Analogue Transverse First Law

• Analogue Transverse First Law is a quasi-local deformation identity defined on closed marginally trapped surfaces, linking Hawking energy to transverse null variations.
• It departs from classical black-hole first laws by using intrinsic surface and normal data to partition energy variations into heat and work contributions.
• The framework unifies gravitational and thermodynamic principles across diverse regimes, extending to Weyl-transverse gravity, Einstein–aether theory, and localized quantum field scenarios.

The expression analogue transverse first law denotes a family of first-law-like relations formulated outside the standard setting of stationary black-hole mechanics. In the most explicit use represented here, it is a codimension-two, quasi-local balance law attached directly to an individual closed marginally trapped surface (MTS), where the Hawking energy plays the role of internal energy, an invariant effective surface gravity controls the thermal term, and the variation is taken in a transverse null direction rather than along a preferred horizon worldtube (Torres, 12 Mar 2026). Closely related literature uses the same motif more broadly: extra transverse or boundary charges can modify first-law structures in Lorentz-violating gravity, Weyl-transverse gravity admits a Noether-charge derivation of black-hole and causal-diamond first laws, localized quantum-field processes satisfy a first-law statement only in a restricted moment sense, and thermodynamic no-go arguments rule out steady transverse vacuum thrust in magnetoelectric media (Arata et al., 25 Jun 2026).

1. Conceptual scope and defining features

In its strictest sense, the analogue transverse first law is a deformation identity on a codimension-two surface rather than a phase-space law comparing neighboring stationary solutions. The relevant variation is the ingoing-null derivative

$\delta \equiv l^{-\alpha}\nabla_\alpha,$

so the law is attached to a single surface and probes how its quasi-local energy changes under a transverse null deformation (Torres, 12 Mar 2026).

This distinguishes the construction from the classical first law for stationary Kerr–Newman black holes,

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$

which is an equilibrium relation between neighboring stationary solutions with global conserved charges. The transverse formulation instead uses only intrinsic and normal data of a closed spacelike two-surface and does not require a preferred horizon history (Torres, 12 Mar 2026).

A broader reading, which is interpretive rather than terminological, is that “transverse first law” names a recurring structure in which first-law-like balances are controlled by quasi-local, null-boundary, or localized spacetime data. This suggests a unifying theme across otherwise distinct settings: codimension-two black-hole geometry, Weyl-transverse symmetry, asymptotic aether alignment, null-boundary charges in holographic complexity, and spacetime-localized thermodynamics in quantum field theory (Alonso-Serrano et al., 2022).

2. Closed marginally trapped surfaces and the codimension-two law

The central explicit formulation is developed for a closed spacelike codimension-two surface $\mathcal S$ embedded in a four-dimensional, causally orientable spacetime $(\mathcal V,g)$ with signature $\{-,+,+,+\}$. With tangent vectors

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$

the induced metric is

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$

Two future-directed null normals $l^\pm_\mu$ satisfy

$$l^\pm_\mu e^\mu_A=0,\qquad l^\pm_\mu l^{\pm\mu}=0,\qquad l^+_\mu l^{-\mu}=-1.$$

The null second fundamental forms $K^\pm_{AB}$ have traces $$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$0, the outgoing and ingoing null expansions (Torres, 12 Mar 2026).

A future marginally trapped surface is characterized by

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$1

The mean curvature vector is

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$2

and the dual expansion vector is

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$3

On an MTS, $$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$4 becomes null and coincides with $$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$5 (Torres, 12 Mar 2026).

The internal energy is taken to be the Hawking energy

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$6

with areal radius

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$7

The associated “energy concentration” is

$$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$8

On a future MTS, because $$\delta M=\frac{\kappa}{8\pi}\,\delta A+\Omega_H\,\delta J+\Phi_H\,\delta Q,$$9 and $\mathcal S$0,

$\mathcal S$1

The dependence on the Euler characteristic shows that topology enters directly into the energy concentration (Torres, 12 Mar 2026).

The resulting balance law is

$\mathcal S$2

or equivalently

$\mathcal S$3

with

$\mathcal S$4

The law is therefore explicitly split into a generalized heat contribution and a total work contribution (Torres, 12 Mar 2026).

3. Effective surface gravity, work densities, and thermal interpretation

A central ingredient is the effective surface gravity

$\mathcal S$5

with mean value

$\mathcal S$6

The construction is described as geometrically invariant and independent of the normalization or parametrization of $\mathcal S$7, once a surrounding family of surfaces has been chosen (Torres, 12 Mar 2026).

The work term is built from both matter and geometry. The auxiliary tangent vector

$\mathcal S$8

enters the null variation identity

$\mathcal S$9

After removing the divergence term on a closed MTS, one obtains

$(\mathcal V,g)$0

where

$(\mathcal V,g)$1

The work sector is thus intrinsically split into a matter work density and a geometric work density (Torres, 12 Mar 2026).

The heat term admits a second, explicitly thermodynamic rewriting: $(\mathcal V,g)$2 so that

$(\mathcal V,g)$3

With the local temperature assignments

$(\mathcal V,g)$4

the generalized heat exchange becomes

$(\mathcal V,g)$5

The paper interprets this as the behavior of a spatially extended, non-isothermal thermodynamic medium, where the heat term measures entropy-weighted temperature inhomogeneity over the marginally trapped surface (Torres, 12 Mar 2026).

The law is explicitly not an Iyer–Wald phase-space first law. It depends on a chosen transverse foliation $(\mathcal V,g)$6, and the interpretation of $(\mathcal V,g)$7 is stated to be physically meaningful only in slowly varying or near-equilibrium settings (Torres, 12 Mar 2026).

4. Spherical symmetry, semiclassical regimes, and Kerr surfaces

For round spheres in spherically symmetric spacetimes, the framework reproduces familiar results. In advanced Eddington–Finkelstein coordinates,

$(\mathcal V,g)$8

the outgoing expansion is

$(\mathcal V,g)$9

so $\{-,+,+,+\}$0 is a marginally trapped surface. For round spheres,

$\{-,+,+,+\}$1

and therefore

$\{-,+,+,+\}$2

so the Hawking energy reduces to the Misner–Sharp mass (Torres, 12 Mar 2026).

On the apparent horizon,

$\{-,+,+,+\}$3

and in spherical symmetry $\{-,+,+,+\}$4. Moreover,

$\{-,+,+,+\}$5

so the work sector is purely matter work (Torres, 12 Mar 2026).

The semiclassical applications separate equilibrium and evaporation. In the equilibrium regime, with a Schwarzschild-like black hole in a reflecting cavity and the Hartle–Hawking vacuum $\{-,+,+,+\}$6, regularity implies

$\{-,+,+,+\}$7

and therefore

$\{-,+,+,+\}$8

Spherical symmetry implies

$\{-,+,+,+\}$9

so the balance reduces to

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$0

In the evaporating regime, using the Unruh vacuum $$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$1, the contraction

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$2

remains well defined, and again symmetry gives

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$3

The paper emphasizes that the transverse law depends on $$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$4, rather than on the divergent outgoing flux $$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$5, so it bypasses problematic longitudinal evaporation fluxes (Torres, 12 Mar 2026).

The non-spherical Kerr analysis shows that the framework is not tied to spherical symmetry. The marginally trapped surfaces occur at

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$6

The effective surface gravity depends on latitude, and the twist term no longer vanishes: $$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$7 Rotation therefore opens a genuinely geometric work channel. The dual expansion vector determines privileged observers with angular velocity

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$8

and on the outer MTS,

$$\vec e_A = e^\mu_A \frac{\partial}{\partial x^\mu}\Big|_{\mathcal S} = \frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial}{\partial x^\mu}\Big|_{\mathcal S},$$9

This aligns the quasi-local construction with the usual corotating horizon picture at the marginally trapped surface (Torres, 12 Mar 2026).

5. Gravitational generalizations: Weyl-transverse gravity and Einstein–aether theory

A closely related development appears in Weyl transverse gravity (WTG), a theory invariant under Weyl transformations and transverse diffeomorphisms. In this setting, the Noether-charge formalism yields the black-hole first law, the causal-diamond first law, and variable-$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$0 thermodynamic relations. The vacuum action simplifies to

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$1

and the Noether charge tensor for transverse diffeomorphisms is

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$2

The Hamiltonian variation contains a bulk term proportional to $$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$3,

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$4

which the paper identifies as a structural difference from general relativity. In unimodular gauge, the Schwarzschild–AdS first law takes the black-hole-chemistry form

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$5

with

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$6

The same work derives the first law of causal diamonds in vacuum and interprets the whole construction as an analogue transverse first law because the thermodynamic structure is reproduced in a framework based on WTDiff rather than full Diff invariance (Alonso-Serrano et al., 2022).

In Einstein–aether theory, the static, spherically symmetric, asymptotically AdS sector with $$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$7 exhibits an additional term in the universal-horizon first law whenever the aether is misaligned with the timelike Killing vector at infinity. The paper identifies this term as the boundary manifestation of a previously unnoticed symmetry of the reduced action,

$$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$8

where the parameter $$\gamma_{AB}=g_{\mu\nu}\Big|_{\mathcal S}\frac{\partial \Phi^\mu}{\partial \lambda^A}\frac{\partial \Phi^\nu}{\partial \lambda^B}.$$9 obeys the elliptic constraint

$l^\pm_\mu$0

The associated current is

$l^\pm_\mu$1

with charge

$l^\pm_\mu$2

Although $l^\pm_\mu$3 on a hypersurface orthogonal to the aether, the physically relevant effect is a flux across a timelike boundary. The aligned limit

$l^\pm_\mu$4

is precisely the limit in which the aether-charge flux vanishes, recovering the clean first law

$l^\pm_\mu$5

The paper describes this as the Einstein–aether analogue of the transverse first-law mechanism previously identified in Hořava–Lifshitz gravity: asymptotic alignment corresponds to the ensemble in which the extra aether-charge contribution is frozen to zero (Arata et al., 25 Jun 2026).

