Compute the Lie derivative of a tensorial expression - Maple Programming Help

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Physics[LieDerivative] - Compute the Lie derivative of a tensorial expression

Calling Sequence

LieDerivative[v, ...](T)

LieDerivative(T, v, ...)

Parameters

T

-

an algebraic expression, or a relation, or a list, set, Matrix or Array of them

v

-

a contravariant vector as a tensor or tensor function, passed with or without one free spacetime contravariant index (prefixed by ~), with respect to which the derivative is being taken

...

-

optional, more contravariant vectors as v to perform higher order differentiation

...

-

optional, the last argument can be the non-covariant operator d_ to be used instead of the covariant D_

Description

• 

The LieDerivative[mu] command computes the Lie derivative of a tensorial expression T - say with one contravariant and one covariant spacetime indices as in Tba - according to the standard definition

vTba=vc𝒟cTbaTbc𝒟cva+Tca𝒟bvc

  

where Einstein's summation convention is used, a,b and c represent spacetime indices, v is a contravariant vector field, a tensor with one index, and 𝒟 is the covariant derivative operator D_. When the expression T has no free indices, the Lie derivative is equal to only the first term of the right-hand-side, and when T has more than one contravariant or covariant tensor indices, there is a term like the second one and another like the third one respectively for each contravariant and covariant free indices in T.

• 

From this definition it is clear that the LieDerivative is a tensor with regards to the free indices of the derivand, but not with regards to the free index of the indexing vector field (vc in the above), whose index appears in the definition contracted with the differentiation operator D_ (or d_). In other words, the index c of vc enters the Lie derivative as a dummy. If this dummy is found in the indices of the derivand, it is automatically replaced by another spacetime dummy index. You can also pass V without an index, with our with functionality, as in V or VX.

• 

The vector vc indexing in LieDerivative[v[~c]](T) is expected to be defined as a tensor such using Define, and ~c (entered suffixed by ~) to be a spacetime contravariant index. Because this index is a dummy index, you can also pass v without indexation, possibly as a function too, as in LieDerivative[v](T) (case of a constant v) or LieDerivative[v(x)](T).

• 

The indexation of LieDerivative can also consists of a sequence of spacetime vectors like v, in which case a higher order LieDerivative is computed (orderly differentiating from left to right).

• 

This single differentiating vector, or a sequence of them, can also be passed after the first argument, as in other Maple differentiation commands.

• 

When the spacetime is Galilean, so all the Christoffel symbols are zero, the operator d_ is used instead of the covariant D_. Also, when the spacetime is non-Galilean, due to the symmetry of the Christoffel symbols under permutation of their 2nd and 3rd indices, all the terms involving Christoffel symbols cancel so that a mathematically equivalent result can be obtained replacing D_ by d_. To obtain a result directly expressed using d_, pass d_ as the last argument.

Examples

In the examples that follow, as well as in the context of tensor computations with the Physics package, Einstein's summation convention for repeated indices is used.

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Set a system of coordinates - say X

Setupcoordinates=X

* Partial match of 'coordinates' against keyword 'coordinatesystems'

Default differentiation variables for d_, D_ and dAlembertian are: X=x1,x2,x3,x4

Systems of spacetime Coordinates are: X=x1,x2,x3,x4

coordinatesystems=X

(2)

Define some tensors for experimentation

Definev,ξ,T

Defined objects with tensor properties

T,v,ξ,γμ,σμ,Xμ,μ,gμ,ν,δμ,ν,εα,β,μ,ν

(3)

Compute the Lie derivative of a scalar fX: it is the same as the directional derivative in the direction of vμ

LieDerivative[v[`~mu`]X]fX

vμμXμfX

(4)

Because the spacetime at this point in the worksheet is flat, the output above involves d_, not the covariant D_. Set the spacetime to any nongalilean value, for instance (see g_):

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

The Schwarzschild metric in coordinates r,θ,φ,t

Parameters: m

gμ,ν=rr+2m0000r20000r2sinθ20000r2mr

(5)

Use the declare facility of PDEtools to avoid redundant display of functionality and have derivatives displayed with compact indexed notation

PDEtools:-declarev,ξ,TX

vr,θ,φ,twill now be displayed asv

ξr,θ,φ,twill now be displayed asξ

Tr,θ,φ,twill now be displayed asT

(6)

The Lie derivatives of the covariant and contravariant vectors xialpha and ξα differ in the sign of the second term, but not just in that: note the different ways in which the index μ is contracted

LieDerivative[v[`~mu`]X]ξ[α]X

vμμ𝒟μξα+ξμ𝒟αvμμ

(7)

LieDerivative[v[`~mu`]X]ξ[`~alpha`]X

vμμ𝒟μξααξμμ𝒟μvαα

(8)

The index μ in vμ enters the Lie derivative as a dummy. If this dummy is found in the indices of the derivand, it is automatically replaced by another spacetime index. For example: pass V and xi with the same index alpha

LieDerivative[v[`~alpha`]X]ξ[`~alpha`]X

vμμ𝒟μξααξμμ𝒟μvαα

(9)

You can also pass v without indices, with or without functionality depending on whether v is constant, in which case only the part involving the Christoffel symbols remains after computing the covariant derivative:

LieDerivative[vX]ξ[`~alpha`]X

vμμ𝒟μξααξμμ𝒟μvαα

(10)

LieDerivative[v]ξ[`~alpha`]X

vμμ𝒟μξααξμμΓαμ,ναμ,νvνν

(11)

When the derivand is a contravariant vector field, the Lie derivative is equal to the the LieBracket between the indexing vector field v in the above) and the derivand ξα. For that reason, you can also pass the first argument to LieBracket with or without the index

LieBracketv[`~mu`]X,ξ[`~alpha`]X

vμμ𝒟μξααξμμ𝒟μvαα

(12)

So in LieBracket the free index of the first vector field is also a dummy and the index of the second one is the free index of the result.

The Lie derivative of a Tβα contains three terms: first there is the term equivalent to a directional derivative, then another with a minus sign related to the contravariant index, then another one related to the covariant index

LieDerivative[v[`~mu`]X]T[`~alpha`,β]X

vμμ𝒟μTαβαβTμβμβ𝒟μvαα+Tαμαμ𝒟βvμμ

(13)

The Lie derivative is essentially the change in form of the derivand under transformations generated by the indexing vector field (v in the examples above). In turn the Killing vectors generate transformations that leave the form of the spacetime metric g_ invariant. So equating to zero the Lie Derivative of the metric (currently Schwarzschild) results in a system of partial differential equations defining the Killing vectors

LieDerivative[ξ[`~mu`]X]g_α,β=0

gμ,β𝒟αξμμ+gα,μ𝒟βξμμ=0

(14)

Simplify

𝒟βξα+𝒟αξβ=0

(15)

The array of Killing equations behind this tensorial expression can be obtained with TensorArray, KillingVectors or the Library command TensorComponents:

TensorArray

2ξ1r2mξ1rr+2m=0ξ1θ2ξ2r+ξ2r=0ξ1φ2ξ3r+ξ3r=0ξ1t+2mξ4rr+2m+ξ4r=0ξ1θ2ξ2r+ξ2r=02ξ2θ2r+2mξ1=0ξ2φ2cosθξ3sinθ+ξ3θ=0ξ2t+ξ4θ=0ξ1φ2ξ3r+ξ3r=0ξ2φ2cosθξ3sinθ+ξ3θ=02ξ3φ2r+2msinθ2ξ1+2sinθcosθξ2=0ξ3t+ξ4φ=0ξ1t+2mξ4rr+2m+ξ4r=0ξ2t+ξ4θ=0ξ3t+ξ4φ=02ξ4t+2r+2mmξ1r3=0

(16)

These equations can be solved using pdsolve

pdsolve

ξ1=0,ξ2=_C2sinφ+_C3cosφr2,ξ3=r2_C4cos2θ2+sinθ_C2cosφ_C3sinφcosθ+_C42,ξ4=r2m_C1r

(17)

Alternatively, the same equations components of the tensorial expression (15) can be obtained directly with KillingVectors that in addition performs a reduction to involutive form (canonical form) for the DE system, and can as well compute and solve the equations all in one go

KillingVectorsξ

ξμμ=0,0,0,1,ξμμ=0,sinφ,cosφtanθ,0,ξμμ=0,cosφ,sinφtanθ,0,ξμμ=0,0,1,0

(18)

Yet another manner of obtaining the same result is using Physics:-Library:-TensorComponents - that computes similar to TensorArray - but returns a list of equations instead, where the ordering is ascending with respect to value of the free indices -

Library:-TensorComponents

2ξ1r2Γα1,1α1,1ξα=0,ξ1θ2Γα1,2α1,2ξα+ξ2r=0,ξ1φ2Γα1,3α1,3ξα+ξ3r=0,ξ1t2Γα1,4α1,4ξα+ξ4r=0,ξ1θ2Γα1,2α1,2ξα+ξ2r=0,2ξ2θ2Γα2,2α2,2ξα=0,ξ2φ2Γα2,3α2,3ξα+ξ3θ=0,ξ2t2Γα2,4α2,4ξα+ξ4θ=0,ξ1φ2Γα1,3α1,3ξα+ξ3r=0,ξ2φ2Γα2,3α2,3ξα+ξ3θ=0,2ξ3φ2Γα3,3α3,3ξα=0,ξ3t2Γα3,4α3,4ξα+ξ4φ=0,ξ1t2Γα1,4α1,4ξα+ξ4r=0,ξ2t2Γα2,4α2,4ξα+ξ4θ=0,ξ3t2Γα3,4α3,4ξα+ξ4φ=0,2ξ4t2Γα4,4α4,4ξα=0

(19)

Expanding the sum over the repeated indices,

SumOverRepeatedIndices

2ξ1r2mξ1rr+2m=0,ξ1θ2ξ2r+ξ2r=0,ξ1φ2ξ3r+ξ3r=0,ξ1t+2mξ4rr+2m+ξ4r=0,ξ1θ2ξ2r+ξ2r=0,2ξ2θ2r+2mξ1=0,ξ2φ2cosθξ3sinθ+ξ3θ=0,ξ2t+ξ4θ=0,ξ1φ2ξ3r+ξ3r=0,ξ2φ2cosθξ3sinθ+ξ3θ=0,2ξ3φ2r+2msinθ2ξ1+2sinθcosθξ2=0,ξ3t+ξ4φ=0,ξ1t+2mξ4rr+2m+ξ4r=0,ξ2t+ξ4θ=0,ξ3t+ξ4φ=0,2ξ4t+2r+2mmξ1r3=0

(20)

pdsolve

ξ1=0,ξ2=_C2sinφ+_C3cosφr2,ξ3=r2_C4cos2θ2+sinθ_C2cosφ_C3sinφcosθ+_C42,ξ4=r2m_C1r

(21)

See Also

Christoffel, Coordinates, d_, D_, Define, g_, KillingVectors, Library[TensorComponents], LieBracket, PDEtools[declare], pdsolve, Physics, Physics conventions, Physics examples, Setup, Simplify, SumOverRepeatedIndices, TensorArray

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.

  

Weinberg, S. Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity, John Wiley & Sons, Inc, 1972.


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