Thermodynamics-GR fields, currents, reference normalization and units

Use a smooth, oriented and time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$ with its metric-selected Levi-Civita connection $\mathrm{\nabla }$, as defined in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. A chart is a smooth coordinate map; an observer congruence is a smooth future-directed unit timelike vector field $u$ with $g(u,u)=-1$; its integral curves, an adapted chart, and a tetrad field are distinct objects. The exact regularity and rest-space conventions of Rest fluid state, frame and geometric current definitions and the chart/congruence distinction of Local flow boxes, adapted charts and the Frobenius criterion for rest distributions apply. An orthogonal family of rest slices is not available for every congruence.

For physical unit assignments use length coordinates $x^0=c{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ and spatial coordinates in metres, where $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$. In this convention the metric components, a unit observer $u$, and a length-normalized Killing field $\xi$ are dimensionless. Other coordinate choices transform component units by their coordinate Jacobians. State maps and geometric currents have the types and units in Rest fluid state, frame and geometric current definitions: a formal marker density $n$ is $\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}{\mathrm{m}}^{-3}$; entropy density $s$ is $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$; a $C^1$ entropy current $S^a$ has components $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$; and its physical flux is $c{S}^a$ in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. Its geometric divergence ${\mathrm{\nabla }}_a{S}^a$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-4}$, so $c{\mathrm{\nabla }}_a{S}^a$ is entropy production per physical volume per second. The symmetric stress tensor ${\mathsf{T}}^{ab}$ has components in pascals in length coordinates; it is a tensor, not an observer-independent array of energy densities. The state-function energy density $\epsilon$ and the observer projection ${\mathsf{T}}^{ab}{u}_a{u}_b$ remain distinct unless a material model identifies them.

The constant $c$ supplies the spacetime length/time conversion. In a branch with dynamical Einstein gravity, $G$ is an independent coupling parameter in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$. The Boltzmann constant $k_B$ has units $\mathrm{J}{\mathrm{K}}^{-1}$ and is used when a statistical entropy/temperature calibration is adopted. Planck's constant $h_{\mathrm{P}}=2\pi \mathrm{\hslash }$ has units $\mathrm{J}\mathrm{s}$; $\mathrm{\hslash }$ is used only in a quantum branch. Under the current SI definition, $c$, $h_{\mathrm{P}}$ and $k_B$ have exact defining numerical values; this metrological fact does not make them derived predictions of a thermodynamic or gravitational model. $G$ is experimentally adjusted and carries measurement uncertainty. The roles of these quantities as primitive model inputs are framework-specific. The local energy function, pressure, temperature and chemical potential in Rest fluid state, frame and geometric current definitions are derived from the supplied state function; no particle mass or universal equation of state is derived from gravity alone.

For a smooth Killing field $\xi$ on an open domain, meaning ${\mathcal{L}}_{\xi }g=0$ by Killing fields of a Lorentzian metric, restrict attention to a domain where $\xi$ is future timelike and choose a positive normalization at infinity or at a specified wall/observer. Define the lapse $N_{\xi }:=\sqrt{-g(\xi ,\xi )}>0$ and unit observer $u_{\xi }:=\xi \mathrm{/}{N}_{\xi }$. For a supplied proper-temperature scalar $T>0$ in kelvins, define the Killing-normalized temperature field $T_{\xi }:=N_{\xi }T$. It is not assumed constant; if a separate equilibrium argument proves constancy on its stated domain, denote that value by $T_K$. The finite matter charge and its side-flux convention are those in Finite Killing energy current and reference normalization and Finite spacelike slab and normal flux charge; their charge-balance clause retains its explicitly conditional finite-slab hypotheses. This charge is not a general gravitational energy.

Keep covector types separate. A mechanical momentum covector $p_a$ has units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$; its Killing energy is $E_{\xi }(p):=-c{p}_a{\xi }^a$ in joules. An energy-valued covector ${\pi }_a$ already has units joules and pairs as $E_{\xi }(\pi ):=-{\pi }_a{\xi }^a$, without another factor of $c$. An inverse-length wave covector $k_a$ has units ${\mathrm{m}}^{-1}$ and pairs as $E_{\xi }(k):=-\mathrm{\hslash }c{k}_a{\xi }^a$. For a specified charged stationary gauge, charge $q$ has coulombs and a potential covector $A_a$ in length coordinates has units $\mathrm{V}\mathrm{s}{\mathrm{m}}^{-1}$; when a global stationary potential is actually defined, ${\mathrm{\Phi }}_{\xi }:=-c{A}_a{\xi }^a$ has volts and $q{\mathrm{\Phi }}_{\xi }$ has joules. A charged energy or chemical-potential convention requires its stated gauge and reference; no global potential, electrochemical equilibrium, or constancy is asserted here.

