Coarse-Grained Vacuum Boundaries in Gravitational Collapse: A Timelike Thin-Shell Balance Framework

1. Introduction

Gravitational collapse is usually formulated in terms of bulk geometry, global charges, trapped surfaces, and late-time exterior fields. A complementary description is obtained by focusing on the boundary at which an interior high-density or vacuum-like region is matched to an exterior gravitational field. At such a boundary, geometric matching, surface stress, redshift, quasilocal energy, and thermodynamic bookkeeping are tied to one and the same hypersurface.

This boundary-centered perspective is suggested by several developments in gravitational physics. Black-hole thermodynamics assigns entropy and temperature to horizon structures. Hawking radiation and the Unruh effect relate thermality to horizons, acceleration, and observer-dependent field decompositions [1,2]. Jacobson’s thermodynamic derivation of the Einstein equation shows that local causal boundaries, heat flow, and area-proportional entropy can organize gravitational dynamics [3]. These results identify boundaries as natural carriers of the relation between geometry, thermodynamics, entropy, and observability.

The present work applies this boundary logic to a finite-radius timelike hypersurface rather than to a null horizon. The system considered here is a spherically symmetric timelike thin shell separating an effective interior vacuum reference sector from an exterior Schwarzschild or Schwarzschild–de Sitter region. The shell carries surface stress, quasilocal energy, tangential pressure, an areal radius, and a redshift map to exterior static observers. It is therefore a classical quasilocal boundary on which interior reference-state change can be converted into exterior-accessible response.

Timelike thin-shell methods provide the geometric basis for this construction. They allow two spacetime regions to be joined across a timelike hypersurface, with the jump in extrinsic curvature encoded in a surface stress tensor through the Israel junction conditions [4,5,6]. In spherical symmetry the shell is described by its induced metric, areal radius, surface energy density, tangential pressure, normal orientation, and one-sided extrinsic curvatures. These data make the interface a well-defined dynamical and quasilocal object.

A previous timelike thin-shell construction established the underlying classical geometry for a regular de Sitter-type interior matched to a Schwarzschild or Schwarzschild–de Sitter exterior [7]. That analysis treated the shell motion through an effective potential, identified the static-patch domain, obtained bounded curvature scalars in the shell-supported region, and derived a mass-scaled frequency bound for persistent near-shell modes. The present work uses that geometric setting as the background for a different question: how a finite timelike boundary registers and redistributes changes of an interior vacuum reference state.

The interior sector is described by a coarse-grained reference density ${\rho }_{\mathrm{ref}}\left(\chi \right)$, where $\chi$ labels a macroscopic ordering or relaxation parameter. The corresponding reference energy inside the shell is

$$E_{\mathrm{ref}}(R,\chi )={\displaystyle \frac{4\pi }{3}}{R}^3{\rho }_{\mathrm{ref}}\left(\chi \right).$$

Its proper-time variation contains two distinct contributions: the intrinsic relaxation of ${\rho }_{\mathrm{ref}}$ and the geometric change of the volume enclosed by the moving boundary. Reference-energy descent occurs when the relaxation of ${\rho }_{\mathrm{ref}}$ dominates the geometric volume increase. This condition defines the effective quasilocal boundary input

$${\mathsf{\Phi }}_{\mathrm{eff}}:=-{\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}}.$$

Thus ${\mathsf{\Phi }}_{\mathrm{eff}}>0$ is the boundary signal of a descending interior reference energy.

The main result of the paper is the corresponding flux-coupled shell balance. The shell partitions the effective input into quasilocal storage, outward release, and pressure–area work:

$${\dot{E}}_{\Sigma }=A_{\Sigma }({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}})-P{\dot{A}}_{\Sigma }.$$

Here $E_{\Sigma }=\sigma A_{\Sigma }$ is the quasilocal shell energy, ${\mathsf{\Phi }}_{\mathrm{out}}$ is the outward release flux into exterior-accessible channels, and $P{\dot{A}}_{\Sigma }$ is the tangential pressure–area work. This equation is the macroscopic boundary balance developed in this manuscript. It expresses how a change of the interior reference sector is recorded at the timelike shell through storage, release, mechanical response, and area evolution.

The boundary response is supplemented by a local temperature variable $T_{\Sigma }$ and a trajectory-dependent response coefficient

$$C_{\mathrm{eff}}:={\displaystyle \frac{d{E}_{\mathrm{ref}}}{d{T}_{\Sigma }}}.$$

On a negative-response branch, $C_{\mathrm{eff}}<0$, reference-energy descent raises the local shell temperature. Area growth is treated as a finite local response channel, and the entropy-like variable

$$S_{\Sigma }=\alpha A_{\Sigma }$$

records the associated macroscopic surface growth. Local shell temperatures and near-boundary mode frequencies are mapped to exterior static observers through the exterior lapse. Spatial infinity enters only in the asymptotically Schwarzschild case; for Schwarzschild–de Sitter exteriors, observer-accessible quantities are defined relative to finite static observers inside the exterior static patch.

The framework is spherical, static-patch based, and finite-response. The bulk metrics are used as local static-patch representatives adjacent to the shell, while the boundary variables evolve with the shell proper time $\tau$. The time dependence of ${\rho }_{\mathrm{ref}}\left(\chi \left(\tau \right)\right)$ is not assigned directly to exterior observables. It is registered through ${\mathsf{\Phi }}_{\mathrm{eff}}$, the shell balance, pressure–area work, area response, and redshifted quantities.

Recent analyses have shown that relativistic self-gravitating membranes and thin shells can develop dynamical non-radial instabilities. Yang, Bonga, and Pan identified high-angular-momentum warping instabilities, while Pitre, Schneider, and Poisson found unstable modes on all angular scales for a static perfect-fluid thin shell [8,9]. These results set the stability benchmark for the present boundary formulation. Here the perturbative problem is the stability of a regulated boundary response: shell deformations, surface-stress variations, reference-density variations, effective input, exterior release, and area response have to be varied as a coupled system.

2. Timelike Thin-Shell Boundary

Units $c=\hslash =k_{\mathrm{B}}=1$ are used, while G is kept explicit. The construction is based on a spherically symmetric timelike hypersurface $\Sigma$ that separates an effective interior reference region ${\mathcal{M}}_{-}$ from an exterior gravitational region ${\mathcal{M}}_{+}$. The shell is parametrized by its proper time $\tau$ and areal radius $R\left(\tau \right)$. Its induced metric is

$$d{s}_{\Sigma }^2=-d{\tau }^2+R^2\left(\tau \right)d{\mathsf{\Omega }}^2,$$

(1)

and the corresponding shell area is

$$A_{\Sigma }=4\pi R^2.$$

(2)

All shell quantities used below are defined either intrinsically on $\Sigma$ or as one-sided limits from ${\mathcal{M}}_{\pm }$ to the shell.

