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Tensors and Relativity: Chapter 6

Covariant derivatives and Christoffel symbols

Covariant derivatives and Christoffel symbols

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In Minkowski spacetime with Minkowski coordinates (ct,x,y,z) the derivative of a vector tex2html_wrap_inline3473 is just

equation2701

since the basis vectors do not vary. In a general spacetime with arbitrary coordinates, tex2html_wrap_inline3475 with vary from point to point so

equation2703

Since tex2html_wrap_inline3477 is itself a vector for a given tex2html_wrap_inline3434 it can be written as a linear combination of the bases vectors:

equation2705

The tex2html_wrap_inline3481 's are called Christoffel symbols [ or the metric connection  ]. Thus we have:

equation2707

so

equation2709

Thus we can write

equation2711

where

equation2713

Let us now prove that tex2html_wrap_inline3483 are the components of a 1/1 tensor. Remember in section 3.5 we found that tex2html_wrap_inline3487 was only a tensor under Poincaré transformations in Minkowski space with Minkowski coordinates. tex2html_wrap_inline3483 is the natural generalization for a general coordinate transformation.

Writing tex2html_wrap_inline3491 , we have:

eqnarray2717

Now

equation2738

therefore

equation2745

so we obtain:

eqnarray2760

Now using tex2html_wrap_inline3493 , tex2html_wrap_inline3495 and tex2html_wrap_inline3497 we obtain:

eqnarray2806

so

equation2824

We have shown that tex2html_wrap_inline3483 are indeed the components of a 1/1 tensor. We write this tensor as

equation2830

It is called the covariant derivative  of tex2html_wrap_inline3398 . Using a Cartesian basis, the components are just tex2html_wrap_inline3487 , but this is not true in general; however for a scalar tex2html_wrap_inline3392 we have:

equation2836

since scalars do not depend on basis vectors.

Writing tex2html_wrap_inline3509 , we can find the transformation law for the components of the Christoffel symbols .

eqnarray2840

This is just

equation2869

We can calculate the covariant derivative of a one- form  tex2html_wrap_inline3511 by using the fact that tex2html_wrap_inline3513 is a scalar for any vector tex2html_wrap_inline3398 :

equation2882

We have

eqnarray2884

Since tex2html_wrap_inline3517 and tex2html_wrap_inline3483 are tensors, the term in the parenthesis is a tensor with components:

equation2886

We can extend this argument to show that

eqnarray2888