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Potential ambiguity changes the current by a corner term

Statement

For a fixed smooth two-parameter family of commuting variations, replace the Lagrangian and its selected potential by

$L^{'} = L + d_{M} μ, θ^{'} (ϕ; δ ϕ) = θ (ϕ; δ ϕ) + δ μ (ϕ; δ ϕ) + d_{M} Y (ϕ; δ ϕ),$

where $μ$ is a smooth local spacetime three-form and $Y (ϕ; δ ϕ)$ is a smooth local spacetime two-form, linear in the variation and its spacetime derivatives. Their local coefficients depend smoothly on the spacetime coordinates, finitely many derivatives of $ϕ$ and any fixed smooth background fields, and $Y$ depends linearly on finitely many derivatives of $δ ϕ$ . Then

$ω^{'} - ω = d_{M} (δ_{2} Y (δ_{1} ϕ) - δ_{1} Y (δ_{2} ϕ)) .$

Consequently, for a finite oriented smooth singular three-chain $Σ$ , the integrated form changes by the integral of this two-form over $\partial Σ$ .

Statement references

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.3

Covariant phase space with boundaries, §2.2

Covariant presymplectic current

Covariant field families and variations

Stokes theorem for smooth singular chains

The exterior derivative squares to zero

Facts & Assumptions

Given: The chosen decomposition and its smooth commuting two-parameter variation family, together with the smooth local forms $μ$ and $Y$ satisfying the finite-order and linearity conditions in the Statement.

[A1]

$ω (δ_{1}, δ_{2}) = δ_{2} θ (δ_{1}) - δ_{1} θ (δ_{2})$ . Covariant presymplectic current

[A2]

Mixed variations commute on the given smooth two-parameter family. Covariant field families and variations

[F1]

The exterior derivative squares to zero: $d_{M}^{2} β = 0$ . The exterior derivative squares to zero

[F2]

The exterior derivative commutes with finite-chain integration: $\int_{Σ} d_{M} β = \int_{\partial Σ} β$ . Stokes theorem for smooth singular chains

Proof

Given: As in [A1]–[A2].

1.1A1A2F1algebra

Varying $L^{'}$ gives $δ L^{'} = δ L + d_{M} (δ μ)$ because parameter differentiation commutes with spacetime differentiation on the smooth family. Since $d_{M}^{2} Y = 0$ by [F1], the stated $θ^{'}$ is a valid selected potential for $L^{'}$ .

2.1A1A2step 1.1algebra

Substitute $θ^{'}$ into [A1] for both ordered pairs. The added $δ_{2} δ_{1} μ - δ_{1} δ_{2} μ$ terms cancel by [A2], while the remaining terms are $d_{M} (δ_{2} Y (δ_{1} ϕ) - δ_{1} Y (δ_{2} ϕ))$ .

3.1F2step 2.1∎

Integrate step 2.1 over $Σ$ and apply [F2]. This gives exactly the integral of the displayed two-form over $\partial Σ$ ; if $\partial Σ = 0$ , that contribution vanishes.

Remarks

The exact corner term need not vanish on a hypersurface with boundary. Covariance alone supplies no boundary condition that removes it.

The sign in the displayed corner term follows from the specified current ordering and the explicit variation in step 2.1. Iyer--Wald's equations (43)--(44) give the potential ambiguity and this current ordering, but their printed integrated ambiguity formula (47) has the opposite sign under those conventions; equation (47) is therefore not used for the sign here.

Potential ambiguity changes the current by a corner term

Statement

Statement references

Facts & Assumptions

Proof

Remarks

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