2.1. Adjacent Static-Patch Geometries

On both sides of the shell, static, spherically symmetric line elements are used as local representatives of the geometry adjacent to $\Sigma$,

$$d{s}_{\pm }^2=-f_{\pm }\left(r\right)d{t}_{\pm }^2+{\displaystyle \frac{d{r}^2}{f_{\pm }\left(r\right)}}+r^{2}d{\mathsf{\Omega }}^2,$$

(3)

with

$$f_{\pm }\left(r\right)=1-{\displaystyle \frac{2G{M}_{\pm }}{r}}-{\displaystyle \frac{{\mathsf{\Lambda }}_{\pm }{r}^2}{3}}.$$

(4)

The interior mass parameter is set to zero,

$$M_{-}=0,$$

(5)

so that the interior reference geometry is de Sitter-type. The exterior region is characterized by

$$M_{+}=M,$$

(6)

and by the exterior cosmological parameter ${\mathsf{\Lambda }}_{+}$. The asymptotically Schwarzschild case is recovered by setting ${\mathsf{\Lambda }}_{+}=0$.

The shell embedding is

$$x_{\pm }^{\mu }(\tau ,\theta ,\varphi )=\left(t_{\pm }\left(\tau \right),R\left(\tau \right),\theta ,\varphi \right).$$

(7)

The shell four-velocity and unit normal are denoted by $u_{\pm }^{\mu }$ and $n_{\pm }^{\mu }$, respectively. They satisfy

$$u_{\pm }^{\mu }{u}_{\mu }^{\pm }=-1,n_{\pm }^{\mu }{n}_{\mu }^{\pm }=1,u_{\pm }^{\mu }{n}_{\mu }^{\pm }=0.$$

(8)

The normal is oriented from ${\mathcal{M}}_{-}$ toward ${\mathcal{M}}_{+}$, and jumps are written as

$${\left[X\right]}_{-}^{+}:=X^{+}-X^{-}.$$

(9)

The sign-sensitive relation between this orientation, the Israel surface stress tensor, and the one-sided Brown–York data is collected in Appendix A.

2.2. Surface Stress and Quasilocal Shell Energy

The shell carries a surface stress tensor $S_{ab}$. In spherical symmetry the surface stress is written as

$$S_b^a=\mathrm{diag}(-\sigma ,p,p),$$

(10)

where $\sigma \left(\tau \right)$ is the surface energy density and $p\left(\tau \right)$ is the isotropic tangential surface pressure. The quasilocal shell energy is

$$E_{\Sigma }=\sigma A_{\Sigma }.$$

(11)

The Israel junction condition relates $S_{ab}$ to the discontinuity of the extrinsic curvature across the shell,

$${\left[K_{ab}\right]}_{-}^{+}-h_{ab}{\left[K\right]}_{-}^{+}=-8\pi G{S}_{ab}.$$

(12)

Here $h_{ab}$ is the induced metric on $\Sigma$, and $K_{ab}^{\pm }$ are the one-sided extrinsic curvatures. The branch-reduced square-root form of Eq. (12), including the definition of $\kappa =4\pi G\sigma R$, is given in Appendix A. The main text uses only the quasilocal variables

$$R,A_{\Sigma },\sigma ,p,E_{\Sigma },h_{ab},u^{\mu },n^{\mu },$$

(13)

together with the exterior lapse $f_{+}\left(R\right)$, which determines the redshift map from local shell quantities to exterior static observers.

2.3. Static-Patch and Finite-Response Regime

The boundary construction is restricted to the static-patch domain

$$f_{\pm }\left(R\right)>0.$$

(14)

This condition keeps the shell timelike on both sides and fixes the regime in which local shell variables can be redshifted to exterior static observers.

The bulk metrics in Eq. (3) are used as local static-patch representatives of the geometry adjacent to the shell. The proper time $\tau$ labels the boundary history. Along this history, the interior reference sector may vary through a coarse-grained variable ${\rho }_{\mathrm{ref}}\left(\chi \left(\tau \right)\right)$, or equivalently through ${\mathsf{\Lambda }}_{-}\left(\chi \left(\tau \right)\right)$. This variation is registered at the boundary by the reference energy and the effective input introduced in the next sections.

The finite-response regime is defined by

$$R\left(\tau \right)\ge R_{min}>0,\left|\sigma \left(\tau \right)\right|<\infty ,\left|p\left(\tau \right)\right|<\infty .$$

(15)

Within this regime the shell can store quasilocal energy, sustain finite surface stress, perform pressure–area work, mediate exchange with the adjacent regions, and map local boundary variables to exterior-accessible quantities by redshift. All later references to redshifted temperatures, redshifted frequencies, and exterior static observers are understood within the domain defined by Eq. (14).

3. Interior Vacuum Reference Sector

The interior region ${\mathcal{M}}_{-}$ is used as an effective vacuum reference sector. Its macroscopic state is encoded by a coarse-grained reference density ${\rho }_{\mathrm{ref}}$, or equivalently by an interior cosmological parameter ${\mathsf{\Lambda }}_{-}$. The reference sector supplies the interior state relative to which boundary-registered changes are defined.

3.1. Coarse-Grained Reference Density

The interior reference density is written as

$${\rho }_{\mathrm{ref}}={\rho }_{\mathrm{ref}}\left(\chi \right),$$

(16)

where $\chi$ is a macroscopic ordering or relaxation parameter of the reference sector. The associated interior cosmological parameter is

$${\mathsf{\Lambda }}_{-}\left(\chi \right)=8\pi G{\rho }_{\mathrm{ref}}\left(\chi \right).$$

(17)

The interior lapse therefore takes the form

$$f_{-}(r;\chi )=1-{\displaystyle \frac{{\mathsf{\Lambda }}_{-}\left(\chi \right)r^2}{3}},$$

(18)

with $M_{-}=0$ as specified above.

Along the shell worldvolume, the relaxation history is described by $\chi =\chi \left(\tau \right)$. Hence

$${\dot{\rho }}_{\mathrm{ref}}={\displaystyle \frac{d{\rho }_{\mathrm{ref}}}{d\chi }}\dot{\chi }.$$

(19)

The function ${\rho }_{\mathrm{ref}}\left(\chi \left(\tau \right)\right)$ is the constitutive input that specifies the macroscopic reference-sector history. The shell does not access this history directly as an exterior observable. It registers the corresponding change through the reference energy and the effective boundary input constructed below.

