Thermodynamics, Sound Speed, and Causality

The pressure of a relativistic fluid is not an independently adjustable force: it is a derivative of an energy function with specified entropy and composition. That thermodynamic structure determines the characteristic sound cone. Linearizing the Euler equations in a local rest frame shows that the adiabatic derivative $c_s^2=(\partial p/\partial\varepsilon)_{s,Y_i}$ is both the squared sound speed and the coefficient controlling local hyperbolicity. Stability requires it to be nonnegative, while relativistic causality requires it not to exceed unity.

Required background. Hilbert Stress–Energy from Metric Variation supplies stress–energy conservation; Events, Intervals, and Minkowski Geometry supplies local inertial frames and causal cones; and Differential Equations supplies linearization and hyperbolicity.

Helpful background. The Einstein Equation fixes the gravitational coupling; Relativistic Matter and Compact Objects gives the volume-level setting; and Computational Relativity uses these characteristic speeds in evolution schemes.

First law and enthalpy

Let $n\gt0$ be baryon number density, $s$ entropy per baryon, and $Y_i$ independent composition fractions. Assume positive enthalpy density $w=\varepsilon+p$. A convenient fundamental relation is $\varepsilon=\varepsilon(n,s,Y_i)$. For fixed composition, the local first law reads

$$d\!\left(\frac{\varepsilon}{n}\right) =T ds-p d\!\left(\frac1n\right),$$

or, equivalently,

$$d\varepsilon=\frac{\varepsilon+p}{n}dn+nT ds.$$

Here $w=\varepsilon+p$ is enthalpy density and $h=w/n$ is enthalpy per baryon. At fixed entropy per baryon,

$$p=n\left(\frac{\partial\varepsilon}{\partial n}\right)_s-\varepsilon.$$

This identity is a stringent consistency check for an equation-of-state table: interpolating $p$ and $\varepsilon$ independently can violate it even when both columns look smooth. With composition, add $n\sum_i\mathcal A_i\,dY_i$, where $\mathcal A_i=(\partial(\varepsilon/n)/\partial Y_i)_{n,s,Y_{j\ne i}}$ is the chemical potential combination conjugate to the independent fraction $Y_i$. It can be a difference of species chemical potentials when conservation constraints eliminate other fractions. The dynamical derivative must declare whether composition is frozen or follows chemical equilibrium. The underlying first law and composition constraints are developed in Andersson and Comer 2021, §§ 2.1–2.2, Eqs. (2.1)–(2.32).

Sound waves from the Euler system

Consider a homogeneous equilibrium in a local Minkowski frame with $u^a=(1,0,0,0)$, constant $n_0$, $p_0$, and $\varepsilon_0$. This is an exact flat-spacetime calculation, or a local short-wavelength calculation neglecting background gradients and gravitational perturbations. Take adiabatic, fixed-composition perturbations proportional to $e^{i(kx-\omega t)}$. Linearized number and momentum conservation give

$$-i\omega\,\delta n+i k n_0\delta v=0,$$ $$-i\omega(\varepsilon_0+p_0)\delta v+i k\,\delta p=0.$$

The first law at fixed $s$ gives $\delta\varepsilon=(\varepsilon_0+p_0)\delta n/n_0$, while the equation of state gives $\delta p=c_s^2\delta\varepsilon$. Eliminating $\delta n$ and $\delta v$ yields

$$\omega^2=c_s^2 k^2, \qquad c_s^2=\left(\frac{\partial p}{\partial\varepsilon}\right)_{s,Y_i}.$$

This derivation checks the normalization: in units with $c=1$, $c_s$ is dimensionless. Restoring units would compare the physical sound speed with $c$, not change the thermodynamic derivative after energy density and pressure are expressed in the same units. The same local sound-wave calculation is given in Andersson and Comer 2021, § 7.2, Eqs. (7.20)–(7.34).

In a frame where the background fluid moves with one-dimensional velocity $v$, special-relativistic velocity addition gives the two acoustic characteristic speeds

$$\lambda_\pm=\frac{v\pm c_s}{1\pm v c_s}.$$

They reduce to $\pm c_s$ in the rest frame and approach the light cone only as $c_s\to1$. The material characteristic is $\lambda_0=v$.

