Gravitational boundary mechanics

Manuscript framing

The study of how gravitational action and energy assignments respond when data on a region’s boundary change. In these papers the central variables are the cut’s area and the selected normal rate. The force is a response to acceleration length, not an additional bulk material force law. The underlying field equations remain Einstein’s.

Related: Action and Einstein–Hilbert action · Source and conjugate response · Scope of the canonical theorem

Source: Seven: §§1–3. Full: Part I; §§I–II.

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Cut area and area element

Established concept

A spacelike cut has a two-metric $q_{AB}$ and area element $\epsilon_S=\sqrt q\,d^2x$; $A$ is its integral. Area is intrinsic transverse geometry. For a round sphere, $A=4\pi R^2$, but a Rindler patch need not be spherical. $A/4=\pi R^2$ does not establish a physical disk as more fundamental than the cut.

$$A=\int_S\epsilon_S$$

Related: Areal radius · Boost charge or boost momentum

Source: Seven: §§2–3. Full: §II.A; §V.A.

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Surface gravity and its normalization

Established concept

The acceleration-valued rate $\kappa$ assigned to a chosen normal evolution. On a Killing horizon it is the clock-normalized horizon inaffinity multiplied by $c^2$. It is not generally the proper acceleration of a freely falling observer. Clock choice and the horizon prescription matter. The regular variable is $\kappa$, even where its reciprocal length diverges.

$$\kappa=c^2\widehat\kappa$$

Related: Signed and positive acceleration length · Cai–Kim instantaneous calibration · Signed Kodama–Hayward rate

Source: Seven: §2; §7. Full: §§II–III.

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Signed and positive acceleration length

Manuscript construction

The reciprocal-rate length selected as the mechanical source coordinate. $L_s$ retains the sign of $\kappa$; $L_\kappa$ is its magnitude. It is a light-travel length associated with the timescale $c/|\kappa|$. It equals a proper horizon distance exactly in Rindler, $2R_H$ for standard Schwarzschild, and $R_A$ by the Cai–Kim prescription. It is not a universal proper distance.

$$L_s=c^2/\kappa,\qquad L_\kappa=|L_s|=c^2/|\kappa|$$

Related: Signed branch and zero surface gravity · Proper distance · Physical clock and normalization

Source: Seven: §2. Full: Part I; §§II.B, V.

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Energy-valued boundary potential

Established concept

The scalar area–rate function used throughout the papers. It has energy units and a horizon boundary-Hamiltonian role, but need not be the total energy. For Schwarzschild it is $Mc^2/2$; for Cai–Kim FRW it equals the horizon Misner–Sharp energy. Rindler supplies a finite-patch value without an enclosed source mass. Do not confuse this function with the symplectic potential.

$$U_\partial=\frac{Ac^2\kappa}{8\pi G}=\frac{Ac^4}{8\pi G L_s}$$

Related: The mechanical boost-product dictionary · Hamiltonian generator · Complete boundary differential

Source: Seven: §§1–2. Full: Part I.

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Acceleration-length force

Manuscript construction

The negative fixed-area derivative of the boundary potential with respect to signed acceleration length. In the selected null ensemble it is also the response per unit boundary time to that source. It has force units. A material traction or force on an instrument additionally requires an operational displacement and clock protocol.

$$F_L=-\left(\frac{\partial U_\partial}{\partial L_s}\right)_A=\frac{A\kappa^2}{8\pi G}=M_\partial\kappa$$

Related: Normal-scale pressure · Boundary mass equivalent · Momentum conjugate to acceleration length

Source: Seven: §§2–3. Full: §II.B; §IV.B.

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Normal-scale pressure

Manuscript construction

The acceleration-length force divided by transverse area. Its mechanical work is $P_LA\,dL_s$. It is distinct from the two-dimensional surface stress and the cosmic fluid pressure. Squaring removes the branch sign but not dependence on the selected physical clock.

$$P_L=F_L/A=\kappa^2/(8\pi G)=c^4/(8\pi G L_s^2)$$

Related: Normal-volume one-form · Membrane paradigm and tangential balance · Scope of the canonical theorem

Source: Seven: §2. Full: Part I; §IV.A–B.

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Surface pressure or tension

Established concept

The area coefficient $\Pi_\partial$ of the boundary potential. It has energy-per-area or force-per-length units, unlike $P_L$. It is the type of two-dimensional stress used in membrane equations. Its oriented sign depends on conventions. The mechanical representation resolves it as pressure times acceleration length.

$$\Pi_\partial=\frac{c^2\kappa}{8\pi G}=P_LL_s,\qquad d\Pi_\partial=-P_L\,dL_s$$

Related: Membrane paradigm and tangential balance · Pullback · Normal force balance and Young–Laplace form

Source: Seven: §2. Full: §IV.A,C–D.

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Complete boundary differential

Manuscript identity

Changing both area and rate gives two terms in one exact differential. The first changes transverse geometry; the second changes the normal scale. They are not two independent stored energies. Along a physical horizon family both variables may change together. The usual mass first law varies a different Hamiltonian.

$$dU_\partial=\Pi_\partial\,dA-F_L\,dL_s$$

Related: The mechanical boost-product dictionary · Euler products and homogeneity · Integrability and separate scale energy

Source: Seven: §§2,4. Full: Part I; §§III.B,V.C.

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Signed and magnitude boost rate

Established concept

For the stationary normal flow, $\omega_s$ is the signed boost-angle rate and $\Omega_\kappa$ its magnitude. They have inverse-time units. Multiplying the action-valued boost charge by the rate gives the energy-valued boundary potential. This is a mechanical definition before any thermal conversion.

$$\omega_s=\kappa/c,\quad\Omega_\kappa=|\kappa|/c,\quad L_s=c/\omega_s$$

Related: Normal boost angle or rapidity · Physical clock and normalization

Source: Seven: §2; §3.1. Full: §§II.C,IV.E.

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Boost charge or boost momentum

Established concept

The gravitational momentum conjugate to the relative normal boost angle in the Einstein corner structure. It has action units. It is not ordinary rotational angular momentum and can be nonzero in Schwarzschild. A charge labels a generator or response; it is not automatically conserved when area changes. Its normalization also underlies Wald/Einstein horizon entropy.

$$J_\eta=\frac{Ac^3}{8\pi G}$$

Related: Gravitational charge and Noether charge · Boundary corner or joint · The mechanical boost-product dictionary

Source: Seven: §3.1. Full: §II.C.

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The mechanical boost-product dictionary

Explanatory consolidation

The identity in the user’s screenshot makes the action scale behind the energy–length product explicit. At fixed area, changing the normal rate changes energy and reciprocal length inversely. With signed variables the product is $U_\partial L_s$; with magnitudes it is $|U_\partial|L_\kappa$. This consolidates existing equations, rather than introducing another conserved energy.

$$U_\partial=\omega_sJ_\eta,\qquad U_\partial L_s=cJ_\eta,\qquad dU_\partial=\omega_s\,dJ_\eta+J_\eta\,d\omega_s$$

Related: Energy-valued boundary potential · Boost charge or boost momentum · Mechanical-to-thermal dictionary

Source: Seven: §§2–4 combined. Full: §II.C and original Eq. (84); explicit in revision.

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Boundary mass equivalent

Manuscript definition

Energy divided by $c^2$ for the selected boundary potential. It is not generally total black-hole mass, particle rest mass, or enclosed fluid mass. The identity $F_L=M_\partial\kappa$ gives an exact mass-times-rate representation, not a new equation of motion for a body.

$$M_\partial=U_\partial/c^2$$

Related: Misner–Sharp energy · Schwarzschild realization

Source: Seven: §2. Full: Part I.

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Action and Einstein–Hilbert action

Established concept

The action is an energy-times-time functional of fields and a spacetime region. The Einstein–Hilbert term contains the Ricci scalar; matter, boundary, and corner terms complete the stated problem. Its first variation gives equations in the interior and source–response terms on the boundary. Boundary terms do not generally vanish merely because the bulk equations hold.

Related: On shell and off shell · Symplectic potential · GHY, null boundary, and corner terms

Source: Seven: §3. Full: §II; Appendix C.

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Cut, cross-section, and codimension

Established concept

A cut is a two-dimensional spacelike cross-section of a three-dimensional boundary hypersurface in four-dimensional spacetime. It therefore has codimension two. The boundary’s history is a worldtube, not the cut itself. A condition on each cut need not be a condition along the actual generator of the entire worldtube.

Related: Null hypersurface · Cut area and area element · Apparent horizon and marginal sphere

Source: Seven: §3. Full: §II.A; §VI.C.

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Null hypersurface

Established concept

A hypersurface whose normal is null and also tangent to it. Its ruling curves are lightlike generators. Its induced three-metric is degenerate, while a spacelike cut has a nondegenerate two-metric. A hypersurface is not necessarily a horizon simply because it is null. The main canonical calculation is restricted to such a surface and appropriate variations.

Related: Evolution vector and null generator · Null expansion · Nonexpanding null sector

Source: Seven: §3. Full: §II.A.

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Evolution vector and null generator

Established concept

The vector field specifying how points or cuts are followed. For a null boundary it is tangent to its null generators. The physical vector and its normalization are part of the boundary prescription. Holding a vector fixed in field space does not in general hold its inaffinity fixed, since the spacetime connection can change.

$$\nabla_\xi\xi^a=\widehat\kappa_\xi\xi^a$$

Related: Inaffinity and affine parameter · Physical clock and normalization · Auxiliary gauge versus physical clock change

Source: Seven: §3. Full: §III.A–B.

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Inaffinity and affine parameter

Established concept

Inaffinity measures failure of the chosen null-generator parameter to be affine. The derivative of its tangent is proportional to the tangent itself. It is not the proper acceleration of a null observer: no null observer has proper time. With a length parameter, $\widehat\kappa$ has inverse-length units; multiplying by $c^2$ gives the paper’s surface-gravity units.

$$\nabla_\xi\xi^a=\widehat\kappa\xi^a$$

Related: Surface gravity and its normalization · Evolution vector and null generator · Physical clock and normalization

Source: Seven: §3. Full: §II.A.

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Null expansion

Established concept

Fractional change of the cut’s area element along the selected null generator. Positive expansion means neighboring generators spread; negative expansion means contraction. Its units depend on the generator parameter; the canonical formula uses inverse length. A vanishing expansion for one null normal on a marginal cut does not make the entire apparent-horizon worldtube nonexpanding.

$$\theta=\frac{1}{\sqrt q}\mathcal L_\xi\sqrt q$$

Related: Shear and the trace-free tensor sector · Mixed null scalar source · Nonexpanding null sector

Source: Seven: §3. Full: §II.A; §VI.C.

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Shear and the trace-free tensor sector

Established concept

Shape distortion of a bundle of generators, separated from its fractional area change. The trace-free part of the cut metric pairs with densitized shear in the null symplectic potential. It is a distinct canonical contribution which cannot be discarded on a general evolving boundary just because the scalar rate has been reparameterized.

Related: Null expansion · Scalar, vector, and tensor sectors · Symplectic potential

Source: Full: §II.A; needed for the scope of seven §3.

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Scalar, vector, and tensor sectors

Established concept

A decomposition of the boundary response by its geometric character on a cut. The scalar sector contains area and the mixed normal/expansion rate; the vector sector involves generator/twist data; the trace-free tensor sector involves conformal shape and shear. These are not additional types of matter or extra unconstrained bulk graviton species.

Related: Mixed null scalar source · Shear and the trace-free tensor sector · Normal connection and twist data

Source: Seven: §3. Full: §II.A.

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Mixed null scalar source

Established concept

The scalar rate appearing in the four-dimensional Einstein null canonical pair. It combines inaffinity and half the expansion. It is a derivative-containing geometric source, not material pressure and not a quantum mixed state. On the nonexpanding sector it reduces to inaffinity; on a generic expanding boundary that reduction is unavailable.

$$\mu=\widehat\kappa+\frac12\theta$$

Related: Area and rate polarization · Nonexpanding null sector · Expanded scalar response beyond nonexpansion

Source: Seven: §3. Full: §II.A–B.

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Physical clock and normalization

Established concept

The rule fixing the parameter of the selected normal evolution. Rindler uses observer proper time; standard Schwarzschild uses time normalized at infinity. It determines the numerical rate, energy, length, and pressure. This is different from simultaneously rescaling an auxiliary null vector and compensating its multiplier so the physical evolution vector remains unchanged.

Related: Boundary lapse · Auxiliary gauge versus physical clock change · Signed and magnitude boost rate

Source: Seven: §§1–3. Full: §III.A.

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Variation, differential, and derivative

Established concept

$\delta$ compares nearby fields or source assignments; $d$ differentiates a displayed state function or a spacetime form, with context identifying which. A partial derivative fixes the subscripted variables. A physical path can link quantities that are independent virtual source coordinates. Confusing these operations can turn a fixed-area work coefficient into an incorrect total derivative.

Related: Source and conjugate response · Pullback · Integrability and separate scale energy

Source: Seven: §§2–4. Full: §IV.B; §VII.A.

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Source and conjugate response

Established concept

A source is boundary data controlled in the selected variational problem. A response is the coefficient multiplying its variation. In rate polarization the source is $\mu$ or its allowed reciprocal coordinate; the area density supplies the conjugate coefficient. “Source” does not mean a new bulk stress tensor or a material source creating all the gravity.

$$\delta I_{\mathrm{os}}=\int\text{response}\times\delta(\text{source})$$

Related: Permitted or admissible source variation · Hamilton–Jacobi boundary response · Area and rate polarization

Source: Seven: §3. Full: §II.A–B.

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Symplectic potential

Established concept

A one-form on field space obtained from the action variation. It is linear in the variation supplied to it, like $p\delta q$ in mechanics. It is not an energy potential. Its field-space derivative gives the canonical two-form. Boundary terms can change its polarization; corner contributions must be retained or controlled.

$$\boldsymbol\Omega=\delta\Theta$$

Related: Action and Einstein–Hilbert action · One-form and response one-form · Energy-valued boundary potential

Source: Seven: §3. Full: §II.

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Symplectic or presymplectic two-form

Established concept

The antisymmetric pairing of two variations that encodes conjugate variables. It is presymplectic before gauge degeneracies are removed. In the reduced scalar sector, the area/rate and length/force expressions describe the same pair, not two extra independent canonical pairs. Its symbol is distinct from the boost rate.

$$\boldsymbol\Omega/\Delta\tau=\delta A\wedge\delta\Pi_\partial=\delta L_s\wedge\delta F_L$$

Related: Wedge product and exterior derivative · Area and rate polarization · Auxiliary gauge versus physical clock change

Source: Full: §II.B, original Eq. (26).

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Area and rate polarization

Established concept

A choice of which member of a canonical pair is treated as the displayed configuration/source. Adding the exact variation of the area–rate product exchanges $-\mu\delta\epsilon_S$ for $\epsilon_S\delta\mu$ in the stated conventions. The canonical two-form does not change. This choice is meaningful only together with the full action, corners, and remaining boundary data.

$$-p\,\delta q+\delta(pq)=q\,\delta p$$

Related: Boundary Legendre transform · Counterterms and reference subtraction · Source and conjugate response

Source: Seven: §3. Full: §II.A.

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Boundary Legendre transform

Established concept

Adding or subtracting a product of conjugate variables from the boundary action to exchange the controlled source. It changes the variational problem’s presentation. The area-to-rate exchange is such a transform. The invertible substitution $L_s=c^2/\kappa$ afterward is only a coordinate change within that selected rate polarization.

Related: Area and rate polarization · Rescaled timelike stress source

Source: Seven: §3. Full: §II.A–B; Appendix C.

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Permitted or admissible source variation

Established concept

A variation compatible with the stated boundary class, clock prescription, cut identification, matter data, and remaining source restrictions. The null theorem uses variations tangent to the nonexpanding sector. Across neighboring boundary problems the source can vary; within one conservative fixed-source problem it is fixed. Algebra alone does not establish solution existence for arbitrary boundary assignments.

Related: On shell and off shell · Nonexpanding null sector · Hamilton–Jacobi boundary response

Source: Seven: §3. Full: §II.A–B.

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Nonexpanding null sector

Established concept

The selected null-boundary class with zero expansion, together with variations keeping it zero. Both conditions are needed to reduce the varied mixed source to surface gravity. Stationary Rindler and black-hole horizons realize this setting. A sequence of marginal FRW cuts need not.

$$\theta=0,\qquad\delta\theta=0$$

Related: Apparent horizon and marginal sphere · Scope of the canonical theorem · Expanded scalar response beyond nonexpansion

Source: Seven: §3. Full: §II.B.

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On shell and off shell

Established concept

On-shell fields satisfy the equations of motion. Off-shell fields need not. The on-shell action is evaluated on solutions as a functional of their allowed boundary data. It may be nonzero and may respond when the prescribed boundary data change. Stationarity within a fixed-source problem does not set all source-response derivatives to zero.

Related: Action and Einstein–Hilbert action · Hamilton–Jacobi boundary response

Source: Full: §II.B; Appendix C. Background to seven §3.

