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 optionally you indicate the indices (not their range) over which the summation is to be performed. 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 = ...
Examples
withPhysics:
Setupmathematicalnotation=true
mathematicalnotation=true
Consider the complete contraction of indices between the Riemann tensor and its dual R2 in a Schwarzschild spacetime in spherical coordinates
R*α,β,μ,ν=12Εμ,ν,ρ,σRα,βρ,σ
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
g_sc
_______________________________________________________
Systems of spacetime coordinates are:X=r,θ,φ,t
Default differentiation variables for d_, D_ and dAlembertian are:X=r,θ,φ,t
Setting lowercaselatin_is letters to represent space indices
The Schwarzschild metric in coordinates r,θ,φ,t
Parameters: m
Signature: - - - +
gμ,ν=r2m−r0000−r20000−r2sinθ20000r−2mr
Enter the dual of the Riemann tensor
R2≔12LeviCivitaα,β,ρ,σRiemann~rho,~sigma,μ,ν
R2≔εα,β,ρ,σRρ,σμ,νρ,σμ,ν2
Multiply both
R2Riemann~alpha,~beta,~mu,~nu
εα,β,ρ,σRρ,σμ,νρ,σμ,νRα,β,μ,να,β,μ,ν2
Check the indices
Check,all
The repeated indices per term are: ...,...,..., the free indices are: ...
α,β,μ,ν,ρ,σ,∅
Perform the summation over these 6 indices
SumOverRepeatedIndices
0
So (4) is zero; this term enters the computation of the 1st of the Riemann scalars, S1=148Rα,β,μ,νRα,β,μ,ν−iRα,β,μ,νR*α,β,μ,ν
and S2=196Rα,β,μ,νRα,β,ρ,σRρ,σμ,ν+iRα,β,μ,νRα,β,ρ,σR*ρ,σμ,ν
Riemannscalars
S1=m2r6,S2=−m3r9
and actually for both scalars only the first term in these formulas is different from zero:
Riemannα,β,μ,νRiemann~alpha,~beta,~mu,~nu
Rα,β,μ,νRα,β,μ,να,β,μ,ν
SumOverRepeatedIndices,simplifier=simplify
48m2r6
Riemannα,β,μ,νRiemann~alpha,~beta,~rho,~sigmaRiemannρ,σ,~mu,~nu
Rα,β,μ,νRα,β,ρ,σα,β,ρ,σRρ,σμ,νρ,σμ,ν
−96m3r9
See Also
Check, Coordinates, Define, g_, LeviCivita, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, 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.
Compatibility
The Physics[SumOverRepeatedIndices] command was introduced in Maple 16.
For more information on Maple 16 changes, see Updates in Maple 16.
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