Killing Fields and Conserved Matter Currents

A symmetric conserved stress tensor becomes an ordinary conserved current when contracted with an exact Killing field. Stress-energy conservation removes one term in the divergence; the Killing equation removes the other. Turning that local identity into a conserved charge additionally requires oriented hypersurfaces, a suitable connecting region, and accounted boundary flux.

Required background. Killing fields supply the metric symmetry, stress-energy supplies the symmetric conserved tensor, and differential forms and integration supply oriented fluxes.

Helpful background. Special relativity fixes the energy-sign example, and tensor operations organize the contractions. Presymplectic charges and gravitational boundary terms explain why total gravitational charges require more than a matter current.

The local current

Let $T^{ab}=T^{ba}$ satisfy $\nabla_aT^{ab}=0$, and let $\xi^a$ obey the Killing equation

$$\nabla_{(a}\xi_{b)}=0.$$

Define

$$J^a=T^{ab}\xi_b.$$

Its divergence is

$$\begin{aligned} \nabla_aJ^a &=(\nabla_aT^{ab})\xi_b+T^{ab}\nabla_a\xi_b\\ &=T^{ab}\nabla_{(a}\xi_{b)}=0. \end{aligned}$$

Symmetry of $T^{ab}$ is essential: it kills the antisymmetric part of $\nabla_a\xi_b$. Reall develops Killing constants and their stress-tensor analogue in the geometry of symmetries Reall 2013, § 7.2, pp. 71–76, PDF.

If $\xi^a$ is not Killing, the same calculation gives

$$\nabla_aJ^a=T^{ab}\nabla_{(a}\xi_{b)}.$$

This source measures work associated with the changing geometry generated by $\xi^a$. For a conformal Killing field it reduces to a trace term, so traceless matter can still produce a conserved current under additional conditions.

From current to charge

Let $\Sigma$ be an oriented spacelike hypersurface with future-directed unit normal $n^a$ and induced volume $d\Sigma$. A conventional charge is

$$Q_\xi[\Sigma]=\int_\Sigma J^a n_a\,d\Sigma.$$

For a future timelike translation this convention gives positive energy when the dominant energy condition holds: $J^an_a=T_{ab}n^a\xi^b\ge0$. For example, in an inertial Minkowski frame with $\xi^a=n^a=(1,0,0,0)$, one has $J^0=-T^{00}$ but $n_0=-1$, so $J^an_a=T^{00}$. Lowering the normal supplies the second minus sign. If one instead defines the future-directed energy current $j^a=-J^a$, the same positive charge is $Q_\xi=-\int_\Sigma j^an_a\,d\Sigma$.

Let two slices $\Sigma_1$ and $\Sigma_2$ bound a region together with a timelike side boundary $\mathcal B$ of outward unit spacelike normal $s^a$. The directed surface element is $-n_a\,d\Sigma$ on the future slice and $+n_a\,d\Sigma$ on the past slice. Stokes’ theorem therefore gives

$$Q_\xi[\Sigma_2]-Q_\xi[\Sigma_1] =\int_{\mathcal B}J^a s_a\,d\mathcal B =-\int_{\mathcal B}j^a s_a\,d\mathcal B.$$

An outward positive energy flux thus decreases the energy remaining between the slices, as it should. A null side boundary uses its directed null surface element in place of $s_a\,d\mathcal B$; there is no unit null normal. The slice charge is equal only if the side flux vanishes, falls off sufficiently fast, or is included in the balance law. Local current conservation alone does not guarantee a finite or hypersurface-independent global charge.

Particle constants and continuum currents

For an affinely parameterized geodesic, let $p^a=dx^a/d\lambda$ be its tangent, so $p^b\nabla_b p^a=0$. An exact Killing field then gives

$$\frac{d}{d\lambda}(p_a\xi^a) =p^ap^b\nabla_{(a}\xi_{b)}=0.$$

For a massive particle one may choose $\lambda=\tau$, in which case the physical four-momentum is $P^a=mp^a$. Under a general affine rescaling $\lambda=c\tau+d$ with $c>0$, one instead has $P^a=mc\,p^a$; this constant factor only rescales the conserved quantity. The stress current is the continuum analogue: each flow of matter carries the symmetry-weighted energy or momentum, and its divergence records exchange through boundaries.

A stationary timelike Killing field yields a Killing energy; an axial Killing field yields angular momentum. Their numerical normalization matters. Rescaling $\xi^a$ rescales the charge, so stationary energy requires a chosen time normalization, often fixed at an asymptotic boundary.

MIT’s treatment relates Killing symmetries to geodesic constants and conservation laws Guth–Turner 2017, § 22.1, pp. 97–98, Eqs. (22.8)–(22.13), PDF. With the present future-directed timelike $\xi^a$, the positive particle energy is $-P_a\xi^a$. Exact symmetry, rather than a special coordinate name, is the invariant premise.

Matter current versus gravitational charge

$J^a=T^{ab}\xi_b$ is a covariant local matter current on a prescribed or dynamical geometry. It does not include a local gravitational energy tensor. In a fully gravitating system, a Hamiltonian or asymptotic symmetry charge can include boundary contributions from the metric. Those require the action’s surface structure and falloff conditions.

This distinction is visible in vacuum: $T^{ab}=0$ makes the matter current vanish, yet a stationary isolated spacetime can possess a nonzero total mass measured at infinity. The missing quantity is not another local stress tensor; it is encoded in gravitational boundary data.

Exercises

Show that for a conformal Killing field satisfying $\nabla_{(a}\xi_{b)}=(1/D)(\nabla\cdot\xi)g_{ab}$, the current $T^{ab}\xi_b$ is conserved when $T^a{}_a=0$.

Solution

Assuming $\nabla_aT^{ab}=0$,

$$\nabla_aJ^a=T^{ab}\nabla_{(a}\xi_{b)} =\frac{1}{D}(\nabla\cdot\xi)T^{ab}g_{ab}.$$

The final factor is the stress-tensor trace, so it vanishes for traceless matter. The result is local; a conserved finite charge still requires controlled boundary flux.

References

• Guth, Alan H., lectures with notes by Andrew P. Turner. 8.962 General Relativity, Spring 2017. MIT (2017). PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.