calculate the conformal Killing vectors for a given metric - Maple Help

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Tensor[ConformalKillingVectors] - calculate the conformal Killing vectors for a given metric

Calling Sequences

     ConformalKillingVectors(g, options )

Parameters

  g         - a metric tensor on a manifold

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

Description

• 

A vector field X  is called a conformal Killing vector for the metric g if ℒX g = λ g, where ℒX denotes Lie differentiation with respect to X and λ is a function (proportional to the divergence of X ). If  Xj= gjk Xk and denotes covariant differentiation with respect to the given metric, then this equation can be written as 2(iXj) = λ gij . The Killing vectors of g are the solutions to this equation with λ =0. The set of all conformal Killing vectors forms a Lie algebra of vector fields.

• 

The program ConformalKillingVectors generates the defining system of 1st order PDE for a conformal vector field and uses pdsolve to find the solutions to these PDE.

• 

The command ConformalKillingVectors returns a sequence of two lists. The first list contains the non-trivial conformal Killing vectors and the second the Killing vectors.  If there are no non-trivial conformal Killing vector fields then the first list is empty.

• 

The keyword argument coefficientvariables = x1, x2, ... , xk allows the user to specify the coefficients functions in the conformal Killing vector Xas functions of the variables x1 , x2, ... , xk .

• 

The exact form of the conformal Killing vector X can be specified with the keyword argument ansatz = X. For example, if the coordinates on the underlying manifold are x, y, z and X1, X2 are defined vectors, then one may solve for conformal Killing vectors of the form X = fy, zX1 + gy,zX2 . 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 conformal Killing equations.

• 

If the metric g 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 conformal Killing vectors changes, are calculated.

• 

With keyword argument output = pde, the defining partial differential equations for the conformal Killing vectors are returned. The option output = general returns the general conformal Killing vector in terms of a number of arbitrary constants _C1, _C2 ,... . The option output = list returns a list of vectors 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 ConformalKillingVectors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-ConformalKillingVectors(...).

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

We find the conformal Killing vectors for the Euclidean metric g in 3 dimensions.

DGsetupx,y,z,M

frame name: M

(2.1)
M > 

g:=evalDGdx &t dx+dy &t dy+dz &t dz

g:=dxdx+dydy+dzdz

(2.2)

 

There are a total of 10 conformal Killing vectors, 6 of which are Killing vectors.

M > 

C,K:=ConformalKillingVectorsg

C,K:=xzD_x+yzD_y+12z212x212y2D_z,xyD_x+12z2+12y212x2D_y+yzD_z,xD_x+yD_y+zD_z,12y212z2+12x2D_x+xyD_y+xzD_z,D_y,D_x,zD_y+yD_z,D_z,zD_x+xD_z,yD_x+xD_y

(2.3)
M > 

nopsC,nopsK

4,6

(2.4)

 

We can check this result by calculating the Lie derivative of the metric with respect to these vector fields (see LieDerivative) . We see that the vector fields C are conformal Killing tensors and that the vector fields K are Killing vectors.

M > 

LieDerivativeC,g

2zdxdx+2zdydy+2zdzdz,2ydxdx+2ydydy+2ydzdz,2dxdx+2dydy+2dzdz,2xdxdx+2xdydy+2xdzdz

(2.5)
M > 

LieDerivativeK,g

0dxdx,0dxdx,0dxdx,0dxdx,0dxdx,0dxdx

(2.6)

 

We can use the LieAlgebraData command in the LieAlgebras package to calculate the structure equations for the Lie algebra Γ of conformal Killing vectors.

M > 

Γ:=opC,opK

Γ:=xzD_x+yzD_y+12z212x212y2D_z,xyD_x+12z2+12y212x2D_y+yzD_z,xD_x+yD_y+zD_z,12y212z2+12x2D_x+xyD_y+xzD_z,D_y,D_x,zD_y+yD_z,D_z,zD_x+xD_z,yD_x+xD_y

(2.7)
M > 

LD:=LieAlgebras:-LieAlgebraDataΓ,CVF

LD:=e1,e3=e1,e1,e5=e7,e1,e6=e9,e1,e7=e2,e1,e8=e3,e1,e9=e4,e2,e3=e2,e2,e5=e3,e2,e6=e10,e2,e7=e1,e2,e8=e7,e2,e10=e4,e3,e4=e4,e3,e5=e5,e3,e6=e6,e3,e8=e8,e4,e5=e10,e4,e6=e3,e4,e8=e9,e4,e9=e1,e4,e10=e2,e5,e7=e8,e5,e10=e6,e6,e9=e8,e6,e10=e5,e7,e8=e5,e7,e9=e10,e7,e10=e9,e8,e9=e6,e9,e10=e7

(2.8)

 

This output shows, for example, that the Lie bracket of the first and third vector fields in Γ is minus the first vector field.

