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.

 

[JimWhoKnew]

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.

Your remark in #2 may be correct after all. The apparent discrepancy is probably due to a minus sign obscured by notation juggling.

The paper is from 1978. I don't think there's a preprint on an arxiv.
I'll try to summarize the relevant equations here. The authors explicitly say that they follow section 13.6 in MTW. I'll drop the hats (^) off the indices, since all equations presented here are in the observer's proper reference frame.

They start from eqs. 13.60-13.62 in MTW. Defining$$\mathbf{b}:=\nabla_\mathbf{u}\mathbf{a} \quad,\quad \mathbf{\eta}:=\nabla_\mathbf{u}\mathbf{\omega} \quad, \tag{1}$$ eqs 13.60-13.62 lead to$$b^0=\vec{a}\cdot\vec{a} \quad,\quad \vec{b}=\frac{d\vec{a}}{d\tau}+\vec{\omega}\times\vec{a} \tag{2}$$$$\eta^0=\vec{\omega}\cdot\vec{a} \quad,\quad \vec{\eta}=\frac{d\vec{\omega}}{d\tau} \quad.\tag{3}$$Now they say (adaptations of numbering are mine): "Differentiating eqs. 13.69a and 13.69b (MTW) along the trajectory with respect to $\tau$ and using eqs. (1)-(3) we have"$$\Gamma^0{}_{00,0}=\Gamma^\alpha{}_{jk,0}=0 \tag{4}$$$$\Gamma^0{}_{j0,0}=b^j(\tau)+\epsilon^{jkl}a^k(\tau)\omega^l(\tau) \tag{5}$$$$\Gamma^i{}_{j0,0}=-\eta^k(\tau)\epsilon^{ijk} \quad. \tag{6}$$

If I got the signs right this time, the RHS of (5) is just $~d a^j/d\tau~$.

Thanks, @PeterDonis.

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

I'll be surprised if it appears anywhere in the book.

Edit:
Corrected a typo in equation (2). Thanks to @PeterDonis for spotting it.

 

[PeterDonis]

Now they say (adaptations of numbering are mine): "Differentiating eqs. 13.69a and 13.69b (MTW) along the trajectory with respect to $\tau$ and using eqs. (1)-(3) we have"$$\Gamma^0{}_{00,0}=\Gamma^\alpha{}_{jk,0}=0 \tag{4}$$$$\Gamma^0{}_{j0,0}=b^j(\tau)+\epsilon^{jkl}a^k(\tau)\omega^l(\tau) \tag{5}$$$$\Gamma^i{}_{j0,0}=-\eta^k(\tau)\epsilon^{ijk} \quad. \tag{6}$$

This doesn't look like they're differentiating with respect to $\tau$, but with respect to coordinate time $x^0$, at least on the LHS. Do they make some argument for why the two are (at least to some approximation) the same on the LHS?

If I got the signs right this time, the RHS of (5) is just $~d a^j/d\tau~$.

Wouldn't that just be $b^j (\tau)$, by definition (your equation 2)?

 

[JimWhoKnew]

This doesn't look like they're differentiating with respect to $\tau$, but with respect to coordinate time $x^0$, at least on the LHS. Do they make some argument for why the two are (at least to some approximation) the same on the LHS?


Wouldn't that just be $b^j (\tau)$, by definition (your equation 2)?

The way in which the observer's coordinates are constructed (Lorentzian metric on worldline, $~u^\mu=\delta^\mu_0~$), $~\frac{\partial}{\partial x^0}~$ does the same thing as $~\frac{d}{d\tau}~$ in this case (doesn't it?).

I got a typo in equation (2) of #8. I'll correct it now. Thanks.

 

[PeterDonis]

The way in which the observer's coordinates are constructed (Lorentzian metric on worldline, $~u^\mu=\delta^\mu_0~$), $~\frac{\partial}{\partial x^0}~$ does the same thing as $~\frac{d}{d\tau}~$ in this case (doesn't it?).

I think so. I'm wondering whether the paper talks about this explicitly.

 

[JimWhoKnew]

I think so. I'm wondering whether the paper talks about this explicitly.

They don't. The sentence quoted in #8 is their entire explanation.

Their results for the metric are summarized (and partially re-derived) on Appendix B here. With the additional terms of the metric at hand, Ni and Zimmermann go on to interpret them from the geodesic equation (as done in exercise 13.14 of MTW).

 

[PeterDonis]

I got a typo in equation (2) of #8. I'll correct it now. Thanks.

Ah, I see it now.

 

[PeterDonis]

Their results for the metric are summarized (and partially re-derived) on Appendix B here.

That at least explains their $D / d \tau$ notation--it's the same as what MTW call $\nabla_\mathbf{u}$. So it's not quite the same as an ordinary derivative of a function of one variable, at least not in the general case; but with properly chosen coordinates, it can be done that way.

I'm thinking MTW has a discussion of this point somewhere, but I can't find it right now.

Note, though, that their notation is not entirely consistent; in their (B4), they define the proper acceleration using a small $d$ in $d / d \tau$, when for consistency with their (B1), it should be a capital $D$.

 

[PeterDonis]

their notation is not entirely consistent

Also, their use of the term "Fermi-Walker transport" is not consistent with other literature. They say their equation (B1) defines Fermi-Walker transport, but in other literature (such as MTW), that's only the case if $\omega = 0$, i.e., if there is no rotation of the spatial axes. Their math is clear enough (modulo the notation issue I described in my previous post), but I think one should be cautious about their terminology.