Perfect sound characteristic and actual stable causal linear branch

Fix $c>0$ and take $M={\mathbb{R}}^4$ with global length coordinates $(x^0,x^1,x^2,x^3)=(ct,\mathbf{x})$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and future causal cone $\{V\in {\mathbb{R}}^4:V^0\ge 0,{\eta }_{ab}{V}^a{V}^b\le 0\}$. For $V=(V^0,\mathbf{V})\in {\mathbb{R}}^4$ with $V^0>0$, define its coordinate velocity by $\mathbf{v}(V):=c\mathbf{V}\mathrm{/}{V}^0$. Since $\eta (V,V)=(V^0{)}^2(\mathrm{\mid }\mathbf{v}(V){\mathrm{\mid }}^2\mathrm{/}{c}^2-1)$, membership in this cone is equivalent to $\mathrm{\mid }\mathbf{v}(V)\mathrm{\mid }\le c$, with equality on its nonzero null boundary. The zero vector belongs to the cone but has no coordinate velocity under this definition. The constant rest observer is $\bar{u}=(1,0,0,0)$.

Let $X\subset (0,\mathrm{\infty })\times \mathbb{R}$ be open, let $e\in C^2(X;\mathbb{R})$, and fix $(v_0,{\sigma }_{p0})\in X$. Define $n_0:=1\mathrm{/}{v}_0$, ${ϵ}_0:=n_{0}e(v_0,{\sigma }_{p0})$, $p_0:=-e_v(v_0,{\sigma }_{p0})$, $T_0:=e_{{\sigma }_p}(v_0,{\sigma }_{p0})$, and $h_0:={ϵ}_0+p_0$. Assume $n_0,T_0,h_0,e_{vv}>0$, with $e_{vv}$ evaluated at fixed ${\sigma }_p$. A first variation is defined by a family of state maps $(v_{\lambda },{\sigma }_{p,\lambda }):M\to X$ and fields $u_{\lambda }:M\to {\mathbb{R}}^4$, jointly smooth in dimensionless $\lambda$ near $0$ and in $x$, with the displayed reference state and $\bar{u}$ as their constant $\lambda =0$ values and with $\eta (u_{\lambda },u_{\lambda })=-1$, $u_{\lambda }^0>0$. For any such field $f_{\lambda }$, write $\delta f:={{\mathrm{\partial }}_{\lambda }{f}_{\lambda }\mid }_{\lambda =0}$; put $r:=\delta ϵ$, $w:=(\delta u^1,\delta u^2,\delta u^3)$, and impose the adiabatic first-variation condition $\delta {\sigma }_p=0$. The linearized equations mean the first variations at $\lambda =0$ of the no-force balances ${\mathrm{\partial }}_a{T}_{\lambda }^{ab}=0$ and ${\mathrm{\partial }}_a{J}_{\lambda }^a=0$, with $n_{\lambda }=1\mathrm{/}{v}_{\lambda }$, ${ϵ}_{\lambda }=n_{\lambda }e(v_{\lambda },{\sigma }_{p,\lambda })$, $p_{\lambda }=-e_v(v_{\lambda },{\sigma }_{p,\lambda })$, ${\mathrm{\Delta }}_{\lambda }^{ab}={\eta }^{ab}+u_{\lambda }^a{u}_{\lambda }^b$, $T_{\lambda }^{ab}={ϵ}_{\lambda }{u}_{\lambda }^a{u}_{\lambda }^b+p_{\lambda }{\mathrm{\Delta }}_{\lambda }^{ab}$, and $J_{\lambda }^a=n_{\lambda }{u}_{\lambda }^a$; no member with $\lambda \mathrm{\ne }0$ is required to solve the nonlinear balances. Differentiating $\eta (u_{\lambda },u_{\lambda })=-1$ gives $\delta u^0=0$.

The local Gibbs identity of Local Gibbs, Euler, parcel first law and response identities gives

$a_s:={\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }ϵ}\right)}_{{\sigma }_p}=\frac{e_{vv}(v_0,{\sigma }_{p0})}{n_0{h}_0},$

and assume additionally $a_s\le 1$. In the adopted units, $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $r$ and $h_0$ have units Pa, and $w$ and $a_s$ are dimensionless. Since $x^0=ct$ and $\bar{u}={\mathrm{\partial }}_{x^0}$, the first variations of the no-force perfect-fluid projection equations of Perfect stress and particle balances with entropy identity are

$r_t+c{h}_0\mathrm{div}w=0,w_t+\frac{c{a}_s}{h_0}\mathrm{\nabla }r=0.$

For initial data $U_0=(r_0,w_0)$ whose real components belong componentwise to $\mathcal{S}({\mathbb{R}}^3)$ as defined by Schwartz space and its seminorms, or for smooth periodic data on ${\mathbb{T}}_L^3$ with $L>0$, these equations have the unique global smooth solution supplied by Complete symmetrized constant-coefficient evolution. The positive quadratic energy density $\mathcal{E}=\frac{1}{2}[(a_s\mathrm{/}{h}_0)r^2+h_0\mathrm{\mid }w{\mathrm{\mid }}^2]$ has flux $c{a}_{s}rw$, and the solution is contractive in the corresponding weighted $L^2$ and Sobolev norms. Its characteristic speeds are $0$ and $\pm c\sqrt{a_s}$; for positive $a_s$, this propagation cone lies inside the Minkowski reference cone exactly when $a_s\le 1$. These are statements about this frozen linear system only; they do not assert a nonlinear or curved-spacetime evolution theorem.

For the formal coefficient comparison in [A3], if the same linear equations are assigned $a_s>1$, their plane-wave speed is $c\sqrt{a_s}>c$. If they are assigned $a_s<0$, smooth periodic modes grow as $e^{c\sqrt{-a_s}\mathrm{\mid }k\mathrm{\mid }t}$ at arbitrarily large wave numbers. The case $a_s=0$ is degenerate: the displayed positive energy matrix is no longer positive definite, so the stability proof above does not apply.