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Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ , let $U \subseteq M$ be a nonempty open subset and let $n$ be a smooth observer congruence on $U$ , with acceleration $a = \nabla_{n} n$ , expansion $θ = \nabla_{a} n^{a}$ and along-congruence derivative $D$ as in Acceleration, expansion and the along-congruence derivative of an observer congruence. Then:

(a) The acceleration is spatial. $a$ is smooth and

$g_{p} (a_{p}, n_{p}) = 0 for every p \in U,$

that is, $a_{p} \in n_{p}^{⊥}$ ; equivalently $h (a) = a$ for the projection field $h_{p} : = h_{n_{p}}$ of Observers, rest spaces and the spatial projection at an event clause (c), namely $h_{p} (v) = v + g_{p} (n_{p}, v) n_{p}$ for $v \in T_{p} M$ .

(b) Acceleration of the observer curves. Let $γ : I \to U$ be an observer curve of the congruence, that is, an integral curve of $n$ (Observer congruences, their observer curves and adapted charts clause (b)), and let $D_{s}$ denote the covariant derivative along $γ$ (Covariant derivative along a curve). Then

$D_{s} γ^{'} (s) = a_{γ (s)} for every s \in I .$

Consequently every observer curve is a geodesic in its flow parameter if and only if $a = 0$ on $U$ (Geodesic of an affine connection).

(c) Divergence form of the expansion. In every smooth chart of $U$ , with $g_{a b}$ the metric components and $w : = \sqrt{∣ \det (g_{a b}) ∣} > 0$ ,

$θ = w^{- 1} \partial_{a} (w n^{a}),$

and $θ = g^{a b} \nabla_{a} n_{b} = h^{a b} \nabla_{a} n_{b}$ with $h^{a b} = g^{a b} + n^{a} n^{b}$ . In particular $θ$ is smooth.

(d) Derivation property. For smooth functions $f, h : U \to R$ , $D (f h) = f D h + h D f$ , and $D$ annihilates constants.

Facts & Assumptions

Given: The time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , the open subset $U$ , the smooth observer congruence $n$ on $U$ , and the fields $a$ , $θ$ , $D$ of Acceleration, expansion and the along-congruence derivative of an observer congruence.

[L1]

$\nabla$ is torsion-free and $g$ -compatible: for all smooth vector fields $X, Y, Z$ on $M$ , $X (g (Y, Z)) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z)$ (Metric compatibility, torsion freedom and the metric-selected connection class clause (a), Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

$\nabla_{X} Y$ is a smooth vector field for smooth $X, Y$ , so $\nabla_{n} n$ is smooth on $U$ ; $\nabla_{X} Y$ is additive and $C^{\infty} (M)$ -linear in $X$ , and additive in $Y$ (Affine connection on a smooth manifold).

[L3]

For an integral curve $γ$ of $n$ , $γ^{'} = n \circ γ$ is the pullback of the supplied smooth field $n$ . The pullback connection defining the curve derivative gives $D_{s} (n \circ γ) = \nabla_{γ^{'}} n = (\nabla_{n} n) \circ γ$ (Covariant derivative along a curve, Geodesic of an affine connection, Observer congruences, their observer curves and adapted charts).

[L4]

$g_{p} (n_{p}, n_{p}) = - 1$ for every $p \in U$ (Observers, rest spaces and the spatial projection at an event, Observer congruences, their observer curves and adapted charts).

[L5]

In every smooth chart of $U$ , for every smooth vector field $J$ on $U$ , $\nabla_{a} J^{a} = w^{- 1} \partial_{a} (w J^{a})$ with $w = \sqrt{∣ \det (g_{a b}) ∣}$ , and the chart formula represents the chart-independent scalar $\nabla_{a} J^{a}$ (The derivative of the metric density and the chart formula for the divergence of a vector field clause (c), Christoffel symbols of an affine connection).

[L6]

Define $h_{p} : = h_{n_{p}}$ by $h_{p} (v) = v + g_{p} (n_{p}, v) n_{p}$ (Observers, rest spaces and the spatial projection at an event clause (c)). This projection field satisfies $h_{p} (a_{p}) = a_{p}$ for $a_{p} \in n_{p}^{⊥}$ and annihilates multiples of $n_{p}$ (The observer rest-space decomposition and the spatial projection, Observers, rest spaces and the spatial projection at an event).

[L7]

For smooth functions $f$ on $U$ , the induced tensor derivative has $\nabla_{a} f = \partial_{a} f$ , so $D f = n (f)$ is the action of the smooth vector field $n$ as a derivation on functions (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), A smooth vector field is a smooth section of the tangent bundle).

Proof

technique · direct; differentiate the normalization $g(n,n)=-1$, identify the covariant derivative along observer curves, and express the divergence of $n$ in a chart

1.1L1L2L4L6algebra

(Spatiality of $a$ .) By [L2], $a = \nabla_{n} n$ is a smooth vector field on $U$ . For any smooth vector field $X$ on $U$ , [L1] gives $X (g (n, n)) = g (\nabla_{X} n, n) + g (n, \nabla_{X} n) = 2 g (\nabla_{X} n, n)$ ; by [L4] the left-hand side is $X (- 1) = 0$ , so $g (\nabla_{X} n, n) = 0$ for every $X$ . Taking $X = n$ gives $g_{p} (a_{p}, n_{p}) = 0$ at every $p \in U$ ; hence $a_{p} \in n_{p}^{⊥}$ and $h_{p} (a_{p}) = a_{p}$ by [L6]. This proves (a).

2.1L3step 1.1algebra

(Acceleration of the observer curves.) Let $γ$ be an integral curve of $n$ ; then $γ^{'} = n \circ γ$ , and [L3] gives $D_{s} γ^{'} (s) = (\nabla_{n} n)_{γ (s)} = a_{γ (s)}$ for every $s \in I$ . If $a = 0$ on $U$ , then $D_{s} γ^{'} = 0$ , so each observer curve is a geodesic in its parameter (Geodesic of an affine connection); conversely, if every observer curve satisfies $D_{s} γ^{'} = 0$ , then $a_{γ (s)} = 0$ for every $s$ and every observer curve, and since every point of $U$ lies on an observer curve (Observer congruences, their observer curves and adapted charts clause (b)), $a = 0$ on $U$ . This proves (b).

2.2L5L1L4step 1.1algebra

(Divergence form.) Applying [L5] to $J : = n$ gives $θ = \nabla_{a} n^{a} = w^{- 1} \partial_{a} (w n^{a})$ ; in particular $θ$ is smooth on $U$ , since the right-hand side is smooth. The trace identities follow from $h^{a b} = g^{a b} + n^{a} n^{b}$ and $n^{b} \nabla_{a} n_{b} = \frac{1}{2} \nabla_{a} (n \cdot n) = 0$ , the last by [L1] and [L4]: $h^{a b} \nabla_{a} n_{b} = g^{a b} \nabla_{a} n_{b} + n^{a} n^{b} \nabla_{a} n_{b} = θ + 0$ . This proves (c).

3.1L7algebra∎

(Derivation property.) For smooth $f, h$ , [L7] gives $D = n$ as a derivation on functions, so $D (f h) = n^{a} \nabla_{a} (f h) = n^{a} (f \nabla_{a} h + h \nabla_{a} f) = f D h + h D f$ ; for a constant function $c$ one has $\nabla_{a} c = 0$ , so $D c = 0$ . This proves (d).

Remarks

Dependence on the congruence only through $n$ and $\nabla$ . The acceleration and expansion are computed from $n$ and the metric-selected connection alone; no adapted chart, tetrad or foliation is required, and no global existence statement for such structures is used.
Choice convention. The smooth structure of the tangent bundle underlying $n$ assumes ${A C}_{ω}$ through Observer congruences, their observer curves and adapted charts; no other choice principle is used here.

Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources