Equivalent condition for conjugate points (Wald's "General Relativity" Chapter 9.3)

Question

I am currently dealing with chapter 9.3 in Wald's "General Relativity". In particular, I'm interested in the equivalence $$detA_{\nu}^{\mu} = 0 \text{ at }q \iff q\text{ is conjugate to }p$$ He begins as follows:
Let $\gamma$ be a timelike geodesic with tangent $\xi^{a}$ and let $p \in \gamma$. Consider the congruence of all timelike geodesics passing through $p$. We shall show that a point $q \in \gamma$ lying to the future of $p$ is conjugate to $p$ if and only if the expansion, $\theta$, of this congruence approaches $-\infty$ at $q$. For this purpose, it is convenient to introduce an orthonormal basis of spatial vectors $e_{1}^{a}$, $e_{2}^{a}$, $e_{3}^{a}$ orthogonal to $\xi^{a}$ and parallely propagated along $\gamma$.

I do not see the need to choose such a "special" basis. As I see it, the same result can also be achieved with any basis for the tangential space $V_{p}$ together with the associated dual basis of $V_{p}^{*}$. Am I missing something at any point here? I would be very grateful for help with this.

Answer 1

Incidentally, I just saw this question. I would recommend Hawking and Ellis, since most of the concepts discussed are very mathematically clear and rigorous, albeit in a thick bold print. Not sure about what Wald wrote (a picture of the page will help), but a point $q$ on a curve $\gamma (\lambda )$ is conjugate to a point $p$ if there exists a Jacobi field that vanishes at $p$ and $q$. The matrix $A_{\mu \nu }$ you have mentioned is a matrix that vanishes at the conjugate point. For a better idea of this, just note that the determinant of $A$ defines the expansion. A conjugate point (I usually used the word caustic, but ok.) can be better visualised by taking a spacelike surface $\sigma $. Now, you have four null $\sigma $-orthogonal congruences -- a future and past outward pointing congruence and a future and past inward pointing congruence. Call them $\mathcal{K}^{\pm }$ and $\mathcal{L}^{\pm }$ respectively. $\mathcal{L}^{+}$ would point inward if initially the expansion $\theta _{\mathcal{L}^{+}}|_{\partial \sigma }<0$ (respectively for the past congruence). Now, if we assume the null energy condition, which says that for null vectors $X^{\mu }$, $T_{\mu \nu }X^{\mu }X^{\nu }\geq 0$, then the Raychaudari equation becomes $\frac{d\theta }{d\lambda }=-\frac{1}{2}\theta ^{2}-8\pi GT_{\mu \nu }X^{\mu }X^{\nu }$ (dropping vorticity and shear terms). Now, you can see that the expansion $\theta $ will be negative till a certain point $P$ on the boundary of the future (see https://arxiv.org/abs/1711.06689 on this), and this compact null hypersurface from $\sigma $ to $P$ is called a light-sheet, which finds applications in the covariant entropy bound (since the domain of dependence of this light-sheet is the same as that of the compact surface, due to which only so much information can pass through each) and generalized second law things. Now, if you follow the congruence after $P$, the expansion goes wildly divergent, $\theta \to +\infty $. $P$ is a conjugate point. For a nicer detail of this, see proposition 4.4.1 in Hawking and Ellis (basically section 4.4 is a very invaluable discussion on geodesics and conjugate points). I believe the set of new basis introduced is to essentially capture this.