computes and solves the Killing equations - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Physics : Physics Package : Physics/KillingVectors

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 right-hand-side can be equations, solutions (synonym: componentsolutions) or vectorsolutions (default), to return the Killing equations or attempt returning their solutions for each of the Killing vector components, or a vector solution with the Killing vector on the left-hand side and a list with its components on the right-hand side

parameters = ...

-

optional - the right-hand-side 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

• 

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

𝒟νVα+𝒟αVν=0

  

where 𝒟 is the covariant derivative operator D_, and V entering Vα is the name representing the Killing vectors, provided, with or without a spacetime index, as first argument. As it is the case of the other commands in Physics, KillingVectors operates in tensorial notation. To perform the same operation using differential forms notation see DifferentialGeometry[Tensor][KillingVectors].

• 

When the name V is provided without an index, the equations computed are satisfied by the covariant components Vα. To request the equations or solutions for the contravariant components Vα, pass it with a contravariant index (prefixed by a ~), say as in V[~alpha]. Note that when the spacetime is non-galilean, the covariant and contravariant components of the Killing vectors are not the same, and while the system may succeed in computing solutions for the covariant components, depending on the metric g_ it may fail in computing solutions for the contravariant ones, or vice-versa.

• 

By default, KillingVectors automatically computes the equations and attempts solving them in one step. When successful, KillingVectors returns a vector solution with the Killing vector on the left-hand side and a list with its components on the right-hand side. When no solution is found, it returns NULL, the same way the PDE solver pdsolve does. -When passing the optional argument output = solutions, instead of returning a vector equation, the output is a set of solution equations, with each Killing vector component on the left-hand side and the corresponding solution on the right-hand side. This is useful, for instance, for verification purposes: you can test the solutions computed against the output of KillingVectors using the option output = equations by using the pdetest command. -When passing the optional argument output = equations, the Killing 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.

• 

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.

• 

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 right-hand-side 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.

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(1)

Set a system of Coordinates

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 a tensor with one index to represent the Killing vector

DefineV

Defined objects with tensor properties

V,γμ,σμ,Xμ,μ,gμ,ν,δμ,ν,εα,β,μ,ν

(3)

The covariant components  Vα

KillingVectorsV

Vμ=_C1x2+_C2x3+_C3x4+_C4,_C1x1+_C5x3+_C6x4+_C7,_C2x1_C5x2+_C8x4+_C9,_C3x1_C6x2_C8x3+_C10

(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

g_[]

gμ,ν=1000010000100001

(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 non-galilean spacetime, for instance set the Schwarzschild metric as the current metric (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

(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 VX repeatedly use the compact notation available of the PDEtools package

PDEtools:-declareVX

Vr,θ,φ,twill now be displayed asV

(7)

KillingVectorsV`~alpha`,output=equations

V33r=0,V33θ=V22φsinθ2,V33φ=cosθV22sinθ,V33t=0,V22φ,φ=V22,V22r=0,V22θ=0,V22t=0,V44r=0,V44θ=0,V44φ=0,V44t=0,V11=0

(8)

where derivatives are displayed indexed due to the use of declare. Compare with

KillingVectorsV`~alpha`,output=equations,integrabilityconditions=false

2r2V44trm+V44tr2+mV11=0,8mr2rmr2V11r+mV112=0,4mr22V44rV11tr2=0,r2mV44φr3V33tsinθ2=0,r2mV44θr3V22t=0,2V22θr2V11=0,2mr2sinθ2rV33rV11φ=0,V33θcosθ2V33θV22φ=0,2V22rmrV22rr2V11θ=0,2V33φrsinθ+2cosθV22r+2V11sinθ=0

(9)

The contravariant components of the Killing vectors all in one step

KillingVectorsV`~alpha`

Vαα=0,_C2sinφ+_C3cosφ,_C2cosφ_C3sinφtanθ+_C4,_C1

(10)

Taking m as parameter, there is another solution (from the point of view of differential algebra this other solution is a singular solution). These solutions can also be expressed as solution equations with the vector components on the left-hand sides by using the optional argument output = solutions:

KillingVectorsV`~alpha`,parameters=m,output=solutions

m=0,V11=_C3t+_C6cosφ+sinφ_C2t+_C5sinθ+cosθ_C1t+_C4,V22=_C5sinφ+_C6cosφcosθ_C4sinθr+_C8sinφ+_C9cosφ2_C1sinθ_C3cosφθ_C3cosφ+θ_C2sinφ+θ_C2sinφθt2r,V33=_C10rsinθ+cosφ_C2t+_C5_C3t+_C6sinφtanθ+rsinθ_C8cosφ_C9sinφrsinθtanθ,V44=r_C2sinφ+_C3cosφsinθ+_C1cosθr+_C7,m=m,V11=0,V22=_C2sinφ+_C3cosφ,V33=_C2cosφ_C3sinφtanθ+_C4,V44=_C1

(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.


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam