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Boost normalization and affine focusing

Physical scope

Pure mathematical normalization example; no field equation or empirical claim is used.

Statement

Assume Countable Choice and Dependent Choice for the Lorentzian and causal suppliers. Let $(M, g, N)$ be a smooth time-oriented Lorentzian four-manifold, and let $S \subset M$ be a smooth embedded two-manifold without boundary with positive induced metric. Let $U \subseteq S$ be nonempty and open and carry a smooth normalized future null normal pair $(ℓ_{0}, k_{0})$ , so $g (ℓ_{0}, ℓ_{0}) = g (k_{0}, k_{0}) = 0$ and $g (ℓ_{0}, k_{0}) = - 2$ . Let $F : U \times J \to M$ be a smooth null geodesic hypersurface congruence from $U$ with initial tangent $\partial_{λ} F (p, 0) = ℓ_{0} (p)$ . Write $θ_{0} (p) : = θ (p, 0)$ for the initial optical expansion along the generator from $p$ . For $f \in C^{\infty} (U, R)$ , define $\hat{f} (p, λ) : = f (p)$ on the parameter domain and set $ℓ^{'} : = e^{\hat{f}} \partial_{λ} F$ ; the initial companion is $k_{0}^{'} = e^{- f} k_{0}$ . Then ${\hat{θ}}_{0} = e^{f} θ_{0}$ , so the sign of the initial expansion is independent of this positive normalization. Along the generator from $p$ , the new affine parameter is $λ^{'} = e^{- f (p)} λ$ ; optical expansion, shear and Ricci contraction scale so that the null Raychaudhuri equation is unchanged. Define $I^{+} (S) : = ⋃_{s \in S} I^{+} (s)$ , $J^{+} (S) : = ⋃_{s \in S} J^{+} (s)$ , and $E^{+} (S) : = J^{+} (S) ∖ I^{+} (S)$ , where $I^{+} (s)$ and $J^{+} (s)$ use future timelike and future causal reachability. A focal event and the exclusion from $E^{+} (S)$ of points sufficiently close after the event are independent of the chosen constant-along-generator boost. On any generator with $θ_{0} (p) < 0$ and $Ric (ℓ, ℓ) \geq 0$ throughout the comparison interval, the bound $λ_{*} (p) : = 2 / ∣ θ_{0} (p) ∣$ rescales to $λ_{*}^{'} = e^{- f (p)} λ_{*} (p)$ .

Statement references

Null normal pairs and screen geometry

Null Raychaudhuri and focal-point bound

Null focal points exclude later horismos generators

Facts & Assumptions

Given: Countable Choice and Dependent Choice; the time-oriented Lorentzian manifold, spacelike surface $S$ , nonempty open section $U \subseteq S$ , smooth normalized future null normal pair, its smooth null geodesic hypersurface congruence, and the smooth boost factor $f$ extended constantly along each generator as in the Statement.

[F1]

Under a normal-frame boost, $χ_{ℓ}$ , $θ_{ℓ}$ and $σ_{ℓ}$ scale by $e^{f}$ ; the optical map and expansion are defined on the positive screen (Null normal pairs and screen geometry).

[F2]

The affine screen equation is $θ^{'} = - θ^{2} / 2 - ∣ σ ∣^{2} - Ric (ℓ, ℓ)$ and the bound is $2 / ∣ θ (0) ∣$ when $θ (0) < 0$ and $Ric (ℓ, ℓ) \geq 0$ , provided the congruence and generator exist through that comparison parameter (Null Raychaudhuri and focal-point bound).

[F3]

The focal event is geometric and every point of the normal generator sufficiently just beyond it lies in $I^{+} (S)$ , hence outside $E^{+} (S)$ (Null focal points exclude later horismos generators).

[A1]

$f$ is constant as a function of affine parameter on each generator; without this restriction $e^{f} ℓ$ need not be affine.

[A2]

Countable Choice and Dependent Choice are assumed, retaining the hypotheses of the focal-boundary supplier used in [F3] (Null focal points exclude later horismos generators).

Proof

Given: [F1]–[F3] and [A1]–[A2].

1.1A1F1algebra

Since $e^{f} > 0$ , the boost preserves the future null rays, and $g (ℓ^{'}, k^{'}) = e^{f} e^{- f} g (ℓ, k) = - 2$ . The boost law in [F1] gives ${\hat{θ}}_{0} = e^{f} θ_{0}$ , so ${\hat{θ}}_{0}$ has the same sign as $θ_{0}$ . The new tangent is $ℓ^{'} = d / d λ^{'}$ , while $ℓ = d / d λ$ ; solving $d λ / d λ^{'} = e^{f}$ with both parameters zero at the initial section gives $λ^{'} = e^{- f} λ$ because $f$ is constant along the generator.

2.1F1F2step 1.1algebra

On the open parameter domain where $e^{f (p)} λ^{'} \in J$ , put $\hat{F} (p, λ^{'}) : = F (p, e^{f (p)} λ^{'})$ . For $X \in T_{p} S$ , differentiation at fixed $λ^{'}$ gives $d_{p} \hat{F} (X) = d_{p} F (X) + e^{f (p)} λ^{'} (X f) ℓ .$ The added term vanishes in the screen quotient, so at corresponding points ${\hat{A}}_{λ^{'}} = A_{e^{f (p)} λ^{'}}$ . The screen metric is unchanged because the null line is unchanged. Along the fixed generator, $f (p)$ is constant, and differentiation with the screen connection gives ${\hat{A}}_{λ^{'}}^{'} = e^{f (p)} A_{e^{f (p)} λ^{'}}^{'}, \hat{B} = {\hat{A}}^{'} {\hat{A}}^{- 1} = e^{f (p)} B .$ Taking trace and trace-free part yields $\hat{θ} = e^{f} θ$ and $\hat{σ} = e^{f} σ$ . Therefore $d \hat{θ} / d λ^{'} = e^{2 f} d θ / d λ$ and $∣ \hat{σ} ∣^{2} = e^{2 f} ∣ σ ∣^{2}$ . Tensoriality gives $Ric (ℓ^{'}, ℓ^{'}) = e^{2 f} Ric (ℓ, ℓ)$ . Multiplying the Raychaudhuri equation in [F2] by $e^{2 f}$ proves its form is unchanged. For a generator from $p$ with $θ_{0} (p) < 0$ and $Ric (ℓ, ℓ) \geq 0$ through the comparison interval, the bound rescales as $λ_{*}^{'} (p) = \frac{2}{∣ {\hat{θ}}_{0} (p) ∣} = e^{- f (p)} \frac{2}{∣ θ_{0} (p) ∣} = e^{- f (p)} λ_{*} (p) .$

3.1A2F1F3step 1.1step 2.1algebra∎

By step 2.1, ${\hat{A}}_{λ^{'}} = A_{e^{f (p)} λ^{'}}$ , so its rank-loss event is the same point of $M$ and its parameter value is multiplied by $e^{- f (p)}$ . Under [A2], [F3] applies: the geometric statement that points just beyond that event lie in $I^{+} (S)$ is unchanged by relabeling the affine parameter. Thus only the numerical affine label changes; the causal-boundary conclusion does not.

Remarks

This is a normalization calculation for a constant-along-generator boost. No field equation, energy condition, or empirical input is used.

Boost normalization and affine focusing

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources