Separate no-conserved-number radiation fluid branch

Use the smooth time-oriented Lorentzian geometry, metric-selected Levi-Civita connection $\mathrm{\nabla }$, and length-coordinate conventions of Thermodynamics-GR fields, currents, reference normalization and units, with signature $(-,+,+,+)$. On an open region $O$ of a smooth four-manifold without boundary, take a smooth positive temperature field $T:O\to (0,\mathrm{\infty })$ and a smooth future unit observer-congruence field $u$ satisfying $g(u,u)=-1$. Adopt the separate entropy-only state maps of Separate entropy-only radiation state maps with a fixed parameter $a_R>0$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$:

$\epsilon =a_R{T}^4,p=\frac{1}{3}{a}_R{T}^4,s=\frac{4}{3}{a}_R{T}^3,h:=\epsilon +p\mathrm{.}$

The fields $\epsilon ,p,h$ have units $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{P}\mathrm{a}$ and $s$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$. Adopt the perfect stress tensor and its source-free balance

${\mathsf{T}}^{ab}:=h{u}^a{u}^b+p{g}^{ab},{\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$

on the specified smooth background. In the proper-time velocity convention $U^a:=c{u}^a$, this is equivalently ${\mathsf{T}}^{ab}=(h\mathrm{/}{c}^2)U^a{U}^b+p{g}^{ab}$ with $g(U,U)=-c^2$. This branch has no conserved particle-number coordinate, particle-number current, or chemical-potential field. It adopts an effective equation of state and perfect-stress balance; it does not determine $a_R$ from quantum statistics, assert that a physical system realizes these maps, report a measurement, or derive an Einstein-field equation.