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Hamilton–Jacobi boundary response

Established concept

The derivative of the action evaluated on solutions with respect to boundary data. In the selected length-source ensemble, the functional derivative is a force because the response is integrated over boundary time. A constant-source derivative over a fixed interval is instead force times duration. This is an action response, not automatically a material-force measurement.

$$\delta I_{\mathrm{os}}=-\int d\tau F_L\delta L_s,\qquad F_L=-\delta I_{\mathrm{os}}/\delta L_s(\tau)$$

Related: Momentum conjugate to acceleration length · Source and conjugate response · On shell and off shell

Source: Full: §II.B, original Eq. (24); Appendix C.

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Boundary corner or joint

Established concept

A codimension-two intersection of boundary pieces, such as an initial/final cut meeting a side boundary. The relative orientation of the boundary normals contributes to the gravitational action. The area–boost pair lives here. Ignoring endpoint variations requires an explicit restriction; it is not automatic for moving boundaries.

Related: Normal boost angle or rapidity · Boost charge or boost momentum · GHY, null boundary, and corner terms

Source: Seven: §3.1. Full: §II.C; Appendix C.

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Normal boost angle or rapidity

Established concept

The Lorentzian hyperbolic angle relating normal frames in the two-plane perpendicular to a cut. The corner variable is a relative frame angle, not necessarily the naive velocity rapidity of a moving cosmological sphere. For stationary boost flow its accumulated value equals the boost rate times the prescribed interval.

$$\eta=\omega_s\Delta\tau=\kappa\Delta\tau/c$$

Related: Boost charge or boost momentum · Momentum conjugate to acceleration length · Normal time–radial plane and normal bundle

Source: Seven: §3.1. Full: §II.C; §VI.G.

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Momentum conjugate to acceleration length

Established concept

The coefficient of $\delta L_s$ when the corner one-form $J_\eta\delta\eta$ is expressed at fixed stationary interval. It has momentum units, not pressure units. Dividing its magnitude by the prescribed time interval gives the force. It is an endpoint/source conjugate, not automatically the momentum of a moving material shell.

$$p_L=-\frac{c\Delta\tau J_\eta}{L_s^2}=-\Delta\tau F_L$$

Related: Normal boost angle or rapidity · Acceleration-length force · Hamilton–Jacobi boundary response

Source: Full: §II.C, original Eq. (31); seven §3.1.

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Gravitational charge and Noether charge

Established concept

A quantity associated with a specified gravitational transformation or evolution. A surface Noether charge is constructed from the action and the generator. It is not automatically a conserved total energy or an integrable Hamiltonian. The transformation parameter determines its natural units: boost angle gives an action-valued charge; time evolution gives energy.

Related: Boost charge or boost momentum · Hamiltonian generator · Noether current and surface-charge form

Source: Seven: §§1,4. Full: §III.B; §VI.E.

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Hamiltonian generator

Established concept

A state function whose variation generates the selected evolution through the symplectic form. Its existence depends on integrability, boundary conditions, and flux treatment. The complete gravitational Hamiltonian variation includes more than the variation of a Noether surface term. In particular, $U_\partial$ is not universally the ADM mass energy.

Related: Gravitational charge and Noether charge · Integrability and separate scale energy · Standard black-hole mass first law

Source: Seven: §4. Full: §III.B; §VII.A.

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Standard black-hole mass first law

Established concept

The Hamiltonian mass relation for a specified horizon evolution. In Schwarzschild it is $d(Mc^2)=T\,dS$. This is not obtained by simply differentiating $U_\partial=TS$. The latter is $Mc^2/2$ in that sector, and its full differential contains both area and scale contributions. Adding its scale term to the mass law would double-count the constrained variation.

Related: Smarr relation · Complete boundary differential · Schwarzschild realization

Source: Seven: §4. Full: §III.B; §V.C.

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Normal-volume one-form

Manuscript definition

The volume-valued work displacement $A\,dL_s$. On the independent source space it is not generally the differential of a global volume function. At fixed area it is exact; along CK spherical data it equals the areal-volume differential. Do not replace it by $d(AL_s)$ while leaving the area coefficient unchanged.

$$\vartheta_V=A\,dL_s,\quad d\vartheta_V=dA\wedge dL_s,\quad d(AL_s)=A\,dL_s+L_s\,dA$$

Related: Product volume · Areal versus proper spatial volume · Integrability and separate scale energy

Source: Seven: §2. Full: Part I; §IV.B.

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Euler products and homogeneity

Established concept

The boundary potential is linear in area and inverse-linear in acceleration length. Multiplying each source derivative by its coordinate gives the same potential, with the appropriate sign. These equal products are not independently variable energy reservoirs; their differential contributions differ.

$$U_\partial=\Pi_\partial A=F_LL_s$$

Related: Smarr relation · The mechanical boost-product dictionary · Complete boundary differential

Source: Seven: §2. Full: Part I; §IV.E.

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Rindler horizon and finite patch

Established concept

The observer-dependent horizon of eternal uniform acceleration in Minkowski spacetime. The observer’s proper time calibrates $\kappa=a$ and the exact proper horizon distance is $c^2/a$. A finite transverse patch supplies an area. No spherical center, enclosed mass, or spacetime curvature is needed for the boundary response.

$$L_\kappa=c^2/a,\qquad P_L=a^2/(8\pi G)$$

Related: Proper distance · FGP local energy and proper-distance prescription · Scope of the canonical theorem

Source: Seven: §5.1. Full: §V.A.

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Schwarzschild realization

Established concept

For time normalized at infinity, the stationary black-hole radius is $R_H=2GM/c^2$ and $L_\kappa=2R_H$. The boundary potential is half the total mass energy. The generalized force is conjugate to $2R_H$; its radius-work coefficient is twice as large. This dictionary is sector-specific.

$$\kappa=\frac{c^2}{2R_H},\quad F_L=\frac{c^4}{8G},\quad U_\partial=\frac{Mc^2}{2}$$

Related: Smarr relation · FGP local energy and proper-distance prescription · Radius-work coefficient

Source: Seven: §§4,5.2. Full: §V.

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De Sitter stationary cosmological horizon

Established concept

The constant-positive-vacuum-energy cosmology has a stationary horizon in its static-patch description. It is a special case where the cosmological apparent and event horizons coincide. With matched normalization, the CK and Hayward magnitudes agree and $L_\kappa=R_A$. It provides a stationary spherical overlap, not a proof for every evolving FRW screen.

Related: Cai–Kim instantaneous calibration · Signed Kodama–Hayward rate · Scope of the canonical theorem

Source: Full: §V.E; §VI.D; Appendix C. Revised scope discussion.

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Smarr relation

Established concept

An integrated gravitational scaling relation. In Schwarzschild it becomes the exact equality shown below, with both products equal to the same boundary potential. Rotation and charge introduce additional terms in their respective black-hole mass relations. The equality alone does not prove independent kinetic and potential energies.

$$Mc^2=TS+F_LL_\kappa,\qquad TS=F_LL_\kappa=Mc^2/2$$

Related: Euler products and homogeneity · Standard black-hole mass first law · Virial analogy and formal free energy

Source: Seven: §5.2. Full: §V.C.

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FRW or FLRW cosmology

Established concept

A homogeneous, isotropic spacetime with scale factor $a(t)$ and spatial curvature parameter $k$. The Hubble rate is $H=\dot a/a$. Its apparent-horizon sphere is quasi-local and need not be a permanent causal barrier. Homogeneity is crucial to identifying the Misner–Sharp horizon energy with total bulk density times areal volume using Einstein’s equations.

Related: Apparent horizon and marginal sphere · Cai–Kim instantaneous calibration · Total bulk energy density

Source: Seven: §5.3. Full: §VI; Appendix B.

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Apparent horizon and marginal sphere

Established concept

A marginal spherical cut has one vanishing null expansion. The locus of such cuts defines an apparent-horizon history, which need not be null or nonexpanding along its actual tangent. In FRW the radius depends on expansion and spatial curvature. It is not always the Hubble radius.

$$R_A=\frac{c}{\sqrt{H^2+kc^2/a^2}}$$

Related: Boundary worldtube and causal character · Nonexpanding null sector · Scope of the canonical theorem

Source: Seven: §5.3. Full: §VI.A,C.

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Cai–Kim instantaneous calibration

Established concept

The separately specified apparent-horizon radius prescription used for the clean FRW screen-state realization. It sets a positive normal rate and temperature parameter. It is not generally the full dynamical Hayward surface gravity or a comoving observer’s acceleration. The associated mechanical screen identities can be exact without an exact stationary KMS state.

$$\kappa_{\mathrm{CK}}=c^2/R_A,\quad L_{\mathrm{CK}}=R_A,\quad k_BT_{\mathrm{CK}}=\hbar c/(2\pi R_A)$$

Related: Positive FRW screen pressure · Signed Kodama–Hayward rate · Screen-state law versus matter-flux law

Source: Seven: §5.3. Full: §VI.C.

