perform summation over the repeated indices of a tensorial expression

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Physics[SumOverRepeatedIndices] - perform summation over the repeated indices of a tensorial expression

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

Parameters

expression	-	any algebraic tensorial expression having spacetime repeated indices implying summation
alpha, beta, ...	-	optional, the repeated indices to be summed, if not given, all the spacetime repeated indices of expression are summed
simplifier = ...	-	optional - indicates the simplifier to be used instead; default is none

Description

•	The SumOverRepeatedIndices performs the summation over the repeated indices of expression implied when using the Einstein summation convention. The summation takes into account the covariant and contravariant character of each contracted index.

•

The summation is performed from 1 to the dimension of spacetime, and you only indicated the indices over which the summation is to be performed, not their range. The summation indices are indicated in sequence after expression. If no indices are indicated then summation is performed over all the repeated indices of expression.

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

•	By default, the summation is performed without simplifying the result; to have the result simplified before returning, indicate the simplifier on the right-hand-side of the optional argument simplifier = ...

Compatibility

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

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

Examples

(1)

Consider the complete contraction of indices between the Riemann tensor and its dual with the Schwarzschild metric in spherical coordinates

For that 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 = 18446790986282449366)

(2)

Enter the dual of the Riemann tensor

R2 := (1/2)*LeviCivita[alpha, beta, rho, sigma]*Riemann[`~rho`, `~sigma`, mu, nu]

(1/2)*Physics:-LeviCivita[alpha, beta, rho, sigma]*Physics:-Riemann[`~rho`, `~sigma`, mu, nu]

(3)

Multiply both

(1/2)*Physics:-LeviCivita[alpha, beta, rho, sigma]*Physics:-Riemann[`~rho`, `~sigma`, mu, nu]*Physics:-Riemann[`~alpha`, `~beta`, `~mu`, `~nu`]

(4)

Check the indices

(5)

Perform the summation over these 6 indices

(6)

So (4) is zero; this term enters the computation of the 1st of the Riemann scalars,

and

S[1] = m^2/r^6, S[2] = -m^3/r^9

(7)

and actually for both scalars only the first term in these formulas is different from zero:

(8)

32*m^2/r^6-16*m^2*sin(theta)^2/(r^6*(-1+cos(theta)^2))

(9)

48*m^2/r^6

(10)

(11)

-96*m^3/r^9

(12)

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