construct an Array that can be indexed to return the values of a tensorial expression

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Physics[TensorArray] - construct an Array that can be indexed to return the values of a tensorial expression

	Calling Sequence
	TensorArray(expression, alpha, beta, ...)

Parameters

expression	-	any algebraic tensorial expression having spacetime free indices possibly having also repeated indices implying summation
performsumoverrepeatedindices	-	optional, can be true (default) or false, to perform the sum over repeated tensor indices in the returned result
simplifier = ...	-	optional - indicates the simplifier to be used instead; default is none

Description

•	The TensorArray receives a tensorial expression having n free indices, typically involving sums and products, and returns a corresponding n dimensional Array, which can be indexed as a single object to return the values of the tensorial expression for given values of its free indices.

•	To check and determine the free and repeated indices of an expression use Check.

•	The returned Array is constructed taking into account the covariant and contravariant character of each free index in expression. To compute the values of expression you index this array giving values between 1 and the spacetime dimension to the indices.

•

By default, in the returned result, summation is explicitly performed over all the repeated indices found in expression, taking into account the covariant/contravariant character of each index. To avoid performing this summation and keep repeated indices not summed pass the optional argument performsumoverrepeatedindices = false.

•	By default, the Array is constructed without simplifying its components; to have them simplified indicate the simplifier on the right-hand-side of the optional argument simplifier = .... A frequently convenient simplification is achieved with simplifier = `simplify/size`@simplify.

Compatibility

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

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

Examples

(1)

Set the spacetime metric to be the Schwarzschild metric and construct an Array for the product of the metric tensor g_ contracted and multiplied with an arbitrary tensor as in

For this purpose, set first the metric and the coordinates -you can use Setup for that, or because the Schwarzschild metric is known to the system you can directly pass the keyword or an abbreviation of it to the metric g_ to do all in one step

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

Physics:-g_[mu, nu] = Matrix(%id = 18446790986281085398)

(2)

Define now an arbitrary tensor

${A, Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(3)

Construct now the tensorial expression mentioned

(4)

Check the indices

(5)

Construct now the tensor-array

Matrix(%id = 18446790986300516342)

(6)

In the above, the sum over is performed. With performsumoverrepeatedindices = false the sum is not performed:

Matrix(%id = 18446790986300473342)

(7)

Rewrite the Riemann tensor with all its indices covariant in terms of Christoffel symbols and their derivatives and construct a tensor-array for the resulting tensorial expression; in view of the presence of trigonometric functions, use the simplifier option

(8)

(9)

Array(%id = 18446790986281114670)

(10)

Verify the result comparing , constructed with the definition of Riemann in terms of Christoffel symbols, with the Riemann tensor itself

-sin(theta)^2*m/(-r+2*m)

(11)

-sin(theta)^2*m/(-r+2*m)

(12)

Compare all the nonzero values of the two arrays: for Riemann, pass the option nonzero, for use ArrayElems; all the nonzero components are same:

(13)

(14)

(15)

A contraction of the Riemann tensor over two of its indices

(16)

(17)

Matrix(%id = 18446790986343456886)

(18)

	See Also
	Check, Coordinates, Define, g_, Physics, Physics conventions, Physics examples, Setup

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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