3.2. Reference Energy Inside the Shell

The coarse-grained reference energy assigned to the interior sector bounded by $\Sigma$ is defined by

$$E_{\mathrm{ref}}(R,\chi )={\displaystyle \frac{4\pi }{3}}{R}^3{\rho }_{\mathrm{ref}}\left(\chi \right).$$

(20)

Using Eq. (17), this can equivalently be written as

$$E_{\mathrm{ref}}(R,\chi )={\displaystyle \frac{R^3}{6G}}{\mathsf{\Lambda }}_{-}\left(\chi \right).$$

(21)

For a de Sitter-type reference sector, this has the same volume-scaling form as the Misner–Sharp energy associated with a vacuum density inside the areal radius R [10].

Taking the derivative along the shell history gives

$${\dot{E}}_{\mathrm{ref}}=4\pi R^2\dot{R}{\rho }_{\mathrm{ref}}+{\displaystyle \frac{4\pi }{3}}{R}^3{\dot{\rho }}_{\mathrm{ref}}.$$

(22)

With $A_{\Sigma }=4\pi R^2$, this becomes

$${\dot{E}}_{\mathrm{ref}}=A_{\Sigma }\left({\rho }_{\mathrm{ref}}\dot{R}+{\displaystyle \frac{R}{3}}{\dot{\rho }}_{\mathrm{ref}}\right).$$

(23)

Equation (23) separates two effects. The first term is the geometric volume contribution generated by the motion of the boundary through a region with reference density ${\rho }_{\mathrm{ref}}$. The second term is the intrinsic reference-sector contribution generated by the relaxation of ${\rho }_{\mathrm{ref}}$ at fixed shell radius.

3.3. Reference-Energy Descent

The reference energy descends when

$${\dot{E}}_{\mathrm{ref}}<0.$$

(24)

Using Eq. (23), this condition is equivalent to

$${\dot{\rho }}_{\mathrm{ref}}<-3{\displaystyle \frac{\dot{R}}{R}}{\rho }_{\mathrm{ref}}.$$

(25)

Thus, during outward shell motion, a decrease of the reference density produces a descending reference energy only when it dominates the geometric increase of the enclosed volume.

The two contributions in Eq. (23) will be kept separate in the boundary-flux construction. Their difference determines the effective input registered at the shell. This input is defined in the next section as the areal rate associated with reference-energy descent.

4. Effective Boundary Input from Reference-Energy Descent

Starting from Eq. (23), reference-energy descent is converted into an effective areal input at the timelike boundary. Positive input is counted as energy supplied to $\Sigma$ per unit shell area and per unit shell proper time. The construction separates the intrinsic relaxation of the reference density from the geometric change of the volume enclosed by the moving shell.

Equation (23) may be written as

$${\dot{E}}_{\mathrm{ref}}=A_{\Sigma }\left({\rho }_{\mathrm{ref}}\dot{R}+{\displaystyle \frac{R}{3}}{\dot{\rho }}_{\mathrm{ref}}\right).$$

(26)

The two terms on the right-hand side define the ordering and geometric contributions to the boundary input.

4.1. Ordering Contribution

The intrinsic reference-sector contribution is obtained by varying $E_{\mathrm{ref}}$ at fixed shell radius:

$${\left({\dot{E}}_{\mathrm{ref}}\right)}_R={\displaystyle \frac{4\pi }{3}}{R}^3{\dot{\rho }}_{\mathrm{ref}}.$$

(27)

The corresponding areal input associated with reference-density relaxation is defined by

$${\mathsf{\Phi }}_{\mathrm{ord}}:=-{\displaystyle \frac{1}{A_{\Sigma }}}{\left({\dot{E}}_{\mathrm{ref}}\right)}_R=-{\displaystyle \frac{R}{3}}{\dot{\rho }}_{\mathrm{ref}}.$$

(28)

Thus

$${\dot{\rho }}_{\mathrm{ref}}<0⟹{\mathsf{\Phi }}_{\mathrm{ord}}>0.$$

(29)

The quantity ${\mathsf{\Phi }}_{\mathrm{ord}}$ measures the areal input generated by relaxation of the interior reference density at fixed boundary radius.

4.2. Geometric Volume Contribution

The geometric contribution is generated by the motion of the shell through a region with reference density ${\rho }_{\mathrm{ref}}$. It is

$${\left({\dot{E}}_{\mathrm{ref}}\right)}_{\mathrm{geo}}=4\pi R^2\dot{R}{\rho }_{\mathrm{ref}}=A_{\Sigma }{\rho }_{\mathrm{ref}}\dot{R}.$$

(30)

The corresponding areal rate is

$${\mathsf{\Phi }}_{\mathrm{geo}}:={\rho }_{\mathrm{ref}}\dot{R}.$$

(31)

For outward motion, $\dot{R}>0$, and positive reference density, ${\rho }_{\mathrm{ref}}>0$, the geometric term increases the reference energy because the boundary encloses a larger volume. In the net boundary input it therefore competes with the ordering contribution.

The proper-time variation of the reference energy can now be written as

$${\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}}={\mathsf{\Phi }}_{\mathrm{geo}}-{\mathsf{\Phi }}_{\mathrm{ord}}.$$

(32)

4.3. Effective Boundary Input

The effective boundary input generated by reference-energy descent is defined as

$${\mathsf{\Phi }}_{\mathrm{eff}}:=-{\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}}.$$

(33)

Using Eq. (32), this gives

$${\mathsf{\Phi }}_{\mathrm{eff}}={\mathsf{\Phi }}_{\mathrm{ord}}-{\mathsf{\Phi }}_{\mathrm{geo}}=-{\displaystyle \frac{R}{3}}{\dot{\rho }}_{\mathrm{ref}}-{\rho }_{\mathrm{ref}}\dot{R}.$$

(34)

Thus

$${\mathsf{\Phi }}_{\mathrm{eff}}>0⟺{\dot{E}}_{\mathrm{ref}}<0.$$

(35)

Equivalently,

$${\mathsf{\Phi }}_{\mathrm{eff}}>0⟺{\dot{\rho }}_{\mathrm{ref}}<-3{\displaystyle \frac{\dot{R}}{R}}{\rho }_{\mathrm{ref}}.$$

(36)

The effective input ${\mathsf{\Phi }}_{\mathrm{eff}}$ is the quasilocal input term by which the proper-time descent of the coarse-grained reference energy is registered at $\Sigma$. It carries the reference-sector contribution into the shell balance and fixes the amount of boundary input available for shell storage, exterior release, and pressure–area work.

