Insights Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes

[ergospherical]

Introduction
The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief summary. Consider a 1-parameter family of timelike geodesics $\gamma_s(\lambda)$, where $s$ labels each geodesic in the family whilst $\lambda$ is an affine parameter along each $\gamma_s$. Then the vector field $\xi \equiv \partial / \partial s$ is tangent to curves of constant $\lambda$ and is interpreted as a deviation vector between neighbouring geodesics.
In some neighbourhood of the family, $(s,\lambda, x^2, x^3)$ is a coordinate chart satisfying $\xi = \partial/\partial s$ and $u = \partial/\partial \lambda$. By the equality of mixed partial derivatives, the commutator of $u$ and $\xi$ is zero (i.e. $\xi$ is Lie transported along $u$),\begin{align*}
0 = [u, \xi]^a = (L_{u} \xi)^a = \xi^b \nabla_b u^a – u^b \nabla_b \xi^a
\end{align*}which implies that ##\dfrac{D\xi^a}{d\lambda} = u^b...

Continue reading...

 

[cianfa72]

Hi,

congratulation for the job done. I would like to point out some topic already discussed in PF so far.

I noted you use both Latin and Greek indices even if not together in the same formula ! So for example in the first part of the Introduction you use Latin indices (i.e. Abstract Index Notation) while in FRW section Greek ones (Ricci calculus notation).

My understanding, as discussed so far and in line with the first insight's reference (H. Reall, Part 3 General Relativity section 1.6), is that the second one (Greek indices) actually involves objects's components in a given basis (namely a given basis for the vector space and its associated dual-vector basis).

So, from a general point of view, a Greek indices equation valid in a particular/specific basis is not true in other bases.

That said, I believe the reason since you employed Greek indices in the rest of the insight is that you were assuming specific coordinate charts for each of the spacetimes discussed.

 

[ergospherical]

Yeah as far as possible I tried to use Latin indices (or no indices) for coordinate independent expressions and Greek indices when evaluating the components in a particular basis. (That's the convention of Wald and Reall.)

 

[cianfa72]

To be onest, I'm often in trouble with expression like that in FRW section, namely $\delta_t^{\mu}$. Here $\mu$ as Greek index has the role of "which component" whereas $t$ is the name of a fixed given component (coordinate name).

Do you think there is a way to get rid of that (and similar) ambiguity ? Thank you.

 

[ergospherical]

Can you clarify what is confusing you? In the coordinates $(t,r,\theta, \phi)$, the vector $u = \partial/\partial t$ has a $t$ component of $1$ and the rest of the components 0, i.e. $u^{\mu} = \delta^{\mu}_t$. Recall that the symbol $\delta^{\mu}_{\nu}$ is $1$ if $\mu = \nu$ and 0 if $\mu \neq \nu$.

Alternatively, you can write $u^{\mu} = dx^{\mu}(u) = dx^{\mu} \left( \dfrac{\partial}{\partial t} \right) = \dfrac{\partial x^{\mu}}{\partial t} = \delta^{\mu}_t$.

 

[cianfa72]

Can you clarify what is confusing you? In the coordinates $(t,r,\theta, \phi)$, the vector $u = \partial_t$ has a $t$ component of $1$ and the rest of the components 0, i.e. $u^{\mu} = \delta^{\mu}_t$.

Recall that the symbol $\delta^{\mu}_{\nu}$ is $1$ if $\mu = \nu$ and 0 if $\mu \neq \nu$.

No no it makes sense. IMO the point to be highlighted is that the letter $t$, actually, is not Greek so in this case there is no problem.

My point is broader in the sense that many times mixing Greek and Latin index names turns out to be confusing.

 

[ergospherical]

Ohhh, haha, okay. I wouldn't lose sleep over that.

 

[PAllen]

Another common convention is for latin letters to be 3-tensors, and greek letters 4-tensors. It always helps to have a section making explicit any conventions that are not universal. An unobtrusive way to do it is to have an appendix on this, with one sentence in the intro referring to the appendix for conventions.

 

[vanhees71]

Yes, and also all the sign conventions about the curvature and Ricci tensor are also nice to have ;-).