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Speed bounds along geodesics in a chart, and small initial velocity of every geodesic segment contained in a small ball

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class) and geodesics as in The geodesic initial-value problem and the exponential map of the metric-selected connection. Let $(U, x)$ be a chart, $B_{s} = {y \in R^{4} : ∣ y ∣_{e} < s}$ and let $K \subseteq U$ be compact with $∥ Γ ∥_{K} < \infty$ as in Convexity of the squared Euclidean radius along small contained geodesics: segments with endpoints in a small ball stay in the ball. Let $γ : [0, 1] \to U$ be a geodesic, write $s \mapsto w (s) : = ∣ \dot{x} (s) ∣_{e}$ for the Euclidean norm of its coordinate velocity, and let $v_{0} : = w (0) .$

(a) Two-sided speed bounds. If $0 \leq δ \leq 1$ , $4 ∥ Γ ∥_{K} v_{0} δ \leq 1$ , and the restriction of $γ$ to $[0, δ]$ has image in $K$ , then, for $t \in [0, δ]$ , $w (t) \leq \frac{4}{3} v_{0}, ∣ \dot{x} (t) - \dot{x} (0) ∣_{e} \leq \frac{4}{9} v_{0} .$ Consequently, when $v_{0} > 0$ , the Euclidean coordinate displacement has component at least $\frac{5}{9} v_{0} δ$ along the unit vector $\dot{x} (0) / v_{0}$ . When $v_{0} = 0$ , the geodesic is constant and the displacement is zero, so no unit initial direction is selected.

(b) Contained segments have small velocity. Let $c_{0} : = \frac{1}{100}$ . Fix $r > 0$ with $r \leq \frac{1}{4}$ and $∥ Γ ∥_{K} r \leq c_{0}$ . If the image of the geodesic $γ : [0, 1] \to U$ is contained in $K \cap x^{- 1} (B_{r})$ , then $∣ \dot{x} (0) ∣_{e} \leq 4 r .$

Facts & Assumptions

Given: A chart $(U, x)$ , a compact $K \subseteq U$ , a geodesic $γ : [0, 1] \to U$ whose image on the relevant subintervals lies in $K$ , and $v_{0} = ∣ \dot{x} (0) ∣_{e}$ .

[L1]

The coordinate velocity satisfies ${\dot{w}}^{2} = 2 \dot{x} \cdot \ddot{x} = - 2 \dot{x} \cdot Γ (x) [\dot{x}, \dot{x}] \leq 2 ∥ Γ ∥_{K} w^{3}$ at parameters where the image lies in $K$ ; hence $w^{'} \leq ∥ Γ ∥_{K} w^{2}$ (Coordinate geodesic equation, Christoffel symbols of an affine connection, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L2]

On any interval where $w > 0$ , the reciprocal $1 / w$ is smooth. If $(1 / w)' \geq - C$ , the fundamental theorem gives $1 / w (t) - 1 / w (0) = \int_{0}^{t} (1 / w)' (s) d s \geq - C t$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); this needs no two-sided derivative bound.

[L3]

For $t$ in a parameter interval on which the image lies in $K$ , $∣ \dot{x} (t) - \dot{x} (0) ∣_{e} = ∣ \int_{0}^{t} \ddot{x} (s) d s ∣_{e} \leq \int_{0}^{t} ∥ Γ ∥_{K} w (s)^{2} d s$ by the geodesic equation and the fundamental theorem of calculus for continuous integrands (Coordinate geodesic equation, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ for the componentwise statement).

[L4]

If two points of a curve lie in the open coordinate ball $x^{- 1} (B_{r})$ , then the Euclidean norm of their coordinate displacement is at most $2 r$ , by the triangle inequality in $R^{4}$ (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ for the Euclidean norm; the statement is elementary).

Proof

technique · direct

1.1L1L2givenalgebra

(Speed bound.) Assume $v_{0} > 0$ . The geodesic velocity never vanishes: if it vanished at any parameter, uniqueness with that initial datum would make the geodesic constant, contradicting $v_{0} > 0$ (The geodesic initial-value problem and the exponential map of the metric-selected connection). Thus $w > 0$ on $[0, 1]$ , and let $t$ range over $[0, δ]$ , whose image is assumed to lie in $K$ . By [L1], $w^{'} \leq ∥ Γ ∥_{K} w^{2}$ and hence $(1 / w)^{'} = - w^{'} / w^{2} \geq - ∥ Γ ∥_{K}$ there; by [L2], $1 / w (t) \geq 1 / v_{0} - ∥ Γ ∥_{K} t$ and therefore $w (t) \leq v_{0} / (1 - ∥ Γ ∥_{K} v_{0} t) \leq \frac{4}{3} v_{0}$ for $4 ∥ Γ ∥_{K} v_{0} t \leq 1$ . If $v_{0} = 0$ , the constant curve has the same initial data as $γ$ , so uniqueness of the geodesic initial-value problem gives $w \equiv 0$ . If $∥ Γ ∥_{K} = 0$ , the geodesic equation gives $\ddot{x} = 0$ on the subinterval in $K$ , so $w = v_{0}$ and the velocity change is zero. These cases use no reciprocal of $∥ Γ ∥_{K} v_{0}$ .

2.1step 1.1L3algebra

(Velocity-change bound.) On $[0, δ]$ with $0 \leq δ \leq 1$ and $4 ∥ Γ ∥_{K} v_{0} δ \leq 1$ , steps 1.1 and [L3] give $∣ \dot{x} (t) - \dot{x} (0) ∣_{e} \leq \int_{0}^{t} ∥ Γ ∥_{K} (4 v_{0} / 3)^{2} d s \leq ∥ Γ ∥_{K} v_{0}^{2} t \frac{16}{9} \leq \frac{4}{9} v_{0}$ for $4 ∥ Γ ∥_{K} v_{0} t \leq 1$ .

2.2step 1.1L3algebra

(Displacement along the initial direction.) Assume $v_{0} > 0$ , write $u_{0} : = \dot{x} (0)$ and $e : = u_{0} / v_{0}$ , and put $φ (t) : = e \cdot x (t)$ . Then $φ^{'} (0) = v_{0}$ and, by step 1.1, $φ^{''} (t) = e \cdot \ddot{x} (t) \geq - ∥ Γ ∥_{K} w (t)^{2} \geq - \frac{16}{9} ∥ Γ ∥_{K} v_{0}^{2}$ for $t \leq δ$ . Integrating twice on $[0, δ]$ with $0 \leq δ \leq 1$ and $4 ∥ Γ ∥_{K} v_{0} δ \leq 1$ , $φ (δ) - φ (0) \geq v_{0} δ - \frac{8}{9} ∥ Γ ∥_{K} v_{0}^{2} δ^{2} = v_{0} δ (1 - \frac{8}{9} ∥ Γ ∥_{K} v_{0} δ) \geq \frac{7}{9} v_{0} δ .$ If $v_{0} = 0$ the displacement bound is trivial.

3.1step 2.1step 2.2L4algebra

(Contained segments have small velocity.) Let $r > 0$ satisfy $∥ Γ ∥_{K} r \leq \frac{1}{100}$ and $r \leq \frac{1}{4}$ , let $γ : [0, 1] \to U$ have image in $x^{- 1} (B_{r})$ , and put $v_{0} = ∣ \dot{x} (0) ∣_{e}$ . Suppose first that $∥ Γ ∥_{K} v_{0} \leq \frac{1}{4}$ . Then $δ = 1$ is admissible in steps 2.1-2.2, and also admissible trivially if $v_{0} = 0$ . Both $γ (0)$ and $γ (1)$ lie in $B_{r}$ , so by [L4] $∣ x (γ (1)) - x (γ (0)) ∣_{e} \leq 2 r$ ; on the other hand the displacement's component along $u_{0}$ (the case $v_{0} = 0$ is trivial) is at least $\frac{7}{9} v_{0}$ by step 2.2, so $v_{0} \leq \frac{18}{7} r < 4 r$ . Suppose now that $∥ Γ ∥_{K} v_{0} > \frac{1}{4}$ and $v_{0} > 0$ ; then $δ = (4 ∥ Γ ∥_{K} v_{0})^{- 1} < 1$ is admissible, the points $γ (0)$ and $γ (δ)$ both lie in $B_{r}$ , so their displacement has norm at most $2 r$ , while step 2.2 gives a component along $u_{0}$ of at least $\frac{7}{9} v_{0} δ = \frac{7}{36 ∥ Γ ∥_{K}} > 2 r,$ the strict inequality because $∥ Γ ∥_{K} r \leq \frac{1}{100} < \frac{7}{72}$ . This contradiction excludes the second case, so always $v_{0} < 4 r$ . This proves (b).

4.1step 1.1step 2.1step 2.2step 3.1∎

Combining steps 1.1, 2.1, 2.2 and 3.1 proves (a) and (b) respectively.

Remarks

The constants are not optimal. The proof only needs that contained segments have velocity bounded by a quantity tending to zero with $r$ ; the threshold $4 r$ and the constant $c_{0} = \frac{1}{100}$ are fixed so that Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points can choose $r$ smaller than any prescribed velocity radius of the joint exponential diffeomorphism.
Uniqueness of contained geodesics. Combined with The joint exponential map: smoothness, invertible differential at the diagonal, and the small-velocity two-point inverse, step 3.1 says: two geodesic segments contained in a sufficiently small coordinate ball and joining the same pair of points carry initial velocities of size at most $4 r$ , hence lie in the joint inverse domain when $4 r$ is smaller than its velocity radius, and therefore coincide.

Speed bounds along geodesics in a chart, and small initial velocity of every geodesic segment contained in a small ball

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources