Causality, Stability, and Maximum Stiffness

A relativistic equation of state must resist compression without supporting signals faster than light. For a cold one-parameter perfect fluid, the local test is $0\leq dp/d\varepsilon\leq1$: the lower bound prevents mechanical instability and the upper bound keeps the adiabatic sound cone inside the metric null cone. The stiffest causal barotropic continuation therefore has slope one. It is a useful extremal construction, but its stellar predictions are bounds conditioned on a matching density, not a microscopic model of matter.

Required background. Hilbert Stress–Energy from Metric Variation supplies $p$, $\varepsilon$, and relativistic stress.

Helpful background. The Einstein Equation supplies stellar equilibrium; Relativistic Matter and Compact Objects gives the equation-of-state hierarchy; and Computational Relativity supplies numerical stellar and dynamical use.

Local stability and the sound cone

Assume a cold barotrope $p=p(\varepsilon)$ with positive enthalpy $\varepsilon+p\gt0$. In a small locally inertial patch of uniform matter, energy and longitudinal momentum conservation give

$$\partial_t\delta\varepsilon+(\varepsilon+p)\nabla\cdot\delta\boldsymbol v=0, \qquad (\varepsilon+p)\partial_t\delta\boldsymbol v+\nabla\delta p=0.$$

Eliminate the velocity and insert $\delta p=(dp/d\varepsilon)\delta\varepsilon$ to obtain the wave equation. For $e^{i\boldsymbol k\cdot\boldsymbol x-i\omega t}$,

$$\omega^2=c_s^2k^2, \qquad c_s^2=\frac{dp}{d\varepsilon}.$$

If $c_s^2\lt0$, short-wavelength perturbations grow rather than oscillate. If $c_s^2\gt1$, the ideal-fluid acoustic characteristic lies outside the metric null cone. Thus

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

is the basic local stability and causality interval. The derivative is dimensionless because pressure and energy density have the same units.

At finite temperature or with composition, the derivative must follow the dynamical constraint: fixed entropy per baryon and frozen composition for a wave faster than reactions, or an equilibrium path when reactions are fast. Elastic, magnetic, superfluid, and dissipative matter has additional modes; checking one scalar derivative is then insufficient. Thermodynamic constraints and equilibrium assumptions are summarized in Oertel et al. 2017, §§II.A–II.B.

Conditions that should not be conflated

The dominant energy condition $|p|\leq\varepsilon$ constrains the value of the stress tensor. The sound-speed bound constrains its derivative. Neither implies the other. A curve can lie below $p=\varepsilon$ but have a locally superluminal slope, or have causal slope while violating an energy condition because of an offset.

Thermodynamic convexity likewise tests the Hessian of a potential, including density, temperature, and composition directions. A beta-equilibrated one-dimensional slice can hide an unstable transverse composition mode. Near a first-order transition, a density jump or mixed phase is described by phase equilibrium; differentiating a smoothed numerical splice can manufacture a negative or superluminal sound speed.

Finally, the hydrodynamic characteristic speed is not automatically the microscopic front velocity of an arbitrary underlying quantum theory. The barotropic bound is necessary for the stated effective fluid. Establishing microscopic causality requires the relevant retarded response and high-frequency limit. Conversely, dispersive group velocities above unity over a finite frequency interval do not by themselves imply acausal front propagation. The physical claim must name the model and regime.

Causal high-density continuation

Suppose a trusted low-density equation of state ends at $(\varepsilon_m,p_m,n_m)$. A constant-sound-speed extension is

$$p(\varepsilon)=p_m+c_*^2(\varepsilon-\varepsilon_m), \qquad 0\leq c_*^2\leq1.$$

The maximally stiff choice is $c_*^2=1$. Pressure and energy density are continuous at the match, though the derivative can jump. Thermodynamic consistency also requires a baryon-density continuation. At zero temperature the first law gives

$$\frac{dn}{n}=\frac{d\varepsilon}{\varepsilon+p(\varepsilon)},$$

so

$$n(\varepsilon)=n_m \exp\!\left[ \int_{\varepsilon_m}^{\varepsilon} \frac{d\varepsilon'}{\varepsilon'+p(\varepsilon')} \right].$$

Then $\mu=(\varepsilon+p)/n$ and $p=n\mu-\varepsilon$ remain compatible. Patching only $p(\varepsilon)$ without $n$ may suffice for a Tolman–Oppenheimer–Volkoff mass–radius curve, but it cannot consistently supply baryon mass, chemical potential, or merger thermodynamics.

For a constant $s=c_*^2$, the integral can be performed explicitly:

$$n=n_m\left(\frac{\varepsilon+p}{\varepsilon_m+p_m}\right)^{1/(1+s)}, \qquad \mu=\mu_m\left(\frac{n}{n_m}\right)^s.$$

At $s=0$, pressure and chemical potential are constant while density changes, as expected for the idealized coexistence construction. At $s=1$, $n\propto\sqrt{\varepsilon+p}$, the maximally stiff limit. The causal stellar bounds and the effect of a known low-density segment are discussed in Lattimer and Prakash 2016, §2.2, Eqs. (6)–(12), and §4.2.

What maximum stiffness can and cannot show

Integrating stellar equilibrium with the causal-limit extension gives the greatest support obtainable within the chosen perfect-fluid class above the matching point. Raising the matching density leaves less freedom and tightens the bound; lowering it assumes ignorance over a larger region and weakens the physical specificity. The result also depends on the crust and lower-density equation of state.

Such an extension can answer a conditional question: could any causal barotrope joined here support a given mass? If even the maximally stiff continuation fails, the lower-density assumptions or model class are incompatible with that mass. If it succeeds, it does not demonstrate that real matter achieves $c_s^2=1$ or identify its degrees of freedom.

Rotation, temperature, pressure anisotropy, magnetic fields, and modified gravity change the stellar problem. A nonrotating perfect-fluid bound cannot be applied to a rapidly rotating or anisotropic configuration without recomputation. Similarly, a constant-sound-speed quark model with $c_*^2\lt1$ is a physical parameterization, while the $c_*^2=1$ envelope is primarily extremal.

Checks and failure cases

At every table or interpolation point, verify positive enthalpy, nonnegative compressibility, causal derivatives, and continuity of the declared thermodynamic variables. Compute derivatives along both the evolution constraint and nearby independent directions. Vary interpolation order and node spacing because a causal set of nodes can produce an acausal spline between them.

One failure mode is using $dp/d\rho$ with rest-mass density in place of $dp/d\varepsilon$ and interpreting it as a relativistic signal speed. Others include smoothing a phase jump, extrapolating a saturation expansion, neglecting a derivative discontinuity at the match, and reporting a causal-limit stellar mass without its matching density and low-density input. These are model-definition errors, not merely numerical tolerance issues.

Exercises

For the maximally stiff continuation, derive the baryon-density relation in closed form.

Solution

With $p=p_m+\varepsilon-\varepsilon_m$,

$$\varepsilon+p=2\varepsilon+p_m-\varepsilon_m.$$

Integrating the first law gives

$$\ln\frac{n}{n_m} =\frac12\ln\frac{\varepsilon+p}{\varepsilon_m+p_m},$$

and therefore

$$n=n_m\sqrt{\frac{\varepsilon+p}{\varepsilon_m+p_m}}.$$

The formula is continuous at the match and preserves $d\varepsilon=(\varepsilon+p)dn/n$.

References

• Lattimer, James M., and Madappa Prakash. “The Equation of State of Hot, Dense Matter and Neutron Stars.” Physics Reports 621, 127–164 (2016). Open preprint.
• Oertel, Micaela, Matthias Hempel, Thomas Klähn, and Stefan Typel. “Equations of State for Supernovae and Compact Stars.” Reviews of Modern Physics 89, 015007 (2017). DOI.