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The derivative of the metric density and the chart formula for the divergence of a vector field

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Christoffel symbols of an affine connection, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $(U, x)$ be a smooth chart, and define the local metric density by $w : = \sqrt{∣ \det (g_{a b}) ∣}$ on $U$ , using the matrix of metric components in this chart. This definition requires no orientation or volume form. Write $g^{a b}$ for the inverse-metric components (The inverse metric, musical isomorphisms and index raising) and $Γ^{a}_{b c}$ for the Christoffel symbols of $\nabla$ in the chart.

(a) Derivative of the density. At every point of $U$ ,

$\partial_{a} w = \frac{1}{2} w g^{b c} \partial_{a} g_{b c} .$

(b) Trace of the Christoffel symbols. At every point of $U$ ,

$Γ^{b}_{b a} = w^{- 1} \partial_{a} w .$

(c) Divergence formula. For every smooth vector field $J$ on $M$ , at every point of $U$ ,

$\nabla_{a} J^{a} = \partial_{a} J^{a} + Γ^{a}_{a b} J^{b} = w^{- 1} \partial_{a} (w J^{a}),$

where $\nabla_{a} J^{a} = g^{a b} \nabla_{b} J_{a}$ is the divergence of $J$ contracted with the metric and $J^{a}$ are the components of $J$ (A smooth vector field is a smooth section of the tangent bundle).

The identity of (c) is a chart formula for a chart-independent scalar: the right-hand side is independent of the chart, as the left-hand side is.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , a smooth chart $(U, x)$ , the locally defined density $w = \sqrt{∣ \det (g_{a b}) ∣}$ , and a smooth vector field $J$ .

[L1]

In a chart the Christoffel symbols are $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ and satisfy $Γ^{a}_{b c} = Γ^{a}_{c b}$ (Christoffel symbols of an affine connection, Torsion freedom and metric compatibility in chart components, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

The extension of $\nabla$ to tensor fields has coordinate components $J^{a}_{; c} = \partial_{c} J^{a} + Γ^{a}_{d c} J^{d}$ for a vector field $J$ , and $\nabla g = 0$ , $\nabla g^{- 1} = 0$ ; the divergence is $\nabla_{a} J^{a} = J^{a}_{; a} = g^{a b} J_{a; b}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), The inverse metric, musical isomorphisms and index raising).

[L3]

The components $g^{a b}$ of the inverse metric are smooth and satisfy $g^{a c} g_{c b} = δ^{a}_{b}$ (The inverse metric, musical isomorphisms and index raising).

[L4]

For a differentiable matrix curve $A$ with invertible values, $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ (The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ ).

[L5]

There is a $g_{p}$ -orthonormal basis of $T_{p} M$ in which the matrix of $g_{p}$ is $diag (- 1, 1, 1, 1)$ ; consequently $\det (g_{a b} (p)) \neq 0$ and in fact $\det (g_{a b} (p)) < 0$ at every $p$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

Proof

technique · direct; differentiate the determinant along a coordinate curve, then substitute the Christoffel formula

1.1L3L4L5algebra

(Determinant derivative.) Fix $p \in U$ with coordinates $x (p)$ and fix an index $a$ . Define the matrix curve $A (s) : = (g_{b c} (x (p) + s e_{a}))$ for small $∣ s ∣$ , where $e_{a}$ is the $a$ -th standard basis vector; its entries are differentiable at $s = 0$ and, since $\det A (0) = \det (g_{b c} (p)) \neq 0$ by [L5] and the determinant is continuous, $A (s)$ is invertible for small $∣ s ∣$ . By [L4], $\partial_{a} \det (g_{b c}) (p) = \det (g_{b c} (p)) tr (A (0)^{- 1} A^{'} (0)) = \det (g_{b c} (p)) g^{b c} (p) \partial_{a} g_{b c} (p)$ , using [L3].

2.1step 1.1L5algebra

(Derivative of the square root.) By [L5], $\det (g_{b c}) < 0$ on $U$ , so $∣ \det (g_{b c}) ∣ = - \det (g_{b c})$ and $w = (- \det (g_{b c}))^{1 / 2}$ . Differentiating the square root and substituting step 1.1, $\partial_{a} w = \frac{1}{2} (- \det (g_{b c}))^{- 1 / 2} (- \partial_{a} \det (g_{b c})) = \frac{1}{2} w g^{b c} \partial_{a} g_{b c}$ , which is claim (a).

3.1step 2.1L1L3algebra

(Christoffel trace.) By [L1], $Γ^{b}_{b a} = \frac{1}{2} g^{b c} (\partial_{a} g_{b c} + \partial_{b} g_{c a} - \partial_{c} g_{b a})$ . Relabelling the dummy indices $b \leftrightarrow c$ in the third term and using the symmetry of $g_{b a}$ and of $g^{b c}$ turns it into $- \frac{1}{2} g^{b c} \partial_{b} g_{c a}$ , which cancels the second term; hence $Γ^{b}_{b a} = \frac{1}{2} g^{b c} \partial_{a} g_{b c} = w^{- 1} \partial_{a} w$ by claim (a).

4.1step 3.1L2algebra∎

(Divergence.) By [L2], $\nabla_{a} J^{a} = \partial_{a} J^{a} + Γ^{a}_{a b} J^{b}$ . Substituting claim (b), $\nabla_{a} J^{a} = \partial_{a} J^{a} + w^{- 1} (\partial_{b} w) J^{b} = w^{- 1} (w \partial_{a} J^{a} + J^{a} \partial_{a} w) = w^{- 1} \partial_{a} (w J^{a})$ , which is claim (c) and completes the proof.

Remarks

Density reading. The function $w$ is not a scalar: for a change from chart $x$ to chart $y$ , put $A = D (x \circ y^{- 1})$ on the overlap. The metric transformation law gives $g_{y} = A^{T} g_{x} A$ and hence $w_{y} = ∣ \det A ∣ w_{x}$ (Chart independence and the transformation law of a smooth covariant 2-tensor field, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ). Writing $J_{x}^{a}$ and $J_{y}^{a}$ for the components of $J$ in the two charts at the same point, the quantities $w J^{a}$ form a vector density, meaning that $w_{y} J_{y}^{a} = ∣ \det A ∣ (A^{- 1})^{a}_{b} w_{x} J_{x}^{b}$ . By claim (c), $\partial_{y^{a}} (w_{y} J_{y}^{a}) = w_{y} \nabla_{a} J^{a} = ∣ \det A ∣ \partial_{x^{b}} (w_{x} J_{x}^{b})$ . Thus this coordinate divergence is a scalar density, meaning a coefficient transforming by $∣ \det A ∣$ , rather than a chart-independent scalar; $w^{- 1} \partial_{a} (w J^{a})$ is the chart-independent scalar. In oriented charts, the invariant top form represented by $\partial_{a} (w J^{a}) d x^{0} \land \dots \land d x^{3}$ is the expression used in Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.
Sign conventions. No orientation is used in this lemma; claims (a)-(c) hold on every chart of every time-oriented smooth Lorentzian four-manifold.

The derivative of the metric density and the chart formula for the divergence of a vector field

Statement

Facts & Assumptions

Proof

Remarks

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