Action, equations, and boundary terms

An action assigns a number with units of action (energy times time) to a history of fields and a spacetime region. Schematically,

$$I[g,\phi]=I_{\mathrm{EH}}[g]+I_{\mathrm{matter}}[g,\phi]+I_{\mathrm{boundary}}+I_{\mathrm{corner}}.$$

The Einstein–Hilbert part contains spacetime curvature. Matter terms describe the material fields. Boundary and corner terms make the variation appropriate to the data specified on the region’s boundary. A corner is a two-dimensional intersection of boundary hypersurfaces, not necessarily a sharp edge in ordinary three-dimensional space.

Varying an action has the schematic structure

$$\delta I=\int_M E\cdot\delta\Phi+\int_{\partial M}\theta(\Phi,\delta\Phi)+\delta I_{\mathrm{boundary/corner}}.$$

$\Phi$ collectively denotes the fields; $E=0$ denotes their equations of motion. The boundary expression depends on variations of the fields and their geometry. This is why the action supplies both bulk equations and boundary response data. A coefficient in the boundary variation does not replace all the bulk equations.

On shell means that the fields satisfy the equations. It does not mean that the action, every boundary contribution, or every variation of the boundary data vanishes. The action is stationary under the variations appropriate to a fixed boundary-value problem. Comparing solutions with different prescribed boundary sources is a different operation.

Symplectic potential, not potential energy

The integrated boundary variation $\Theta$ is called a symplectic potential. It is a one-form on field space: give it a small change of the fields and it returns a linear response. Its name should not be confused with the energy-valued potential $U_\partial$.

The ordinary-mechanics prototype is $p\,\delta q$. The coefficient $p$ is conjugate to the varied coordinate $q$. In field theory there can be one pair at every boundary point, and an integration over the boundary replaces a finite sum. The field-space exterior derivative

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

encodes the canonical pairings. It is a two-form: it compares two independent variations antisymmetrically. Gauge directions can make it degenerate; before those directions are removed, “presymplectic” is the precise term. This $\boldsymbol\Omega$ is not the boost rate $\Omega_\kappa$.

The Einstein null-boundary potential has scalar, vector, and trace-free tensor sectors. These are geometric classifications under transformations within a cut. They are not three newly postulated species of matter and do not imply three unconstrained bulk graviton polarizations. [HF17; HF18]

What is a charge, and when is it a Hamiltonian?

A charge is the quantity associated with a specified transformation. Electric charge generates an internal gauge transformation; an angular momentum generates rotation; a gravitational boost charge generates a change of the normal frame. Its units depend on the parameter being generated. A dimensionless angle naturally has an action-valued conjugate. A time parameter naturally has an energy-valued generator.

In covariant gravity, the Noether current associated with a vector field $\xi$ has the schematic form $\mathcal J_\xi=\theta(g,\mathcal L_\xi g)-\iota_\xi\mathcal L$. On shell it can be written locally as the exterior derivative of a surface-charge form $Q_\xi$, subject to the usual constraints and conventions. Integrating that form over a cut gives a charge contribution.

A Hamiltonian is a generator whose variation is determined by contraction of the symplectic form with the chosen evolution. It exists as a state function only when that variation is integrable with the specified boundaries and flux treatment. A typical adjusted surface expression is

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

with additional transport or flux terms when required. The last term matters when the generator itself depends on the fields. Therefore the Noether charge, the complete Hamiltonian, and $U_\partial$ are not synonyms. A boundary potential can be a definite contribution without being the total energy of the spacetime. [IW94; original full paper §III.B]

The same point distinguishes ADM mass at infinity, Brown–York energy conjugate to a boundary clock, and Misner–Sharp energy in spherical symmetry. They answer related but different geometric questions; identical units are not enough to identify them.

Null hypersurface, cut, and generator

A hypersurface is a three-dimensional subset of four-dimensional spacetime. A null hypersurface has a null normal that is also tangent to it. Its generators are lightlike curves lying within it. A spacelike cut $S$ is a two-dimensional cross-section. The horizon’s area refers to such a cut, not to the three-dimensional worldtube swept out by its history.

Let $q_{AB}$ be the positive two-metric on a cut, $q$ its determinant, and

$$\epsilon_S=\sqrt q\,d^2x,\qquad A=\int_S\epsilon_S.$$

The capital indices here label the two cut coordinates; the index $A$ is not the total area. A selected generator $\xi^a$ tells us how cuts are followed. Its normalization fixes the parameter along the null generators and must be distinguished from an arbitrary auxiliary null-frame convention.

Inaffinity and surface gravity

With a generator parameter measured in length units, define

$$\nabla_\xi\xi^a=\widehat\kappa\,\xi^a,\qquad \kappa=c^2\widehat\kappa.$$

The operator $\nabla_\xi=\xi^b\nabla_b$ differentiates along the generator. If its parameter is affine, the left side is zero. A nonzero $\widehat\kappa$ measures failure of that chosen parameter to be affine: the derivative points along the same null direction rather than perpendicular to it. A null generator has no proper time and no ordinary four-acceleration of a massive observer. Surface gravity is not automatically a local accelerometer reading.

$\widehat\kappa$ has units of inverse length; $\kappa$ has units of acceleration. On a stationary Killing horizon, the prescribed Killing normalization supplies the usual surface gravity. For Rindler, observer proper time fixes the normalization so $\kappa=a$. For a Schwarzschild horizon, the standard clock is normalized at infinity. A generic null generator can be reparameterized, which changes its inaffinity; the physical clock prescription is therefore essential.

Expansion, shear, and the scalar combination

The expansion is the fractional change of area along the null generators:

$$\theta=\frac{1}{\sqrt q}\mathcal L_\xi\sqrt q,\qquad \mathcal L_\xi\epsilon_S=\theta\epsilon_S.$$

A small bundle of generators with $\theta>0$ spreads out; $\theta<0$ contracts. Shear describes distortion of the cut’s shape at fixed infinitesimal area. The scalar sector retains area change and normal parameterization, whereas the trace-free tensor sector retains shape change. With the same length parameter, $\theta$ also has units of inverse length.

In four-dimensional Einstein gravity the scalar combination is

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

The one-half is the four-dimensional specialization of $(D-3)/(D-2)$ in the cited decomposition. It comes from the geometric split of the null symplectic potential, not a choice made to obtain $P_L$. The term “mixed” refers to the combination of intrinsic area data and derivative/extrinsic rate data entering the boundary prescription. It does not mean “mixed quantum state.” [HF17; HF18; ORS23]

Area versus rate polarization

In this subsection set $c=1$. The scalar area polarization is

$$\Theta_A^{(0)}=-\frac{1}{8\pi G}\int_{\mathcal N}\mu\,\delta\epsilon_S\,dv.$$

The superscript $(0)$ means scalar or spin zero, not perturbative order. The parameter $v$ runs along the generators. This expression treats area as the displayed configuration variable and the negative rate as its conjugate response.

Choose the boundary Legendre term

$$B_\mu=\frac{1}{8\pi G}\int_{\mathcal N}\mu\epsilon_S\,dv.$$

Its exact variation changes the potential to

$$\Theta_\mu^{(0)}=\Theta_A^{(0)}+\delta B_\mu=\frac{1}{8\pi G}\int_{\mathcal N}\epsilon_S\,\delta\mu\,dv.$$

The elementary identity behind this is $-p\,\delta q+\delta(pq)=q\,\delta p$. Area and rate have exchanged roles as the displayed source. Because $\delta^2B_\mu=0$, the symplectic two-form is unchanged. This is a genuine change of polarization. The subsequent substitution $L_s=c^2/\kappa$ is an invertible source-coordinate change inside the chosen polarization, not another Legendre transform.

These are not two new scalar degrees of freedom. They are different descriptions of the same scalar pairing. Boundary conventions still matter: an additional source-dependent action term can shift a response, and corner terms must be treated consistently. [Original full paper §II.A–B; ORS23]

Why the force extraction is guaranteed once the canonical term is supplied

Restrict to a nonexpanding null sector and to variations remaining in it:

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

Then $\mu=\widehat\kappa$. With the prescribed physical time label $\tau$, the scalar response becomes

$$\Theta_\kappa^{(0)}=\frac{c^2}{8\pi G}\int d\tau\int_S\epsilon_S\,\delta\kappa.$$

For uniform stationary data and a fixed duration,

$$\frac{\Theta_\kappa^{(0)}}{\Delta\tau}=\frac{Ac^2}{8\pi G}\delta\kappa.$$

Yes: this is exactly the normal-rate term in the full differential of the displayed boundary potential. Since

$$\delta\kappa=-\frac{c^2}{L_s^2}\delta L_s,$$

it follows directly that

$$\Theta_L^{(0)}=-\int d\tau\int_S\epsilon_SP_L\delta L_s,\qquad P_L=\frac{c^4}{8\pi G L_s^2}=\frac{\kappa^2}{8\pi G}.$$

For a uniform cut, $F_L=AP_L$ and $\Theta_L^{(0)}=-\int F_L\delta L_s\,d\tau$.