6. Field-theoretic analogues, holographic extensions, and limiting results

In quantum field thermodynamics, a first-law-like statement can be formulated for spacetime-localized unitary processes on a free scalar field. The general unitary is

$l^\pm_\mu$6

and a specialized family uses a switching function $l^\pm_\mu$7 and spatial smearing $l^\pm_\mu$8. The paper identifies three QFT-compatible work distributions—Ramsey scheme, ATMH, and FCS—and proves that for ATMH and FCS,

$l^\pm_\mu$9

Thus the first law holds up to second moments. The same framework satisfies Crooks theorem and Jarzynski equality for KMS states. Because the thermodynamic quantities depend explicitly on the spacetime profile $$l^\pm_\mu e^\mu_A=0,\qquad l^\pm_\mu l^{\pm\mu}=0,\qquad l^+_\mu l^{-\mu}=-1.$$0, the paper presents a localized, spacetime-resolved first-law statement rather than a finite-dimensional quench analogue (Teixidó-Bonfill et al., 2020).

In holography, the first law of holographic complexity in the complexity=action proposal is formulated through the variation of the Wheeler–DeWitt action,

$$l^\pm_\mu e^\mu_A=0,\qquad l^\pm_\mu l^{\pm\mu}=0,\qquad l^+_\mu l^{-\mu}=-1.$$1

The resulting expression is written as a linear functional of perturbations with coefficients tied to null-boundary geometric data and conserved charges. The paper states that this structure is “reminiscent of the first law of thermodynamics,” and in examples the relevant coefficients encode mass/energy and angular momentum. This is not a transverse first law in the codimension-two MTS sense, but it is an explicit first-law-like response formula organized by null-boundary charges (Hashemi et al., 2019).

Several papers also delineate limits on how far transverse first-law reasoning can be pushed. In magnetoelectric media, the claim that the quantum vacuum could exert a permanent transverse thrust—the steady Feigel effect—is shown to be incompatible with both thermodynamics and properly treated boundary conditions. In a realistic finite geometry, the averaged boundary stresses satisfy

$$l^\pm_\mu e^\mu_A=0,\qquad l^\pm_\mu l^{\pm\mu}=0,\qquad l^+_\mu l^{-\mu}=-1.$$2

so

$$l^\pm_\mu e^\mu_A=0,\qquad l^\pm_\mu l^{\pm\mu}=0,\qquad l^+_\mu l^{-\mu}=-1.$$3

The paper argues that any steady vacuum-driven circulation or torque would allow work extraction from the quantum vacuum in a closed cycle with no compensating input, violating the first law. Only unsteady effects, with time-dependent susceptibilities or accelerated media, remain compatible with QED and thermodynamics. In the paper’s own summary, the broader implication is that “steady transverse vacuum thrust is forbidden; only driven, nonsteady analogue effects are physically consistent” (Croze, 2013).

A different limitation appears in the critical transverse-field Ising chain. There, in a basis common to all one-site shift-invariant conserved charges, no eigenstate of a noninteracting local spin-$$l^\pm_\mu e^\mu_A=0,\qquad l^\pm_\mu l^{\pm\mu}=0,\qquad l^+_\mu l^{-\mu}=-1.$$4 chain with half-integer central charge satisfies the area law. For the critical Ising class, the paper concludes that there are no excited states satisfying the area law, and that every quasilocal one-site shift-invariant conserved operator is gapless. The authors explicitly connect this to an “analogue transverse first law” perspective in a negative or limiting sense: any first-law-like description relying on locally distinct area-law eigenstates is ruled out in the critical Ising case because the relevant eigenstates are intrinsically gapless and logarithmically entangled (Bocini et al., 2023).

Taken together, these results indicate that the analogue transverse first law is best understood not as a single universally fixed formula, but as a research program centered on quasi-local or boundary-supported thermodynamic balance laws. Its clearest realization is the codimension-two law for marginally trapped surfaces; its closest analogues arise where extra null, transverse, or asymptotic charges modify first-law structures; and its most important caveat is that any purported steady transverse extraction mechanism must survive both full boundary accounting and ordinary energy balance (Torres, 12 Mar 2026).