The Weyl tensor - Maple Help

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Physics[Weyl] - The Weyl tensor

	Calling Sequence
	Weyl[alpha, beta, mu, nu]

Parameters

alpha, beta, mu, nu

the indices, as names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves

Description

•	The Weyl[alpha, beta, mu, nu], displayed as , is a computational representation for the Weyl tensor, defined in terms of the Riemann, Ricci and the spacetime metric g_ tensors as

From this definition, the Weyl tensor has all the symmetries properties of the Riemann tensor; i.e. it is antisymmetric with respect to interchanging the position of its 1st and 2nd indices, or 3rd and 4th indices, and symmetric with respect to interchanging the positions of the 1st and 2nd pair of indices. In addition, it vanishes when contracted on any pair of indices.

•

When the indices of Weyl 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 Weyl is returned. When the indices have symbolic values Weyl returns unevaluated after normalizing its indices taking into account their symmetry properties.

•

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 Weyl 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 the Christoffel symbols, used to construct the tensors entering the definition of the'Weyl' tensor. For that purpose see Coordinates or Setup.

•	Besides being indexed with four indices, Weyl accepts two keywords:

–

array: (synonym: Array, Matrix, matrix, or no indices whatsoever, as in Weyl[]) 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 Weyl. If this keyword is passed preceded by the tensor indices, that can be covariant or contravariant, the values in the resulting array are computed taking into account the character of the given indices.

–

nonzero: returns a set of equations, with the left-hand-side as a sequence of two 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. If only two of these indices are names, the rest have numerical values (if contravariant then preceded by ~), the returned object is the corresponding 2 x 2 Matrix.

•	Some automatic checking and normalization are carried out each time you enter Weyl[...]. The checking is concerned with possible syntax errors. The automatic normalization takes into account the symmetry properties of the indices of Weyl[mu,nu,alpha] as described in the 1st paragraph.

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

Compatibility

•	The Physics[Weyl] command was introduced in Maple 16.

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

Examples

(1)

Set up a coordinate system to work with - the first one to be set is automatically taken as the differentiation variables for d_, the covariant derivative D_ and the dAlembertian

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x, y, z, t)}$

(2)

When Physics is initialized, the default spacetime metric is of Minkowski type. You can see the metric querying Setup, as in Setup(metric);, or directly entering the metric as g_[], with no indices

Physics:-g_[mu, nu] = Matrix(%id = 18446790986282288902)

(3)

Check the nonzero components of Christoffel, used to construct the Weyl tensor entering the definition of Weyl: because the default spacetime is of Minkowski type, there are none

(4)

Hence

(5)

(6)

and the same is valid for all the general relativity tensors defined in terms of Christoffel and derivatives of the metric g_. To set the scenario as a curved spacetime set the metric using Setup, for instance indicating the square of the spacetime interval. In this example, we also choose to work in spherical coordinates, so consider for instance the metric defined by

(7)

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}$

$[coordinatesystems = {X}, metric = {(1, 4) = -1, (2, 2) = r^2, (3, 3) = r^2*sin(theta)^2, (4, 4) = -2*k(r, t)^2}]$

(8)

Physics:-g_[mu, nu] = Matrix(%id = 18446790986282247822)

(9)

Now when the indices are not numerical, Weyl returns itself after normalizing its indices taking advantage of their symmetry properties, so that different forms of the same tensor enter computations in the same manner, for example, if you interchange the positions as in

(10)

(11)

(12)

To express Weyl in terms of the Christoffel symbols and its derivatives convert to Christoffel

(13)

Check the value of , say for , = 2, = 2, = 4

-(2/3)*k(r, t)*r*(diff(k(r, t), r))-1/6+(1/3)*k(r, t)^2+(1/3)*(diff(k(r, t), r))^2*r^2+(1/3)*k(r, t)*(diff(diff(k(r, t), r), r))*r^2

(14)

Check now the value of , with the 1st index contravariant, for the same values of the indices (note you enter the value of the contravariant index prefixed by ~)

(15)

To compute with a representation for Weyl without actually performing the operation, use the inert form %Weyl. To afterwards perform the operation use value

(16)

(17)

To have a more compact display in the following examples, suppress the display of the dependency of and have the display of derivatives in jet notation, indexed (see Typesetting)

(18)

The nonzero values of and of (note you enter the character of the contravariant indices prefixed by ~)

(19)

(20)

As with all the general relativity tensors of the Physics package, you can obtain the same result for the all covariant case by entering the tensor without indices, as in Weyl[].

This is the array form of (note you enter the character of the contravariant indices prefixed by ~)

Physics:-Weyl[`~alpha`, beta, mu, nu] = Array(%id = 18446790986308965070)

(21)

To use this array, because its components were computed already taking into account the (covariant/contravariant) character of its indices, you do not need to indicate furthermore that character. So for we have

(22)

Compare with the output obtained entering Weyl[~1, 2, 2, 4] in eq (15)

(23)

Verify that the Weyl tensor vanishes when contracted on a pair of indices; take for instance the 1st and 3rd indices, you can see the corresponding matrix (note the first index as contravariant)

Physics:-Weyl[`~1`, beta, 1, nu] = Matrix(%id = 18446790986308971206)

(24)

Add the four matrices completing the contraction of indices

Physics:-Weyl[`~1`, beta, 1, nu]+Physics:-Weyl[`~2`, beta, 2, nu]+Physics:-Weyl[`~3`, beta, 3, nu]+Physics:-Weyl[`~4`, beta, 4, nu] = Matrix(%id = 18446790986339382446)

(25)

Physics:-Weyl[`~1`, beta, 1, nu]+Physics:-Weyl[`~2`, beta, 2, nu]+Physics:-Weyl[`~3`, beta, 3, nu]+Physics:-Weyl[`~4`, beta, 4, nu] = Matrix(%id = 18446790986339372318)

(26)

	See Also
	`.`, Array, ArrayElems, Christoffel, Coordinates, DifferentialGeometry[Tensor][WeylTensor], Einstein, g_, Physics, Physics conventions, Physics examples, Ricci, Riemann, Setup, Typesetting, value

References

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

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