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Misner–Sharp energy

Established concept

The geometric quasi-local energy specific to spherical symmetry. On a marginal sphere its geometric definition gives $c^4R/(2G)$. Equality to the actual homogeneous fluid energy $\rho V$ is the Einstein–FRW matter–geometry relation, not a consequence of dividing an arbitrary geometric energy by volume. It must not be identified with every other boundary energy.

$$E_{\mathrm{MS}}=\frac{c^4R}{2G}\left(1-h^{ab}\partial_aR\partial_bR\right)$$

Related: Total bulk energy density · Areal versus proper spatial volume · Friedmann constraint in pressure variables

Source: Seven: §5.3. Full: §VI.A; source-archive spherical comparison.

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Total bulk energy density

Established concept

Energy density of the cosmological fluid defined through its stress–energy, in energy-per-volume units. Vacuum energy is included when treated as part of the total source. It is not mass density; division by $c^2$ converts units. The standard Einstein–FRW identification connects it to the geometric Misner–Sharp density.

$$E_{\mathrm{MS}}=\rho V_A,\qquad \rho=3P_{\mathrm{scr}}$$

Related: Bulk fluid pressure · Bulk enthalpy density and continuity · Misner–Sharp energy

Source: Seven: §5.3. Full: §VI.A–B.

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Bulk fluid pressure

Established concept

The material pressure $p_m$ appearing in the homogeneous fluid stress–energy, with the same units as total energy density $\rho$. It is distinct from $P_L$, $\Pi_\partial$, and the Brown–York surface stress $p_B$. The equation of state specifies it; it is not always $\rho/3$ just because screen pressure is.

$$w=p_m/\rho$$

Related: Bulk enthalpy density and continuity · Positive FRW screen pressure · Hayward work density

Source: Seven: §5.3. Full: §VI.

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Positive FRW screen pressure

Manuscript construction

The constrained apparent-horizon energy–areal-volume slope, equal to the CK realization of $P_L$. It is not the unconstrained thermal derivative $-(\partial E/\partial V)_S$ or the actual fluid pressure. Its sign is tied to the manuscript’s heat-minus-work screen-state convention. The alternate notation $P_{\mathrm{scr}}$ keeps it separate from the Hayward-rate response.

$$P_{\mathrm{scr}}=\frac{dE_A}{dV_A}=\frac{\rho}{3}=P_L^{\mathrm{CK}}$$

Related: Screen-state law versus matter-flux law · Friedmann constraint in pressure variables · Signed Kodama–Hayward rate

Source: Seven: §5.3. Full: §VI.A–C.

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Areal radius

Established concept

The radius defined by a sphere’s intrinsic area, $R=\sqrt{A/(4\pi)}$. It need not equal radial proper distance or acceleration length. A physical sector supplies any relation between $R$ and $L_s$. The later source-archive discussion makes their variable ratio explicit.

Related: Proper distance · Two-scale ratio and complete radius work · Areal versus proper spatial volume

Source: Seven: §5. Full: §§IV.B,V; later radius sections.

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Areal versus proper spatial volume

Established concept

The spherical areal volume is $4\pi R^3/3$, with $dV=A\,dR$. It is the volume used in the Misner–Sharp FRW relation. When spatial curvature is nonzero, the proper volume of a cosmic-time ball contains the spatial metric factor and generally differs. The simple CK work identity uses the areal volume.

$$V_A=4\pi R_A^3/3,\qquad dV_A=A_A\,dR_A$$

Related: Normal-volume one-form · Areal radius · Friedmann constraint in pressure variables

Source: Seven: §5.3. Full: §VI.A.

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Friedmann constraint in pressure variables

Established concept

The first Einstein–FRW equation relating expansion, spatial curvature, and total density. Replacing the established density by $3P_{\mathrm{scr}}$ gives the pressure form. This exposes a mechanical interpretation but does not independently replace the matter–geometry constraint. The general-curvature term enters through the apparent-horizon radius.

$$H^2+kc^2/a^2=\frac{8\pi G}{c^2}P_{\mathrm{scr}}=\frac{8\pi G}{3c^2}\rho$$

Related: Einstein gravitational coupling · Sphere curvature, normal curvature, and spatial curvature · Misner–Sharp energy

Source: Seven: §5.3. Full: §VI.B; Appendix B.

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Einstein gravitational coupling

Established concept

The field-equation normalization $\kappa_E=8\pi G/c^4$, not surface gravity. It converts the normal-scale pressure to an inverse-square length. In CK FRW that length is the apparent-horizon radius. Distinguishing the subscript $E$ from the bare $\kappa$ avoids confusing a fixed coupling with a geometric rate.

$$\kappa_EP_L=L_\kappa^{-2}$$

Related: Surface gravity and its normalization · Sphere curvature, normal curvature, and spatial curvature

Source: Seven: §2; §5.3. Full: Part I; §VI.B.

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Sphere curvature, normal curvature, and spatial curvature

Established concept

Three different geometric quantities. The horizon sphere has Gaussian curvature $1/R_A^2$; the time–radial two-metric has Ricci scalar $2\ddot a/(ac^2)$ in the manuscript convention; cosmic spatial slices have sectional curvature $k/a^2$. A sphere can be curved inside a spatially flat universe. Their numerical factors should not be identified solely by appearance.

$$R_A^{-2}=\kappa_EP_{\mathrm{scr}},\qquad{}^{(2)}\mathcal R_\perp=-\kappa_E(P_{\mathrm{scr}}+p_m)$$

Related: Normal time–radial plane and normal bundle · Friedmann constraint in pressure variables · Trace and active surface-density representations

Source: Seven: §5.3. Full: §VI.B.

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Bulk enthalpy density and continuity

Established concept

For a homogeneous fluid, enthalpy per volume is $\rho+p_m$. It controls density change under expansion and enters horizon matter-energy supply. With $\rho=3P_{\mathrm{scr}}$ it controls screen-pressure evolution. It differs from the acceleration source $\rho+3p_m$, or equivalently $3(P_{\mathrm{scr}}+p_m)$.

$$\dot\rho+3H(\rho+p_m)=0,\qquad\dot P_{\mathrm{scr}}+H(\rho+p_m)=0$$

Related: Total bulk energy density · Bulk fluid pressure · Screen-state law versus matter-flux law

Source: Full: §VI.B; one-page summary; standard FRW background.

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Screen-state law versus matter-flux law

Established concept

The screen-state law resolves the variation of horizon energy into area and size contributions. The Cai–Kim flux law tracks matter energy crossing a momentarily fixed horizon. They classify change differently and use different variation protocols. Work for the moving screen must not be inserted as a duplicate term in a separately defined flux or mass law.

$$dE_A=T_{\mathrm{CK}}dS_A-P_{\mathrm{scr}}dV_A$$

Related: Cai–Kim instantaneous calibration · Bulk enthalpy density and continuity · Hayward unified first law

Source: Seven: §5.3. Full: §VI.C.

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Signed Kodama–Hayward rate

Established concept

The dynamical spherical rate measuring normal cross-focusing, not generally a stationary boost rate or observer acceleration. In FRW it depends on horizon evolution and on the pressure contrast $P_{\mathrm{scr}}-p_m$. It can vanish in radiation cosmology while the CK pressure remains finite. The reciprocal chart fails at that zero.

$$\kappa_H^s=-\frac{c^2}{R_A}\left(1-\frac{\dot R_A}{2HR_A}\right),\quad\frac{\kappa_H^s}{\kappa_{\mathrm{CK}}}=\frac{3w-1}{4}$$

Related: Cross-focusing · Hayward–Mukohyama–Ashworth potential · Exact CK–Hayward work transformation

Source: Seven: §5.3. Full: §VI.D–F.

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Scope of the canonical theorem

Established concept

The direct canonical proof concerns a clock-prescribed nonexpanding Einstein null sector. Stationary Rindler, Schwarzschild, and a consistently normalized de Sitter horizon realize it. Generic FRW apparent-horizon histories use a separate CK screen calibration; their exact state identities do not constitute the same unrestricted null theorem. This distinguishes provenance, not the algebraic validity of the FRW identities.

Related: Nonexpanding null sector · Moving-boundary and embedding variation · Cai–Kim instantaneous calibration

Source: Seven: §7. Full: §II.B; §VI.C; §IX.A.

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Mechanical-to-thermal dictionary

Established concept

Where Hawking–Unruh/KMS thermality applies, a rate is converted to thermal energy and the conjugate boost charge to dimensionless entropy. The mechanical formulas precede this input. On the positive branch the area leg becomes $T\,dS$ and the rate leg becomes $S\,dT=-F_L\,dL_\kappa$.

$$k_BT=\hbar\Omega_\kappa/(2\pi),\qquad S/k_B=2\pi J_\eta/\hbar$$

Related: The mechanical boost-product dictionary · Hawking–Unruh temperature · Einstein entropy and Wald entropy

Source: Seven: §§4,6. Full: §IV.E.

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Hawking–Unruh temperature

Established concept

The semiclassical thermal scale of a horizon or uniformly accelerated observer when the corresponding state and normalization support it. It is linear in $|\kappa|$, unlike the classical quadratic pressure. Using the formula as an instantaneous cosmological parameter does not prove an exact equilibrium radiation spectrum in every evolving spacetime.

$$k_BT=\frac{\hbar|\kappa|}{2\pi c}=\frac{\hbar c}{2\pi L_\kappa}$$

Related: KMS equilibrium condition · Euclidean angular period and thermal length · Cai–Kim instantaneous calibration

Source: Seven: §6. Full: §IV.E.

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Einstein entropy and Wald entropy

Established concept

The stationary Noether-charge entropy associated with the appropriate gravitational action. In Einstein gravity it is the quarter-area formula with conventional additive normalization. The boost-charge representation is established structure. Higher-curvature theories generally replace the area density by the appropriate Wald density, with further qualifications for dynamics and membrane stresses.

$$S=\frac{k_Bc^3A}{4G\hbar}=\frac{2\pi k_BJ_\eta}{\hbar}$$

Related: Planck area and the entropy factor four · Stationary Wald-channel force · Boost charge or boost momentum

Source: Seven: §6. Full: §IV.E; §VIII.

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Planck area and the entropy factor four

Established concept

The area scale built from $G,\hbar,c$. The Einstein entropy coefficient follows from the ratio of the gravitational area coefficient and Hawking–Unruh temperature normalization. This does not establish literal elementary tiles of area $4\ell_P^2$ or an independent microscopic state count.

$$\ell_P^2=G\hbar/c^3,\qquad S/A=k_B/(4\ell_P^2)$$

Related: Mechanical-to-thermal dictionary · Einstein entropy and Wald entropy · Planck benchmarks and scale crossover

Source: Seven: §6. Full: §IV.E.

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Euclidean angular period and thermal length

Established concept

Analytic continuation of stationary normal time changes boost motion to rotation. Regularity fixes a $2\pi$ angular period and hence the reduced thermal length. This is a mathematical continuation, not real physical motion into an imaginary-time direction. The horizon contribution must not be equated indiscriminately to the full Euclidean action.

$$c\Delta\tau_E=2\pi L_\kappa$$

Related: KMS equilibrium condition · Normal boost angle or rapidity · Thermal-scale ratio and Boltzmann exponent

Source: Seven: §6. Full: §IV.E.

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KMS equilibrium condition

Established concept

The equilibrium analyticity relation for correlation functions with respect to a specified time evolution. It connects imaginary-time periodicity to temperature. A mechanical boundary potential alone does not select a quantum state satisfying this condition. Exact Rindler and suitable stationary black-hole settings supply the standard thermal examples.

Related: Hawking–Unruh temperature · Euclidean angular period and thermal length

Source: Seven: §6. Full: §IV.E.

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Signed branch and zero surface gravity

Established concept

The reciprocal length chart is defined separately on positive and negative nonzero $\kappa$. Pressure is positive in the squared convention, while signed potential, surface stress, and work keep orientation information. At $\kappa=0$ the regular rate remains available but $L_s$ diverges. A finite limiting product does not define a finite displacement at the exact zero.

Related: Signed and positive acceleration length · Physical clock and normalization · Signed Kodama–Hayward rate

Source: Seven: §7. Full: §II.B; Appendix A.

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One-form and response one-form

Established concept

An object linear in a variation or tangent vector. For example, $-F_LdL_s$ returns the scale-work contribution for a small source change. It need not be the differential of one globally defined energy. Whether the form lives on spacetime, the cut, or source/field space must be stated.

Related: Variation, differential, and derivative · Integrability and separate scale energy · Normal-volume one-form

Source: Full: §II.B; §IV.B–C; §VII.A.

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Wedge product and exterior derivative

Established concept

The wedge product antisymmetrizes two differentials. $\delta A\wedge\delta\kappa$ detects two independent variation directions; it vanishes after restriction to a one-parameter curve. The exterior derivative tests whether a one-form is locally closed. This is how the full paper distinguishes a work contribution from an independently stored state energy.

Related: Symplectic or presymplectic two-form · Integrability and separate scale energy · Pullback

Source: Full: §II.B; §VII.A.

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Pullback

Established concept

Evaluating a function or differential form after mapping one space into another. Restricting independent source variables to a physical horizon family is a pullback. Making $L_s$ a field on a cut pulls its source response onto spatial gradients. It is the coordinate-independent chain rule, not a separate physical interaction.

$$D_i\Pi_\partial=-P_LD_iL_s$$

Related: Membrane paradigm and tangential balance · Variation, differential, and derivative · Integrability and separate scale energy

Source: Full: §IV.B–C.

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Lie derivative

Established concept

Change of a tensor field under transport by a vector field. $\mathcal L_\xi$ describes how cut geometry changes along the selected evolution. It is used in expansion, in the action of diffeomorphisms on fields, and in moving-boundary pullbacks. It should not be confused with a partial derivative holding all geometric data fixed.

Related: Null expansion · Evolution vector and null generator · Moving-boundary and embedding variation

Source: Full: §II.A; Appendix C.

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Normal time–radial plane and normal bundle

Established concept

The two-dimensional Lorentzian space perpendicular to a spacelike cut. In spherical symmetry it is the orbit/time–radial geometry $h_{ab}$. Normal frames can be related by boosts, and a normal connection describes their transport. The normal two-dimensional Ricci scalar is not the full four-dimensional spacetime Ricci scalar.

Related: Normal boost angle or rapidity · Normal connection and twist data · Sphere curvature, normal curvature, and spatial curvature

Source: Full: §II.C; §IV.E; §VI.B.

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Normal connection and twist data

Established concept

Geometric data describing how the normal frame changes along or across a cut. It supplies vector-sector information and contributes to membrane and stretched-horizon equations. Some sources use $\eta_A$ or $\omega_A$ for such data; these are not the scalar corner rapidity $\eta$. Its effects need not vanish on an arbitrary moving boundary.

Related: Scalar, vector, and tensor sectors · Normal time–radial plane and normal bundle · Membrane paradigm and tangential balance

Source: Full: §II.A; §IV.D; Appendix C.

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Boundary worldtube and causal character

Established concept

The three-dimensional history traced out by a family of spatial cuts. It may be timelike, spacelike, or null. A timelike worldtube can be followed by material observers; a null one is ruled by null curves. Apparent-horizon marginality of each cut does not fix the tube’s causal character or its expansion along its tangent.

Related: Apparent horizon and marginal sphere · Moving-boundary and embedding variation · Null hypersurface

Source: Full: §VI.C,G; Appendix C.

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Proper distance

Established concept

Distance computed using the induced spatial metric on a specified slice. It depends on the geometric/slicing construction. It is distinct from an areal radius and from a reciprocal rate scale. Rindler makes acceleration length an exact proper horizon distance; the local FGP equality is leading near-horizon, not a global equality in Schwarzschild.

Related: FGP local energy and proper-distance prescription · Signed and positive acceleration length · Areal radius

Source: Full: §V.A–B.

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Auxiliary gauge versus physical clock change

Established concept

An auxiliary rescaling of null frame with compensating multiplier can leave the actual evolution vector unchanged; then the assigned source is unchanged. Rescaling the physical evolution itself changes surface gravity and pressure. Gauge reduction must not discard a boundary mode that has a nonzero canonical response under the prescribed physical setup.

$$\xi=f\ell,\quad\ell\to e^\sigma\ell,\quad f\to e^{-\sigma}f$$

Related: Physical clock and normalization · Edge data and relational cuts · Symplectic or presymplectic two-form

Source: Full: §III.A.

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Edge data and relational cuts

Established concept

Boundary or corner reference data that track how the region and its frames are identified across variations. Relational cuts are specified relative to the chosen physical reference construction rather than allowed to drift without accounting. Keeping these data can distinguish physical boundary transformations from gauge degeneracies.

Related: Auxiliary gauge versus physical clock change · Boundary corner or joint · Moving-boundary and embedding variation

Source: Full: §III.A; §II.C.

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Boundary lapse

Established concept

The factor relating coordinate time to proper time of a timelike boundary observer. It is a separate source in the Brown–York boundary variation; its conjugate is an energy. Multiplying a local surface stress by the lapse expresses its contribution per selected coordinate time. Fixing lapse is not the same as fixing every other rate.

$$d\tau_B=N_B\,dt$$

Related: Brown–York energy and surface stress · Rescaled timelike stress source · Physical clock and normalization

Source: Full: Appendix C.

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GHY, null boundary, and corner terms

Established concept

Terms added to the gravitational action to match a specified boundary problem. Gibbons–Hawking–York applies to non-null boundaries with the appropriate orientation; null boundaries require their own formulation; joints contribute normal-angle data. Their variation determines which source–response pairs remain at the boundary. They cannot be omitted solely because a surface approaches a horizon.

Related: Action and Einstein–Hilbert action · Boundary corner or joint · Counterterms and reference subtraction

Source: Full: §II.A; Appendix C.

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Counterterms and reference subtraction

Established concept

Additional boundary functionals or reference choices that can shift the action and associated response/energy. The papers choose a particular polarization and normalization; they do not claim the displayed coefficient is invariant under arbitrary source-dependent additions. The action/boundary convention must be stated before comparing charges.

Related: Area and rate polarization · Brown–York energy and surface stress · Source and conjugate response

Source: Full: §II.A–B; Appendix C.

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Noether current and surface-charge form

Established concept

The current associated with an infinitesimal diffeomorphism can be expressed on shell through a surface-charge form, with constraints and exact-form choices treated appropriately. Its integral is one ingredient in a Hamiltonian charge. The correction involving the symplectic potential and, for field-dependent evolution, $Q_{\delta\xi}$ cannot generally be dropped.

$$\delta H_\xi=\int_S(\delta Q_\xi-\iota_\xi\theta-Q_{\delta\xi})$$

Related: Gravitational charge and Noether charge · Hamiltonian generator · Integrability and separate scale energy

Source: Full: §III.B; Appendix C.

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Brown–York energy and surface stress

Established concept

Quasi-local quantities obtained from the action response to the induced metric of a timelike boundary. In spherical symmetry lapse variation pairs with energy and area variation with isotropic surface stress. Their values depend on clock, orientation, and reference subtraction. They are not generically the same as the Misner–Sharp energy or the scalar horizon potential.

$$\delta I_D|_B=\int dt[-E_B\delta N_B+N_Bp_B\delta A_B]$$

Related: Rescaled timelike stress source · Boundary lapse · Extrinsic curvature and its spherical eigenvalues

Source: Full: Appendix C.

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Extrinsic curvature and its spherical eigenvalues

Established concept

How a hypersurface bends in the surrounding spacetime, distinguished from curvature measured internally on it. For the timelike spherical boundary, $a_n$ is the normal acceleration and $b_B=n(R_B)/R_B$ the angular extrinsic-curvature eigenvalue. Their sum determines the isotropic Brown–York surface stress in the stated convention.

$$p_B=(a_n+b_B)/(8\pi G)\quad(c=1)$$

Related: Brown–York energy and surface stress · Rescaled timelike stress source · Normal force balance and Young–Laplace form

Source: Full: Appendix C, original Eq. (C10).

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Rescaled timelike stress source

Specified-ensemble construction

A chosen normalization of the lapse-weighted surface stress after the spherical area Legendre transform. The factor $8\pi G$ removes the Einstein coefficient and the sign matches the stated convention. Geometry gives a combination of normal acceleration and transverse extrinsic curvature. It is not defined to be Cai–Kim or Hayward surface gravity.

$$\zeta_B=-8\pi G N_Bp_B=-N_B(a_n+b_B)\quad(c=1)$$

Related: Brown–York energy and surface stress · Reciprocal stress length and auxiliary potential · Timelike FRW rate-matching test

Source: Full: Appendix C, original Eqs. (C5)–(C10).

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Reciprocal stress length and auxiliary potential

Specified-ensemble construction

Appendix C uses $L_B=1/\zeta_B$ to rewrite its chosen stress response. It has a force-like coefficient and an auxiliary area–rate potential, but neither the length nor the potential is automatically the original horizon length/energy. The resemblance of formulas is partly a result of reciprocal-coordinate normalization; physical matching must be demonstrated independently.

$$L_B=1/\zeta_B,\quad F_B=A_B\zeta_B^2/(8\pi G),\quad U_B=A_B\zeta_B/(8\pi G)\quad(c=1)$$

Related: Rescaled timelike stress source · Timelike FRW rate-matching test · Moving-boundary and embedding variation

Source: Full: Appendix C.

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Timelike FRW rate-matching test

Manuscript comparison

The explicit flat-dust screen calculation yields a rescaled Brown–York rate different from both CK and Hayward. This rules out their universal identification in the specified ensemble, not the CK state identities or every possible moving-boundary formulation. Stationary stretched Schwarzschild/de Sitter limits have a different, successful matching.

$$\zeta_B=-2/R_A,\quad\kappa_{\mathrm{CK}}=1/R_A,\quad\kappa_H^s=-1/(4R_A)\quad(c=1)$$

Related: Rescaled timelike stress source · Moving-boundary and embedding variation · Cai–Kim instantaneous calibration

Source: Full: Appendix C, original Eq. (C17).

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Moving-boundary and embedding variation

Established concept

Changing the location or history of a boundary requires variations of the pulled-back fields, the embedding, clock, and often corners. A term like $\delta\phi+\mathcal L_\chi\phi$ follows a moved boundary; fixing a field at the old spacetime point is a different condition. A complete canonical theorem must retain all such responses relevant to the stated prescription.

Related: Boundary worldtube and causal character · Edge data and relational cuts · Stretched-Carrollian boundary structure

Source: Full: Appendix C; §VI.G.

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Stretched-Carrollian boundary structure

Established concept

A geometric framework organizing nearly null/timelike stretched-horizon data, their symmetries, and symplectic responses. It retains clock and stretching contributions as well as area/stress data. It is a relevant framework for matching a full moving-screen response, but the manuscript does not demonstrate that it reduces to one CK or Hayward source in general.

Related: Moving-boundary and embedding variation · Boundary lapse · Scalar, vector, and tensor sectors

Source: Full: Appendix C; Freidel–Jai-akson §5.1.

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Integrability and separate scale energy

Established concept

A response one-form is integrable as a state function only when its curl vanishes on the relevant source space (with appropriate global conditions). The isolated scale leg is not integrable on unrestricted independent area–rate data. The complete $dU_\partial$ is exact. Restriction to a one-dimensional physical family removes the two-form obstruction without proving a universal separate stored scale energy.

$$d(-F_LdL_s)=\frac{c^2}{8\pi G}dA\wedge d\kappa$$

Related: Wedge product and exterior derivative · Complete boundary differential · Hamiltonian generator

Source: Full: §VII.A.

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Product volume

Established concept

The extensive product $V_\partial=AL_\kappa$, used in Euler identities and fixed-area responses. It is not a universal enclosed spatial volume. Its full differential contains an extra area term; replacing the normal-volume one-form by this differential changes the area coefficient.

$$dV_\partial=A\,dL_\kappa+L_\kappa\,dA,\quad dU_\partial=2\Pi_\partial dA-P_LdV_\partial\quad(\kappa>0)$$

Related: Normal-volume one-form · Ideal-gas-form boundary identity · Areal versus proper spatial volume

Source: Full: §IV.B,E–F.

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Radius-work coefficient

Established concept

The coefficient obtained when the length-work contribution is expressed through an areal radius. It includes the Jacobian $dL_s/dR$ and is not generally the fixed-area length force. In Schwarzschild $L_\kappa=2R_H$, so radius work uses $2F_L=M\kappa$. The area leg of the full differential remains.

$$F_L\,dL_s=F_R^{(L)}dR,\qquad F_R^{(L)}=F_L\,dL_s/dR$$

Related: Two-scale ratio and complete radius work · Schwarzschild realization · Complete boundary differential

Source: Full: §IV.B; §V.B.

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Two-scale ratio and complete radius work

Later-source manuscript identity

The later editable source describes spherical transverse geometry and normal acceleration length through their ratio $\alpha_R=L_s/R$. Keeping that ratio variable shows exactly when simple mass-times-rate radius work is sufficient. Dropping its differential is justified only on a fixed-ratio family. This is another chart for the same scalar structure, not another degree of freedom.

$$F_LdL_s=(U_\partial/R)dR+U_\partial\,d\alpha_R/\alpha_R$$

Related: Radius-work coefficient · Areal radius · Spherical completion of Rindler boundary data

Source: Recovered source archive: full Part I; §§II.D,V; retained in full revision.

