The Vanishing of Diffeomorphism Hamiltonians examines a fundamental structural result of diffeomorphism-invariant gauge theories: the Hamiltonian generators of spacetime diffeomorphisms reduce entirely to boundary terms and vanish identically on closed spatial manifolds. Starting from invariance under the infinite-dimensional gauge group Diff(M), the paper traces the logical chain: Diff(M) ⇒ Noether II ⇒ Jξ ≈ dQξ ⇒ Hξ = ∫∂Σ(Qξ − ξ·B) ⇒ ∂Σ = ∅ ⇒ Hξ = 0 The argument proceeds through: Application of Noether’s second theorem to local gauge symmetries Demonstration that the diffeomorphism Noether current is exact on-shell Derivation of the Iyer–Wald boundary formula for the Hamiltonian Proof that, in the absence of boundaries, the Hamiltonian vanishes The result is not a technical artifact but a structural property of generally covariant theories. In General Relativity, this explains: Why the Hamiltonian is a constraint rather than an energy functional Why gravity admits no gauge-invariant local energy density Why ADM mass, angular momentum, and BMS charges arise purely as boundary integrals How the Wheeler–DeWitt equation encodes the “problem of time” Crucially, the problem of time does not invalidate this result. On the contrary, the vanishing of the bulk Hamiltonian sharpens the need for relational or geometrically defined temporal parameters. In this context, York time — defined via the mean extrinsic curvature of spatial slices — emerges not as an ad hoc construction, but as a natural and operationally meaningful variable. The structural fact that Hξ = 0 in the bulk motivates the search for intrinsic geometric clocks, and York time provides precisely such an instrument within canonical gravity. The paper presents a concise geometric synthesis of these ideas within the covariant phase space formalism and clarifies the deep relationship between gauge symmetry, cohomological triviality of currents, boundary charges, and the emergence of relational time.
Batenin et al. (Sat,) studied this question.