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.
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 2nd 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
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,
is also symmetrical wrt alpha and beta
Laa Laa now writes from the above
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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