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Spherical completion of Rindler boundary data

Later-source conditional construction

The later source supplies an area–length relation $A=4\pi L_\kappa^2$ to otherwise independent Rindler patch data. The resulting boundary formulas match the CK spherical dictionary. This is a conditional boundary-data correspondence, not a claim that flat spacetime acquires enclosed Misner–Sharp mass or becomes FRW by a coordinate transformation.

Related: Rindler horizon and finite patch · Two-scale ratio and complete radius work · Misner–Sharp energy

Source: Recovered source archive: full Part I; §V.B; retained in full revision.

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Compliance, compressibility, and modulus

Established concept

Derivatives measuring how the chosen source scale responds to pressure at fixed area, or along a specified spherical family. They are source susceptibilities, not automatic dynamical stability criteria. Fixed area gives a modulus $2P_L$; a spherical FRW family gives $2P_{\mathrm{scr}}/3$ because its area also changes.

$$\chi_L^{(A)}=1/(2P_L),\qquad B_L^{(A)}=2P_L$$

Related: Source-space Hessian · Product volume · Positive FRW screen pressure

Source: Full: §IV.F; §VI.A.

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Source-space Hessian

Established concept

The matrix of second derivatives of the boundary potential. It describes local source susceptibilities and cross-responses. Its negative determinant in independent area/length coordinates makes the function a saddle in that source space; it does not alone establish instability of a physical horizon, which requires dynamics and loads.

$$\det\,\partial^2U_\partial/\partial(A,L_\kappa)^2=-P_L^2$$

Related: Compliance, compressibility, and modulus · Integrability and separate scale energy

Source: Full: §IV.F.

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Membrane paradigm and tangential balance

Established concept

A representation of horizon/screen equations using a two-dimensional fluid-like stress and momentum balance. The established pressure-gradient term involves $\Pi_\partial$, not directly a new bulk stress tensor. Pulling back the acceleration-length dependence writes the same tangential term as $P_LD_iL_s$. Shear and other contributions remain in the complete equation.

$$-D_i\Pi_\partial=P_LD_iL_s$$

Related: Pullback · Normal force balance and Young–Laplace form · Surface pressure or tension

Source: Full: §IV.A,C.

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Normal force balance and Young–Laplace form

Established concept

A normal projection of the gravitational screen equation, distinct from tangential membrane momentum balance. In the matched static convention, normal pressure difference balances tension times cut mean curvature. The traction is $P_LL_s\mathcal H$, not universally $P_L$. Time-dependent screens also retain expansion, shear, connection, and lapse terms.

$$\Delta p-P_LL_s\mathcal H=0$$

Related: Mean-curvature trace · Membrane paradigm and tangential balance · Extrinsic curvature and its spherical eigenvalues

Source: Full: §IV.D.

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Mean-curvature trace

Established concept

The sum of principal curvatures of the cut within the relevant spatial screen geometry. The manuscript uses the trace convention; a round sphere in Euclidean three-space has $2/R$. This is denoted $\mathcal H$ in the glossary to avoid confusion with the cosmological Hubble rate. The factor $L_s\mathcal H$ is dimensionless.

Related: Normal force balance and Young–Laplace form · Sphere curvature, normal curvature, and spatial curvature

Source: Full: §IV.D.

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FGP local energy and proper-distance prescription

Established concept

The Frodden–Ghosh–Perez construction assigns near-horizon energy to stationary observers at a small proper distance $\ell$. Its local first law uses an observer-distance prescription. Differentiating its leading $A/\ell$ formula in distance yields the same inverse-square shape, but changing observer distance and proving a complete action work response are distinct tasks.

$$E_{\mathrm{FGP}}\simeq Ac^4/(8\pi G\ell),\qquad L_{\mathrm{loc}}\simeq\ell$$

Related: Local versus infinity-normalized horizon rate · Proper distance · Hamilton–Jacobi boundary response

Source: Full: §V.A.

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Local versus infinity-normalized horizon rate

Established concept

A static observer’s proper-time normalization rescales the horizon rate and boundary energy relative to the clock at infinity. In Schwarzschild the corresponding local acceleration length is reduced by the lapse. Equality to the actual radial proper distance holds only near the horizon; the redshifted horizon rate is not exactly the observer proper acceleration at arbitrary radius.

$$\kappa_{\mathrm{loc}}=\kappa_\infty/N,\quad L_{\mathrm{loc}}=NL_\infty=2R_HN$$

Related: FGP local energy and proper-distance prescription · Physical clock and normalization · Schwarzschild realization

Source: Full: §V.A.

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Newtonian field-stress correspondence

Established concept

At the Schwarzschild horizon in the standard infinity-normalized convention, $g_N=GM/R_H^2=\kappa$ and the longitudinal gravitational field-stress magnitude matches $P_L$. This is a sectoral comparison of magnitudes, not a covariant bulk stress tensor interpretation. Rindler’s nonzero boundary response in flat spacetime shows why the construction is broader than source-field stress.

$$P_g^{\parallel}=g_N^2/(8\pi G)$$

Related: Normal-scale pressure · Rindler horizon and finite patch

Source: Full: §V.B.

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Cross-focusing

Established concept

Change of a null expansion when differentiated in the other normal direction. It probes the normal two-geometry and helps define the spherical Hayward rate. It is different from the area expansion along the actual horizon worldtube or a comoving observer’s proper acceleration. In homogeneous four-dimensional FRW it is controlled by the stress–energy trace combination.

Related: Signed Kodama–Hayward rate · Trace and active surface-density representations · Null expansion

Source: Full: §VI.D.

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Kodama flow

Established concept

A preferred spherical evolution vector built from the normal two-metric and the areal-radius gradient. It exists without a stationary Killing vector and is central to Misner–Sharp conservation and the spherical Noether construction. In dynamic settings its rate and clock interpretation must not be silently replaced by a stationary Killing prescription.

Related: Misner–Sharp energy · Hayward–Mukohyama–Ashworth potential · Noether current and surface-charge form

Source: Full: §III.B; §VI.E; Appendix C.

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Hayward work density

Established concept

The scalar formed by the normal two-trace of the matter stress–energy. In homogeneous FRW it is half energy density minus pressure. It is the matter-work coefficient in the projected unified first law, not CK screen pressure and not Brown–York surface stress.

$$W=-\frac12T^{ab}h_{ab}=\frac{\rho-p_m}{2}$$

Related: Hayward unified first law · Bulk fluid pressure · Exact CK–Hayward work transformation

Source: Full: §VI.E; Appendix B.

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Hayward unified first law

Established concept

The spherical Einstein energy balance combining an energy-supply one-form and work density. Projecting it onto a trapping-horizon trajectory yields an area contribution with Hayward surface gravity plus matter work. It is not identical term-by-term to the CK screen-state differential even where the total energy change agrees.

$$d_HE_A=T_H^s\,d_HS_A+W\,d_HV_A$$

Related: Hayward work density · Hayward–Mukohyama–Ashworth potential · Screen-state law versus matter-flux law

Source: Full: §VI.E–F.

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Hayward–Mukohyama–Ashworth potential

Established concept

The area–Hayward-rate Kodama–Noether potential in spherical dynamics. In homogeneous four-dimensional Einstein FRW it equals minus one quarter of the signed trace energy. It is generally not the Misner–Sharp horizon energy. Its derivative includes rate response, but the complete canonical moving-boundary identification requires more than recognizing its area–rate form.

$$\widetilde U_\partial=\frac{A_Ac^2\kappa_H^s}{8\pi G}=\frac{E_A}{2}-\frac32WV_A$$

Related: Signed Kodama–Hayward rate · Trace and active surface-density representations · Integrability and separate scale energy

Source: Full: §VI.E.

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Trace and active surface-density representations

Manuscript representation

The full paper packages integrated homogeneous stress combinations per horizon area. The trace density uses $\rho-3p_m$ and controls the FRW Hayward rate; the active density uses $\rho+3p_m$ and controls relative cosmic acceleration. They are not automatically literal membrane or thin-shell mass densities, and the simple trace identification is not a universal inhomogeneous spherical law.

$$\Sigma_{\mathrm{tr}}=\frac{(\rho-3p_m)V_A}{c^2A_A},\quad\kappa_H^s=-2\pi G\Sigma_{\mathrm{tr}}$$

Related: Cross-focusing · Junction conditions and literal thin shells · Sphere curvature, normal curvature, and spatial curvature

Source: Full: §VI.D,G.