The next section inserts ${\mathsf{\Phi }}_{\mathrm{eff}}$ into the intrinsic shell energy balance. This converts reference-energy descent into a finite macroscopic boundary response.

5. Flux-Coupled Shell Balance

The effective input ${\mathsf{\Phi }}_{\mathrm{eff}}$ defined in Eq. (33) supplies the timelike boundary whenever the coarse-grained reference energy descends. The shell then redistributes this input into three macroscopic channels: quasilocal shell storage, outward release into the exterior region, and pressure–area work.

5.1. Intrinsic Shell Energy Balance

The quasilocal shell energy is

$$E_{\Sigma }=\sigma A_{\Sigma }.$$

(37)

The intrinsic conservation law for the surface stress tensor on $\Sigma$ gives the spherical shell balance

$$\dot{\sigma }+2{\displaystyle \frac{\dot{R}}{R}}(\sigma +p)={\mathsf{\Phi }}_{\Sigma },$$

(38)

where ${\mathsf{\Phi }}_{\Sigma }$ denotes the net areal input into the shell per unit proper time. Equivalently, using $A_{\Sigma }=4\pi R^2$, this can be written as

$${\dot{E}}_{\Sigma }=A_{\Sigma }{\mathsf{\Phi }}_{\Sigma }-p{\dot{A}}_{\Sigma }.$$

(39)

The derivation from the projected conservation law and the corresponding sign convention for the normal input are given in Appendix B.

5.2. Effective Input and Exterior Release

The present boundary model closes the net input by separating the contribution generated by reference-energy descent from the part released toward the exterior:

$${\mathsf{\Phi }}_{\Sigma }={\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}.$$

(40)

Here ${\mathsf{\Phi }}_{\mathrm{eff}}$ is the effective boundary input defined in Eq. (33), and ${\mathsf{\Phi }}_{\mathrm{out}}$ is the outward release flux into exterior-accessible channels. Both quantities are measured per unit shell area and per unit shell proper time. With this convention, ${\mathsf{\Phi }}_{\mathrm{eff}}>0$ increases the energy available at the boundary, whereas ${\mathsf{\Phi }}_{\mathrm{out}}>0$ carries energy from the boundary toward ${\mathcal{M}}_{+}$.

In the isotropic spherical model considered here, the surface pressure entering the area-work term is

$$P\equiv p.$$

(41)

Substitution of Eq. (40) into Eq. (39) gives the flux-coupled shell balance

$$\begin{array}{|c|}\hline {\dot{E}}_{\Sigma }=A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)-P{\dot{A}}_{\Sigma }\\ \hline\end{array}.$$

(42)

This is the central balance law of the framework. It converts reference-energy descent into a finite boundary response and separates the resulting macroscopic channels into shell storage, exterior release, and pressure–area work.

5.3. Storage, Release, and Pressure–Area Work

The pressure–area work rate is

$${\dot{W}}_{\Sigma }=P{\dot{A}}_{\Sigma }.$$

(43)

With the sign convention of Eq. (42), positive $P{\dot{A}}_{\Sigma }$ reduces the part of the net input that remains stored on the shell. For outward evolution, $\dot{R}>0$, one has

$${\dot{A}}_{\Sigma }=8\pi R\dot{R}>0.$$

(44)

A positive tangential surface pressure therefore converts part of the available boundary input into area work.

Equation (42) may also be written as

$$A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)={\dot{E}}_{\Sigma }+P{\dot{A}}_{\Sigma }.$$

(45)

This form displays the partition of the net boundary input into shell storage and pressure–area work.

Using $E_{\Sigma }=\sigma A_{\Sigma }$, the storage term expands as

$${\dot{E}}_{\Sigma }=A_{\Sigma }\dot{\sigma }+\sigma {\dot{A}}_{\Sigma }.$$

(46)

Substitution into Eq. (42) gives

$$A_{\Sigma }\dot{\sigma }+(\sigma +P){\dot{A}}_{\Sigma }=A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right).$$

(47)

For $P=p$, this is equivalent to

$$\dot{\sigma }+2{\displaystyle \frac{\dot{R}}{R}}(\sigma +p)={\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}.$$

(48)

Shell storage increases when the effective input exceeds the sum of exterior release and pressure–area work,

$$A_{\Sigma }{\mathsf{\Phi }}_{\mathrm{eff}}>A_{\Sigma }{\mathsf{\Phi }}_{\mathrm{out}}+P{\dot{A}}_{\Sigma }.$$

(49)

When the release and work channels dominate, the shell loses stored quasilocal energy even though the descending reference sector continues to supply positive boundary input.

Equation (42) is the macroscopic conversion law for the timelike boundary. The next section adds the temperature and area-response variables that regulate this conversion during the relaxation history.

6. Temperature Regulation and Area Response

The flux-coupled balance in Eq. (42) describes how reference-energy descent is partitioned into shell storage, exterior release, and pressure–area work. This section adds the macroscopic response variables that regulate this partition. The local boundary temperature $T_{\Sigma }$ parametrizes the thermal response of the timelike shell, while the area response describes how the boundary redistributes input through surface growth.

6.1. Effective Reference-Boundary Response

Along a prescribed macroscopic relaxation trajectory, the local boundary temperature response is parametrized by

$$C_{\mathrm{eff}}:={\displaystyle \frac{d{E}_{\mathrm{ref}}}{d{T}_{\Sigma }}}.$$

(50)

Here $T_{\Sigma }$ is the temperature variable assigned to observers comoving with the shell. The derivative in Eq. (50) is taken along the coarse-grained reference-sector history $\chi \left(\tau \right)$.

Using the definition of the effective boundary input,

$${\mathsf{\Phi }}_{\mathrm{eff}}=-{\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}},$$

(51)

the temperature response can be written as

$${\dot{T}}_{\Sigma }={\displaystyle \frac{{\dot{E}}_{\mathrm{ref}}}{C_{\mathrm{eff}}}}=-{\displaystyle \frac{A_{\Sigma }}{C_{\mathrm{eff}}}}{\mathsf{\Phi }}_{\mathrm{eff}}.$$

(52)

The negative-response branch is defined by

$$C_{\mathrm{eff}}<0.$$

(53)

On this branch,

$${\mathsf{\Phi }}_{\mathrm{eff}}>0⟹{\dot{T}}_{\Sigma }>0.$$

(54)

Thus reference-energy descent, expressed as positive effective boundary input, raises the local shell temperature on a negative-response trajectory.

