Stress-energy variation and covariant conservation

Varying the matter action with respect to the inverse metric defines a symmetric Hilbert stress-energy tensor. Metric dependence in both the volume element and the matter Lagrangian must be included. For a minimally coupled scalar, for example, changing the inverse metric changes the contraction of its gradients as well as the determinant. Merely multiplying a flat-space energy density by a metric is not a general construction. Nonminimal curvature couplings can move terms between what one calls source and gravitational dynamics.

Under an infinitesimal diffeomorphism, the matter fields and metric change together. Integrating the metric variation by parts and imposing the matter equations yields covariant divergence zero, with appropriate control of the boundary contribution. Independently, the contracted Bianchi identity makes the divergence of the Einstein tensor vanish. These are consistency statements, not ten unrelated energy bookkeeping rules. When several interacting matter components exchange energy, only their total stress-energy need be conserved separately from the gravitational side.

A covariant divergence includes connection terms that account for changing local bases. It is therefore not generally an ordinary coordinate divergence of a tensor’s components. If a symmetric stress-energy tensor is contracted with a Killing vector, however, the resulting vector has zero covariant divergence. The antisymmetric Killing derivative contracts to zero against the symmetric tensor. With a stationary timelike Killing vector, this construction supplies a physically useful energy current and associated charge after normalization and boundary conventions are chosen.

A divergence theorem relates the charge on two hypersurfaces to flux across the side boundary. Equality of the two charges requires vanishing net flux there, not merely a Killing vector somewhere in the chart. A general cosmological geometry has no global timelike Killing symmetry, so the same construction does not provide a universal total matter energy. This is compatible with exact local stress-energy conservation. Observer-dependent measured energy, coordinate components, and a symmetry-generated conserved charge answer different questions and should be named separately.

For an isotropic perfect fluid, let e be rest-frame energy per volume, p isotropic pressure in the same units, and u the four-velocity normalized to minus c squared. The null energy condition becomes e plus p nonnegative. The weak condition also requires nonnegative e; the dominant condition becomes e at least the absolute value of p. The strong condition requires both e plus p and e plus three p to be nonnegative. These equivalent inequalities assume the stated perfect-fluid form; anisotropic sources require the full tensor tests.

None of these inequalities follows merely by writing Einstein’s equation. Classical matter models can satisfy some and violate others, and quantum stress-energy requires further care. Positive Lambda, when moved to an effective source, has positive energy density and equal negative pressure: it saturates the null condition while violating the strong condition. When Lambda remains on the geometric side, calling the nongravitational vacuum source zero is equally consistent. One must not count that contribution twice or infer attractive focusing solely from positive energy density.