When $T_K>0$ is supplied, distinguish the inverse-temperature vector ${\beta }^a:=u_{\xi }^a\mathrm{/}(k_{B}T)={\xi }^a\mathrm{/}(k_B{T}_K)$, with units ${\mathrm{J}}^{-1}$, from the scalar ${\beta }_K:=1\mathrm{/}(k_B{T}_K)$, also in ${\mathrm{J}}^{-1}$. For a quantum statistical model, let $\mathcal{A}$ be a unital complex $C^{\ast }$-algebra: an associative complex algebra with nonzero identity $1$, a complete norm satisfying $\mathrm{\parallel }AB\mathrm{\parallel }\le \mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$, and a conjugate-linear involution $A\mapsto A^{\ast }$ satisfying $(A^{\ast }{)}^{\ast }=A$, $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$, and $\mathrm{\parallel }{A}^{\ast }A\mathrm{\parallel }=\mathrm{\parallel }A{\mathrm{\parallel }}^2$. Its elements represent bounded observables. Let $\{{\alpha }_t:t\in \mathbb{R}\}$ be bijective complex-linear maps ${\alpha }_t:\mathcal{A}\to \mathcal{A}$ preserving multiplication and involution, with ${\alpha }_t(1)=1$, ${\alpha }_0=\mathrm{id}$, and ${\alpha }_{s+t}={\alpha }_s\circ {\alpha }_t$; require $t\mapsto {\alpha }_t(A)$ to be norm-continuous for every $A\in \mathcal{A}$. A state is a bounded complex-linear functional $\omega :\mathcal{A}\to \mathbb{C}$ with $\omega (1)=1$ and $\omega (A^{\ast }A)\ge 0$ for every $A\in \mathcal{A}$. For $b>0$ in the chosen flow-parameter units, define $S_b:=\{z\in \mathbb{C}:0\le \mathrm{Im}z\le b\}$. The state $\omega$ satisfies the KMS condition at width $b$ when for every $A,B\in \mathcal{A}$ there is a function $F_{A,B}:S_b\to \mathbb{C}$ with ${\sup }_{z\in S_b}\mathrm{\mid }{F}_{A,B}(z)\mathrm{\mid }<\mathrm{\infty }$, continuous on $S_b$ and complex-differentiable at every point of its interior, such that

$F_{A,B}(t)=\omega (A{\alpha }_t(B)),F_{A,B}(t+ib)=\omega ({\alpha }_t(B)A)(t\in \mathbb{R})\mathrm{.}$

This all-observable boundary-value definition uses no separately declared analytic subalgebra. With the quantum statistical calibration ${\beta }_K=(k_B{T}_K{)}^{-1}$, a physical-time parameter $t$ in seconds has KMS width $b_t=\mathrm{\hslash }{\beta }_K$ seconds. If the flow parameter is length $\lambda =ct$, use the reparameterized group ${\tilde{\alpha }}_{\lambda }:={\alpha }_{\lambda \mathrm{/}c}$; its KMS width is $b_{\lambda }=c\mathrm{\hslash }{\beta }_K$ metres. These are unit and parameter-convention assignments only. This definition introduces no Hilbert-space operator $H_K$ and does not identify the abstract automorphism flow or its strip width with the finite matter charge $H_{\xi }$ of Finite Killing energy current and reference normalization. This item asserts neither the existence of an observable algebra or flow for a given thermodynamic-gravity model nor the existence of a KMS state or equilibrium.

On a specified smooth embedded null hypersurface $\mathcal{H}\subset M$, write $i:\mathcal{H}\hookrightarrow M$ for the inclusion. A generator $\chi$ here is a smooth section of $i^{\ast }TM$ with ${\chi }_p\in T_p\mathcal{H}$ nonzero and null at each $p\in \mathcal{H}$. For a smooth tangent field $X$ on $\mathcal{H}$ and a smooth vector field $V$ along it, define ${\mathrm{\nabla }}_{X}V:=(i^{\ast }\mathrm{\nabla }{)}_{X}V$ by Pullback connection, applied to the ambient tangent bundle and its metric-selected connection. The prescription is independent of frames by Pullback connection is well defined and functorial. In particular, ${\mathrm{\nabla }}_{\chi }\chi$ is a vector field along $\mathcal{H}$ and needs no chosen ambient extension. If $\chi$ is the restriction of a smooth ambient field $\tilde{\chi }$, the same theorem gives ${\mathrm{\nabla }}_{\chi }\chi =({\mathrm{\nabla }}_{\tilde{\chi }}\tilde{\chi }){\mathrm{\mid }}_{\mathcal{H}}$; thus this convention also covers the supplied ambient Killing generators. The generator may admit a scalar ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ defined by ${\mathrm{\nabla }}_{\chi }\chi ={\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}\chi$. When the flow parameter has length units, ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ has units ${\mathrm{m}}^{-1}$; the corresponding acceleration-unit quantity ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}:=c^2{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ has units $\mathrm{m}{\mathrm{s}}^{-2}$. This definition does not assert a horizon exists, that $\kappa$ is constant, or that a temperature law follows. The formula $T_K=\mathrm{\hslash }c{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}\mathrm{/}(2\pi k_B)$ requires a specified quantum/semiclassical model, and an area-entropy formula requires a separate thermodynamic identification. In particular, equal units do not identify local material entropy, finite statistical entropy, and area entropy, and no generalized second law is asserted here.

