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

Calling Sequence

LieDerivative[v, ...](T)

LieDerivative(T, v, ...)




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



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_



The LieDerivative[v] 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



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  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.


The vector vc indexing in LieDerivative[v[~c]](T) is expected to be defined as a tensor 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).


The 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.


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.





Set a system of coordinates - say X


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




Define some tensors for experimentation


Defined objects with tensor properties



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




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_):



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: - - - +




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


vr,θ,φ,twill now be displayed asv

ξr,θ,φ,twill now be displayed asξ

Tr,θ,φ,twill now be displayed asT


The Lie derivatives of the covariant and contravariant vectors ξα 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







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 ξ with the same index α




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:







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




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




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







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




These equations can be solved using pdsolve




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




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 -




Expanding the sum over the repeated indices,







See Also

Christoffel, Coordinates, d_, D_, Define, g_, KillingVectors, Library[TensorComponents], LieBracket, PDEtools[declare], pdsolve, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, Simplify, SumOverRepeatedIndices, TensorArray



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.