The all covariant Christoffel[mu, alpha, beta], displayed as , is a computational representation for the Christoffel symbols, defined in terms of derivatives of the spacetime metric as

where the spacetime differentiation operator  is implemented in Maple as d_[mu]. From this definition, Christoffel is symmetric with respect to interchanging the second and third indices.

Christoffel can also have contravariant indices, typically the first one. When any of the indices of Christoffel is contravariant, the definition for its components is according to  and  and not by raising the indices in the terms on the right-hand-side of the definition above for . Note also that, despite computing with them as tensors, the Christoffel symbols are not actual tensors in that, when you transform the coordinates, they do not transform like the coordinate differentials (regarding their contravariant indices) nor like the derivatives of a scalar (regarding their covariant indices).

To express  or  using this definition in terms of derivatives of the spacetime metric use convert to g_. Sometimes it is also convenient to rewrite tensorial expressions the other way around, in terms of the Christoffel symbols and its derivatives. For example: non-covariant derivatives of the metric or any of the general relativity tensors that can be expressed in terms of them. For this purpose you can use convert(expression, Christoffel) - see the Examples section.

When the indices of Christoffel assume integer values they are expected to be between 0 and the spacetime dimension, prefixed by ~ when they are contravariant, and the corresponding value of Christoffel is returned. When the indices have symbolic values Christoffel returns unevaluated after normalizing its indices taking into account the symmetry property of the 2nd and 3rd indices.

Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices - see `.` and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.

During a Maple session, the value of any component of  is automatically determined by the value of the spacetime metric at the moment. When Physics is loaded, the spacetime is set to Minkowski type, and so all the elements of Christoffel are automatically zero. To set the spacetime metric to something different use Setup. Also, at least one system of coordinates must be set in order to compute the derivatives entering the definition of Christoffel. For that purpose see Coordinates or Setup.

Besides being indexed with three indices, Christoffel accepts two keywords:

array: (synonym: Array, or no indices whatsoever, as in Christoffel[]) returns an Array that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of Christoffel. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting array takes into account the character of the indices. If one of these indices has a numerical value, if contravariant then preceded by ~, the returned object is the corresponding 2 x 2 Matrix.

nonzero: returns a set of equations, with the left-hand-side as a sequence of three positive numbers identifying the element of  and the corresponding value in the right-hand-side. Note that this set is actually the output of the ArrayElems command when passing to it the Array obtained with the keyword array.

Some automatic checking and normalization are carried out each time you enter Christoffel[...]. The checking is concerned with possible syntax errors. The automatic normalization takes into account the symmetry of Christoffel[mu, alpha, beta] with respect to interchanging the positions of the indices alpha and beta.

The %Christoffel command is the inert form of Christoffel, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

The Physics[Christoffel] command was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.

Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.