There is no second algebraic hurdle. A correctly identified canonical one-form transforms by the chain rule, and its two-form transforms consistently. What is not supplied by the chain rule is the choice of physical boundary, clock, polarization, or allowed source space. Differentiating an arbitrary energy-valued function would not establish those ingredients. The canonical derivation establishes the match for the stated ensemble; it does not certify every other horizon prescription by the same notation.

The complete $dU_\partial$ is exact; the selected rate leg is not generally exact when $A$ and $L_s$ vary independently. Identifying one leg of $dU_\partial$ with the source response is not identifying the entire differential with the symplectic potential. [Original full paper §§II.B and VII.A]

What makes a source variation permitted?

“Permitted” means compatible with the specified variational problem, not independently assumed to be a physical displacement of matter. In this construction the boundary remains null; the variations remain tangent to the chosen nonexpanding sector; cuts and generator normalization are identified by the prescribed clock/reference construction; matter and the other boundary responses are fixed or separately accounted for; and the reciprocal chart stays on a nonzero branch.

There are two different uses of fixed data. Within one conservative boundary-value problem, a prescribed source has $\delta\mu=0$. To measure the response to that source, one compares nearby problems with different assigned $\mu$ while controlling the remaining data. The response is a derivative across that family, just as a susceptibility compares solutions at neighboring applied fields.

A fixed physical clock prescription does not mean $\delta\kappa=0$. The geometry and its connection can change while the convention defining the time generator remains fixed. Conversely, rescaling the physical clock is not automatically a harmless auxiliary-gauge change.

Not every freely chosen boundary function is guaranteed to admit a compatible solution of the gravitational constraints. The manuscript’s variational identity is conditional on the admitted source space; it is not a theorem of existence and uniqueness for every proposed boundary problem. [Original full paper §II.A–B; Appendix C]

Hamilton–Jacobi meaning and units

In ordinary mechanics the action evaluated on a classical solution is a function of boundary data. Differentiating with respect to an endpoint coordinate yields its momentum; differentiating with respect to a time endpoint yields energy with the appropriate sign. Hamilton–Jacobi theory carries this idea to an action as a function, or functional, of boundary sources.

For the selected mixed gravitational action, its local source response reads

$$\delta I_{\mathrm{os}}\big|_{\text{other sources}}=-\int d\tau\,F_L(\tau)\delta L_s(\tau),\qquad F_L(\tau)=-\frac{\delta I_{\mathrm{os}}}{\delta L_s(\tau)}.$$

The derivative is functional because $L_s(\tau)$ is a function of time, not merely one number. The integration measure is part of the definition: action divided by time and length has force units. For a time-independent source over a prescribed interval,

$$-\frac{\partial I_{\mathrm{os}}}{\partial L_s}=\Delta\tau F_L.$$

This establishes an action response. It does not by itself supply a freely falling material body experiencing that force, nor a protocol for extracting work. The location and sign of the response also depend on the chosen boundary orientation and action terms. [Original full paper Eq. (24)]

Boost angle and its momentum

The two directions normal to a spacelike cut form a Lorentzian plane. Two choices of normal frame can differ by a Lorentz boost. Its rapidity $\eta$ is the hyperbolic-angle analogue of an ordinary rotation angle; for two suitable timelike observers in one local plane, relative speed is $v=c\tanh\eta$. At a boundary joint the relevant quantity is the relative boost of the boundary normals. It need not equal a cosmological screen’s elementary velocity rapidity.

