Jacobi fields, geodesic variations and conjugate instants along a geodesic

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, 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), and let $R$ be its curvature (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $\gamma :I\to M$ be an affinely parametrized geodesic with $D_s{\gamma }^{\mathrm{\prime }}=0$, written $T:={\gamma }^{\mathrm{\prime }}$ (Geodesic of an affine connection); the parameter is the affinely parametrized one of that definition and need not be proper time.

(a) Fields along $\gamma$ and the derivative $D_s$. A field along $\gamma$ is a section $J$ of the pullback bundle ${\gamma }^{\ast }TM$, and $D_s$ denotes the covariant derivative along $\gamma$ of Covariant derivative along a curve. A field is smooth when its coefficients in a smooth local frame of $M$ around each point of $\gamma$ are $C^{\mathrm{\infty }}$ on the corresponding parameter interval. At an included endpoint of $I$ all derivatives below are one-sided.

(b) Jacobi field. A Jacobi field along $\gamma$ is a smooth field $J$ along $\gamma$ satisfying the Jacobi equation $D_s^{2}J=R(T,J)T(s\in I),$ that is $D_s^{2}J=R({\gamma }^{\mathrm{\prime }},J){\gamma }^{\mathrm{\prime }}$. The assignment $J\mapsto R(T,J)T$, which associates with a field along $\gamma$ another field along $\gamma$ and is $\mathbb{R}$-linear, is the Jacobi operator of $\gamma$. Whether the field is smooth only piecewise is stated whenever a statement uses piecewise fields; the unqualified term means globally smooth.

(c) Geodesic variations. A geodesic variation with affine parameter $s$ is a smooth map $F:J_0\times I\to M$, where $J_0\subseteq \mathbb{R}$ is an open interval containing $0$, such that for every $r\in J_0$ the curve $s\mapsto F(r,s)$ is a geodesic with affine parameter $s$. Its variation field is the field $s\mapsto {\mathrm{\partial }}_{r}F(0,s)$ along the curve $s\mapsto F(0,s)$. The variation field of a geodesic variation is a Jacobi field along that curve (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields ↗ clause (b)).

(d) Conjugate instants and multiplicity. Let $s_0<s_1$ be points of $I$. The event $\gamma (s_1)$ is conjugate to $\gamma (s_0)$ along $\gamma$ when there is a nonzero Jacobi field $J$ along $\gamma$ with $J(s_0)=J(s_1)=0$. In that case $s_1$ is called a conjugate instant of $\gamma$ (with respect to $s_0$, when the base point needs to be named), and the multiplicity of the conjugate instant is the dimension of the real vector space of Jacobi fields vanishing at $s_0$ and $s_1$; the fields vanishing at $s_0$ form a $4$-dimensional space and the multiplicity is at most $4$, since a Jacobi field is determined by its value and derivative at $s_0$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields ↗ clause (c)). For a geodesic with $\gamma (s_0)=p$ and ${\gamma }^{\mathrm{\prime }}(s_0)=v$, the corresponding possibility of conjugacy is measured by the derivative of the exponential map at $(s_1-s_0)v$, provided this vector lies in its domain (The differential of the exponential map and conjugacy along a geodesic ↗ applied to $t\mapsto \gamma (s_0+t)$ on $[0,s_1-s_0]$); that equivalence is proved, not assumed here. No conjugate instant is asserted to exist.

(e) The normal bundle. If $\gamma$ is future timelike with $g(T,T)=-1$, define its normal bundle to be the vector subbundle $N_{\gamma }:={⨆}_{s\in I}{N}_{\gamma }(s)\subseteq {\gamma }^{\ast }TM,N_{\gamma }(s):=\{v\in T_{\gamma (s)}M:g_{\gamma (s)}(v,T(s))=0\}\mathrm{.}$ A normal field is a smooth section $J$ of $N_{\gamma }$, equivalently a smooth field along $\gamma$ satisfying $g(J,T)=0$. Each fibre $N_{\gamma }(s)$ has dimension three, the restriction of $g_{\gamma (s)}$ to it is positive definite, and $D_s$ maps normal fields to normal fields, since $\frac{d}{ds}g(J,T)=g(D_{s}J,T)+g(J,D_{s}T)=g(D_{s}J,T)$ and $D_{s}T=0$; these facts are proved in The normal index form before the first conjugate instant is positive definite ↗ clause (b). The norm of a normal field is $\mathrm{\mid }J(s)\mathrm{\mid }:=\sqrt{g_{\gamma (s)}(J(s),J(s))}$. For any field $V$ along $\gamma$, its normal component is the field $V^{\perp }(s):=V(s)+g_{\gamma (s)}(V(s),T(s))T(s)\mathrm{.}$ Indeed $g(T,T)=-1$ implies $g(V^{\perp },T)=0$ and $V=V^{\perp }-g(V,T)T$.

• What is and is not fixed. This definition fixes the vocabulary of the deviation and conjugate-point results on this page: Jacobi field, variation field, conjugate instant and multiplicity. It asserts nothing about the existence of Jacobi fields with prescribed initial data, about conjugacy for any particular geodesic, or about maximality; those are theorems proved separately. The sign convention of the Jacobi operator is the one induced by the curvature sign convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors: $D_s^{2}J=R(T,J)T$, not its negative. Sources using the opposite sign or the opposite curvature convention state the equation with a minus sign; no formula is taken from another source without converting its convention.

• Relation to proper time. When $\gamma$ is future timelike with $g(T,T)=-c^2$ and the affine parameter is proper time $\tau$, the equation above is the same as $D_{\tau }^{2}J=R(U,J)U$ for the four-velocity $U$; the conversion factor is a positive constant and does not change which fields vanish at which parameter values (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent).