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Matterwave

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Two similar threads merged as requested

I have some questions with regards to conjugate points on a congruence of time-like geodesics (will be referring to Wald 9.3 throughout). First, we define ##\gamma## to be a time-like geodesic with tangent ##\xi^a## parametrized by ##\tau## and with ##p\in\gamma##. We consider the "congruence of all timelike geodesics passing through ##p##" and we want to show that ##q\in\gamma## is a conjugate point of ##p## if and only if the expansion, ##\theta##, defined as ##\theta=\nabla_a \xi^a##, approaches ##-\infty## at ##q##.

Anyways, the Jacobi fields are defined as solutions ##\eta^a## to the equation ##\xi^a\nabla_a(\xi^b\nabla_b\eta^c)=-R_{abd}^{~~~~~~~c}\eta^b\xi^a\xi^d##. And at least one particular Jacobi field will vanish, by definition, at conjugate points. We define a set of orthogonal spatial vectors ##e^a_i## which are orthogonal to ##\xi^a## (with ##\xi^a## itself serving as our time-direction basis vector, as it is parametrized by the proper time, it is normalized to -1). Furthermore, for simplicity in notation we define ##B_{ab}=\nabla_b\xi_a##. This is just background information.

Now on to my question. Specifically, in equation 9.3.5 Wald makes the following derivations:

$$\frac{d\eta^\mu}{d\tau}=\xi^a\nabla_a\eta^\mu=\xi^a\nabla_a[(e_\mu)_b\eta^b]$$ $$\frac{d\eta^\mu}{d\tau}=(e_\mu)_b\xi^a\nabla_a\eta^b$$ $$\frac{d\eta^\mu}{d\tau}=(e_\mu)_b B^b_{~a}\eta^a$$ $$\frac{d\eta^\mu}{d\tau}=B^\mu_{~\alpha}\eta^\alpha$$

My question is in going from the second line to the third line (counting the top two equalities as just 1 line).

I believe the full version of that line requires the following deduction: ##\xi^a\nabla_a\eta^b=\eta^a\nabla_a \xi^b=\eta^a B^b_{~a}##. Of course, for this to be true, it must be that ##\mathcal{L}_\xi \eta^a=0##.

So far in Wald, ##\mathcal{L}_\xi\eta^a=0## has been said to be true, but in slightly different contexts than here, which is where I am getting confused (Wald presented equation 9.3.5 without justifications for the steps, leaving me to work out the equalities myself). If I can recall correctly, the relation ##\mathcal{L}_\xi\eta^a=0## was said to be true (originally in chapter 3) because ##\xi^a## and ##\eta^a## were coordinate basis vectors ##\xi^a=(\partial t)^a## and ##\eta^a=(\partial s)^a##(in chapter 3) on the 2-dimensional submanifold ##\Sigma## which is foliated by the congruence ##\gamma_s(t)## (wherein each ##s## denotes a particular geodesic).

Therein lies my problem. For if ##p## is a point through which many different geodesics pass, then the congruence is singular at ##p##. If the congruence is singular at ##p##, then I can't create this ##\Sigma## submanifold with the congruence ##\gamma_s(t)## since the map ##(t,s)\rightarrow \gamma_s(t)## will no longer be bijective at p. In addition, my coordinate ##s## is no good at ##p## since the coordinate basis vector ##\eta^a=(\partial s)^a## in fact vanishes at p by construction. With the inclusion of this singularity in my congruence, and thus my inability to construct the submanifold ##\Sigma##, how can I be sure that ##\mathcal{L}_\xi \eta^a=0## holds everywhere on my congruence? How do I justify going from line 2 to line 3 in equation 9.3.5?

Am I over thinking this? Is it possible Wald did not have to use ##\mathcal{L}_\xi\eta^a=0## in that step? Or is this property still rigorously valid even with the presence of the singularity in the congruence at ##p##?

EDIT: I should add that later, Wald basically asserts that ##\mathcal{L}_\xi \eta^a=0## is "true everywhere" without proof. On page 227 he basically says "...and, as usual, we have ##\mathcal{L}_\xi \eta^a=\eta^b\nabla_b \xi^a-\xi^b\nabla_b\eta^a=0## everywhere" - although here he has switched to using ##T^a## and ##X^a## instead of ##\xi^a## and ##\eta^a## I think because he removed the geodesic requirement on ##\gamma## (which he has switched to ##\lambda## here). I don't know how he can just say this is true now that there's a singularity at p though.

Anyways, the Jacobi fields are defined as solutions ##\eta^a## to the equation ##\xi^a\nabla_a(\xi^b\nabla_b\eta^c)=-R_{abd}^{~~~~~~~c}\eta^b\xi^a\xi^d##. And at least one particular Jacobi field will vanish, by definition, at conjugate points. We define a set of orthogonal spatial vectors ##e^a_i## which are orthogonal to ##\xi^a## (with ##\xi^a## itself serving as our time-direction basis vector, as it is parametrized by the proper time, it is normalized to -1). Furthermore, for simplicity in notation we define ##B_{ab}=\nabla_b\xi_a##. This is just background information.

Now on to my question. Specifically, in equation 9.3.5 Wald makes the following derivations:

$$\frac{d\eta^\mu}{d\tau}=\xi^a\nabla_a\eta^\mu=\xi^a\nabla_a[(e_\mu)_b\eta^b]$$ $$\frac{d\eta^\mu}{d\tau}=(e_\mu)_b\xi^a\nabla_a\eta^b$$ $$\frac{d\eta^\mu}{d\tau}=(e_\mu)_b B^b_{~a}\eta^a$$ $$\frac{d\eta^\mu}{d\tau}=B^\mu_{~\alpha}\eta^\alpha$$

My question is in going from the second line to the third line (counting the top two equalities as just 1 line).

I believe the full version of that line requires the following deduction: ##\xi^a\nabla_a\eta^b=\eta^a\nabla_a \xi^b=\eta^a B^b_{~a}##. Of course, for this to be true, it must be that ##\mathcal{L}_\xi \eta^a=0##.

So far in Wald, ##\mathcal{L}_\xi\eta^a=0## has been said to be true, but in slightly different contexts than here, which is where I am getting confused (Wald presented equation 9.3.5 without justifications for the steps, leaving me to work out the equalities myself). If I can recall correctly, the relation ##\mathcal{L}_\xi\eta^a=0## was said to be true (originally in chapter 3) because ##\xi^a## and ##\eta^a## were coordinate basis vectors ##\xi^a=(\partial t)^a## and ##\eta^a=(\partial s)^a##(in chapter 3) on the 2-dimensional submanifold ##\Sigma## which is foliated by the congruence ##\gamma_s(t)## (wherein each ##s## denotes a particular geodesic).

Therein lies my problem. For if ##p## is a point through which many different geodesics pass, then the congruence is singular at ##p##. If the congruence is singular at ##p##, then I can't create this ##\Sigma## submanifold with the congruence ##\gamma_s(t)## since the map ##(t,s)\rightarrow \gamma_s(t)## will no longer be bijective at p. In addition, my coordinate ##s## is no good at ##p## since the coordinate basis vector ##\eta^a=(\partial s)^a## in fact vanishes at p by construction. With the inclusion of this singularity in my congruence, and thus my inability to construct the submanifold ##\Sigma##, how can I be sure that ##\mathcal{L}_\xi \eta^a=0## holds everywhere on my congruence? How do I justify going from line 2 to line 3 in equation 9.3.5?

Am I over thinking this? Is it possible Wald did not have to use ##\mathcal{L}_\xi\eta^a=0## in that step? Or is this property still rigorously valid even with the presence of the singularity in the congruence at ##p##?

EDIT: I should add that later, Wald basically asserts that ##\mathcal{L}_\xi \eta^a=0## is "true everywhere" without proof. On page 227 he basically says "...and, as usual, we have ##\mathcal{L}_\xi \eta^a=\eta^b\nabla_b \xi^a-\xi^b\nabla_b\eta^a=0## everywhere" - although here he has switched to using ##T^a## and ##X^a## instead of ##\xi^a## and ##\eta^a## I think because he removed the geodesic requirement on ##\gamma## (which he has switched to ##\lambda## here). I don't know how he can just say this is true now that there's a singularity at p though.

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