Einstein’s corner structure contains, in the adopted orientation, the one-form

$$\Theta_{\mathrm{corner}}=J_\eta\delta\eta,\qquad J_\eta=\frac{Ac^3}{8\pi G}.$$

This is the same sense of conjugacy as angular momentum times angle. $J_\eta$ is action-valued and is a gravitational boost charge, not electric charge, rest mass, or ordinary black-hole spin. Its value changes when the area changes; calling it a charge does not imply conservation under every physical process. [CT95; BW12]

The length momentum explicitly

For a stationary boost flow over a fixed interval,

$$\eta=\frac{\kappa\Delta\tau}{c}=\frac{c\Delta\tau}{L_s}.$$

Consequently

$$p_L\equiv J_\eta\frac{\partial\eta}{\partial L_s}=-\frac{c\Delta\tau J_\eta}{L_s^2}=-\Delta\tau\frac{A\kappa^2}{8\pi G}=-\Delta\tau F_L,$$

and $J_\eta\delta\eta=p_L\delta L_s$.

$p_L$ has ordinary momentum units (force times time); $F_L$ is the response per unit boundary time. They are not the same object and neither is the pressure $P_L$. The symbols $p_L$, $P_L$, $p_m$, and $p_B$ name different quantities. The stationary interval is essential to this simple integrated expression. For nonstationary boost data the relation is an integral with additional time/geometry variations as required; one cannot insert a changing cosmological rate into this stationary corner identity and call the result a complete moving-screen derivation.

Why $U_\partial L_s=cJ_\eta$ is revealing

The screenshot’s relation follows directly from $U_\partial=\omega_sJ_\eta$ and $L_s=c/\omega_s$:

$$\boxed{\frac{U_\partial L_s}{c}=J_\eta=\frac{Ac^3}{8\pi G}.}$$

It says that energy multiplied by its reciprocal-rate time $L_s/c$ is the normal boost charge. At fixed area, changing the clock-normalized rate changes $U_\partial$ and $L_s$ inversely while their product retains the area-normalized action. Under a constant physical-clock rescaling, that product is unchanged. This does not make $F_L$ or $P_L$ separately clock invariant.

The full paper already states $S/k_B=2\pi U_\partial L_\kappa/(\hbar c)$ on the positive branch (Part I, Eq. S26; main Eq. 84), and both papers state $J_\eta=Ac^3/(8\pi G)$. Their equivalence was not foregrounded as this compact purely mechanical equality. The boost-Hamiltonian interpretation is established literature; the concise acceleration-length packaging is useful exposition of the manuscript’s framework, not an independently new entropy law.

The relation uses signed $U_\partial,L_s$, or positive magnitudes $|U_\partial|,L_\kappa$. At exactly $\kappa=0$, the reciprocal coordinate is undefined; a finite branchwise limit is not permission to multiply zero by infinity at the endpoint.

Only afterward: the thermal dictionary

Where a regular Hawking–Unruh/KMS interpretation applies,

$$k_BT=\frac{\hbar\Omega_\kappa}{2\pi},\qquad \frac{S}{k_B}=\frac{2\pi J_\eta}{\hbar}.$$

For positive $\kappa$, this gives

$$\omega_s\,dJ_\eta=\Pi_\partial\,dA=T\,dS,\qquad J_\eta\,d\omega_s=-F_L\,dL_\kappa=S\,dT.$$

The word “thermal” belongs here, not to the preceding charge–rate identity. The factor $2\pi$ is associated with the regular Euclidean angular period; the gravitational normalization supplies $J_\eta$. The quarter-area Einstein entropy follows with its conventional zero additive constant. No planar disk, microscopic tiling, or new thermal postulate has been derived by canceling the fours in $A=4\pi R^2$.

A marginal cut is not a nonexpanding null worldtube

At an apparent-horizon cut, one of the two null expansions vanishes. To find the next apparent-horizon cut, however, one follows the locus of marginal spheres, not necessarily that same family of null generators. The worldtube formed by those spheres can be timelike, spacelike, or null. Its area can change even though every constituent sphere is marginal in one null direction.

For flat dust FRW, $a(t)\propto t^{2/3}$ gives $R_A=3ct/2$. Relative to the comoving flow, the horizon has radial speed $\dot R_A-HR_A=c/2$. Its worldtube is timelike and its area increases. The null, nonexpanding hypotheses therefore fail. The velocity rapidity is constant, although the Hayward rate is nonzero. This is why a simple worldtube rapidity derivative does not supply the required horizon boost rate. [Original full paper §VI.G; Faraoni]

There are special null cases, but nullness alone is not enough: the expansion must vanish along the actual generator of the tube. Exact de Sitter supplies the stationary, nonexpanding example. A radiation apparent horizon in flat FRW can be null while the relevant tube expansion is not zero.

