Generalized Misner-Sharp Energy

• Generalized Misner-Sharp energy is a quasilocal mass definition that extends the classical Misner-Sharp formulation to modified gravities, cosmological horizons, and nonstandard geometries.
• It unifies methods such as the inverse unified first law and Kodama flow to derive a conserved scalar energy under strict integrability or conservation conditions.
• The framework establishes actionable thermodynamic relations in theories like Einstein, Lovelock, scalar-tensor, and massive gravity, providing clear insights into matter-energy dynamics.

Generalized Misner-Sharp energy denotes a class of quasilocal energy constructions built around the Misner-Sharp-Hernandez mass of spherically symmetric general relativity and extended to modified gravities, cosmological apparent horizons, and related quasilocal frameworks. In spherical symmetry the baseline definition is

$$E_{\mathrm{MSH}}(R)=\frac{R}{2G}\left(1-g^{ab}\nabla_aR\nabla_bR\right),$$

while for a Robertson-Walker geometry

$$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$

so that at the apparent horizon $E_{\mathrm{MS}}(r_A)=r_A/(2G)$ (Côté et al., 2019, Gohar, 18 May 2026). In the later literature, “generalized” may refer to modified field equations, additional curvature invariants, effective couplings such as $G_{\mathrm{eff}}$, or to generalized geometries in which the classical Misner-Sharp formula is retained but evaluated on nonstandard areal-radius functions (Akbarieh et al., 4 Jun 2025, Moreira et al., 27 Jul 2025).

1. Geometric content and the GR baseline

The standard geometric setting is a $2+2$ split,

$ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$

with $r(x)$ the areal radius. In this form, the Misner-Sharp energy depends only on the scalar $r$ and the norm of its gradient, which is why it is gauge-independent in spherical symmetry (Côté et al., 2019). In Friedmann-Lemaître-Robertson-Walker spacetimes, the same definition reproduces the familiar homogeneous energy inside a sphere, and at the apparent horizon it collapses to the compact relation $E_{\mathrm{MS}}=r_A/(2G)$ (Gohar, 18 May 2026).

The geometric role of the Misner-Sharp construction is sharpened by its relation to other quasilocal masses. In spherical symmetry, the Hawking-Hayward mass reduces to the Misner-Sharp-Hernandez mass for a $2$-sphere of symmetry, and the Hawking quasilocal mass provides the natural extension beyond spherical symmetry (Faraoni, 2015). In that extension, the quasilocal mass admits a split into a matter term and a pure Weyl term, and only the electric part of the Weyl tensor contributes; the magnetic part does not contribute (Faraoni, 2015). This is the basis for the claim that the Misner-Sharp-Hernandez mass, and its Hawking generalization, retain a “Newtonian” character in the specific sense isolated in that analysis.

A recurrent point in later work is that the baseline Misner-Sharp relation remains the reference expression even when the generalized theory is presented in a very different language. In scalar-tensor gravity, for example, one frequently rewrites the generalized energy either as a geometric identity involving $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$0 or as a horizon relation of the form $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$1 (Cembranos et al., 19 Dec 2025, Gohar, 18 May 2026). This does not remove the geometric core; it dresses it with non-Einstein couplings.

2. Unified first law, Kodama flow, and existence conditions

A central organizing principle is Hayward’s unified first law,

$$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$2

with

$$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$3

This form is used directly in GR, Lovelock gravity, scalar-tensor gravity, $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$4 gravity, $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$5 gravity, cubic gravity, and massive gravity (0704.0793, Zhang et al., 2014, Akbarieh et al., 4 Jun 2025). The generalized Misner-Sharp energy is then defined by asking whether the right-hand side can be integrated to a scalar quasilocal energy.

Two construction routes dominate the literature. The first is the inverse unified first law method, which starts from $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$6 and imposes the integrability condition that the associated one-form be closed. The second is the conserved charge method based on the Kodama vector,

$$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$7

with conserved current $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$8 when the relevant constraint holds (Zhang et al., 2014, Akbarieh et al., 4 Jun 2025). In $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$9-dimensional $E_{\mathrm{MS}}(r_A)=r_A/(2G)$0 gravity, Zhang, Hu, and Li showed that the two approaches are equivalent, which are bridged by a constraint, and that this constraint determines the existence of a well-defined Misner-Sharp mass (Zhang et al., 2014). In $E_{\mathrm{MS}}(r_A)=r_A/(2G)$1 gravity, the same pair of methods again yields the same generalized energy when the integrability and current-conservation conditions are satisfied (Akbarieh et al., 4 Jun 2025).

The status of the existence condition depends strongly on the theory. In $E_{\mathrm{MS}}(r_A)=r_A/(2G)$2 and $E_{\mathrm{MS}}(r_A)=r_A/(2G)$3 gravity, gradient terms in $E_{\mathrm{MS}}(r_A)=r_A/(2G)$4 or $E_{\mathrm{MS}}(r_A)=r_A/(2G)$5 can obstruct a path-independent integral unless additional conditions are met (Zhang et al., 2014, Akbarieh et al., 4 Jun 2025). By contrast, in the four-dimensional massive gravity model with a widely used singular reference metric, the existence of the Misner-Sharp mass does not lead to extra constraint for the massive gravity, and the derived mass can be confirmed by the conserved charge method because the stress energy is conserved in that case (Hu et al., 2015). This distinction is structurally important: some generalized theories admit a quasilocal energy only conditionally, while others preserve the unified-first-law machinery more directly.

3. Realizations in modified gravity

In $E_{\mathrm{MS}}(r_A)=r_A/(2G)$6-dimensional $E_{\mathrm{MS}}(r_A)=r_A/(2G)$7 gravity, the generalized Misner-Sharp mass acquires three characteristic pieces: a dressed GR core proportional to

$E_{\mathrm{MS}}(r_A)=r_A/(2G)$8

a term involving $E_{\mathrm{MS}}(r_A)=r_A/(2G)$9, and a potential term involving $G_{\mathrm{eff}}$0, together with possible integration terms depending on derivatives of $G_{\mathrm{eff}}$1 (Zhang et al., 2014). In static spacetimes those extra pieces simplify substantially, and for constant curvature they reduce to the expected cosmological-energy correction. In the pure GR limit $G_{\mathrm{eff}}$2, the generalized mass collapses to the standard $G_{\mathrm{eff}}$3-dimensional Misner-Sharp result (Zhang et al., 2014).

The extension to $G_{\mathrm{eff}}$4 gravity retains the same logic but adds explicit Gauss-Bonnet-sector contributions. For static spherical symmetry, the effective energy contains the GR factor $G_{\mathrm{eff}}$5, a potential term proportional to $G_{\mathrm{eff}}$6, gradient corrections involving both $G_{\mathrm{eff}}$7 and $G_{\mathrm{eff}}$8, and an integral term that vanishes under the conditions stated in the paper (Akbarieh et al., 4 Jun 2025). In this framework the Gauss-Bonnet sector is dynamical because the theory uses a nontrivial $G_{\mathrm{eff}}$9 rather than a constant coefficient multiplying the four-dimensional topological invariant (Akbarieh et al., 4 Jun 2025).

Scalar-tensor theory supports two complementary generalizations. In the Einstein-frame hydrodynamical formulation for collapse, the total quasi-local energy is

$2+2$0

with

$2+2$1

This formulation makes the split into matter and scalar-field energies explicit, and the scalar sector modifies both equilibrium and dynamics through $2+2$2, $2+2$3, and the anisotropic term $2+2$4 (Cembranos et al., 19 Dec 2025). A distinct scalar-tensor construction identifies

$2+2$5

so that generalized entropy functionals determine $2+2$6, hence $2+2$7, and thereby the Misner-Sharp energy (Gohar, 18 May 2026).

The generalized Rastall theory alters the construction in a different way. For static metrics with $2+2$8, the generalized Misner-Sharp energy contains the Einstein term and a curvature-coupling correction weighted by the spacetime-dependent Rastall parameter $2+2$9:

$ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$0

For constant $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$1 this reproduces the original Rastall expression, while the limit $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$2, $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$3 gives the Einstein result (Moradpour et al., 2019). Here the generalization is driven by modified conservation rather than by higher-curvature densities.

4. Cosmological apparent horizons and horizon thermodynamics

Cosmology is the setting in which generalized Misner-Sharp energy is most tightly linked to thermodynamics. Gong and Wang introduced a masslike function that has dimension of energy and equals to the Misner-Sharp mass at the apparent horizon, and showed that the first law of thermodynamics of the apparent horizon $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$4 can be derived from the Friedmann equation in Einstein, Lovelock, nonlinear, and scalar-tensor theories (0704.0793). In this framework, the apparent horizon

$ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$5

is the preferred thermodynamic screen, the heat flow is the projection of the unified first law onto the horizon generator, and the entropy is theory-dependent: $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$6 in Einstein gravity, the Jacobson-Myers/Wald form in Lovelock gravity, $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$7 in nonlinear gravity, and $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$8 in scalar-tensor gravity (0704.0793).

A particularly explicit realization appears in quasi-topological cosmology. In the $ds^2=h_{ab}\,dx^a dx^b+r^2(x)\,d\Omega^2,$9-dimensional FRW analysis with primary focus on the lowest-dimensional quasi-topological case $r(x)$0 ($r(x)$1), the unified first law yields a well-defined Misner-Sharp energy

$r(x)$2

and at the apparent horizon one obtains

$r(x)$3

The paper emphasizes that the generalized Misner-Sharp energy equals the matter energy inside the apparent horizon in quasi-topological cosmology, firmly derived rather than assumed (Chu et al., 19 Feb 2025). The same analysis identifies thermodynamic pressure and thermodynamic volume through the extra terms required by the unified first law, derives an equation of state, and finds $r(x)$4-$r(x)$5 criticality with critical exponents

$r(x)$6

provided $r(x)$7 and $r(x)$8 (Chu et al., 19 Feb 2025). In the Einstein limit, the cosmological equation of state

$r(x)$9

has no $r$0-$r$1 phase transition in the FRW case (Chu et al., 19 Feb 2025).

Not all generalized theories remain in equilibrium form. In $r$2 gravity, the generalized Misner-Sharp energy can still be written on the FLRW apparent horizon, but the thermodynamic relation generically takes the non-equilibrium form

$r$3

because the geometric sector contributes effective fluxes involving $r$4 and $r$5 (Akbarieh et al., 4 Jun 2025). By contrast, in the generalized Rastall theory for a flat FRW universe one has

$r$6

with

$r$7

and the apparent-horizon entropy is nondecreasing for $r$8 and $r$9 (Moradpour et al., 2019). The cosmological literature therefore exhibits both equilibrium and non-equilibrium realizations, depending on whether the modified gravity corrections can be absorbed into an effective quasilocal energy without residual entropy production.

5. Dynamical spacetimes, radiation, and collapse

The generalized Misner-Sharp construction is equally prominent in dynamical, nonstationary geometries. In five-dimensional cubic gravity, generalized Vaidya solutions admit a quasilocal energy

$E_{\mathrm{MS}}=r_A/(2G)$0

which in the generalized Vaidya metric becomes

$E_{\mathrm{MS}}=r_A/(2G)$1

The same theory supports a Clausius relation on the apparent horizon, a first law of the form $E_{\mathrm{MS}}=r_A/(2G)$2, and a second law under the null energy condition (Ruan, 2015). The novelty here is that the cubic Lovelock term vanishes identically in five dimensions, while the “new cubic term” remains non-vanishing and dynamical (Ruan, 2015).

In $E_{\mathrm{MS}}=r_A/(2G)$3-dimensional dRGT massive gravity with a singular reference metric, generalized Vaidya and generalized Vaidya-like solutions again admit a generalized Misner-Sharp mass. The covariant expression is

$E_{\mathrm{MS}}=r_A/(2G)$4

and the usual Clausius relation $E_{\mathrm{MS}}=r_A/(2G)$5 holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state (Hu et al., 2016). The paper also finds that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution (Hu et al., 2016). In the related four-dimensional massive gravity construction, the generalized Misner-Sharp energy contains the Einstein part, a cosmological-constant term, and the $E_{\mathrm{MS}}=r_A/(2G)$6 and $E_{\mathrm{MS}}=r_A/(2G)$7 sectors, and the FRW universe can be in thermodynamic equilibrium (Hu et al., 2015).

In scalar-tensor collapse, the hydrodynamical Einstein-frame formulation gives a generalized Misner-Sharp relation

$E_{\mathrm{MS}}=r_A/(2G)$8

so that the total quasi-local energy is again $E_{\mathrm{MS}}=r_A/(2G)$9 (Cembranos et al., 19 Dec 2025). This split enters directly into the generalized Tolman-Oppenheimer-Volkoff balance and the dynamical acceleration equation. For a homogeneous interior one finds

$2$0

because the gradient terms vanish in that limit (Cembranos et al., 19 Dec 2025). A notable implication, stated in the paper, is that for a canonical scalar with nonnegative potential, $2$1, so $2$2 is nonnegative and adds to the gravitational mass (Cembranos et al., 19 Dec 2025).

6. Transformations, generalized geometries, and recurrent ambiguities

Several later studies clarify what is and is not being generalized. In spherical symmetry, the Misner-Sharp-Hernandez mass is gauge-independent because it is expressed entirely in terms of the scalar areal radius and the spacetime metric, whereas the Brown-York energy is gauge-dependent because changing the slicing alters the extrinsic curvature (Côté et al., 2019). Under conformal rescalings,

$2$3

the Misner-Sharp-Hernandez mass transforms with a compact law,

$2$4

whereas the Brown-York energy acquires a nontrivial square-root dependence on the transformed metric coefficient $2$5 (Côté et al., 2019). Under generalized Kerr-Schild mappings the contrast persists: the Misner-Sharp-Hernandez mass shifts additively by $2$6, while Brown-York again transforms nonlinearly (Côté et al., 2019).

This does not settle the conformal issue, because the hydrodynamic origin of the Misner-Sharp mass behaves differently from the purely geometric definition. In the gravitational-collapse analysis of conformal mapping, the geometric Misner-Sharp mass in GR does not transform like a usual mass, but the original hydrodynamic definition recovers the familiar scaling

$2$7

up to an integral correction interpreted as work required to compress or dilate the matter along with spacetime (Hammad, 2016). In homogeneous FLRW and point-mass cases that correction vanishes, so the pure $2$8 scaling is recovered (Hammad, 2016). The same paper argues that this difference reflects the fact that the widely used geometric definition is fundamentally different from the original conception of the Misner-Sharp mass when conformal transformations are considered (Hammad, 2016).

Another recurring ambiguity concerns the word “generalized” itself. In the generalized black-bounce literature, the Hernandez-Misner-Sharp quasi-local mass is the standard GR construct

$2$9

and the generalization lies in the geometry via

$$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$00

and in the deformation-controlled mass functions $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$01 (Moreira et al., 27 Jul 2025). The resulting quasi-local mass is positive, regular, and monotone from $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$02 at $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$03 to $$E_{\mathrm{MS}}^{\mathrm{FLRW}}=\frac{r^3}{2G}\left(H^2+\frac{k}{a^2}\right),\qquad r_A=\frac{1}{\sqrt{H^2+k/a^2}},$$04 at infinity, but the mass functional itself is not modified (Moreira et al., 27 Jul 2025). A common misconception is therefore that every “generalized Misner-Sharp energy” introduces a new definition; some papers generalize the dynamics or the geometry while keeping the classical formula.

A final conceptual extension is provided by the Hawking quasilocal mass. Beyond spherical symmetry, it is the Hawking mass—not the Misner-Sharp mass in its strict form—that carries the generalization, and its purely gravitational content is exclusively electric in the Weyl decomposition (Faraoni, 2015). This suggests a useful taxonomy: some generalizations preserve spherical symmetry and modify the energy functional; some preserve the functional and modify the geometry; and some replace the spherical-symmetry framework by a broader quasilocal construction while retaining the same physical intuition about enclosed energy and horizon thermodynamics.