6.2. Redshifted Boundary Temperature

The temperature $T_{\Sigma }$ is local to the shell. Exterior static observers measure the corresponding redshifted quantity. For a static observer at radius $r_{\mathrm{obs}}$ in the exterior static patch, the quasi-static redshift map is [11]

$$T_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{T}_{\Sigma },$$

(55)

with

$$f_{+}\left(r\right)=1-{\displaystyle \frac{2GM}{r}}-{\displaystyle \frac{{\mathsf{\Lambda }}_{+}{r}^2}{3}}.$$

(56)

This relation is evaluated within the static-patch domain $f_{+}\left(R\right)>0$, $f_{+}\left(r_{\mathrm{obs}}\right)>0$. It maps the shell-local response to the exterior observer frame.

The relation is the quasi-static gravitational redshift between the shell-local frame and exterior static observers. Doppler corrections associated with rapid shell motion are neglected. The approximation is valid when the radial shell motion is slow relative to the exterior static frame, schematically

$$|\dot{R}|\ll \sqrt{f_{+}\left(R\right)}.$$

(57)

For faster motion, the redshift map must be supplemented by the corresponding Doppler factor.

For an asymptotically Schwarzschild exterior, ${\mathsf{\Lambda }}_{+}=0$, one may take $r_{\mathrm{obs}}\to \infty$, so that $f_{+}\left(r_{\mathrm{obs}}\right)\to 1$. In that case,

$$T_{\infty }=\sqrt{f_{+}\left(R\right)}{T}_{\Sigma }.$$

(58)

For Schwarzschild–de Sitter exteriors, the observable temperature is defined relative to finite static observers inside the exterior static patch.

If the exterior observer is held at fixed $r_{\mathrm{obs}}$, and M and ${\mathsf{\Lambda }}_{+}$ are constant over the interval considered, differentiation of Eq. (55) gives

$${\displaystyle \frac{{\dot{T}}_{\mathrm{obs}}}{T_{\mathrm{obs}}}}={\displaystyle \frac{{\dot{T}}_{\Sigma }}{T_{\Sigma }}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\dot{f}}_{+}\left(R\right)}{f_{+}\left(R\right)}},$$

(59)

where

$${\dot{f}}_{+}\left(R\right)=\left({\displaystyle \frac{2GM}{R^2}}-{\displaystyle \frac{2{\mathsf{\Lambda }}_{+}R}{3}}\right)\dot{R}.$$

(60)

Thus the exterior temperature response is controlled by the local boundary heating and by the changing exterior redshift factor.

6.3. Exterior Reference Temperature at the Shell

The area response is driven by the local temperature difference between the shell and the exterior reference state represented at the shell. This shell-local exterior reference temperature is denoted by $T_{\mathrm{out},\Sigma }$. The corresponding exterior-observer value is obtained by the same redshift map,

$$T_{\mathrm{out},\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{T}_{\mathrm{out},\Sigma }.$$

(61)

Therefore

$$T_{\mathrm{obs}}-T_{\mathrm{out},\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right),$$

(62)

and the sign of the temperature excess is preserved inside the same exterior static patch.

The quantity $T_{\mathrm{out},\Sigma }$ fixes the shell-local exterior reference level against which the boundary temperature is measured. In an asymptotically Schwarzschild exterior this reference level can be chosen as the shell-local image of an exterior bath temperature, including the zero-temperature limit. In a Schwarzschild–de Sitter static patch it is the corresponding finite static-patch reference level. The response law depends only on the local difference $T_{\Sigma }-T_{\mathrm{out},\Sigma }$.

6.4. Area Response

The shell area provides a macroscopic channel through which boundary input can be redistributed. In the linear local-response regime the area rate is written as

$${\dot{A}}_{\Sigma }={\mathsf{\Gamma }}_A\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right),{\mathsf{\Gamma }}_A>0.$$

(63)

Equivalently, the response satisfies the monotonic condition

$$T_{\Sigma }>T_{\mathrm{out},\Sigma }⟹{\dot{A}}_{\Sigma }>0.$$

(64)

The coefficient ${\mathsf{\Gamma }}_A$ is a macroscopic response coefficient of the boundary sector.

Combining Eq. (63) with the pressure–area work term in Eq. (42) gives

$$P{\dot{A}}_{\Sigma }=P{\mathsf{\Gamma }}_A\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right).$$

(65)

Thus a positive local temperature excess can drive area growth and convert part of the available boundary input into pressure–area work.

It is useful to introduce the shell energy density per unit area,

$$u_{\Sigma }:={\displaystyle \frac{E_{\Sigma }}{A_{\Sigma }}}=\sigma .$$

(66)

Differentiating and using Eq. (42) yields

$${\dot{u}}_{\Sigma }={\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}-\left(P+u_{\Sigma }\right){\displaystyle \frac{{\dot{A}}_{\Sigma }}{A_{\Sigma }}}.$$

(67)

For $P+u_{\Sigma }>0$, outward area growth dilutes the shell energy density. In this sense the area channel acts as a macroscopic regulator of the boundary state.

Combining the reference-energy descent condition, the negative response branch, and the monotonic area response gives the chain

$${\dot{E}}_{\mathrm{ref}}<0⟺{\mathsf{\Phi }}_{\mathrm{eff}}>0,C_{\mathrm{eff}}<0⟹{\dot{T}}_{\Sigma }>0,T_{\Sigma }>T_{\mathrm{out},\Sigma }⟹{\dot{A}}_{\Sigma }>0.$$

(68)

This chain summarizes the regulated boundary response: descent of the coarse-grained reference energy supplies the shell, the negative-response branch raises the local boundary temperature, and the resulting temperature excess drives area growth, pressure–area work, exterior release, and redshifted observer-accessible response.

7. Boundary Data and Exterior Accessibility

The previous sections established the timelike shell as a finite-response boundary. Reference-energy descent defines the effective input ${\mathsf{\Phi }}_{\mathrm{eff}}$, the shell balance partitions this input into storage, exterior release, and pressure–area work, and the local temperature response regulates the associated area evolution. The boundary data collected in this section are the macroscopic variables through which this response becomes exterior-accessible.