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Exact CK–Hayward work transformation

Manuscript identity

The scale-work relation between distinct spherical rate prescriptions. With $\chi=\kappa_H^s/\kappa_{\mathrm{CK}}$, varying the ratio contributes an additional term. Dropping it is justified only at fixed ratio. The balances remain compatible but their work allocations and area coefficients differ.

$$F_L^H\,dL_H^s=\chi P_{\mathrm{scr}}\,dV_A-E_A\,d\chi,\qquad\chi=(3w-1)/4$$

Related: Signed Kodama–Hayward rate · Cai–Kim instantaneous calibration · Hayward work density

Source: Full: §VI.F.

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Junction conditions and literal thin shells

Established concept

A physical distributional shell requires matching conditions and a discontinuity in appropriate metric derivatives, producing a surface stress distribution. A smooth FRW apparent horizon has no such shell merely because a trace combination has surface-density units. The glossary’s membrane and trace interpretations do not postulate one.

Related: Trace and active surface-density representations · Membrane paradigm and tangential balance

Source: Full: §VI.G.

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Expanded scalar response beyond nonexpansion

Conditional scalar identity

The same mixed scalar one-form can retain separate inaffinity and expansion variations. The pressure term remains a partial coefficient while an expansion term remains. Defining the reciprocal of $\mu$ absorbs the whole scalar source but changes the physical length. These algebraic identities do not by themselves complete a dynamical worldtube theorem or remove other boundary sectors.

$$\Theta_\mu^{(0)}=-\int d\tau dA\,P_L\delta L_s+\int d\tau dA\,\frac{c^4}{16\pi G}\delta\theta$$

Related: Mixed null scalar source · Moving-boundary and embedding variation · Scope of the canonical theorem

Source: Full: §II.B, with explicit explanatory extension in revision.

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APS dynamical-horizon projection and finite flux law

Established concept

The manuscript uses the Ashtekar–Paraizo–Shu charge–flux construction with an equilibrium Kerr projection assigned to dynamical black-hole cuts. The full projected potential changes by finite gravitational and matter flux. In the nonrotating four-dimensional sector its length force is constant. This is not an exact KMS state or a proper material displacement during arbitrary evolution.

$$\Delta U_{\mathrm{APS}}=F_L^{\mathrm{APS}}\Delta L_{\mathrm{APS}},\qquad F_L^{\mathrm{APS}}=c^4/(8G)$$

Related: Hamiltonian generator · Scope of the canonical theorem · Matter and gravitational symplectic flux

Source: Full: §VII.B.

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Matter and gravitational symplectic flux

Established concept

Transport across a boundary, including material stress–energy and radiative gravitational degrees of freedom. Flux affects conservation and charge integrability. It should not be identified separately with either the area or scale term merely because their sum gives a potential change. The screen-state decomposition sorts by variables, not by flux carrier.

Related: Screen-state law versus matter-flux law · Shear and the trace-free tensor sector · Hamiltonian generator

Source: Full: §§II,VI.C,VII.B.

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Stationary Wald-channel force

Conditional stationary extension

The fixed-Wald-entropy contribution $S_WdT$ can be expressed through reciprocal thermal length. It reduces to the Einstein force when the entropy density is the Einstein area density. Fixed area need not imply fixed Wald entropy in another theory, and the corresponding membrane coefficient can differ.

$$F_L^{(W)}=TS_W/L_\kappa,\qquad P_L^{(W)}=TS_W/(AL_\kappa)$$

Related: Einstein entropy and Wald entropy · Lovelock and other higher-curvature distinctions · Mechanical-to-thermal dictionary

Source: Full: §VIII.

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Lovelock and other higher-curvature distinctions

Established concept

Higher-curvature actions alter the relevant entropy functional and boundary stresses. In the stated pure-order Lovelock membrane sector, a theory/dimension factor separates membrane pressure from the Wald thermal channel. The Einstein coincidence should not be assumed in all theories. Generic varying couplings and dynamical entropy introduce further terms.

$$\Pi_\partial^{(m)}A/T=\frac{D-2m}{D-2}S_W^{(m)}$$

Related: Stationary Wald-channel force · Counterterms and reference subtraction · Einstein entropy and Wald entropy

Source: Full: §VIII.

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Ideal-gas-form boundary identity

Manuscript identity

The Euler identity can be written like $PV=Nk_BT$ using product volume and entropy in Boltzmann units. This is not a molecular model or an additional equation of state. For an actual CK spherical volume, which is one third of $AL$, the effective entropy count is also divided by three.

$$P_LV_\partial=N_Sk_BT,\qquad N_S=S/k_B$$

Related: Product volume · Euler products and homogeneity · Mechanical-to-thermal dictionary

Source: Full: §IV.E.

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Boundary inverse-energy or Compton-equivalent length

Manuscript definition

The exact inverse-energy length formed from the boundary potential. It has the form of a reduced Compton wavelength, but its mass equivalent need not be particle rest mass. Interpreting it as a quantum phase scale additionally requires a boundary Hamiltonian realization; the macroscopic construction does not provide a Hilbert space or a propagating boundary particle.

$$\bar\lambda_\partial=\hbar c/|U_\partial|$$

Related: Thermal-scale ratio and Boltzmann exponent · Boundary equality of Bekenstein form · Planck benchmarks and scale crossover

Source: Full: Part I; §IV.E.

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Thermal-scale ratio and Boltzmann exponent

Established concept

Where the thermal dictionary applies, entropy in Boltzmann units equals the boundary potential measured in thermal energy units. Equivalently it is the ratio of Euclidean thermal circumference to the reduced inverse-energy length. It is not automatically the complete on-shell gravitational action or a count of independent spatial cells.

$$\frac{S}{k_B}=\frac{|U_\partial|}{k_BT}=\frac{2\pi L_\kappa}{\bar\lambda_\partial}$$

Related: Boundary inverse-energy or Compton-equivalent length · Euclidean angular period and thermal length · The mechanical boost-product dictionary

Source: Full: §IV.E.

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Boundary equality of Bekenstein form

Manuscript representation

The entropy identity has the algebraic energy–length form familiar from a Bekenstein bound. For the present boundary variables it is an equality. In a generic sector the energy need not be total system energy and the length need not enclose the system, so this is not a new proof of a universal entropy bound.

$$S/k_B=2\pi |U_\partial|L_\kappa/(\hbar c)$$

Related: The mechanical boost-product dictionary · Boundary inverse-energy or Compton-equivalent length · Einstein entropy and Wald entropy

Source: Full: Part I; §IV.E.

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Matsubara frequency and reduced thermal scale

Established concept

Discrete imaginary-time frequencies in an equilibrium thermal description. The first nonzero bosonic frequency equals the magnitude boost rate with the standard Hawking–Unruh normalization. This identifies the reciprocal mechanical scale with the reduced thermal scale; it is not a claim about the Lorentzian wavelength of a typical emitted photon.

$$\omega_1^E=2\pi k_BT/\hbar=\Omega_\kappa$$

Related: KMS equilibrium condition · Euclidean angular period and thermal length · Signed and magnitude boost rate

Source: Full: §IV.E.

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Planck benchmarks and scale crossover

Established concept

The full paper propagates the classical Einstein and semiclassical thermal coefficients to several specified scale comparisons. Equality of inverse-energy and acceleration lengths fixes a crossover area; further choices fix a normal scale. These are benchmarks, not established minimum lengths, maximum accelerations, or predictions that unchanged formulas survive quantum gravity.

$$\bar\lambda_\partial=L_\kappa\ \Longrightarrow\ A=8\pi\ell_P^2$$

Related: Boundary inverse-energy or Compton-equivalent length · Planck area and the entropy factor four · Two-scale ratio and complete radius work

Source: Full: §IV.E.

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Virial analogy and formal free energy

Established concept

Schwarzschild’s equal Euler products motivate comparison with a scaling/virial balance. The exact formal on-shell Helmholtz value equals $F_LL_\kappa$. This does not establish separately measurable kinetic/potential energies or a stable canonical ensemble; Schwarzschild has the familiar thermodynamic instability in the unrestricted asymptotically flat canonical setup.

$$Mc^2-TS=F_LL_\kappa=Mc^2/2$$

Related: Smarr relation · Euler products and homogeneity · Standard black-hole mass first law

Source: Full: §V.C.

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Quantum-atmosphere diagnostics

Established concept

The full paper compares acceleration length to model- and observer-dependent locations where WKB adiabaticity or other diagnostics peak outside Schwarzschild. Fractions such as $3L_\kappa/4$ and $5L_\kappa/6$ refer to different calculations. They do not establish a universal emission shell or an exact sequence of horizon layers.

Related: Schwarzschild realization · Matsubara frequency and reduced thermal scale

Source: Full: §V.B.

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