Differentiation of the connections

[JimWhoKnew]

I'm reading this paper where the authors implicitly differentiate connections along a worldline$$\frac{D~\Gamma^\mu{}_{\nu\rho}}{d\tau} \quad. \tag{1}$$So far, it seems that I can recover their results if I calculate "$\Gamma^\mu{}_{\nu\rho;\sigma}$" as if the connections were components of an ordinary (1,2) tensor. I'm not confident it's OK to do so, and couldn't find (so far) any example in literature.

What is the correct way to evaluate (1) when the metric and connections on the worldline are known as functions of the proper time $\tau$, and why?

 

[PeterDonis]

What is the correct way to evaluate (1) when the metric and connections on the worldline are known as functions of the proper time $\tau$, and why?

Um, just differentiate the known functions of $\tau$ with respect to $\tau$?

 

[JimWhoKnew]

Um, just differentiate the known functions of $\tau$ with respect to $\tau$?

First, by doing so I fail to recover the results of the mentioned paper.
Second, there are basis vectors and covectors involved. Why are we allowed to ignore their changes?
As I understand your reply, you suggest$$\frac{D~\Gamma^\mu{}_{\nu\rho}}{d\tau} = \frac{\partial~\Gamma^\mu{}_{\nu\rho}}{\partial\tau}\quad.$$right?

Edit:
Section 13.6 in MTW discusses "detector's proper frame". The linked paper extends the results to second order in $x^\mu$ (proper deviations from the relevant point on the worldline).

 

[PeterDonis]

there are basis vectors and covectors involved

Why? You said we know everything as known functions of $\tau$ only. Differentiating a known function of $\tau$ with respect to $\tau$ should be straightforward.

Can you give a specific example of such known functions?

 

[PeterDonis]

As I understand your reply, you suggest$$\frac{D~\Gamma^\mu{}_{\nu\rho}}{d\tau} = \frac{\partial~\Gamma^\mu{}_{\nu\rho}}{\partial\tau}\quad.$$right?

Wrong. If you can only take a partial derivative with respect to $\tau$, then you do not know everything as a function of $\tau$ only.

Note that, in actual GR, we never take partial derivatives with respect to $\tau$; $\tau$ is an affine parameter along a worldline, not a coordinate. If you know something as a function of $\tau$, you know it as a function of $\tau$ alone.

Further comments in a follow-up post.

 

[PeterDonis]

Section 13.6 in MTW discusses "detector's proper frame".

Yes, that's a good reference.

Note that in that discussion, the known functions of $\tau$ are the coordinates $x^\mu$. Differentiating those known functions with respect to $\tau$ gives the components of the 4-velocity in the chart $x^\mu$. The differentiation there is simple ordinary differentiation of a function of one variable.

Note, however, that, while the differentiation just described gives us the components of the 4-velocity as known functions of $\tau$ only, the proper acceleration is not obtained by the same kind of differentiation process. MTW express the proper acceleration as $\nabla_\mathbf{u} \mathbf{u}$, which in component notation becomes $u^a \nabla_a u^b$. I think there are other sources in the literature that write this, sloppily, as $d \mathbf{u} / d\tau$, but it's not just simple ordinary differentiation.

Note also that no such expression as $D \Gamma / d\tau$ appears anywhere in that section.

 

[PeterDonis]

by doing so I fail to recover the results of the mentioned paper.

The paper is behind a paywall so I can't read it. Is there an arxiv link to a preprint?

Or, if you can post a relevant equation here, that would help.