7.1. Entropy-Like Area Variable

The geometric surface variable of the shell is its area,

$$A_{\Sigma }=4\pi R^2.$$

(69)

The area response in Eq. (64) motivates the entropy-like macroscopic area variable

$$S_{\Sigma }:=\alpha A_{\Sigma },\alpha >0.$$

(70)

Equivalently,

$$S_{\Sigma }=4\pi \alpha R^2.$$

(71)

The coefficient $\alpha$ fixes the normalization of the coarse-grained area variable. The boundary-balance structure developed here uses only the area scaling; a statistical closure would determine $\alpha$ and the associated entropy-production law.

Differentiating Eq. (70) gives

$${\dot{S}}_{\Sigma }=\alpha {\dot{A}}_{\Sigma }=8\pi \alpha R\dot{R}.$$

(72)

During an outward response interval, $\dot{R}>0$, this implies

$${\dot{S}}_{\Sigma }>0.$$

(73)

The variable $S_{\Sigma }$ records the macroscopic surface growth associated with the regulated boundary response. In terms of $S_{\Sigma }$, the central shell balance in Eq. (42) becomes

$${\dot{E}}_{\Sigma }=A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)-{\displaystyle \frac{P}{\alpha }}{\dot{S}}_{\Sigma }.$$

(74)

Thus the entropy-like area variable enters the same macroscopic balance as reference-sector input, outward release, shell storage, and pressure–area work.

7.2. Exterior-Accessible Mode Frequencies

Near-boundary modes are represented first by local shell quantities. A mode with shell-local frequency ${\omega }_{\Sigma }$ is seen by an exterior static observer at radius $r_{\mathrm{obs}}$ as

$${\omega }_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{\omega }_{\Sigma },$$

(75)

where

$$f_{+}\left(r\right)=1-{\displaystyle \frac{2GM}{r}}-{\displaystyle \frac{{\mathsf{\Lambda }}_{+}{r}^2}{3}}.$$

(76)

This is the same static-patch redshift map used for the boundary temperature. For an asymptotically Schwarzschild exterior, ${\mathsf{\Lambda }}_{+}=0$, one may take $r_{\mathrm{obs}}\to \infty$, and Eq. (75) reduces to

$${\omega }_{\infty }=\sqrt{f_{+}\left(R\right)}{\omega }_{\Sigma }.$$

(77)

For Schwarzschild–de Sitter exteriors, the exterior-accessible frequency is defined relative to finite static observers inside the exterior static patch.

A minimal localization condition for shell-supported modes is

$$k_{\Sigma }R\lesssim \xi ,\xi =O\left(1\right),$$

(78)

where $k_{\Sigma }$ is the proper wavenumber measured in the shell frame and $\xi$ parametrizes the localization criterion. For relativistic near-shell modes, ${\omega }_{\Sigma }\simeq k_{\Sigma }$. Equation (75) then gives

$${\omega }_{\mathrm{obs}}\lesssim \sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{\displaystyle \frac{\xi }{R}}.$$

(79)

In the asymptotically Schwarzschild case, this becomes

$$f_c\lesssim {\displaystyle \frac{\xi }{2\pi R}}\sqrt{1-{\displaystyle \frac{2GM}{R}}},$$

(80)

where $f_c={\omega }_{\infty }/\left(2\pi \right)$. With the Schwarzschild radius

$$R_S=2GM,$$

(81)

one obtains

$$f_c{R}_S\lesssim {\displaystyle \frac{\xi }{2\pi }}{\displaystyle \frac{R_S}{R}}\sqrt{1-{\displaystyle \frac{R_S}{R}}}.$$

(82)

The right-hand side is maximal at

$$R={\displaystyle \frac{3}{2}}{R}_S,$$

(83)

which yields

$$f_c{R}_S\le {\displaystyle \frac{\xi }{3\sqrt{3}\pi }}.$$

(84)

In the present formulation, this bound is the exterior-accessibility condition for persistent near-boundary spectral features in the asymptotically Schwarzschild case. It follows from two ingredients: localization on the timelike boundary and redshift through the exterior lapse. The coefficient $\xi$ encodes the localization scale of the near-shell mode.

7.3. Macroscopic Boundary Data

The finite-response boundary is characterized by the data

$$\left\{A_{\Sigma },S_{\Sigma },E_{\Sigma },P,T_{\Sigma },{\mathsf{\Phi }}_{\mathrm{eff}},{\mathsf{\Phi }}_{\mathrm{out}}\right\}.$$

(85)

These variables are tied together by the reference-energy descent condition, the effective input definition, the flux-coupled shell balance, the local temperature response, the area-response law, and the exterior redshift map.

The local shell data are defined quasilocally on $\Sigma$. Exterior static observers access redshifted quantities such as

$$T_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{T}_{\Sigma },{\omega }_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{\omega }_{\Sigma }.$$

(86)

The exterior representation therefore depends on both the boundary state and the observer position inside the static patch.

This is the operational boundary role of the TTS. Interior reference-sector change is not assigned directly to exterior observables; it is registered at $\Sigma$ through ${\mathsf{\Phi }}_{\mathrm{eff}}$, redistributed by the shell balance, regulated by the temperature and area response, and mapped to exterior static observers by the redshift factor. The timelike shell thereby provides the finite-radius surface on which reference-state change becomes exterior-accessible boundary response.

8. Stability-Relevant Response Structure

The boundary balance developed above defines a finite-response shell system. Its stability is not determined by the radial background dynamics alone, but by the coupled response of the shell geometry, surface stress, reference-sector input, exterior release, and area channel.

Recent analyses have shown that relativistic self-gravitating membranes and thin shells can develop dynamical non-radial instabilities. Yang, Bonga, and Pan identified high-angular-momentum warping instabilities, while Pitre, Schneider, and Poisson found unstable modes on all angular scales for a static perfect-fluid thin shell [8,9]. These results set the stability benchmark for the present boundary formulation.

In the present framework, the perturbative problem is the stability of the regulated boundary response. The relevant variations include

$$\delta R,\delta h_{ab},\delta \sigma ,\delta p,\delta {\rho }_{\mathrm{ref}},\delta {\mathsf{\Phi }}_{\mathrm{eff}},\delta {\mathsf{\Phi }}_{\mathrm{out}},\delta {\dot{A}}_{\Sigma }.$$

(87)

These variables are linked by the perturbed form of the central balance law,

$$\delta {\dot{E}}_{\Sigma }=\delta \left[A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)\right]-\delta \left(P{\dot{A}}_{\Sigma }\right).$$

(88)

Since

$${\mathsf{\Phi }}_{\mathrm{eff}}=-{\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}},$$

(89)

a perturbation of the reference sector changes the effective input through

$$\delta {\mathsf{\Phi }}_{\mathrm{eff}}=-\delta \left({\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}}\right).$$

(90)

Thus perturbations of the shell radius and perturbations of ${\rho }_{\mathrm{ref}}$ enter the same input channel.

