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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