The source scaffold also discusses relativistic kinetic partition integrals and finite-regulator generalized entropy. Their detailed preparation, existence, convergence, and moment claims require their own exact definitions and proofs on the later planned pages; they are not additional consequences of this units definition. The sources are used for conventions and independent treatments, not as substitutes for those arguments or as evidence that a selected model describes every material.

For a selected fixed-composition, single-material branch, adopt a local-equilibrium description on an open state domain $D\subset (0,\mathrm{\infty })\times \mathbb{R}$ with a supplied function $e\in C^2(D,\mathbb{R})$ and $T:={\mathrm{\partial }}_{{\sigma }_p}e>0$. Here $v$ is the rest volume per selected material marker, ${\sigma }_p$ is entropy per marker, and $e$ is energy per marker, including the rest-energy contribution; their types and SI units are those in Rest fluid state, frame and geometric current definitions and Thermodynamics-GR fields, currents, reference normalization and units. Here “comoving” means that local state values are referred to the observer-congruence rest spaces of Rest fluid state, frame and geometric current definitions; no orthogonal rest-slice foliation or finite-region volume integral is presupposed. For an admissible local continuum parcel, let $N_p>0$ be its real marker amount, $V>0$ its supplied proper-volume parameter, and $S\in \mathbb{R}$ its supplied total entropy, with $(V\mathrm{/}{N}_p,S\mathrm{/}{N}_p)\in D$. The quantities $V\mathrm{/}{N}_p$ and $S\mathrm{/}{N}_p$ have the per-marker state meanings and units of Rest fluid state, frame and geometric current definitions. This model assigns energy $E=N_{p}e(V\mathrm{/}{N}_p,S\mathrm{/}{N}_p)$. Treating the real marker amount as a particle count is part of the fixed-composition continuum model, not a claim about discrete microscopic transitions or a consequence of the mathematical marker definition.

This is an adopted constitutive/local-equilibrium premise for the stated material and regime. It does not derive $e$ microscopically, assert a measured equation of state, impose a spacetime conservation equation, imply a universal gas law, or make the local parcel energy additive with the energy of an arbitrary self-gravitating region. Rest energy belongs in $e$; if a rest mass $m$ is specified, its contribution is $m{c}^2$, not $m$ in SI units. No entropy-maximization, quantum Gibbs representation, or global gravitating thermodynamic limit is assumed.

On an open region of the smooth spacetime of Thermodynamics-GR fields, currents, reference normalization and units, take a separately specified material/current model in its declared smooth regime. Its entropy current is a supplied $C^1$ vector field $S^a$ of the type and units in Rest fluid state, frame and geometric current definitions. Adopt the local admissibility condition

${\mathrm{\nabla }}_a{S}^a\ge 0$

throughout that region. This inequality is a physical restriction on the chosen model and current; it is not a construction of an undefined constitutive current. Each application must state its current, regime, boundary/exchange data, and any source or sink terms it includes.

The inequality is not deduced from Lorentzian geometry, the local state function, or finite data. It does not assert statistical thermalization, a universal microscopic $H$-theorem, entropy-area identification, or a generalized second law. Romatschke's cited calculation is restricted to a specified conformal-fluid model and itself records the absence of experimental verification known to that author; that source limitation does not turn this postulate into an observed law. No experimental observation or uncertainty is supplied by this item.

Use the smooth time-oriented Lorentzian geometry, positively oriented chart and product slab $K=[x_{-}^0,x_{+}^0]\times \mathrm{\Omega }$ with open spatial rectangle $\mathrm{\Omega }$ and compact closure $\bar{K}$ of Finite spacelike slab and normal flux charge, with nonempty rectangular factors, spacelike slices, future unit normal $n$, induced volume $d{V}_h$, and a $C^1$ current defined on a neighborhood of $\bar{K}$. Let $S^a$ be the entropy current selected by Local entropy-current admissibility in a separately defined model and put $\mathrm{\Pi }:={\mathrm{\nabla }}_a{S}^a\ge 0$ on $K$. Define the proper slice entropy

$Q_S(x^0):={\int }_{\mathrm{\Omega }}[-g(S,n)]d{V}_h$

and let ${\mathrm{\Phi }}_S(B)$ be the net outward flux through the union $B$ of the six spatial side faces, with the sign convention of Finite spacelike slab and normal flux charge. Conditional on the exact current statement and hypotheses of the draft mathematical theorem Complete finite-slab production and outward-flux balance,

$Q_S(x_{+}^0)-Q_S(x_{-}^0)+{\mathrm{\Phi }}_S(B)={\int }_K\mathrm{\Pi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g\ge 0.$

Every term has units $\mathrm{J}{\mathrm{K}}^{-1}$ in the length-coordinate convention of Thermodynamics-GR fields, currents, reference normalization and units. If the net outward side flux is zero, then the endpoint charges satisfy $Q_S(x_{+}^0)\ge Q_S(x_{-}^0)$. This is a deduction conditional on both the adopted local inequality and the actual draft supplier interface; it is not an observation, a proof approval of that mathematical supplier, or a generalized second law.