From Mathematics wiki
Christoffel symbols are also known as connection coefficients and sometimes the Levi-Civita connection (see below). Historically the symbols come in two forms, known as the second kind (more commonly used) and first kind (less commonly used). Christoffel symbols of the second kind are used to define the affine connection and the covariant derivative.
(Note that the differentiation index comes last on .)
The Christoffel symbols of the first kind are defined by
It is important to note that these symbols are not tensors as they do not transform as tensors do.
Compatibility with the metric
Although the Christoffel symbols that define the covariant derivative may be chosen with some degree of freedom, we can choose them in a way that is compatible with the metric . Consider parallel transporting two vectors and along a curve with tangent vector . We first require that and . Since the dot product of and , should remain unaltered by the parallel transport, we then require . Then:
Since was arbitrarily chosen, it must be true that . Using this equation, along with the no-torsion condition, can be determined in terms of :
First, we expand the components of :
We can then permute the indices , , and , and then adding/subtracting gives:
so that the metric compatible connection is
We can again permute the indices , , and , and then adding/subtracting gives:
so that zero torsion implies that
where are the commutation coefficients. Then
or , or more symmetrically,
Transformation under diffeomorphism
Under a change of coordinates from to , we find that
showing that does not transform as a tensor.
transforms as a tensor. This can also be seen from Lifshitz' formula.
which gives Lifshitz' formula:
This also implies that
back to covariant derivative on to divergence
- ↑ E. Lifshitz (1946). "On the Gravitational Stability of the Expanding Universe". J. Phys. (USSR) 10: 116.