A conditional regulated generalized second law

Fix a smooth time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$, with its metric-selected connection $\mathrm{\nabla }$. Let $\mathcal{H}\subset M$ be a smooth embedded null hypersurface, meaning the induced metric on each $T_p\mathcal{H}$ has a one-dimensional radical. Supply a smooth nonzero future-directed null generator field $k$ in this radical with ${\mathrm{\nabla }}_{k}k=0$. Choose nonempty compact smooth embedded spacelike two-surfaces ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2\subset \mathcal{H}$ transverse to $k$, so $T_p{\mathrm{\Sigma }}_i\oplus \mathbb{R}{k}_p=T_p\mathcal{H}$ for every $p\in {\mathrm{\Sigma }}_i$. Their supplied family labels $t_1<t_2$ order the cuts; these geometric data use the horizon-section and affine-generator conventions of Specified regulated generalized entropy, but no global event-horizon property is assumed.

At a named finite regulator, take nonzero finite-dimensional complex inner-product spaces $H_2,H_E$ as the model outside Hilbert factors at ${\mathrm{\Sigma }}_2$ and the additional region between the cuts, respectively, and use the ordered factorization $H_1=H_2\otimes H_E$ at ${\mathrm{\Sigma }}_1$. Supply dimensionless density operators ${\rho }_1,{\sigma }_1$ on $H_1$ and ${\rho }_2,{\sigma }_2$ on $H_2$ such that ${\rho }_2={\mathrm{Tr}}_E{\rho }_1$ and ${\sigma }_2={\mathrm{Tr}}_E{\sigma }_1$. Require both ${\sigma }_1$ and ${\sigma }_2$ to be positive definite; the ${\rho }_i$ may have zero eigenvalues.

For each $i=1,2$, define $K_i:=-\log {\sigma }_i$ by finite-dimensional spectral calculus and $s(\tau ):=-\mathrm{tr}(\tau \log \tau )$ in nats, with $0\log 0:=0$. Set ${\mathrm{\Delta }}_i:=\mathrm{tr}(({\rho }_i-{\sigma }_i)K_i)$. Supply finite real geometric-entropy data $a_i$ in nats and adopt the balance premise

$a_2-a_1={\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2\mathrm{.}$

With the physical Boltzmann constant $k_B>0$ in SI units $\mathrm{J}{\mathrm{K}}^{-1}$, define the model's reference-subtracted finite generalized entropy by

$S_{\mathrm{g}\mathrm{e}\mathrm{n},i}:=k_B[a_i+s({\rho }_i)-s({\sigma }_i)]\mathrm{.}$

For a named sequence of regulators, impose these finite-dimensional premises term by term under one fixed subtraction and finite-normalization scheme. This adoption supplies no convergence claim. The scalar $a_i$ is model data; it is not thereby identified with an area term or with a calibrated gravitational entropy.

Specified regulated generalized entropy

Finite relative entropy equals modular energy minus entropy

Aron C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices

H. S. Reall, Part III Black Holes

John Watrous, The Theory of Quantum Information

The matrices and balance relation are adopted finite-regulator premises. In particular, the balance equation is not derived from classical area increase, stationary area calibration, a semiclassical Einstein equation, or Wall's continuum proof. The finite-regulator definition in Specified regulated generalized entropy supplies fixed-scheme and unit conventions but does not itself impose this reference subtraction or balance. The present $S_{\mathrm{g}\mathrm{e}\mathrm{n},i}$ is the expressly adopted reference-subtracted model value; $a_i$ is not thereby identified with $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},i}\mathrm{/}{k}_B$ or with an area term. The quantity $K_i=-\log {\sigma }_i$ is dimensionless modular data, not a physical Hamiltonian.

For every finite-regulator data set satisfying the nested balance postulate, set $D_i:=D({\rho }_i\mathrm{\parallel }{\sigma }_i)$ for $i=1,2$. Then

$S_{\mathrm{g}\mathrm{e}\mathrm{n},2}-S_{\mathrm{g}\mathrm{e}\mathrm{n},1}=k_B[D({\rho }_1\mathrm{\parallel }{\sigma }_1)-D({\rho }_2\mathrm{\parallel }{\sigma }_2)]\ge 0.$

If a sequence of such data sets satisfies the postulate term by term under one fixed scheme, and both endpoint sequences converge to finite real values $S_{\mathrm{g}\mathrm{e}\mathrm{n},i,n}\to L_i$, then $L_2\ge L_1$.

Nested finite regulator and gravitational balance premises

Finite relative entropy equals modular energy minus entropy

Aron C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices

H. S. Reall, Part III Black Holes

John Watrous, The Theory of Quantum Information

The balance identity is an adopted premise. The proof uses finite-dimensional partial-trace data processing and real-sequence convergence only; it does not establish that the balance premise holds in a continuum horizon model or that either limit exists.