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Tensor[KillingYanoTensors] - calculate the Killing-Yano tensors for a given connection or a given metric

Calling Sequences

     KillingYanoTensors(g ,p ,options)

     KillingYanoTensors(Γ, p ,options)

Parameters

  g         - a metric tensor on a manifold M

  Γ         - an affine connection on M 

    options   - any of the following keywords arguments: ansatz, unknowns, auxiliaryequations, coefficientvariables, parameters, output

 

Description

Examples

Description

• 

Let denote covariant differentiation with respect to the given connection Γ, or with respect to the Christoffel connection defined by the metric g. A degree p Killing-Yano tensor T is a p-form, or equivalently, a skew-symmetric covariant tensor such that (i Tj)h ... k=0. 

• 

The program KillingYanoTensors generates the defining system of 1st order PDE for a Killing-Yano tensor and uses pdsolve to find the solutions to these PDE.

• 

The keyword argument coefficientvariables = x1 , x2, ... , xk allows the user to specify the coefficient functions in the Killing-Yano tensor Tas functions of the variables x1 , x2, ... , xk .

• 

The exact form of the tensor T can be specified with the keyword argument ansatz = T. For example, if the coordinates on the underlying manifold are x, y, z and T1, T2, T3 are defined p-forms, then one may solve for Killing-Yano tensors of the form T = xfy,zT1 + x2 T2 + gy,zT3. In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns = fy,z, gy,z.

• 

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations = EqList. Here EqList is a list of the auxiliary equations to be added to the Killing-Yano equations.

• 

If the metric g or connection Γ depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters= ParList,where ParList is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing-Yano tensors changes, are calculated.

• 

With keyword argument output = pde, the defining partial differential equations for the Killing-Yano tensors are returned. The option output = general returns the general solution in terms of a number of arbitrary constants _C1, _C2, ... while the option output = list returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output = list.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form KillingYanoTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KillingYanoTensors(...).

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

We find the Killing-Yano tensors of degree 2 and 3 for following metric g. 

DGsetupu,v,x,y,M

frame name: M

(2.1)
M > 

gevalDGexp2axdu&tdu+du&tdv+dv&tdu+dx&tdx+dy&tdy

g:=_DGtensor,M,cov_bas,cov_bas,,1,1,ⅇ2ax,1,2,1,2,1,1,3,3,1,4,4,1

(2.2)

 

There are 2 Killing-Yano tensors of degree 2.

M > 

K2KillingYanoTensorsg,2

K2:=_DGform,M,2,1,4,1,_DGform,M,2,1,3,1

(2.3)

 

There are 2 Killing-Yano tensors of degree 3.

M > 

K3KillingYanoTensorsg,3

K3:=_DGform,M,3,1,3,4,1,_DGform,M,3,1,2,3,1

(2.4)

 

We can use the CovariantDerivative and SymmetrizeIndices commands to verify that the differential forms K2 and K3 satisfy the Killing-Yano equation. First we need the Christoffel connection for the metric g.

M > 

CChristoffelg

C:=_DGconnection,M,con_bas,cov_bas,cov_bas,,2,1,3,aⅇ2ax,2,3,1,aⅇ2ax,3,1,1,aⅇ2ax

(2.5)

 

To check the Killing-Yano equation we take the covariant derivative of one of the tensors and symmetrize on the last two indices (the DifferentialGeometry convention is to place the index or slot for the covariant derivative last)

M > 

SymmetrizeIndicesCovariantDerivativeK21,C,2,3,Symmetric

_DGtensor,M,cov_bas,cov_bas,cov_bas,,1,1,1,0

(2.6)

 

M > 

SymmetrizeIndicesCovariantDerivativeK31,C,3,4,Symmetric

_DGtensor,M,cov_bas,cov_bas,cov_bas,cov_bas,,1,1,1,1,0

(2.7)

 

 

Example 2.

We can use the keyword arguments ansatz and unknowns to find a subset of the Killing-Yano tensors for this metric, say, the 2-forms, which are independent of dx and have coefficients which are functions of the variable u alone.

Sym > 

TevalDGrudu&wdy+sudu&wdv+tudv&wdy

T:=_DGform,M,2,1,2,su,1,4,ru,2,4,tu

(2.8)
M > 

varsru,su,tu

vars:=ru,su,tu

(2.9)
M > 

KillingYanoTensorsg,2,ansatz=T,unknowns=vars

_DGform,M,2,1,4,1

(2.10)

 

With the keyword argument output =pde, the defining differential equations for the Killing-Yano tensors are returned.

M > 

KillingYanoTensorsg,2,ansatz=T,unknowns=vars,output=pde

0,aⅇ2axsu,aⅇ2axtu,12aⅇ2axsu,12aⅇ2axtu,12ⅆⅆuru,12ⅆⅆusu,12ⅆⅆutu,12ⅆⅆutu,ⅆⅆuru,ⅆⅆusu

(2.11)

 

Example 3.

Consider the following metric which depends upon an arbitrary function fy.

M > 

DifferentialGeometry:-DGsetupu,v,y,z,M

frame name: M

(2.12)
M > 

gevalDGevalDG32du&tdvy232dv&tduy23fydv&tdv+3dy&tdyy2+3dz&tdzy3

g:=_DGtensor,M,cov_bas,cov_bas,,1,2,32y2,2,1,32y2,2,2,3fy,3,3,3y2,4,4,3y3

(2.13)

 

With the keyword argument parameters we can identity those special values of fy for which the metric admits a rank 2 Killing-Yano tensor.

M > 

KYKillingYanoTensorsg,2,parameters=fy

KY:=_DGform,M,2,1,2,1y3,,fy=_C1y2,fy=fy

(2.14)

We see that there are no Killing-Yano tensors for generic choices of fy and 1 Killing-Yano tensor when fy=_C1y2.

See Also

DifferentialGeometry

Tensor

ConformalKillingVectors

KillingVectors

KillingTensors

KillingSpinors