Perfect-Fluid Collapse

A relativistic perfect fluid adds the physical ingredient that dust suppresses: pressure communicates between neighboring shells. Pressure gradients can delay or reverse contraction, while pressure itself gravitates and compression can increase the mass-energy inside a comoving shell. The resulting collapse problem is governed not just by compactness but also by an equation of state, sound characteristics, regularity at the center, and the possible formation of shocks.

Required background. Use The Einstein Equation for the coupling of fluid energy and pressure to spherical geometry.

Helpful background. Black-Hole Geometry and Mechanics supplies trapping diagnostics, Black-Hole Perturbation Theory supplies stability language, Formulations of General Relativity supplies the constrained initial-value problem, Relativistic Matter and Compact Objects develops detailed equations of state and microphysics, and Computational Relativity treats shock-capturing evolution.

The Misner–Sharp fluid system

The perfect-fluid stress tensor is

$$T_{ab}=(\rho+p)u_a u_b+p\,g_{ab},$$

where $\rho$ is the rest-frame total energy density and $p$ is the isotropic pressure. In comoving coordinates write

$$ds^2=-\alpha(t,r)^2dt^2+B(t,r)^2dr^2+R(t,r)^2d\Omega^2.$$

Define derivatives along and orthogonal to the flow within the radial two-space,

$$D_t=\alpha^{-1}\partial_t, \qquad D_r=B^{-1}\partial_r,$$

together with the areal velocity $U=D_tR$ and radial factor $\Gamma=D_rR$. The Misner–Sharp mass $m$ is fixed invariantly by

$$\Gamma^2=1+U^2-\frac{2Gm}{R}.$$

Two Einstein equations take the especially useful form

$$D_rm=4\pi\rho R^2\Gamma, \qquad D_tm=-4\pi pR^2U.$$

The first is the radial mass constraint. The second is a relativistic pressure-work law for a comoving shell. The radial Euler equation and acceleration equation may be written

$$D_rp=-(\rho+p)D_r\ln\alpha,$$ $$D_tU=-\frac{\Gamma}{\rho+p}D_rp -\frac{Gm}{R^2}-4\pi GpR.$$

The three acceleration terms have distinct meanings: the pressure gradient can point outward, while both accumulated mass and local pressure contribute inward gravitational acceleration. This is why “pressure supports” is incomplete. A positive pressure with too small an outward gradient strengthens gravitational attraction.

At a smooth center, $R\sim r$, $m\sim(4\pi/3)\rho_cR^3$, $U=O(R)$, and radial gradients of scalar fluid variables vanish. A marginal sphere still satisfies $2Gm/R=1$ when the outward areal derivative becomes null. The original invariant formulation and collapse equations are developed in Misner and Sharp 1964, pp. B571–B576.

Equation of state and sonic structure

The acceleration equation contains the static equilibrium equation as a check. Set $U=0$, $D_tU=0$, and use $R$ as radial coordinate, so $\Gamma^2=1-2Gm/R$ and $D_r=\Gamma\,d/dR$. Then

$$\frac{dp}{dR} =-\frac{G(\rho+p)(m+4\pi pR^3)} {R^2(1-2Gm/R)}.$$

This is the Tolman–Oppenheimer–Volkoff balance. Its numerator contains both the inertial density $\rho+p$ and the pressure contribution $4\pi pR^3$ to active gravity. Recovering it checks the pressure prefactor in the dynamical equation. Conversely, collapse is not just a sequence of static equilibria: when $U\neq0$, pressure work changes $m$ and sound takes time to communicate a disturbance.

The stress tensor does not close the system. A barotropic relation $p=p(\rho)$ supplies a local sound speed

$$c_s^2=\frac{dp}{d\rho}.$$

Hyperbolicity and causal material propagation require $0\leq c_s^2\leq1$ in units with $c=1$. The scale-free family $p=k\rho$, with $0<k<1$, is especially useful for collapse because it introduces no material length. It is nevertheless an idealization; nuclear transitions, temperature, composition, viscosity, magnetic fields, and radiation transport belong to the material model, not to spherical geometry alone.

Sound characteristics tilt relative to the fluid worldlines. In the comoving chart their radial slopes are $dr/dt=\pm\alpha c_s/B$, whereas radial light rays have $dr/dt=\pm\alpha/B$. Thus pressure can influence a neighboring shell only inside the narrower sound cone when $c_s\lt1$. A self-similar flow generally encounters a sonic point where a similarity surface is characteristic for the fluid equations. The reduced ordinary differential equations then have a vanishing characteristic coefficient; a smooth solution must also make the corresponding numerator vanish. This regularity condition selects special self-similar solutions. Failure of that condition can produce a kink or shock rather than the smooth critical solution one intended to study.

Pressure changes the possible histories. A sufficiently weak or outward-moving configuration can disperse. A bound configuration can oscillate or settle only if an allowed mechanism removes kinetic energy; a perfect, nondissipative model does not provide arbitrary relaxation. Strong inward data can form a trapped region. Near the threshold, the radiation fluid $p=\rho/3$ supplied an early continuously self-similar example with black-hole mass scaling Evans and Coleman 1994, pp. 1782–1785. The critical exponent depends on $k$, so fluid critical behavior is not one matter-independent number Gundlach and Martín-García 2007, §4.2, pp. 25–27.

Shocks, surfaces, and the dust limit

Even smooth initial data can steepen. At a shock, $\rho$, $p$, and $u^a$ cease to be classically differentiable; the correct statement is conservation of flux in the weak sense, with Rankine–Hugoniot jump conditions and an entropy condition selecting the physical branch. Continuing to use the smooth acceleration equation pointwise through a shock is an error. Numerical calculations require a conservative formulation and a demonstrated shock-capturing resolution study.

A material surface also needs care. For a finite star joined directly to vacuum without a surface layer, pressure must vanish at the boundary. Some simple scale-free equations of state have density behavior that makes such a compact surface inconvenient, so they are often used for localized collapse or critical solutions rather than a complete stellar model.

Choosing the pressureless equation of state $p(\rho)\equiv0$ removes the pressure-work and pressure-gradient terms and recovers dust mass conservation along each comoving shell. It also gives $c_s=0$, eliminating pressure communication between shells. Merely reaching a state with $p=0$ does not require $dp/d\rho=0$. Shell crossing in dust is the zero-pressure counterpart of multistreaming; an arbitrarily small but nonzero pressure need not resolve it smoothly, because gradients and shocks may become large. The dust limit is a check on the equations, not proof that dust and fluid endpoints vary continuously for all data.

Physical checks and limitations

A well-posed collapse statement must specify the equation of state and its valid density range, the initial density and velocity profiles, center and outer-boundary conditions, and whether shocks are admitted as weak solutions. It should monitor the Hamiltonian mass constraint, baryon number when defined, shock jumps, and the invariant compactness $2Gm/R$. Statements about a naked or covered endpoint additionally require a causal-geodesic analysis and a declared differentiability class.

Spherical perfect fluids contain no gravitational-wave channel and no rotation. Moreover, a perfect fluid omits shear viscosity, heat flow, and anisotropic stress. These restrictions make the model analytically revealing, but astrophysical inference requires importing a justified closure from Relativistic Matter and Compact Objects and validating its evolution numerically.

Exercises

Consider a collapsing comoving shell with $U<0$ and positive pressure. Use the pressure-work equation to determine the sign of $D_tm$. Contrast it with dust and explain the energy transfer.

Solution

Because

$$D_tm=-4\pi pR^2U,$$

$p>0$ and $U<0$ imply $D_tm>0$. Compression does positive work on the material inside the comoving shell, increasing its enclosed total energy. For dust, $p=0$ and therefore $D_tm=0$: each shell retains a constant Misner–Sharp mass until the dust description fails. This comparison also shows why pressure cannot be represented only as an outward force; it contributes to the mass-energy budget and to the inward term $-4\pi GpR$.

References

• Evans, Charles R., and Jason S. Coleman. “Critical Phenomena and Self-Similarity in the Gravitational Collapse of Radiation Fluid.” Physical Review Letters 72 (1994): 1782–1785. doi:10.1103/PhysRevLett.72.1782.
• Gundlach, Carsten, and José M. Martín-García. “Critical Phenomena in Gravitational Collapse.” Living Reviews in Relativity 10 (2007): 5. doi:10.12942/lrr-2007-5. Open PDF.
• Misner, Charles W., and David H. Sharp. “Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse.” Physical Review 136 (1964): B571–B576. doi:10.1103/PhysRev.136.B571.