Energy Conservation in General Relativity: Killing Fields, Conformal Charges, and the Scope of Noether's Theorems

We give a self-contained account of when energy is conserved in general relativity. From Noether's two theorems we obtain ^ T_=0 along two routes—the contracted Bianchi identity with Einstein's equation, and the Diff (M) Noether identity with the matter field equations—which are one identity applied to the two sectors of the total action. The relation holds on every solution, yet is no conservation law: it yields no scalar charge, and no local, covariant gravitational energy density exists. The central result is an equivalence: on a globally hyperbolic spacetime, the charge _ T^_ n_\, d³y is the same on every Cauchy slice, for every *transient source*—a symmetric, covariantly conserved T whose support meets the slab between any two nearby Cauchy slices in a compact set—if and only if is a Killing field. That class, and no larger one, is what the proof may quantify over: a conserved dust persists for all time, and none has compact support in spacetime. Narrowing the quantifier to sources obeying the dominant energy condition costs nothing; narrowing it to the *tracefree* ones yields only a conformal Killing field, whose charge is no energy. Minkowski spacetime and the Schwarzschild exterior realize the first case; a Friedmann–Lemaître–Robertson–Walker cosmology carrying a curvature invariant of nowhere-vanishing time derivative realizes the second, with conformal charge the familiar a⁴. We also treat the Komar, ADM and Bondi masses; the compatibility of energy non-conservation with general relativity's empirical success; the ADM and Wheeler–DeWitt formalism; and PT-symmetric quantum mechanics, a contrasting dynamical mechanism. Prerequisites: classical mechanics, electrodynamics, special relativity, and the elements of pseudo-Riemannian geometry.