, clarifying why the spherical Misner–Sharp–Hernandez mass often appears to encode only the Newtonian-type gravitational content (Faraoni, 2015).

Taken together, these developments fix the present encyclopedic meaning of the term. The linear effective Misner–Sharp–Hernandez mass function is best understood as a noncanonical but technically meaningful label for spherical quasi-local mass constructions in which the enclosed mass is either exactly linear in areal radius, as in the quantum dust core, or decomposes into a linear leading term plus theory-dependent corrections. Its precise interpretation is controlled by the surrounding framework—quantum collapse, PBH perturbation theory, spacetime mappings, or modified gravity—and by the distinction between the geometric Misner–Sharp–Hernandez mass itself and the effective mass variables built from, or compared to, that geometric quantity.