perform substitution of covariant/contravariant tensor indices in tensorial expressions - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Physics : Physics/SubstituteTensorIndices

Physics[SubstituteTensorIndices] - perform substitution of covariant/contravariant tensor indices in tensorial expressions

Calling Sequence

SubstituteTensorIndices(mu = .., ... expression)

Parameters

expression

-

any algebraic tensorial expression typically having some free and some repeated indices

mu = ...

-

the substitution equation, or a set {...} or list [...] of them

evaluatetensor = ...

-

optional - can be true (default) or false, to indicate whether to evaluate the tensors after substituting on them

evaluateexpression = ...

-

optional - can be true or false (default), to indicate whether to evaluate expression after substituting the indices in its tensors

covariantandcontravariant = ...

-

optional - can be true (default) or false, to indicate whether to substitute both covariant and contravariant indices when only one of them is given on the left-hand sides of the substitution equations

Description

• 

The SubstituteTensorIndices substitutes indices in tensors - the ones displayed when you enter Define(); see Define to define one. Nowhere else are the indices substituted, and the substitution can be performed on a covariant index, the corresponding contravariant one, or on both.

• 

The first argument is a substitution equation with an index on the left-hand side, or a set or list of them. All the substitution equations are expected in this first argument. The second argument is the target, where the substitutions are to be performed.

• 

The tensors where indices are substituted are re-evaluated after substitution; this re-evaluation can optionally be suppressed giving the argument evaluatetensor = false. The expression where these re-evaluated tensors are introduce is by default not re-evaluated; you can change that passing the optional argument evaluateexpression = true.

• 

By default both covariant and contravariant indices are substituted, even if the substitution equation has only one of them on the left-hand side; this behavior is convenient both when substituting contracted or free indices. To substitute only the kind of index found on the left-hand side of the substitution equation use covariantandcontravariant = false.

• 

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

• 

When substituting indices by numerical values, contravariant values are prefixed with ~, say as in ~1; otherwise, say as in 1, the value is interpreted as covariant. Hence, the equation ~nu = 1 will also transform ~nu into covariant. Substituting the natural way, say as in nu = 1, will automatically substitute both nu = 1 and also ~nu = ~1 keeping the covariant/contravariant character unchanged.

• 

When the right-hand side of a substitution equation is a contravariant index, so suffixed with ~, say as in ~nu = ~1, note the separation between = and ~; that separation is important because ~ is also the element-wise operator and so =~ has other specific meaning.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Define a couple of arbitrary spacetime tensors for exploration purposes

DefineA,B

Defined objects with tensor properties

A,B,γμ,σμ,μ,gμ,ν,εα,β,μ,ν

(2)

Enter, for example, this tensorial expression, with a contravariant free index ρ

g_α,μA`~mu`g_`~alpha`,`~nu`Bν,σ,`~rho`

gα,μAμμgα,να,νBν,σρν,σρ

(3)

To check the repeated and free indices in an expression use Check

Check,all

The repeated indices per term are: ...,...,..., the free indices are: ...

α,μ,ν,σ,~rho

(4)

So (3) has α and the contravariant ρ as free indices. Substitute now α=β: the standard subs command will only substitute the covariant α

subsα=β,

gβ,μAμμgα,να,νBν,σρν,σρ

(5)

Consequently, the resulting expression is not equivalent to (3): it now has four free indices

Check,all

The repeated indices per term are: ...,...,..., the free indices are: ...

μ,ν,β,σ,~alpha,~rho

(6)

To substitute both covariant and contravariant repeated indices obtaining an expression equivalent to original one use

SubstituteTensorIndicesα=β,

gβ,μAμμgβ,νβ,νBν,σρν,σρ

(7)

Check,all

The repeated indices per term are: ...,...,..., the free indices are: ...

β,μ,ν,σ,~rho

(8)

Substitute now the contravariant ρ by τ; the standard subs command will fail because contravariant indices are prefixed by ~, so this returns (3) as given

subsρ=β,

gα,μAμμgα,να,νBν,σρν,σρ

(9)

To make it work with subs you would need to substitute ~rho = ~beta instead. Using SubstituteTensorIndices you get the desired result regardless of this subtlety

SubstituteTensorIndicesρ=β,

gα,μAμμgα,να,νBν,σβν,σβ

(10)

Check,all

The repeated indices per term are: ...,...,..., the free indices are: ...

α,μ,ν,σ,~beta

(11)

To substitute several indices at once, enclose the substitution equations in a set or a list

SubstituteTensorIndicesρ=β,α=gamma,

gγ,μAμμgγ,νγ,νBν,σβν,σβ

(12)

After substituting, tensors are re-evaluated; in this example, ν becomes 1 and the normalized form of g_ is presented with it in the first place, so switching places with μ

g_μ,ν

gμ,ν

(13)

SubstituteTensorIndicesν=1,

g1,μ

(14)

To avoid this re-evaluation of tensors in the result use evaluatetensor = false

SubstituteTensorIndicesν=1,,evaluatetensor=false

gμ,1

(15)

See Also

Check, Define, element-wise operator, g_, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, SubstituteTensor

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[SubstituteTensorIndices] command was introduced in Maple 16.

• 

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