Stability, hyperbolicity, and causality

For this ideal, first-order fluid system with $w\gt0$, $c_s^2\gt0$ makes the acoustic frequencies real and supplies distinct sound characteristics away from degeneracies. If $c_s^2\lt0$, a Fourier mode grows as $e^{\sqrt{-c_s^2}\,\lvert k\rvert t}$, signaling local mechanical instability and loss of hyperbolicity. At $c_s^2=0$ the acoustic fields become degenerate and require separate analysis. This local test does not exclude gravitational instabilities such as long-wavelength Jeans growth.

Causality imposes

$$0\leq c_s^2\leq1.$$

The upper bound ensures that the local sound cone lies within or on the metric null cone. It is a necessary local test, but it does not by itself establish that a complicated constitutive theory is causal. Multi-fluid, elastic, magnetized, or dissipative systems have additional characteristic modes and relaxation variables. Conversely, a tabulated derivative slightly above unity can arise from interpolation error rather than the underlying model; one must distinguish a numerical defect from genuine superluminal stiffness.

Thermodynamic convexity supplies related but not identical conditions. Positive heat capacity, positive susceptibility matrices, and positive isentropic compressibility constrain different directions in state space. A one-parameter barotrope can hide unstable composition directions. Moreover, the dominant energy condition $|p|\leq\varepsilon$ does not imply $dp/d\varepsilon\leq1$, and the derivative bound does not imply the dominant energy condition. They test different properties.

Regimes and limiting checks

For a cold polytropic relation $p=K n^\Gamma$, with $K,m\gt0$ and $\Gamma\gt1$, take total energy density

$$\varepsilon=mn+\frac{p}{\Gamma-1},$$

the sound speed is

$$c_s^2=\frac{\Gamma p}{\varepsilon+p}.$$

At low pressure this becomes $c_s^2\simeq\Gamma p/(mn)$, the Newtonian result in $c=1$ units. At extremely high pressure the formal limit is $c_s^2\to\Gamma-1$; therefore a polytrope with $\Gamma\gt2$ eventually becomes acausal if extrapolated indefinitely. This is an extrapolation failure, not evidence that the low-density fit was invalid where calibrated.

For an ultrarelativistic gas with $p=\varepsilon/3$, one recovers $c_s^2=1/3$. Pressureless dust has $c_s=0$ and admits caustics rather than pressure-supported waves. A first-order phase transition can contain a constant-pressure coexistence interval with $c_s^2=0$ in a Maxwell construction. These distinct limits show why “soft” and “unstable” are not synonyms.

Exercises

For the cold polytrope above, derive $c_s^2=\Gamma p/(\varepsilon+p)$ and determine the condition on $\Gamma$ for its high-pressure extrapolation to remain causal.

Solution

Differentiate with respect to $n$:

$$\frac{dp}{dn}=\frac{\Gamma p}{n}, \qquad \frac{d\varepsilon}{dn}=m+\frac{\Gamma p}{(\Gamma-1)n} =\frac{\varepsilon+p}{n}.$$

Their ratio gives $c_s^2=(dp/dn)/(d\varepsilon/dn)=\Gamma p/(\varepsilon+p)$. When $p\gg mn$, $\varepsilon\simeq p/(\Gamma-1)$, so $c_s^2\to\Gamma-1$. The extrapolated model respects $c_s^2\leq1$ only for $\Gamma\leq2$; equality is the causal limiting value.

Can a barotrope obey the dominant energy condition over an interval while having superluminal sound? Test $p=2(\varepsilon-\varepsilon_*)$ for $\varepsilon_*>0$ and $\varepsilon_*\leq\varepsilon\leq2\varepsilon_*$.

Solution

On this interval $0\leq p\leq\varepsilon$, so the perfect-fluid dominant energy condition holds. Nevertheless $dp/d\varepsilon=2$, giving $c_s^2=2$. The stress inequalities constrain values; causality constrains a derivative. This provides an explicit counterexample to treating them as the same condition.

References

• Andersson, Nils, and Gregory L. Comer. “Relativistic Fluid Dynamics: Physics for Many Different Scales.” Living Reviews in Relativity 24, 3 (2021). DOI.