Einstein–Vlasov Matter

Einstein–Vlasov matter couples gravity to a collisionless distribution on the future mass shell. Each particle follows a spacetime geodesic, while the second momentum moment of the distribution sources Einstein’s equation. The model retains velocity dispersion and multiple streams, so it remains meaningful after a single fluid velocity would become multivalued. Its central consistency property is exact: geodesic transport makes the kinetic stress tensor divergence-free, as required by the Bianchi identity.

Required background. Hilbert Stress–Energy from Metric Variation supplies the source tensor and its conservation law; Events, Intervals, and Minkowski Geometry supplies the mass-shell geometry; and Differential Equations supplies transport and initial-value concepts.

Helpful background. The Einstein Equation gives the gravitational field equation; Relativistic Matter and Compact Objects locates collisionless matter among other models; and Computational Relativity gives the evolution setting for the coupled system.

Collisionless matter as a geometric flow

Assume uncharged particles of fixed rest mass $m\gt0$ with a nonnegative scalar distribution $f$ on

$$P^+=\{(x,p):g_{ab}p^a p^b=-m^2, p^a\text{ future directed}\}.$$

In local momentum coordinates $p^i$, collisionless transport is

$$p^a\frac{\partial f}{\partial x^a} -\Gamma^i{}_{ab}p^a p^b\frac{\partial f}{\partial p^i}=0.$$

This is the Vlasov equation: $f$ is constant along the geodesic flow lifted to phase space. In a local orthonormal frame, the invariant mass-shell measure is $dP=d^3p/p^{\hat0}$, with $p^{\hat0}=\sqrt{m^2+\lvert\boldsymbol p\rvert^2}$. It defines

$$N^a=\int p^a f dP, \qquad T^{ab}=\int p^a p^b f dP.$$

The full Einstein–Vlasov system is

$$G_{ab}=8\pi G T_{ab}, \qquad \mathcal L[f]=0.$$

Its geometric formulation, coordinate measure, constraint data, and local initial-value statement are presented in Andréasson 2011, § 2, pp. 11–14, Eqs. (25)–(34) and Theorem 1. The normalization of $f$ can absorb spin and $(2\pi)^{-3}$ factors, but it must be the same in the current, stress tensor, and total-mass definition.

Why stress–energy is conserved

The Liouville vector field preserves the natural mass-shell volume. For a compactly supported distribution in momentum, integration by parts on each fiber gives

$$\nabla_aT^{ab}=\int p^b \mathcal L[f] dP.$$

The right side vanishes by the Vlasov equation. Boundary terms vanish because of momentum compactness or sufficiently rapid decay. The same calculation with one fewer power of momentum yields $\nabla_aN^a=0$. This result is not an imposed fluid closure; it follows from geodesic motion and the invariant measure.

Nonnegative $f$ also makes standard energy inequalities transparent. For any future timelike $u^a$, the measured energy density is

$$T_{ab}u^a u^b=\int (p_a u^a)^2 f dP\geq0.$$

The energy flux is a positive superposition of future particle momenta, so massive Vlasov matter satisfies the dominant energy condition under the stated integrability assumptions. None of this makes the stress isotropic: an anisotropic momentum distribution produces unequal radial and tangential pressures.

Initial data and gravitational feedback

On a spacelike hypersurface, one prescribes a spatial metric, extrinsic curvature, and nonnegative distribution over positions and on-shell momenta. They must satisfy the Hamiltonian and momentum constraints with the kinetic energy and momentum densities. The evolution then transports $f$ along geodesics of a metric that is itself determined by the moments of $f$.

This feedback distinguishes Einstein–Vlasov from test-particle dynamics. Test particles move on a fixed metric and do not contribute to curvature. In the coupled problem, changing the momentum distribution changes both $T^{ab}$ and the future geometry. The general local Cauchy result is stated in Rendall 2002, § 1, pp. 1–6, Eqs. (1)–(9) and Theorem 1.1; for spherical symmetry, a momentum-support continuation criterion is given in Andréasson 2011, § 3.2, p. 16, Eq. (45).

