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Null geodesic radial first integral in the Schwarzschild exterior

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum with $s > 0$ and $f (r) = 1 - s / r$ on $r > s$ , with the countable-choice hypothesis of its geometric suppliers. Let $γ : J \to M$ be a smooth affinely parametrized geodesic of its metric-selected connection on a real interval $J$ with nonempty interior, with nonzero tangent and $g (γ^{'}, γ^{'}) = 0$ , whose image lies in the equatorial plane $ϑ = π / 2$ . Orient its affine parameter toward the future, so that

$E : = f (r) \frac{d x^{0}}{d λ} > 0, L : = r^{2} \frac{d φ}{d λ} \in R .$

Both quantities are constant along $γ$ . Reversing the affine parameter changes both signs and does not change $L / E$ ; no sign restriction on $L$ is imposed in the fixed longitude convention. With nonnegative impact parameter $b : = ∣ L ∣ / E$ and $H (r) : = f (r) / r^{2}$ , its radial equation is

${(\frac{d r}{d λ})}^{2} = E^{2} (1 - b^{2} H (r)) .$

Moreover, $H^{'} (r) = (3 s - 2 r) / r^{4}$ , so $H$ has its unique exterior maximum at $r_{p h} = 3 s / 2$ , with $H (r_{p h}) = 4 / (27 s^{2})$ . Thus the critical impact parameter for a radial turning point is $b_{c} = 3 \sqrt{3} s / 2$ .

Facts & Assumptions

Given: The smooth Schwarzschild exterior metric with $r > s$ and the countable-choice hypothesis of its geometric suppliers, its metric-selected connection, and a smooth nonconstant affine null geodesic in the equatorial plane, oriented toward the future with $E > 0$ and signed $L \in R$ .

[F1]

In spherical-polar product charts the Schwarzschild metric is diagonal with $g_{00} = - f$ , $g_{r r} = f^{- 1}$ , $g_{ϑ ϑ} = r^{2}$ , and $g_{φ φ} = r^{2} \sin^{2} ϑ$ , where $f = 1 - s / r$ ; the smooth exterior is Lorentzian and its metric-selected connection exists (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[F2]

A Killing field $ξ$ contracted with the tangent of an affinely parametrized geodesic has constant scalar product $g (ξ, γ^{'})$ (Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (b)); a Killing field is defined by $L_{ξ} g = 0$ (Killing fields of a Lorentzian metric), with the Lie derivative given by its flow pullback derivative (The Lie derivative of a tensor field).

[F3]

An affine geodesic satisfies $D_{λ} γ^{'} = 0$ (Geodesic of an affine connection); the given null condition is $g (γ^{'}, γ^{'}) = 0$ .

[F4]

For a function continuous on a closed interval and differentiable on its interior, the mean value theorem equates its endpoint difference to an interior derivative times the interval length (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ). A continuous real function on a closed bounded interval takes every value between its endpoint values (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ ).

Proof

technique · use the time-translation and axial-rotation symmetries for first integrals, then impose the null condition in the diagonal metric

1.1givenF1F2F3algebra

In each equatorial spherical chart the components in [F1] are independent of $x^{0}$ and $φ$ , and the coordinate components of $\partial_{0}$ and the axial generator $\partial_{φ}$ are constant. The flow pullback derivative in [F2] therefore gives $L_{\partial_{0}} g = L_{\partial_{φ}} g = 0$ , so both fields are locally Killing. By the conservation part of [F2] and the affine geodesic equation [F3], $- g (\partial_{0}, γ^{'}) = f d x^{0} / d λ = E$ and $g (\partial_{φ}, γ^{'}) = r^{2} d φ / d λ = L$ are constant on each chart interval; longitude charts differ by additive constants, so these values agree on overlaps and are constant along $γ$ .

2.1step 1.1F1givenalgebra

At $ϑ = π / 2$ , the null condition and [F1] give $0 = - f {(\frac{d x^{0}}{d λ})}^{2} + f^{- 1} {(\frac{d r}{d λ})}^{2} + r^{2} {(\frac{d φ}{d λ})}^{2} .$ A nonzero null tangent cannot have $d x^{0} / d λ = 0$ , since the remaining quadratic form is positive definite. Its time component has constant sign on $J$ by continuity; the future orientation relative to $N = \partial_{0}$ gives $- g (N, γ^{'}) = E > 0$ . Parameter reversal flips both $E$ and $L$ , leaving their ratio invariant. Substitute $d x^{0} / d λ = E / f$ and $d φ / d λ = L / r^{2}$ , then multiply by $f > 0$ . The result is $(d r / d λ)^{2} = E^{2} [1 - (L / E)^{2} f / r^{2}] = E^{2} [1 - b^{2} H (r)]$ , valid for either sign of $L$ .

3.1step 2.1F4algebra

Differentiating $H (r) = r^{- 2} - s r^{- 3}$ yields $H^{'} (r) = - 2 r^{- 3} + 3 s r^{- 4} = (3 s - 2 r) / r^{4}$ . Since $r^{4} > 0$ , this derivative is positive on $(s, 3 s / 2)$ and negative on $(3 s / 2, \infty)$ ; the mean value theorem in [F4] gives strict increase and decrease on these respective intervals, hence the unique maximum is at $3 s / 2$ . Substitution gives $H (3 s / 2) = 4 / (27 s^{2})$ , and solving $b^{2} H (3 s / 2) = 1$ gives $b_{c} = 3 \sqrt{3} s / 2$ .

4.1step 2.1step 3.1F4givenalgebra∎

The right side of the radial first integral vanishes at a radius precisely when $b^{2} H (r) = 1$ . Since $H$ is positive on $r > s$ , tends to zero at both ends, and has the maximum computed in the preceding step, no root exists for $0 \leq b < b_{c}$ , whereas $b = b_{c}$ gives the root $3 s / 2$ . For $b > b_{c}$ , choose $R > 3 s / 2$ so large that $H (R) < 1 / b^{2} < H (3 s / 2)$ ; continuity and the intermediate value theorem in [F4] on $[3 s / 2, R]$ give an exterior root. Thus the threshold for the existence of an exterior root is exactly $b_{c}$ . This is a first-integral statement only; it asserts no global scattering continuation.

Null geodesic radial first integral in the Schwarzschild exterior

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources