Point Particles and Dust

A point particle is represented by a timelike worldline, while dust is the continuum limit of many noninteracting worldlines sharing one local four-velocity. The invariant point-particle stress–energy is a tensor distribution supported on the worldline. When it is replaced by the smooth dust form $T^{\mu\nu}=\rho u^\mu u^\nu$, covariant conservation separates exactly into rest-mass continuity and geodesic motion wherever $\rho>0$. This model excludes pressure, velocity dispersion, collisions, finite size, and self-force.

Required background. Classical Fields and Their Actions provides action variation and Euler–Lagrange equations; Flat-Background Hilbert Stress-Energy from Metric Variation provides the metric-response definition of stress–energy; Affine Connections and Covariant Derivatives provides covariant differentiation along worldlines.

Helpful background. Events, Intervals, and Minkowski Geometry reviews four-momentum and proper time; Tensor Fields and Natural Operations reviews invariant tensor fields and integration; Continuum Kinematics and Balance Laws supplies continuum mass balance and closure; Presymplectic Potentials and Currents prepares distributional action variations.

Worldline action and invariant support

Let $z^\mu(\tau)$ be a future-directed timelike worldline parametrized by proper time, with $u^\mu=dz^\mu/d\tau$ and $u^\mu u_\mu=-1$. The action of a structureless particle of constant mass $m$ is

$$S_{\rm p}=-m\int d\tau.$$

Under an inverse-metric variation at fixed coordinate curve,

$$\delta d\tau =\frac12d\tau\,u_\mu u_\nu\delta g^{\mu\nu}(z), \qquad \delta S_{\rm p} =-\frac{m}{2}\int d\tau\, u_\mu u_\nu\delta g^{\mu\nu}(z).$$

Define the invariant delta distribution $\delta_g^{(4)}(x,z)$ by

$$\int d^4x\,\sqrt{-g(x)}\, \delta_g^{(4)}(x,z)f(x)=f(z)$$

for every compactly supported scalar test function $f$. Comparing the worldline variation with the metric-response definition gives

$$\boxed{ T^{\mu\nu}_{\rm p}(x) =m\int d\tau\, u^\mu u^\nu\delta_g^{(4)}(x,z(\tau)).}$$

In a coordinate chart, $\delta_g^{(4)}(x,z)=\delta^{(4)}(x-z)/\sqrt{-g(x)}$. The factor $1/\sqrt{-g}$ is not optional: the coordinate delta is a density, whereas the displayed combination makes the integral invariant. For several particles, one sums the same expression over worldlines. Reall’s local relation $P^\mu=m\,u^\mu$ and the observer energy $E=-P_\mu v^\mu$ are developed in Reall 2013, §5.2, pp. 51–52, PDF.

Distributional conservation and motion

The divergence of $T^{\mu\nu}_{\rm p}$ must be interpreted weakly. Pair it with a compactly supported test one-form $f_\nu$:

$$\begin{aligned} \left\langle\nabla_\mu T^{\mu\nu}_{\rm p},f_\nu\right\rangle &=-m\int d\tau\,u^\mu u^\nu\nabla_\mu f_\nu \\ &=m\int d\tau\,a^\nu f_\nu, \end{aligned}$$

where $a^\nu=u^\mu\nabla_\mu u^\nu$ and the endpoint term vanishes because $f_\nu$ has compact support. Therefore $\nabla_\mu T^{\mu\nu}_{\rm p}=0$ as a distribution exactly when the interior of the worldline is geodesic. This is the distributional counterpart of varying the worldline itself.

The result has a controlled domain. It describes a particle on a prescribed smooth geometry or a test-body approximation. The particle’s own gravitational field is singular, and the nonlinear Einstein tensor is not generally defined by inserting an arbitrary delta-function metric source. Extended-body limits, regularization, radiation reaction, and self-force require additional structure.

From a stream of particles to dust

Suppose many particles can be coarse-grained on scales large compared with their separation but small compared with macroscopic gradients. If there is one timelike velocity $u^\mu$ at each event and negligible random motion, pressure, and interaction stress, their smooth stress–energy is

$$T^{\mu\nu}=\rho u^\mu u^\nu,$$

where $\rho$ is the rest-frame mass-energy density. This is a single-stream closure, not the most general continuum tensor.

Expand its divergence:

$$\nabla_\mu T^{\mu\nu} =u^\nu\nabla_\mu(\rho u^\mu) +\rho u^\mu\nabla_\mu u^\nu.$$

Contracting with $u_\nu$ uses $u_\nu u^\nu=-1$ and $u_\nu u^\mu\nabla_\mu u^\nu=\tfrac12u^\mu\nabla_\mu(u_\nu u^\nu)=0$. Hence conservation implies

$$\boxed{\nabla_\mu(\rho u^\mu)=0.}$$

Substitution back into the uncontracted equation gives

$$\boxed{u^\mu\nabla_\mu u^\nu=0}$$

wherever $\rho>0$. The first equation conserves rest mass in the flow; the second says that each dust element follows a timelike geodesic. Pressureless perfect-fluid motion is identified with this limit in Reall 2013, §5.2, p. 54, equations (5.23)–(5.24), PDF.

The continuity equation has a finite-region meaning. For a spacelike hypersurface $\Sigma$ with future-directed unit normal $n^\mu$, the rest mass crossing it is obtained from the current $J^\mu=\rho u^\mu$ as

$$M_\Sigma=-\int_\Sigma J^\mu n_\mu\,d\Sigma.$$

The sign makes the integrand positive when both vectors are future directed. Applying the divergence theorem to a spacetime tube shows that $M_\Sigma$ is unchanged between two slices when no current crosses the timelike side boundary. This is rest-mass conservation, not a claim that an arbitrary curved spacetime possesses a globally conserved energy. The latter would require additional structure, such as a suitable Killing field and boundary behavior.

Breakdown of the dust description

Dust can form caustics: distinct worldlines cross, so a single velocity field ceases to exist even if each worldline remains smooth. A kinetic distribution can then have several momenta at one event; its second moment contains velocity-dispersion stress and no longer has rank one. Collisions, thermal motion, electromagnetic forces, and material stresses likewise require pressure or anisotropic terms. The perfect-fluid page adds isotropic pressure but still excludes viscosity, heat conduction, and multistreaming.

Coarse-graining also fixes what $\rho$ means. It is measured per proper volume in the instantaneous rest frame, not per coordinate volume on an arbitrary slice. A tilted slicing introduces Lorentz factors through both $u^\mu n_\mu$ and the induced volume element. Treating a coordinate number density as the scalar $\rho$ would therefore give the wrong tensor transformation and the wrong flux through $\Sigma$.

Conservation also changes when an external force is present. Charged particles satisfy $u^\mu\nabla_\mu u^\nu=(q/m)F^\nu{}_{\rho}u^\rho$, so their matter tensor alone exchanges four-momentum with the electromagnetic field. Only the combined stress–energy is divergence-free in a closed system. General covariance of the equations, including the geodesic and Lorentz-force limits, is discussed in Carroll 1997, Chapter 4, pp. 105–108, PDF.

Exercise: recover the relativistic particle energy

In Minkowski coordinates let a particle have $u^0=\gamma$ and spatial trajectory $\mathbf z(t)$. Reduce the invariant distribution to a constant-time slice and integrate $T^{00}$ over space.

Solution

Here $\sqrt{-g}=1$ and $dt/d\tau=u^0=\gamma$. Using

$$\delta(t-z^0(\tau))\,d\tau =\frac{1}{u^0}\,\delta(\tau-\tau_t)\,d\tau,$$

the worldline expression becomes

$$T^{00}(t,\mathbf x) =m\frac{(u^0)^2}{u^0} \delta^{(3)}(\mathbf x-\mathbf z(t)) =m\gamma\,\delta^{(3)}(\mathbf x-\mathbf z(t)).$$

Therefore

$$\int d^3x\,T^{00}=m\gamma,$$

the special-relativistic energy. The factor $1/u^0$ from reducing the spacetime delta is essential; omitting it would incorrectly produce $m\gamma^2$.

References

• Carroll, Sean M. Lecture Notes on General Relativity. NSF-ITP/97-147, arXiv:gr-qc/9712019v1, 1997. arXiv. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. PDF.