What already holds in FRW

With arbitrary spatial curvature,

$$R_A=\frac{c}{\sqrt{H^2+kc^2/a^2}},\qquad\kappa_{\mathrm{CK}}=\frac{c^2}{R_A},\qquad L_{\mathrm{CK}}=R_A.$$

Using the Einstein–FRW matter relation $E_{\mathrm{MS}}=\rho V_A=c^4R_A/(2G)$ and areal volume $V_A=4\pi R_A^3/3$,

$$P_{\mathrm{scr}}=\frac{\rho}{3}=P_L^{\mathrm{CK}},\qquad E_{\mathrm{MS}}=U_\partial=F_LL_{\mathrm{CK}}=3P_L^{\mathrm{CK}}V_A.$$

The complete mechanical state differential is

$$dE_{\mathrm{MS}}=\Pi_{\mathrm{CK}}\,dA_A-P_{\mathrm{scr}}\,dV_A.$$

The thermal version uses $\Pi_{\mathrm{CK}}dA_A=T_{\mathrm{CK}}dS_A$. This is a constrained screen-state law, not the matter energy-flux law across a momentarily fixed horizon. The bulk-energy identification is explicit gravitational input, not a second density invented for presentation. No new bulk Friedmann dynamics are claimed.

What can already be said on an expanding null boundary

There is a precise partial extension available from the same scalar one-form. Define $\mu_{\mathrm{acc}}=c^2\mu=\kappa+c^2\theta/2$. With the same fixed clock-measure and polarization conventions, its scalar part is

$$\Theta_\mu^{(0)}=\int d\tau\int_S\epsilon_S\left(\frac{c^2}{8\pi G}\delta\kappa+\frac{c^4}{16\pi G}\delta\theta\right).$$

Using $L_s=c^2/\kappa$ gives the exact scalar coordinate identity

$$\boxed{\Theta_\mu^{(0)}=-\int d\tau\int_S\epsilon_SP_L\delta L_s+\int d\tau\int_S\epsilon_S\frac{c^4}{16\pi G}\delta\theta.}$$

Therefore the familiar surface-gravity response survives as a partial scalar coefficient; an independent expansion response remains. Setting $\delta\theta=0$ removes that displayed term, even at a nonzero fixed expansion, provided the remaining boundary conditions really admit the chosen variations. This last proviso is a geometric/source-space question, not answered by algebra alone.

Alternatively, on $\mu\ne0$ one can use $L_\mu=1/\mu$ and write the whole scalar sector as $-\int\epsilon_SP_\mu\delta L_\mu\,d\tau$ with $P_\mu=c^4/(8\pi G L_\mu^2)$. But $L_\mu$ is generally not $c^2/\kappa$. Neither version removes shear, generator, matter, clock, or corner responses. This is an algebraic consequence of the established scalar potential, added explicitly in this explanatory revision; it is not promoted to a complete dynamical horizon theorem.

What a full moving-boundary theorem would have to establish

“Generic moving null-boundary theorem” is not a particular standard theorem name. Two problems should be distinguished: an evolving but still null hypersurface with expansion/shear, and an apparent-horizon worldtube whose causal type may itself be non-null. The latter requires an appropriate timelike/spacelike or signature-aware treatment, not merely a null formula with time-dependent symbols.

A complete theorem for a selected horizon prescription would specify its geometry, embedding, physical evolution, matter boundary data, polarization, counterterms, and corner conditions. It would calculate the full on-shell boundary variation, then demonstrate which response is paired with the original intended $L_{\mathrm{CK}}$ or $L_H$, with all remaining source and transport terms retained. It must also distinguish a local response coefficient from an integrable Hamiltonian charge.

The papers already establish several pieces, including the general null scalar structure, the stationary boost check, spherical Noether identities, and the explicit timelike calculation. What is not established is their universal identification with the CK or Hayward rate on arbitrary moving FRW screens. The timelike counterexample below disproves that identification for that ensemble, not for all ensembles. The stretched-Carrollian formalism is relevant because it retains clock, stretching, and other responses in a common description. [Full paper Appendix C; Freidel–Jai-akson §5.1]