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Vacuum curvature of a globally static product

Statement

For the static product of the regular exterior data, the equation $Ric (g) = 0$ on $R \times X$ is equivalent to $Δ_{h} V = 0, D^{2} V = V Ric (h) .$ When these equivalent conditions hold, the scalar curvature of $h$ is zero. The displayed spatial equations then extend continuously to $B$ when $h, V$ are $C^{2}$ there, in the multiplication form displayed above.

Statement references

Complete regular static exterior data and its normalized end

Stationary Lorentzian data and tensor conventions

Gradient hessian and divergence connection formulas

Scalar curvature

Ricci curvature

Facts & Assumptions

[F1]

The formulation and calculations use the following exact object, operator and earlier-result interfaces: Complete regular static exterior data and its normalized end Stationary Lorentzian data and tensor conventions Gradient hessian and divergence connection formulas

[F2]

On the smooth interior, the Riemannian Christoffel formula applies to the positive-definite metric $h$ , and the nondegenerate four-metric Christoffel formula applies to $g = - V^{2} d t^{2} + h$ because $V > 0$ there. The ZFC convention of [F1] includes the countable choice assumed by the latter supplier. Christoffel formula for the levi civita connection Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

[F3]

For the smooth Riemannian metric $h$ on $X$ , scalar curvature is the metric trace of its Ricci tensor: $Scal (h) = h^{i j} {Ric}_{i j} (h)$ . The ZFC convention of [F1] includes the countable choice assumed by this definition. Scalar curvature

[F4]

On the smooth Riemannian interior, $Ric (h) (A, C) = tr (Z \mapsto R_{h} (Z, A) C)$ , with $R_{h} (A, C) E = D_{A} D_{C} E - D_{C} D_{A} E - D_{[A, C]} E$ . This curvature sign agrees with the Lorentzian convention in [F1]. Ricci curvature

Proof

Given: The static product of [F1], its Levi–Civita curvature conventions and coordinate Hessian formula, the Christoffel formulas in [F2], the scalar-curvature definition in [F3], and the Riemannian Ricci definition in [F4].

Goal: Derive the spatial equations, rather than presume a static vacuum reduction.

1.1givenF1F2F4

In product coordinates, $g_{t t} = - V^{2}$ , $g_{t i} = 0$ and $g_{i j} = h_{i j}$ . The Christoffel formulas of [F2] give $Γ^{t}_{t i} = V^{- 1} \partial_{i} V$ , $Γ^{i}_{t t} = V h^{i j} \partial_{j} V$ , $Γ^{i}_{j k} = Γ^{i}_{j k} (h)$ , with all other mixed Christoffel symbols zero. Substituting into ${Ric}_{a b} = \partial_{c} Γ^{c}_{a b} - \partial_{b} Γ^{c}_{a c} + Γ^{c}_{c d} Γ^{d}_{a b} - Γ^{c}_{b d} Γ^{d}_{a c}$ yields ${Ric}_{t t} (g) = V Δ_{h} V$ , ${Ric}_{t i} (g) = 0$ , and ${Ric}_{i j} (g) = {Ric}_{i j} (h) - V^{- 1} (\partial_{i} \partial_{j} V - Γ^{k}_{i j} (h) \partial_{k} V)$ . For the last formula, the extra terms are $- \partial_{j} (V^{- 1} \partial_{i} V) + V^{- 1} Γ^{k}_{i j} (h) \partial_{k} V - V^{- 2} \partial_{i} V \partial_{j} V = - V^{- 1} (D^{2} V)_{i j}$ .

2.1givenF1F3step 1.1∎

Since $V > 0$ in $X$ , vanishing of these components is precisely the two equations in the Statement. Their spatial trace, using [F3], is $V Scal (h) = Δ_{h} V = 0$ , giving zero scalar curvature. At $B$ all quantities in the multiplication form $D^{2} V = V Ric (h)$ and in $Δ_{h} V = 0$ are continuous under the stated $C^{2}$ hypotheses, so their zero identities have the continuous boundary limits. This extension does not divide by the vanishing lapse.

Depends on

Used by · 3

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