Physics[KillingVectors]  computes and solves the Killing equations

Calling Sequence


KillingVectors(V, options = ...)


Parameters


V



a name to be used for the vector field (rank 1 spacetime tensor) representing the Killing vector

integrabilityconditions = ...



optional  can be true (default) or false, to reduce or not the determining PDE output system taking into account its integrability conditions

output = ...



optional  the righthandside can be equations or solutions (default), to return the Killing equations or attempt returning their solutions directly

parameters = ...



optional  the righthandside can be a set or a list of names or functions, that parameterize the Killing vectors; the system of equations or solutions will be split according to the different cases of these parameters





Description


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KillingVectors computes and attempts to solve the Killing equations, that is, the PDE system

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By default, KillingVectors automatically computes the equations and attempts solving them in one step. When no solution is found, it returns NULL, the same way the PDE solver pdsolve does. When passing the optional argument output = equations, the equations themselves are returned without attempting solving them. This is useful when KillingVectors fails in finding a solution, or when particular solutions of different forms may be of interest; these solutions can frequently be computed using the symmetry commands of PDEtools like FunctionFieldSolutions and PolynomialSolutions, or mainly InvariantSolutions used together with its various options for particularizing solutions.

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The PDE system returned by KillingVectors with the option output = equations is by default returned as a list of equations and automatically reduced taking into account its integrability conditions. The order of the equations in the list corresponds to a total degree, obtained with an orderly ranking (see details in casesplit). Simplifying the PDE system taking into account integrability conditions is a frequently desired but in some cases expensive mathematical computation. To avoid it, pass the optional argument integrabilityconditions = false, in which case KillingVectors will return faster, a set of equations, not a list, and with no differential elimination simplifications.

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When the metric g_ depends on parameters, either symbols or functions of spacetime variables, the Killing vectors computed using KillingVectors are expected to be valid for arbitrary values of these parameters. It is sometimes of interest, however, to investigate the different kinds of solutions that may exist for different particular values of these parameters. To perform such an investigation, use the optional argument parameters = ... where the righthandside is a set or a list with the parameters, and the system of equations will be split into cases with respect to these parameters using differential algebra techniques. As frequently happens when splitting into cases, the resulting lists of equations may involve inequations as well. Note: passing parameters automatically forces reducing the PDE system using integrability conditions, regardless of the value of the keyword integrabilityconditions.



Compatibility


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The Physics[KillingVectors] command was introduced in Maple 17.



Examples


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 (1) 
Set a system of Coordinates
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 (2) 
Define a tensor with one index to represent the Killing vector
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 (3) 
The covariant components
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 (4) 
To understand this result, recall that the Killing vectors generate transformations that leave the metric g_ invariant in form; these are isometries. In this case, the spacetime loaded with Physics by default is galilean
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 (5) 
The result (3) generates translations along the four axis that leave invariant the distance between two points and thus invariant in form the metric.
Consider now a nongalilean spacetime, for instance set the Schwarzschild metric as the current metric (see g_):
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 (6) 
Compute the Killing vectors again. For illustration purposes, compute the contravariant components and compute first only the equations, with and without simplifying using integrability conditions, then compute everything automatically. First using integrability conditions. To avoid displaying repeatedly use the compact notation available of the PDEtools package
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 (7) 
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 (8) 
where derivatives are displayed indexed due to the use of declare. Compare with
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 (9) 
The contravariant components of the Killing vectors all in one step
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 (10) 
Taking as parameter, there is another solution (from the point of view of differential algebra this other solution is a singular solution):
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 (11) 
>




See Also


casesplit, Coordinates, D_, declare, Define, DifferentialAlgebra, DifferentialGeometry[Tensor]KillingVectors], FunctionFieldSolutions, g_, InvariantSolutions, Library, pdsolve, Physics, Physics conventions, Physics examples, PolynomialSolutions, 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.


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