Regular finite-mass data require more than $f\geq0$. A useful sufficient class on an asymptotically flat slice has $f$ smooth, compactly supported in position and momentum, and constraint data approaching Euclidean geometry with the appropriate falloff. More general decay classes are possible, but the support and regularity assumptions must be stated because they control whether the integrals and asymptotic mass exist.

Newtonian and fluid comparisons

For weak fields and velocities small compared with unity, write $g_{00}=-(1+2\Phi)$ and $p^i\simeq m v^i$. Define the velocity-space distribution $f_{\rm N}(x,v)=m^3 f(x,mv)$ at leading order, so that $f_{\rm N}d^3v=f d^3p$. The system reduces to Vlasov–Poisson,

$$\frac{\partial f_{\rm N}}{\partial t} +v^i\frac{\partial f_{\rm N}}{\partial x^i} -\frac{\partial\Phi}{\partial x^i} \frac{\partial f_{\rm N}}{\partial v^i}=0,$$ $$\nabla^2\Phi=4\pi G\rho, \qquad \rho=m\int f_{\rm N}d^3v.$$

The sign of the force term agrees with $d\boldsymbol v/dt=-\boldsymbol\nabla\Phi$, providing an independent check on the connection convention. The Vlasov–Poisson equations and force sign appear in Andréasson 2011, § 1.2, pp. 7–8, Eqs. (19)–(20), while the rigorous nonrelativistic Einstein–Vlasov limit is summarized in § 3.3, p. 17.

A cold monokinetic distribution concentrated at one momentum at each event formally resembles dust. After streams cross, momentum can still label them separately in phase space. This does not guarantee smoothness of the exactly cold delta-function limit, nor prevent gravitational singularities; regular Vlasov results assume a sufficiently regular phase-space distribution. Collisionless Vlasov evolution also does not drive a general distribution toward local thermal equilibrium.

For a concrete moment comparison, take two cold streams in a local orthonormal frame with equal proper mass density $\rho_0$ and velocities $u_\pm^a=W(1,\pm v,0,0)$. Adding their dust tensors gives

$$T^{\hat a\hat b} =\rho_0u_+^{\hat a}u_+^{\hat b} +\rho_0u_-^{\hat a}u_-^{\hat b} =2\rho_0W^2\operatorname{diag}(1,v^2,0,0).$$

The momentum density vanishes, but the longitudinal pressure is nonzero and the transverse pressures vanish. Choosing the zero-momentum frame has not made the system a perfect fluid. A smooth distribution with two narrow peaks approaches these moments without requiring exact delta functions.

In a static spherically symmetric spacetime, particle energy $E=-p_t$ and squared angular momentum $L$ are constants along geodesics. An ansatz $f=\mathcal F(E,L)$ therefore solves the Vlasov equation automatically; Einstein’s equations still determine which choices yield regular centers, finite radii, and asymptotically flat metrics. Dependence on $L$ permits pressure anisotropy, a feature unavailable to a perfect-fluid equation of state.

Exercises

Explain why a distribution $f=\mathcal F(E,L)$ is stationary in a static, spherically symmetric geometry, and identify what remains to be solved.

Solution

The Vlasov equation states that $f$ is constant along geodesics. Static time-translation symmetry conserves $E=-p_t$, and rotational symmetry conserves $L$. Therefore

$$\mathcal L[f] =\mathcal F_{,E}\mathcal L[E] +\mathcal F_{,L}\mathcal L[L] =0.$$

This solves only the kinetic transport equation. One must still integrate the momentum moments to obtain the density and pressures, solve Einstein’s radial field equations, and check central regularity, nonnegative $f$, finite total mass, and asymptotic flatness.

References

• Andréasson, Håkan. “The Einstein–Vlasov System/Kinetic Theory.” Living Reviews in Relativity 14, 4 (2011). DOI.
• Rendall, Alan D. “The Einstein–Vlasov System.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 231–250. Birkhäuser, 2004. Preprint.