(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Help! I wish to prove the following important statements:

(1) The presence of Christoffel symbols in the covariant derivative of a tensor assures that this covariant derivative can transform like a tensor.

(2) The reason for this is because, under transformation, the Christoffel symbol picks up a term that cancels out a problematic term which would prevent the covariant derivative from transforming as a tensor. That term arises from the transformation of the other part of the covariant derivative, which is the partial derivative of the tensor. (This is why the partial derivative of a tensor does not in general transform as a tensor.)

These statements come from 'Relativity Demystified' (McMahon, p. 68; you can look it up in Amazon book search). The book provides some supporting equations for the assertions, but not a full proof. I would like to derive a proof. Here is what I've gotten so far.

3. The attempt at a solution

Let's start with the covariant derivative of of a vector http://members.aol.com/mlucen/2.bmp...AIN][URL]http://members.aol.com/mlucen/3.bmp, where the Г is the Christoffel symbol.

Our task is to show that this transforms as a tensor; and that the reason it does, is because the Г term transformation cancels out otherwise problematic quantities arising from transformation of the other term. This is what the book says.

A way to do this is presumably as follows:

(a) Write out an equation showing how the covariant derivative would transform if it transformed under ordinary tensor rules.

(b) Transform each term of the covariant derivative separately. Show that this leads to the same result as in (a). And show that it does so because the transformation of the Г term cancels out "problems" arising from transformation of the other other term.

I am now going to put primes after all the indices used above, because this will more closely follow the relevant equations in the book, and I already did it this way so it's easier for me. Transformation will then be to an uprimed coordinate system. First (a), we write an equation showing how the covariant derivative (the one above but primed) would transform if it did so under ordinary tensor rules.

[URL]http://members.aol.com/mlucen/4.bmp[/URL] [COLOR="Green"](Eq. 1)[/COLOR]

Above the first two terms show the basic transformation equation. Subsequent terms show its expansion in terms of the two known parts of the unprimed covariant derivative.

Our next task, as stated in (b), is to recover the same equation by transforming each of the two parts of the original (primed) covariant derivative separately. i.e., transform each of the two halves of

[PLAIN][URL]http://members.aol.com/mlucen/5.bmp.[/URL]

This transformation leads to the following results. For the first half:

[PLAIN][URL]http://members.aol.com/mlucen/6.bmp[/URL] [COLOR="Green"](Eq. 2)[/COLOR]

[Note that the 2nd order derivative term in Eq. 2 that makes it impossible for the partial derivative of a tensor -- alone -- to transform as a tensor. It would clearly do so without that term.]

For the second half:

[PLAIN][URL]http://members.aol.com/mlucen/7.bmp.[/URL] [COLOR="Green"](Eq. 3)[/COLOR]

This skips over quite a bit of calculation, but it's all shown in McMahon and the final answer is lifted straight from there. Note that I have left in some terms that apparently would be easily canceled out. This again follows McMahon's presentation; I think some of these extra terms may be useful for conversions that would complete the proof.

In fact, our proof would in fact be complete if we could show that

[B][COLOR="Purple"](everything after the last = in Eq. 2) [B]+[/B]

(everything after the last = in Eq. 3) [B]=[/B]

(everything after the last = in Eq. 1).[/COLOR][/B]

It looks promising: the terms look vaguely similar, and presumably the second-order derivative terms cancel out. Unfortunately, even the most elaborate index gymnastics that I can think of fail to deliver me the conclusion for which I yearn. The terms just don't seem to add up.

Any help would be greatly appreciated!

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# Homework Help: Covariant derivative

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