Conjugate point to spacelike hypersurface

Question

Suppose you have a smooth spacelike hypersurface $\Sigma$ in some spacetime (four-dimensional Lorentzian manifold). Let $\gamma$ be a timelike geodesic meeting $\Sigma$ orthogonally and let $p$ be a point on $\gamma$. Then $p$ is said to be conjugate to $\Sigma$ iff there exists a Jacobi field $J$ orthogonal to the tangent vector $\gamma'$, which is zero at $p$, nonzero on $\Sigma$, and comes from a variation of $\gamma$ by geodesics intersecting $\Sigma$ orthogonally.

Why is this the definition? Why not say that $p$ is conjugate to $\Sigma$ iff there exists a Jacobi field $J$ which is zero at $p$ and zero on $\Sigma$ (and thus $J$ is orthogonal everywhere to $\gamma$), the same definition as conjugacy along a geodesic? If you do that, you cannot ask of the variation to come from geodesics intersecting $\Sigma$ (if $q$ is the intersection of $\gamma$ and $\Sigma$, then the orthogonal space to the tangent space at $q$ of $\Sigma$, i.e. $(T_q\Sigma)^{\perp}$, is one-dimensional, so any geodesic intersecting $\Sigma$ orthogonally at $q$ must be a reparametrization of $\gamma$), but why does one care about intersecting orthogonally?

EDIT: If you take $\theta$ to be the expansion of the timelike geodesics through $p$, then $p$ is conjugate to $q$ along $\gamma$ iff $\theta$ goes to infinity at $q$.

I would like to see a proof for that in the hypersurface case: $p$ is conjugate to $\Sigma$ along $\gamma$ iff $\theta$, the expansion of the timelike geodesics orthogonal to $\Sigma$, goes to infinity at $p$. In particular, I do not know how to impose the condition that the Jacobi field $J$ arises from a variation of geodesics orthogonal to $\Sigma$.

Answer 1

There are two cases: Jacobi fields defined in terms of a geodesic spray from a point and a geodesic spray from a surface. In both cases the differential equation that defines the Jacobi tensor is the same. It is only the initial conditions that are different. Because of this conjugate points or focal points both occur with the expansion diverges.

The reason for the similarity in the differential equation can be seen from either a mathematical point of view or a physical one. I'll start with the maths.

A causal geodesic is length maximal up to its first conjugate point. Similarly the normal geodesic to a surface is length maximal (the distance from the surface to some point) up to its first focal point. In both cases variation of arc length arguments can be used to show this (via the Morse index theory) and thus in both cases the same differential equation results.

The physical point of view comes from analysing the infinitesimal acceleration of a congruence of geodesics. It turns out that the differential of the exponential map defines a Jacobi tensor. Thus the study of the infinitesimal acceleration of a congruence of geodesics is equivalent to the study of the variation of arc length.

Beem, Erhlich and Easley (BEE) have much to say about both points of view (see Chapter 10 and 12). In particular they carefully describe why it is sufficient to study congruences of geodesics (rather than arbitrary causal curves). If you'd like to see the formula for arbitrary congruences of causal curves they are in Hawking and Ellis.

As to why we want congruences that are normal to the surface in the case of focal points; From the math point of view the components of the Jacobi tensor that are not orthogonal to the geodesic aren't interesting (this in Lemma 10.9 in BEE). By this I mean the non-orthongal components don't provide information about variation of arc-length. So we should only consider Jacobi tensors with compoents normal to the geodesic. The question then becomes do we want the Jacobi tensor to initially have zero volume or non-zero volume. This corresponds to the point and surface cases.

From the physical point of view, in the surface case we wish to describe the behaviour of a volume element of the surface. Thus the initial conditions must ensure non-zero volume. Since the normal to the surface is... well normal to the surface the correct initial conditions correspond to a congruence of geodesics each geodesic of which is normal to the surface.

TL;DR: Read Beem, Erhlich and Easley chapter 10 and 12 and season with some Hawking and Ellis. Make sure to read up about variation of arc length and Morse index theory.