# Christoffel symbol

### 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.

Given coordinate basis vectors , define the **Christoffel symbols of the second kind** through the affine connection:

(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.

## Contents |

## 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 :

### Coordinate basis

Let us assume that the Lie bracket vanishes, i.e., that are a coordinate basis.

First, we expand the components of :

We can then permute the indices , , and , and then adding/subtracting gives:

. |

Now, since the metric is symmetric, , and since we have imposed the torsion free condition that , we have:

- ,

or

- ,

so that the **metric compatible connection** is

.

### Non-coordinate basis

In a non-coordinate basis, such as an orthonormal basis, , and the relevant covariant derivative is

- .

We can again permute the indices , , and , and then adding/subtracting gives:

. |

The metric is of course symmetric, but now . Instead, the torsion is

- ,

so that zero torsion implies that

- ,

where are the commutation coefficients. Then

, |

or , or more symmetrically,

,

where . In particular, in an orthogonal basis, only the commutation coefficients contribute.

## Transformation under diffeomorphism

Under a change of coordinates from to , we find that

showing that *does not* transform as a tensor.

As an aside, note that the difference between two connections does transform as a tensor. In other words, suppose we consider a one-parameter family of connections parametrized by . Then

transforms as a tensor. This can also be seen from Lifshitz' formula.

## Lifshitz' formula

The variation of due to a variation in the metric can be found as follows^{[1]}:

, | |

, | |

, | |

which gives **Lifshitz' formula**:

.

This also implies that

- .

*back to***covariant derivative**

*on to***divergence**

## References

- ↑ E. Lifshitz (1946). "On the Gravitational Stability of the Expanding Universe".
*J. Phys. (USSR)***10**: 116.