The local area response contributes through

$$\delta {\dot{A}}_{\Sigma }=\delta \left[{\mathsf{\Gamma }}_A\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right)\right].$$

(91)

If ${\mathsf{\Gamma }}_A$ is fixed, this reduces to

$$\delta {\dot{A}}_{\Sigma }={\mathsf{\Gamma }}_A\left(\delta T_{\Sigma }-\delta T_{\mathrm{out},\Sigma }\right).$$

(92)

If ${\mathsf{\Gamma }}_A$ is itself a state-dependent response coefficient, its variation provides an additional closure term.

The temperature perturbation is constrained by

$$\delta {\dot{T}}_{\Sigma }=\delta \left(-{\displaystyle \frac{A_{\Sigma }}{C_{\mathrm{eff}}}}{\mathsf{\Phi }}_{\mathrm{eff}}\right).$$

(93)

Therefore the stability problem couples the reference-sector perturbation, the shell stress perturbation, and the temperature-area response.

Equations (88)–(93) define the response structure that must be closed by a perturbative shell model. A complete stability analysis requires constitutive relations for $\delta p$, $\delta {\mathsf{\Phi }}_{\mathrm{out}}$, $\delta C_{\mathrm{eff}}$, and $\delta {\mathsf{\Gamma }}_A$, together with the angular dependence of shell deformations. The finite-response boundary formulation identifies these closure data explicitly.

9. Discussion: Thermodynamic Boundary Interpretation

The construction developed above gives a boundary-level thermodynamic reading of gravitational collapse. The central object is not a null horizon, but a finite-radius timelike shell carrying surface stress, quasilocal energy, temperature response, area response, and redshifted exterior observables. The main result is the conversion of reference-energy descent into the boundary balance

$${\dot{E}}_{\Sigma }=A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)-P{\dot{A}}_{\Sigma }.$$

(94)

This equation places the thermodynamic bookkeeping at the timelike boundary: reference-sector input, exterior release, shell storage, and pressure–area work are defined on the same quasilocal surface.

9.1. Relation to Horizon Thermodynamics

Black-hole thermodynamics established that gravitational systems can assign temperature and entropy to geometric boundary structures. Hawking radiation relates black-hole temperature to horizon structure, while the Unruh effect relates thermality to accelerated observers and the observer-dependent decomposition of fields [1,2]. These results motivate the use of local boundary temperature and redshifted observer quantities as central variables in gravitational settings.

The present construction transfers this boundary logic to a timelike surface. The temperature $T_{\Sigma }$ is assigned locally to the shell, and exterior static observers access the redshifted value

$$T_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{T}_{\Sigma }.$$

(95)

Thus the temperature response is a local boundary property, while its exterior representation depends on the lapse and on the observer position inside the static patch. The same redshift structure controls exterior-accessible near-boundary mode frequencies.

The distinction between shell-local and exterior-accessible quantities is essential for the finite-radius setting. In an asymptotically Schwarzschild exterior, spatial infinity provides a natural reference frame. In a Schwarzschild–de Sitter exterior, the same construction is read relative to finite static observers. The thermodynamic boundary interpretation therefore does not require a preferred asymptotic observer; it uses the static-patch redshift map appropriate to the exterior geometry.

9.2. Boundary Balance and Local Thermodynamic Response

Jacobson’s thermodynamic derivation of the Einstein equation showed that local causal boundaries, heat flow, and area-proportional entropy can organize gravitational dynamics [3]. In the present framework the corresponding organizing principle is the timelike shell balance. The effective input

$${\mathsf{\Phi }}_{\mathrm{eff}}=-{\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}}$$

(96)

translates the descent of the coarse-grained reference energy into an areal boundary input. The shell then partitions this input into storage, exterior release, and pressure–area work.

The role of the temperature response is to regulate this partition. Along the negative-response branch,

$$C_{\mathrm{eff}}<0,{\dot{T}}_{\Sigma }=-{\displaystyle \frac{A_{\Sigma }}{C_{\mathrm{eff}}}}{\mathsf{\Phi }}_{\mathrm{eff}},$$

(97)

a positive effective input raises the local boundary temperature. If the shell temperature exceeds the exterior reference temperature at the shell, the area response

$${\dot{A}}_{\Sigma }={\mathsf{\Gamma }}_A\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right)$$

(98)

converts part of the available input into surface growth and pressure–area work. The thermodynamic reading is therefore not attached only to an area law; it is encoded in the coupled response of input, storage, release, temperature, and area.

9.3. Area Variable and Entropy Interpretation

The entropy-like variable

$$S_{\Sigma }=\alpha A_{\Sigma }$$

(99)

records the macroscopic area response of the boundary. Its normalization is set by $\alpha$, while the present analysis uses only the area scaling. Written in terms of $S_{\Sigma }$, the shell balance becomes

$${\dot{E}}_{\Sigma }=A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)-{\displaystyle \frac{P}{\alpha }}{\dot{S}}_{\Sigma }.$$

(100)

This form identifies the entropy-like area variable as the surface-growth counterpart of pressure–area work.

In relation to entropic-gravity and boundary-information perspectives, such as Verlinde’s proposal [12], the present framework supplies a finite-radius quasilocal balance that any microscopic or statistical boundary closure must match. The relevant constraint is Eq. (100): a closure has to reproduce the partition of reference-sector input into shell storage, exterior release, and pressure–area work, together with the redshift map from local boundary variables to exterior observables.

9.4. Role of the Timelike Boundary

The TTS gives the thermodynamic interpretation a definite geometric support. The boundary variables

$$A_{\Sigma },S_{\Sigma },E_{\Sigma },P,T_{\Sigma },{\mathsf{\Phi }}_{\mathrm{eff}},{\mathsf{\Phi }}_{\mathrm{out}}$$

are all defined on the same timelike hypersurface. The exterior-accessible data are obtained from these local variables by the static-patch redshift map. This makes the shell the finite-radius interface at which reference-state change is converted into observable boundary response.

