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Zero boundary metric value does not kill EH derivative flux

Example

Work in boundaryless ambient $M = R^{4}$ and integrate the local divergence over the half-space subdomain $H = {x^{3} \geq 0}$ . Give $M$ the flat metric $η = diag (- 1, 1, 1, 1)$ , coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , and outward conormal $- d x^{3}$ at its boundary. All coordinates have length units and choose a length $ℓ > 0$ . Take dimensionless cutoffs $χ \in C_{c}^{\infty} (R^{3})$ and $ρ \in C_{c}^{\infty} (R)$ with $χ \geq 0$ , $χ = 1$ near $0 \in R^{3}$ , and $ρ = 1$ near $0 \in R$ . Define

$f (x) : = \frac{x^{3}}{ℓ} χ (x^{0}, x^{1}, x^{2}) ρ (x^{3}), h_{a b} : = f η_{a b} .$

Then $h_{a b} = 0$ on $\partial H$ , but its normal derivative is nonzero there. For the Einstein–Hilbert summand with coefficient $c^{3} / (16 π G_{N})$ , the local variation formula in Compact Einstein–Hilbert variation and SI coupling has a nonzero boundary flux for this variation. The diagram records only the chosen half-space and normal; the proof is the component calculation below.

Facts & Assumptions

Given: The half-space chart and flat metric above, compact cutoffs $χ, ρ$ , and the metric variation $g_{s} = η + s h$ for sufficiently small $s$ .

[F1]

The first variation of the Einstein–Hilbert density is $δ (w R) = w G_{a b} q^{a b} + \partial_{a} (w V^{a})$ , where $V^{a} = g^{b d} δ Γ^{a}_{d b} - g^{a b} δ Γ^{d}_{d b}$ and $δ Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\nabla_{b} h_{c d} + \nabla_{c} h_{b d} - \nabla_{d} h_{b c})$ . (Compact Einstein–Hilbert variation and SI coupling)

[F2]

Curvature and the Einstein tensor are defined from the metric-selected connection and its curvature; for the constant metric $η$ , its Christoffel symbols and curvature vanish. (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors)

[F3]

Nonnegative smooth cutoffs with compact support and a plateau equal to one near a point exist in Euclidean coordinate balls. (A smooth bump between concentric Euclidean balls)

[A1]

The cutoffs are smooth, compactly supported, nonnegative where stated, and equal to $1$ on neighborhoods of the origin and $x^{3} = 0$ respectively. The first variation is interpreted as a compactly supported action difference on the half-space.

[A2]

The geometric supplier chain carries the ${A C}_{ω}$ convention stated in The Axiom of Countable Choice ( ${A C}_{ω}$ ); the local cutoff calculation makes no further choice.

Verification

1.1A1F3algebra

By [F3], choose nonnegative compactly supported dimensionless $χ$ equal to one near the origin and dimensionless $ρ$ equal to one near zero. The factor $x^{3} / ℓ$ gives $h_{a b} ∣_{x^{3} = 0} = 0$ . At the boundary, $\partial_{3} f = χ / ℓ$ and $\partial_{α} f = 0$ for $α = 0, 1, 2$ ; in particular, at the origin $\partial_{3} f = 1 / ℓ$ . Since $f$ is bounded and nonzero, $g_{s} = (1 + s f) η$ has the same Lorentzian signature whenever $∣ s ∣ < 1 / (2 ∥ f ∥_{\infty})$ . Thus the metric value is fixed at the boundary while its normal derivative is not.

2.1A2F1step 1.1algebra

At the flat background, substituting $h_{a b} = f η_{a b}$ into [F1] gives $δ Γ^{a}_{b c} = \frac{1}{2} (δ_{c}^{a} \partial_{b} f + δ_{b}^{a} \partial_{c} f - η_{b c} \partial^{a} f)$ . Contracting the two terms in $V^{a}$ gives $η^{b d} δ Γ^{a}_{d b} = - \partial^{a} f$ and $η^{a b} δ Γ^{d}_{d b} = 2 \partial^{a} f$ , hence $V^{a} = - 3 \partial^{a} f$ . On $x^{3} = 0$ , $V^{3} = - 3 χ / ℓ$ .

3.1A1F1F2step 2.1algebra

Since $G_{a b} = 0$ , the linearized Einstein–Hilbert density is just $\partial_{a} V^{a}$ by [F1]. The tangential integrals vanish because $χ$ is compactly supported, and the upper normal endpoint contributes zero because $ρ$ is compactly supported. The fundamental theorem of calculus on $x^{3} \in [0, \infty)$ therefore gives, after multiplying by the action coefficient, the boundary contribution $- \frac{c^{3}}{16 π G_{N}} \int_{x^{3} = 0} V^{3} d x^{0} d x^{1} d x^{2} = \frac{3 c^{3}}{16 π G_{N} ℓ} \int_{R^{3}} χ d x^{0} d x^{1} d x^{2} > 0$ . This is the outward flux for conormal $- d x^{3}$ .

4.1step 1.1step 3.1algebra∎

The nonzero first-variation boundary term in step 3.1 occurs despite $h_{a b} = 0$ on $\partial H$ . Therefore fixing only boundary metric values does not justify discarding the Einstein–Hilbert derivative flux.

Remarks

This is the local divergence flux on a subdomain of a boundaryless ambient manifold; it is not a boundary value problem for Einstein's equations.
The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02; no independent acceptance is asserted: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.

Zero boundary metric value does not kill EH derivative flux

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources