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Coordinate components of the transformed metric

PDF p. 10: §2.3, first display

Statement

Adopt the stationary vacuum model and regular stationary vacuum spacetime data, including the original maximal exterior section and mass-one normalization. Let $q \in [0, 1]$ , and suppose the Harrison one-form and metric are supplied on an open set $V$ where $D_{q} > 0$ . Thus $f = - g (T, T)$ , $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ , $d_{q} = D_{q} (p)$ at a supplied zero $p$ of $T$ , and $W_{q} (T) = d_{q} - D_{q}$ .

In an original stationary coordinate chart $(x^{0}, x^{1}, x^{2}, x^{3})$ on $V$ , meaning $T = \partial_{0}$ there, the transformed metric has components

$(g_{q})_{00} = - \frac{d_{q}^{2} f}{4 D_{q}}, (g_{q})_{0 i} = \frac{d_{q}}{4} (g_{0 i} - \frac{f (W_{q})_{i}}{D_{q}}),$

$(g_{q})_{i j} = \frac{1}{4} (D_{q} g_{i j} + g_{0 i} (W_{q})_{j} + g_{0 j} (W_{q})_{i} - \frac{f (W_{q})_{i} (W_{q})_{j}}{D_{q}}), i, j \in {1, 2, 3} .$

Statement references

Stationary Harrison one-form and metric

Mass-one convention and Harrison denominator

Stationary Lorentzian data and tensor conventions

Regular non-degenerate stationary vacuum spacetime

Stationary vacuum model and asymptotic normalization

Local frame fields, coordinate frames and tetrad fields

The tensor product of multilinear tensors

Facts & Assumptions

[A1]

The stationary vacuum model and regular spacetime data are assumed, together with the specified maximal exterior section and mass-one convention. The component calculation is conditional on the supplied chart and fields. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Mass-one convention and Harrison denominator

[F1]

Stationary Lorentzian data supply the metric $g$ , its Killing field $T$ , the metric dual $T^{♭} = g (T, \cdot)$ , and $f = - g (T, T)$ . Stationary Lorentzian data and tensor conventions

[F2]

The Harrison normalization defines $D_{q}$ for the supplied potential. Mass-one convention and Harrison denominator The Harrison one-form definition sets $d_{q} = D_{q} (p)$ at the supplied zero $p$ of $T$ . Stationary Harrison one-form and metric

[F3]

For the supplied one-form, $W_{q} (T) = d_{q} - D_{q}$ , and $g_{q} = \frac{1}{4} (D_{q} g + T^{♭} \otimes W_{q} + W_{q} \otimes T^{♭} - f D_{q}^{- 1} W_{q} \otimes W_{q})$ on $V$ . Stationary Harrison one-form and metric

[F4]

A smooth chart supplies the coordinate frame $(\partial_{0}, \partial_{1}, \partial_{2}, \partial_{3})$ . We write $g_{a b} = g (\partial_{a}, \partial_{b})$ , $(g_{q})_{a b} = g_{q} (\partial_{a}, \partial_{b})$ , and $α_{a} = α (\partial_{a})$ for a one-form $α$ . Local frame fields, coordinate frames and tetrad fields

[F5]

For covectors $α, β$ on the same tangent space, their tensor product satisfies $(α \otimes β) (u, v) = α (u) β (v)$ . The tensor product of multilinear tensors

Proof

Given: The model, data, parameter, supplied Harrison one-form and metric, positivity domain, and stationary coordinate chart in the Statement.

Goal: Calculate the three displayed component formulas.

1.1givenF1F3F4

Since $T = \partial_{0}$ , the coordinate components satisfy $g_{00} = g (T, T) = - f$ , $(T^{♭})_{0} = - f$ , $(T^{♭})_{i} = g_{0 i}$ , and $(W_{q})_{0} = W_{q} (T) = d_{q} - D_{q}$ . Positivity of $D_{q}$ permits the displayed divisions.

2.1F3F4F5step 1.1algebra

Set $w = d_{q} - D_{q}$ . Evaluating the tensor formula on $(\partial_{0}, \partial_{0})$ gives $4 (g_{q})_{00} = - f (D_{q} + 2 w + w^{2} / D_{q}) = - f d_{q}^{2} / D_{q}$ . Evaluating it on $(\partial_{0}, \partial_{i})$ gives $4 (g_{q})_{0 i} = (D_{q} + w) g_{0 i} - f (1 + w / D_{q}) (W_{q})_{i} = d_{q} (g_{0 i} - f (W_{q})_{i} / D_{q})$ .

3.1F3F4F5step 1.1step 2.1algebra∎

Evaluating the same tensor formula on $(\partial_{i}, \partial_{j})$ gives the displayed spatial expression directly, because $(T^{♭})_{i} = g_{0 i}$ . These calculations prove all three formulas on the chart.

Source notes

The article citation attributes the component formulas. The proof is the local tensor expansion from the supplied Harrison definition; it does not use the article's assertion as a premise.

Depends on

Used by · 3

Dependency tree · 2 levels

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Sources

The uniqueness of smooth stationary black holes · PDF page 10, §2.3, first display