DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Selected ingoing RN extension and time orientation

Definition

Let $M, q_{e}, q_{m} \in R$ be length parameters. Set $Q^{2} : = q_{e}^{2} + q_{m}^{2}$ and $f (r) : = 1 - \frac{2 M}{r} + \frac{Q^{2}}{r^{2}} .$ The selected ingoing RN manifold is $M_{i n} : = R_{v} \times (0, \infty)_{r} \times S^{2},$ where $v, r$ have units of metres, $S^{2}$ has unit round metric $γ$ and outward area form $ω$ , and the signature is $(-, +, +, +)$ . Orient the product by $d v \land d r \land ω$ . Define $g_{i n} = - f (r) d v^{2} + 2 d v d r + r^{2} γ, F_{i n} = - \frac{q_{e}}{r^{2}} d v \land d r + q_{m} ω .$ These are smooth on every $r > 0$ , including every positive root of $f$ ; the domain excludes $r = 0$ . This is an extension of the static RN metric and field, not a maximal extension or a specification of a source at the excluded centre.

For the time orientation, define the global smooth fields $ℓ : = - \partial_{r}, n : = \partial_{v} + \frac{f (r)}{2} \partial_{r}, N : = ℓ + n = \partial_{v} + (\frac{f (r)}{2} - 1) \partial_{r} .$ The chosen future timelike field is $N$ ; the future cone is the component of the timelike cone containing it. The well-definedness proof shows that $N$ is timelike and that $ℓ, n$ are future null. On $f \neq 0$ , the substitution $v = t + r_{*} (r)$ with $r_{*}^{'} = 1 / f$ identifies the metric and two-form with the static dyonic RN formulas of Dyonic Reissner-Nordström metric and field domains. This coordinate identification does not assert agreement of time orientations on the interior static components: on $f > 0$ the selected orientation agrees with $\partial_{t}$ , while on $f < 0$ it is the opposite of the separate choice $\partial_{r}$ made by that static definition.

Facts & Assumptions

Given: Real length parameters $M, q_{e}, q_{m}$ , the unit round oriented sphere, and signature $(-, +, +, +)$ .

[F1]

Dyonic Reissner-Nordström metric and field domains: on each open $J \subset (0, \infty)$ where $f \neq 0$ , the static metric is $- f d t^{2} + f^{- 1} d r^{2} + r^{2} γ$ and the two-form is $- q_{e} r^{- 2} d t \land d r + q_{m} ω$ ; its independently selected time orientation is $\partial_{t}$ on $f > 0$ and $\partial_{r}$ on $f < 0$ . This is a current draft definition, used only through these exact formulas.

[F2]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model: a smooth Lorentzian metric together with a smooth everywhere timelike vector field specifies a time orientation. Time orientation, future and past timelike/null/causal vectors, and the closed cones defines the future cone as the closure of the timelike component containing that field.

[F3]

Classification of future and past causal vectors by a timelike field: for a nonzero causal vector $Z$ , future-directedness is equivalent to $g (Z, N) < 0$ for the chosen future timelike field $N$ .

[A1]

The coordinate domain is $r > 0$ . No positive-mass or nonzero-charge hypothesis is imposed. The finite component calculations below use no choice principle.

Proof

1.1F1A1algebra

In product coordinates the $v, r$ block is $(\begin{array}{cc} - f & 1 \\ 1 & 0 \end{array}),$ with determinant $- 1$ and inverse $(\begin{array}{cc} 0 & 1 \\ 1 & f \end{array}) .$ Its eigenvalues have opposite signs because their product is negative. The sphere block $r^{2} γ$ is positive definite for $r > 0$ , so $g_{i n}$ is a smooth nondegenerate Lorentzian metric on the stated product. The two-form is smooth since $r^{- 2}$ is smooth on $(0, \infty)$ and $ω$ is global.

1.2F2F3A1algebra

The product fields $\partial_{v}, \partial_{r}$ are global and smooth, hence so are $ℓ, n, N$ . Direct contraction gives $g (ℓ, ℓ) = g (n, n) = 0$ , $g (ℓ, n) = - 1$ , $g (N, N) = - 2$ , and $g (ℓ, N) = g (n, N) = - 1$ . Thus $N$ is an everywhere timelike field and fixes the asserted time orientation; [F3] then makes both nonzero null fields $ℓ, n$ future-directed.

1.3F1A1algebra

On any component where $f \neq 0$ , substitute $d v = d t + f^{- 1} d r$ into $- f d v^{2} + 2 d v d r$ . The mixed terms cancel and the radial coefficient is $f^{- 1}$ , giving $- f d t^{2} + f^{- 1} d r^{2}$ . Also $d v \land d r = d t \land d r$ , so $F_{i n}$ becomes the static two-form in [F1]. The substitution is defined separately on each component where $f \neq 0$ ; it does not extend $r_{*}$ through a root.

2.1F1F3step 1.2algebra∎

On $f > 0$ , $\partial_{t} = \partial_{v}$ and $g (\partial_{v}, N) = - 1 - f / 2 < 0$ , so the exterior orientations agree. On $f < 0$ , the static-coordinate vector is $\partial_{r} ∣_{t} = \partial_{r} ∣_{v} + f^{- 1} \partial_{v}$ , and $g (\partial_{r} ∣_{t}, N) = 1 / 2 - 1 / f > 0$ ; hence the static definition's $\partial_{r} ∣_{t}$ is past-directed under the selected ingoing orientation. This proves the stated convention boundary and makes no identification of the two interior time orientations.

Remarks

The future orientation is chosen by the smooth global field $N$ . The static metric and field formulas agree on every $f \neq 0$ component, while their separate interior time-orientation convention is not inherited across the ingoing extension. No global event-horizon or maximal-extension conclusion is part of this definition.

Selected ingoing RN extension and time orientation

Definition

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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