Let $S$ be a Cauchy hypersurface of a globally hyperbolic spacetime $(\mathcal{M},\mathcal{O},\mathcal{A},g,T)$ with unit normal vector field $n$. Define the exponential map on a neighborhood $U\subseteq \mathbb{R}\times S$ of $\{0\}\times S$ by $\exp(t,p)=c_p(t)$ where $c_p$ is the timelike geodesic which goes through $p$ with tangent vector $n_p$. A regular value of $\exp$ is called conjugate to $S$.

I'm having trouble relating this definition to the intuitive notion where conjugate points to $S$ are points where timelike geodesics starting at nearby points in $S$ intersect. I am sure this has to do with some mathematical theorem I'm not aware of. Can somebody help me understand this?

  • $\begingroup$ Is definition taken from a reference? Title? Author? Page? $\endgroup$
    – Qmechanic
    Dec 9 '17 at 17:25
  • $\begingroup$ Yes Riemannian Geometry with Applications to Mechanics and Relativity $\endgroup$ Dec 9 '17 at 17:44
  • $\begingroup$ That definition, as it stands, says nothing about the existence of converging geodesics. It only says that if $q$ is conjugated to $S$ and $\exp(t,p)=q$, then $\exp$ is a local diffeomorphism from an open neighborhood of that $(t,p)$ to an open neighborhood of $q$. However there can exist different such pairs $(t,p) \in U$ for the given conjugate value $q$. $\endgroup$ Dec 9 '17 at 20:53
  • $\begingroup$ Different such pairs, if exist and therefore there are converging geodesics in that case, are necessarily far from each other. $\endgroup$ Dec 9 '17 at 20:57
  • $\begingroup$ All that is a direct application of the theorem of regular values. $\endgroup$ Dec 9 '17 at 21:03

In this context, I believe we are to consider $\exp(t,p)$ as a family of maps $\exp_t : S \to M$, and take a regular value to be a point $q \in \exp_t(S)$ such that the push-forward $\exp_{t*} : T_{\exp_t^{-1}(q)}S \to T_qM$ is a surjection. Here $t$ is to be considered merely a parameter which tells us which map we are dealing with. In other words, $q$ is a regular value we would have $T_qM = \exp_{t*}(T_{\exp_t^{-1}(q)}S).$ Since $S$ has co-dimension one, this is obviously only possible if the pre-image $\exp_t^{-1}(q)$ contains more than one point. At least this would recover, in some sense, the intuitive notion of conjugate points, and bears some resemblence to the standard definition.


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