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A stationary product rules out a zero Hawking coefficient

Statement

Let $M$ be a smooth manifold and let $D, V \subseteq M$ be open subsets with their restricted smooth structures and $D \cap V \neq \emptyset$ . Let $T$ be a smooth vector field on $M$ whose restriction to $D$ is complete, with global flow $ϕ : R \times D \to D$ .

Let $C$ be a smooth manifold and $i : C \to D$ a smooth map. Equip $R \times C$ with its product smooth structure, and suppose $F : R \times C \to D$ , $F (t, x) = ϕ_{t} (i (x))$ , is a diffeomorphism.

Let $K_{0}$ be a smooth vector field on $V$ , let $λ \in R$ be constant, and set $Z_{0} = T ∣_{V} + λ K_{0}$ . Suppose $Z_{0}$ has a local flow $Ψ : U \to V$ , where $U \subseteq R \times V$ , and there is a constant $P > 0$ such that ${P} \times V \subseteq U$ and $Ψ (P, q) = q$ for every $q \in V$ . Then $λ \neq 0$ . No stationary orbit is treated as a causal curve in this argument.

Statement references

Local and global flows generated by a vector field

Through each point there is a unique maximal integral curve

Complete vector fields

Diffeomorphisms and local diffeomorphisms of manifolds

Products of smooth manifolds have a canonical product smooth structure

An open subset of a smooth manifold has a canonical restricted smooth structure

Smooth manifolds and their smooth charts

Cʳ and smooth maps between smooth manifolds

Derivations at a point and the tangent space

The velocity derivation of a smooth curve

Facts & Assumptions

[F1]

A flow obeys the group law, its trajectories are integral curves, and each local-flow time fibre is an interval containing zero. Integral curves of the same smooth field on the same ambient manifold with the same initial point agree on their common time interval by uniqueness of the maximal integral curve. Local and global flows generated by a vector field Through each point there is a unique maximal integral curve

[F2]

Completeness means that each maximal integral curve is defined on all of $R$ . A diffeomorphism is a bijective smooth map with smooth inverse; in particular it is surjective and injective. Complete vector fields Diffeomorphisms and local diffeomorphisms of manifolds

[F3]

Open subsets carry their canonical restricted smooth structures, and the product of two smooth manifolds carries the canonical product smooth structure. These are the structures used for $D$ , $V$ and $R \times C$ . An open subset of a smooth manifold has a canonical restricted smooth structure Products of smooth manifolds have a canonical product smooth structure

[F4]

Use the local-function definition of a smooth field $X$ on a smooth manifold $N$ : each $X (p) \in T_{p} N$ is a derivation and $X (f) (p) : = X (p) ([f])$ is smooth for every local smooth function $f$ . This is Merry, Proposition 8.2(iii). Manifolds, smooth maps and tangent derivations have their existing meanings. Smooth manifolds and their smooth charts Cʳ and smooth maps between smooth manifolds Derivations at a point and the tangent space

[F5]

An integral curve satisfies $γ^{'} (s) = X (γ (s))$ (Merry, Definition 9.3). With velocity as a derivation this means $(f \circ γ)^{'} (s) = X (f) (γ (s))$ for every local smooth $f$ . The velocity definition at $s$ is the definition at zero applied to $t \mapsto γ (s + t)$ . The velocity derivation of a smooth curve

External input

The terminology in [F4] and [F5] uses only Merry, Proposition 8.2(iii) and Definition 9.3. The local-function formulation does not require constructing a smooth manifold structure on the total tangent bundle.

Proof

Given: The data and hypotheses in the Statement.

Goal: Prove $λ \neq 0$ .

1.1givenF1F2

For any $q \in D$ , surjectivity gives $q = F (t, x)$ for some $(t, x) \in R \times C$ . The flow law gives $ϕ_{P} (q) = ϕ_{t + P} (i (x)) = F (t + P, x)$ . If this equaled $q$ , injectivity of $F$ would give $t + P = t$ , contrary to $P > 0$ . Thus the complete stationary flow has no positive period at any point of $D$ .

2.1givenF1F2F3F4F5step 1.1∎

Suppose $λ = 0$ and choose $q \in V \cap D$ . The curve $α (s) = ϕ_{s} (q)$ is defined for all $s \in R$ and takes values in $D$ . The curve $β (s) = Ψ (s, q)$ is defined on the time fibre $I_{q} = {s : (s, q) \in U}$ , an interval containing $[0, P]$ , and takes values in $V$ . For every local smooth function $f$ on $M$ , restrict $f$ to the relevant open domain. By [F3]-[F5], the two flow-curve equations give $(f \circ α)^{'} (s) = T (f) (α (s))$ and $(f \circ β)^{'} (s) = T (f) (β (s))$ , since $Z_{0} = T ∣_{V}$ . Restricted charts identify the germs and derivations on each open domain with those in $M$ . Thus both curves, viewed in $M$ , are integral curves of the same ambient field $T$ with initial point $q$ . Uniqueness therefore gives $β (s) = α (s)$ throughout $I_{q}$ , without assuming beforehand that $β$ stays in $D$ . In particular $ϕ_{P} (q) = Ψ (P, q) = q$ , contradicting step 1.1. Hence $λ \neq 0$ .

Depends on

Used by · 1

Local Hawking and axial fields, boost and pole

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Will J. Merry, Differential Geometry (2021) · Lecture 8, Proposition 8.2(iii), equations (8.1)-(8.3); Lecture 9, Definition 9.3, equation (9.2).

On a nonempty open manifold domain, a tangent-vector assignment is smooth iff X(f) is smooth for every local smooth f. A smooth curve is an integral curve iff its velocity equals the field along it.

Hypotheses: A smooth manifold M and nonempty open W contained in M; X(p) in T_pM for p in W. For the integral-curve definition, X is a smooth vector field and the curve is smooth on an open real interval.

Reading limits: Read the primary-source indexed excerpts containing Proposition 8.2 and Definition 9.3. Full PDF retrieval failed. Only these formulations are imported; no tangent-bundle second-countability construction or choice-independence theorem is imported.

Sources

Will J. Merry, Differential Geometry (2021) · Lecture 8, Proposition 8.2(iii), equations (8.1)-(8.3); Lecture 9, Definition 9.3, equation (9.2).