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Jacobi fields in flat space are affine, and no geodesic has conjugate instants

Example

Let $M = R^{4}$ carry the identity chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ , the flat Lorentzian metric $g = diag (- 1, 1, 1, 1)$ and the time orientation $N = \partial_{0}$ ; let $\nabla$ be its metric-selected connection and $R$ its curvature (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class). Then:

In the identity chart $\nabla$ is the coordinate connection, $Γ \equiv 0$ , and its geodesics are exactly the affine lines $t \mapsto p + t v$ ; moreover $\exp_{p} (v) = p + v$ and $d (\exp_{p})_{u} = id$ for every $u \in T_{p} M$ .
$R = 0$ : every curvature component vanishes, although the identity chart is global.
Along the geodesic $γ (t) = p + t v$ the covariant derivative is the componentwise derivative in the identity chart, and the Jacobi fields along $γ$ are exactly the affine fields $J (t) = J_{0} + t J_{1}, J_{0}, J_{1} \in R^{4},$ with $D_{t} J (0) = J_{1}$ ; the map $J \mapsto (J (0), D_{t} J (0))$ from this space onto $T_{p} M \oplus T_{p} M$ is the initial-value isomorphism of The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (c), and its inverse sends $(J_{0}, J_{1})$ to the field $t \mapsto J_{0} + t J_{1}$ .
No geodesic of this model has a conjugate instant: if $J (t_{0}) = J (t_{1}) = 0$ for $t_{0} < t_{1}$ , then $J = 0$ . Consistently $d (\exp_{p})_{u} = id$ is invertible for every $u \in T_{p} M$ , because $\exp_{p} (v) = p + v$ is a translation.
For the flat congruence $F (t, r) : = p + t v + r w$ with $p, v, w \in R^{4}$ , $v \neq 0$ , the variation field is the constant field $J = w$ and $D_{t}^{2} J = 0 = R (T, J) T$ : the deviation equation holds with no curvature term, and the relative acceleration of the congruence vanishes identically.

Facts & Assumptions

Given: The flat Lorentzian four-manifold $(R^{4}, g)$ with identity chart $x$ , $g = diag (- 1, 1, 1, 1)$ , and the geodesics $t \mapsto p + t v$ .

[L1]

The constant components $g_{a b} = diag (- 1, 1, 1, 1)$ in the global identity chart define a smooth Lorentzian metric of signature $(-, +, +, +)$ with time orientation $N = \partial_{0}$ , since $g (\partial_{0}, \partial_{0}) = - 1$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L2]

An affine connection is $g$ -compatible if and only if $\partial_{a} g_{b c} = \sum_{d} (Γ^{d}_{a b} g_{d c} + Γ^{d}_{a c} g_{b d})$ in every chart, and torsion-free if and only if $Γ^{a}_{b c} = Γ^{a}_{c b}$ ; the metric-selected connection exists, is unique, and is exactly the torsion-free $g$ -compatible connection (Torsion freedom and metric compatibility in chart components clauses 1-2, 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 clause (a)).

[L3]

In a chart, a geodesic with affine parameter satisfies ${\ddot{x}}^{k} + \sum_{i, j} Γ^{k}_{i j} {\dot{x}}^{i} {\dot{x}}^{j} = 0$ ; the geodesic initial-value problem has a unique maximal solution $γ_{p, v}$ , and $\exp_{p} (v) = γ_{p, v} (1)$ (Coordinate geodesic equation, Geodesic of an affine connection, The geodesic initial-value problem and the exponential map of the metric-selected connection).

[L4]

The curvature components in a chart are $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , and $R$ is a smooth tensor field (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clauses (a)-(b), The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L5]

Along a curve the covariant derivative has, in a chart, the component expression $D_{t} V^{a} = \partial_{t} V^{a} + \sum_{b, c} Γ^{a}_{b c} {\dot{γ}}^{b} V^{c}$ for a field $V = \sum_{a} V^{a} \partial_{a}$ (Covariant derivative along a curve).

[L6]

A Jacobi field along a geodesic $γ$ is a smooth field with $D_{t}^{2} J = R (γ^{'}, J) γ^{'}$ ; for a smooth family of affinely parametrized geodesics the variation field satisfies the same equation (Jacobi fields, geodesic variations and conjugate instants along a geodesic clauses (b)-(c), The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (b)).

[L7]

For a geodesic $γ (t) = \exp_{p} (t v)$ with $v \in T_{p} M$ and $s_{1} > 0$ with $s_{1} v$ in the exponential domain, $d (\exp_{p})_{s_{1} v}$ is injective if and only if no nonzero Jacobi field along $γ$ vanishes at $t = 0$ and at $t = s_{1}$ (The differential of the exponential map and conjugacy along a geodesic clause (b)).

Verification

technique · direct

1.1L1L2L3algebra

(The metric-selected connection and its geodesics.) For the coordinate connection with $Γ \equiv 0$ the symmetry $Γ^{a}_{b c} = Γ^{a}_{c b}$ is immediate and the compatibility identity of [L2] reads $\partial_{a} g_{b c} = 0$ , which holds because the metric components are constant; hence by [L2] this connection is the metric-selected connection and unique. By [L3] its geodesic equation is ${\ddot{x}}^{k} = 0$ , whose solutions are the affine lines, so the maximal geodesic with initial data $(p, v)$ is $t \mapsto p + t v$ defined on all of $R$ ; therefore $\exp_{p} (v) = p + v$ , and differentiating the chart expression of this translation gives $d (\exp_{p})_{u} = id$ for every $u$ .

1.2L4L5algebra

(Vanishing curvature and the componentwise derivative.) With $Γ \equiv 0$ every term of the coordinate formula [L4] contains a Christoffel symbol or a derivative of one, so $R^{a}_{b c d} = 0$ for all $a, b, c, d$ and $R = 0$ . By [L5], for a field $V = \sum_{a} V^{a} \partial_{a}$ along a curve, $D_{t} V = \sum_{a} (\partial_{t} V^{a}) \partial_{a}$ : the identity frame is parallel along every curve, and $D_{t}$ is the componentwise derivative.

2.1step 1.1step 1.2L6algebra

(Jacobi fields are affine.) Let $J = \sum_{a} J^{a} \partial_{a}$ be a smooth field along the geodesic $γ (t) = p + t v$ ; by step 1.1 $γ^{'} = v$ has constant components. By step 1.2, $D_{t}^{2} J = \sum_{a} (J^{a})^{''} \partial_{a}$ and $R (γ^{'}, J) γ^{'} = 0$ , so the Jacobi equation of [L6] is equivalent to $(J^{a})^{''} = 0$ for $a = 0, 1, 2, 3$ , whose solutions are exactly the affine functions $J^{a} (t) = J_{0}^{a} + t J_{1}^{a}$ . Hence the Jacobi fields are exactly the fields $J (t) = J_{0} + t J_{1}$ with $J_{0}, J_{1} \in R^{4}$ , and $D_{t} J (0) = J_{1}$ by step 1.2.

3.1step 2.1step 1.1L7algebra

(No conjugate instants.) Let $t_{0} < t_{1}$ and let $J (t) = J_{0} + t J_{1}$ be a Jacobi field as in step 2.1. If $J (t_{0}) = J (t_{1}) = 0$ , then subtracting the two equations gives $(t_{1} - t_{0}) J_{1} = 0$ , so $J_{1} = 0$ , and then $J_{0} = J (t_{0}) = 0$ : the only Jacobi field vanishing at two instants is the zero field. By [L7] applied to the geodesic $γ (t) = \exp_{p} (t v)$ , whose exponential data lie in the domain $T_{p} M$ by step 1.1, $d (\exp_{p})_{s_{1} v}$ is injective for every $s_{1} > 0$ and every $v$ : no geodesic of the flat model has a conjugate instant relative to any of its points.

4.1step 1.2step 1.1L6algebra∎

(The flat congruence.) Let $F (t, r) : = p + t v + r w$ and $T = \partial_{t} F = v$ , $J = \partial_{r} F = w$ , both with constant components. By step 1.2, $D_{t}^{2} J = 0$ and $R (T, J) T = 0$ , so $D_{t}^{2} J = R (T, J) T$ holds, in accordance with the deviation equation of [L6]; the variation field is parallel and the relative acceleration of the family vanishes everywhere.

Remarks

Flat versus curved. In the flat model the deviation of nearby geodesics is governed by affine initial data alone, and no conjugate instants occur. The contrast with the sphere-factor metric of Curvature, Ricci, scalar and Einstein tensors of a metric with a round two-sphere factor, where a Jacobi field vanishes again at parameter distance $π$ , is computed in An equator geodesic of the round-sphere factor: conjugate instant at distance pi with multiplicity one.
Scope. The statements are about the flat Lorentzian four-manifold with its metric-selected connection; no maximality of geodesics, completeness or causal claim beyond the displayed computations is made.

Jacobi fields in flat space are affine, and no geodesic has conjugate instants

Example

Facts & Assumptions

Verification

Remarks

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