General Relativity For Tellytubbys

The Covariant Derivative

Sir Kevin Aylward B.Sc., Warden of the Kings Ale

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The approach presented here is one of the most direct routes possible. I plagiarized it from a number of sources and added my bit of finesse to it i.e. no introduction to all that superfluous mumbo jumbo that disappears without actually doing anything, but confuses you like no bodies business. Most steps are shown in complete detail, cos I remember I was totally lost when I first learned this stuff.

So, no beating about the bush, onward with a quick derivation of the covariant derivative.

Section 1

Consider a vector or tensor of rank 1, with components

 

or in full notation

 

The covariant derivative is defined by deriving the second order tensor obtained by

 

No mystery at all here, we just have to account for the fact that the basis vectors are not constant by using the usual differentiation of the product rule. Note the ";" to indicate the covariant derivative.

The last term containing the derivative of the basis vector can clearly be expressed as a sum of the basis vectors themselves. This will be written as:

 

Where the big funny R shape is to be determined, and is called the Christoffel symbol of the 2^nd kind, and not to be confused with Close Encounters of the Third Kind, which was crap, and almost as bad as ET it was.

Also note the new introduction of "," to mean ordinary partial derivative.

So now the covariant derivative can be written as:

 

 

 

 

which means that the covariant derivative of the vector, specified only by its components, can now be expressed as

 

Which is indeed a tensor, but we certainly don’t care one iota about proving that it is a tensor, some other fool can do that.

Now lets consider a second order tensor

 

Then calculating its covariant derivative by differentiating by parts, and using the above results, gives

 

or, after expanding and collecting up terms as we did above

 

And if your so inclined, you can go and derive the covariant derivative for a downstairs index as

 

Section 2

Next task my Tellytubbys, is to derive the Christoffel symbols so that we can actually do something.

To start off, the Symmetry of  is first shown

, back to the other pages for  refresher if you've forgotten this

 

 

From our previous result above,

 

then

 

 

is also symmetrical wrt alpha and beta

Laa Laa now writes from the above

 

or

 

 

 

 

 

and due to the symmetry found above, we can also write

 

So adding these last two gives

 

So, by re-differentiating, the following line can immediately be seen to be correct, don’t you just love these ones, ah.

 

 

 

 

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