The discussion also fixes the main closure problems. A perturbative closure must specify the coupled variations of surface stress, reference-density input, exterior release, temperature response, and area response. A statistical or microscopic closure must determine the normalization $\alpha$, the response coefficient $C_{\mathrm{eff}}$, the area coefficient ${\mathsf{\Gamma }}_A$, and the release channel ${\mathsf{\Phi }}_{\mathrm{out}}$. The boundary balance derived here sets the macroscopic structure that these closures have to reproduce.

The boundary formulation also separates substrate and exterior representation. The coarse-grained reference sector is specified by ${\rho }_{\mathrm{ref}}\left(\chi \right)$ and $E_{\mathrm{ref}}$, whereas exterior observers access only the boundary-mediated variables obtained after storage, release, area response, and redshift. In this sense, thermodynamic or field-theoretic exterior signatures are boundary representations of reference-state change. A microscopic or quantum-field-theoretic closure would therefore have to reproduce this conversion from coarse-grained reference dynamics to exterior-accessible observables.

9.5. Cosmological Reference Level and Boundary Growth

The boundary response also admits a cosmological reading when the exterior reference level is tied to an expanding background. In the local response law

$${\dot{A}}_{\Sigma }={\mathsf{\Gamma }}_A\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right),$$

(101)

the quantity $T_{\mathrm{out},\Sigma }$ is the shell-local representation of the exterior reference state. If this reference level decreases with cosmological expansion, the local difference $T_{\Sigma }-T_{\mathrm{out},\Sigma }$ can increase for fixed or rising $T_{\Sigma }$. The same boundary law then converts cosmological cooling of the exterior reference state into outward area response.

This gives the model a local mechanism relevant to current discussions of cosmologically coupled compact objects. Observational studies have reported evidence for black-hole mass growth correlated with the expansion history and have discussed its possible relation to an astrophysical source of dark energy [13]. Theoretical analyses have emphasized that cosmological coupling is naturally formulated in terms of quasilocal mass and can arise for nonsingular compact objects or black-hole mimickers [14]. At the same time, gravitational-wave constraints and population studies provide important tests of the allowed coupling strength [15].

In the present framework this connection is formulated as a local boundary response rather than as a prescribed global mass-scaling relation. The interior reference sector is vacuum-energy-like, through ${\mathsf{\Lambda }}_{-}\left(\chi \right)=8\pi G{\rho }_{\mathrm{ref}}\left(\chi \right)$, while the exterior cosmological environment enters through the shell-local reference level $T_{\mathrm{out},\Sigma }$ and the redshift map. A cosmological closure would therefore have to specify how $T_{\mathrm{out},\Sigma }$, ${\mathsf{\Phi }}_{\mathrm{out}}$, and the exterior parameters evolve with the background expansion. The boundary balance derived here then supplies the local conversion law linking reference-state change, exterior cooling, area response, and exterior-accessible observables.

10. Conclusions

This work has formulated a macroscopic boundary-balance framework for gravitational collapse based on a finite-radius timelike thin shell. The shell separates an effective interior vacuum reference sector from an exterior Schwarzschild or Schwarzschild–de Sitter region and provides the quasilocal surface on which reference-state change is registered.

The interior reference sector was described by a coarse-grained density ${\rho }_{\mathrm{ref}}\left(\chi \right)$ and the associated reference energy

$$E_{\mathrm{ref}}={\displaystyle \frac{4\pi }{3}}{R}^3{\rho }_{\mathrm{ref}}.$$

(102)

Its proper-time variation separates into an intrinsic density-relaxation term and a geometric volume term. Reference-energy descent occurs when the decrease of ${\rho }_{\mathrm{ref}}$ dominates the geometric increase of the enclosed volume. This condition defines the effective boundary input

$${\mathsf{\Phi }}_{\mathrm{eff}}=-{\displaystyle \frac{1}{A_{\Sigma }}}{\dot{E}}_{\mathrm{ref}},$$

(103)

which is positive precisely on the descending-reference branch.

The central result is the flux-coupled shell balance

$${\dot{E}}_{\Sigma }=A_{\Sigma }\left({\mathsf{\Phi }}_{\mathrm{eff}}-{\mathsf{\Phi }}_{\mathrm{out}}\right)-P{\dot{A}}_{\Sigma }.$$

(104)

This equation converts reference-energy descent into a finite boundary response and partitions the available input into quasilocal shell storage, outward release, and pressure–area work. It is the macroscopic conversion law of the timelike boundary.

The response was supplemented by a local boundary temperature $T_{\Sigma }$ and a trajectory-dependent coefficient

$$C_{\mathrm{eff}}={\displaystyle \frac{d{E}_{\mathrm{ref}}}{d{T}_{\Sigma }}}.$$

(105)

On the negative-response branch, $C_{\mathrm{eff}}<0$, positive effective input raises the local shell temperature. A local temperature excess relative to the exterior reference level then drives area response,

$${\dot{A}}_{\Sigma }={\mathsf{\Gamma }}_A\left(T_{\Sigma }-T_{\mathrm{out},\Sigma }\right),$$

(106)

and thereby contributes to pressure–area work. The entropy-like variable

$$S_{\Sigma }=\alpha A_{\Sigma }$$

(107)

records this macroscopic surface growth and enters the same balance through the work term.

Local shell quantities are mapped to exterior static observers by the exterior lapse. For a static observer at radius $r_{\mathrm{obs}}$,

$$T_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{T}_{\Sigma },{\omega }_{\mathrm{obs}}=\sqrt{{\displaystyle \frac{f_{+}\left(R\right)}{f_{+}\left(r_{\mathrm{obs}}\right)}}}{\omega }_{\Sigma }.$$

(108)

Spatial infinity is therefore a special feature of the asymptotically Schwarzschild case. In Schwarzschild–de Sitter exteriors, exterior-accessible quantities are defined relative to finite static observers inside the exterior static patch.

The framework also identifies the stability-relevant response structure. Perturbations of the shell geometry, surface stress, reference density, effective input, exterior release, temperature response, and area response are coupled by the perturbed boundary balance. Known non-radial instabilities of relativistic self-gravitating membranes and thin shells therefore provide a benchmark for the closure problem, while the present formulation specifies the macroscopic variables that such a perturbative analysis must vary.

The resulting picture is a classical quasilocal boundary framework for collapse. Its main content is not the radial motion of the shell alone, but the finite boundary response through which reference-sector change is converted into storage, release, work, area growth, and redshifted exterior-accessible quantities. This provides the macroscopic balance structure required for perturbative, finite-thickness, statistical, or microscopic boundary closures.