M > 

evalDGLieBracketΓ1,Γ3+Γ1

0D_x

(2.9)

 

The Lie algebra of conformal Killing vector fields is a simple Lie algebra, that is, it is indecomposable and semi-simple.

M > 

DGsetupLD

Lie algebra: CVF

(2.10)

 

We check these properties using the Query command from the LieAlgebras package.

CVF > 

LieAlgebras:-QueryIndecomposable

true

(2.11)
CVF > 

LieAlgebras:-QuerySemisimple

true

(2.12)

 

Example 2.

We look for conformal Killing vector fields for the metric g, of the special form specified by the vector X. 

M > 

X:=evalDGax,yD_x+bx,y,zD_y+cx,y,zD_z

X:=ax,yD_x+bx,y,zD_y+cx,y,zD_z

(2.13)
M > 

ConformalKillingVectorsg,ansatz=X,unknowns=ax,y,bx,y,z,cx,y,z

xD_x+yD_y+zD_z,xyD_x+12z2+12y212x2D_y+yzD_z,D_z,zD_y+yD_z,yD_x+xD_y,D_y,D_x

(2.14)

 

Example 3.

We look for conformal Killing vector fields for the metric g which have constant divergence. These are also known as homothetic vector fields. 

M > 

X:=evalDGax,y,zD_x+bx,y,zD_y+cx,y,zD_z

X:=ax,y,zD_x+bx,y,zD_y+cx,y,zD_z

(2.15)
M > 

div:=xax,y,z+ybx,y,z+zcx,y,z

div:=xax,y,z+ybx,y,z+zcx,y,z

(2.16)
M > 

Eq:=xdiv=0,ydiv=0,zdiv=0

Eq:=2x2ax,y,z+2yxbx,y,z+2zxcx,y,z=0,2yxax,y,z+2y2bx,y,z+2zycx,y,z=0,2zxax,y,z+2zybx,y,z+2z2cx,y,z=0

(2.17)
M > 

ConformalKillingVectorsg,ansatz=X,unknowns=ax,y,z,bx,y,z,cx,y,z,auxiliaryequations=Eq

xD_x+yD_y+zD_z,zD_x+xD_z,D_z,zD_y+yD_z,yD_x+xD_y,D_y,D_x

(2.18)

 

Example 4.

We find the general conformal Killing vector for the metric g, depending upon 10 constants.

M > 

ConformalKillingVectorsg,output=general

12_C4y2+y_C2xy_C812_C4z2+z_C1xz_C7+12_C4x2+_C3x+_C10D_x+12_C2z2+zy_C1z_C5+12_C2y2+y_C4x+_C3y12_C2x2+_C8x+_C9D_y+12_C1z2+z_C4x+z_C2y+_C3z12_C1x212_C1y2+_C6+_C7x+_C5yD_z

(2.19)

 

Example 5.

We calculate the conformal Killing vector fields for the metric g5 which depends upon 3 parameters a,b,c, where c 0. For generic values of the parameters there are no non-trivial conformal Killing vectors. However, there are non-trivial conformal Killing vectors in 3 exceptional cases : a=0,b=0,c=c, a=a,b=b,c=14b2a,a=0,b=b,c=c

M > 

g5:=evalDGdx &t dx+dy &t dy+ax2+bx+cdz &t dz

g5:=dxdx+dydy+ax2+bx+cdzdz

(2.20)
M > 

CKV:=ConformalKillingVectorsg5,parameters=a,b,c,auxiliaryequations=c0

CKV:=xzD_xc+yzD_yc+12cz2y2x2D_zc2,xyD_xc12cz2y2+x2D_yc+yzD_zc,xD_xc+yD_yc+zD_zc,12cz2x2+y2D_xc+xyD_yc+xzD_zc,zD_y+yD_zc,D_zc,zD_x+xD_zc,yD_x+xD_y,D_y,D_x,122ax+bD_xa+yD_y,y2ax+bD_xa+144y2a2+b2+4xba+4x2a2D_ya2,cosazD_xa+2sinazD_z2ax+b,sinazD_xa+2cosazD_z2ax+b,D_za,D_y,2bx+cD_xb+2yD_y+zD_z,D_z,D_y,,D_z,D_y,14cosaz4y2a2+b2+4xba+4x2a2D_xa5/2+cosazy2ax+bD_ya3/2+12b2+4xba+4x2a2+4y2a2sinazD_za22ax+b,14sinaz4y2a2+b2+4xba+4x2a2D_xa5/2sinazy2ax+bD_ya3/2+12b2+4xba+4x2a2+4y2a2cosazD_za22ax+b,122ax+bD_xa+yD_y,y2ax+bD_xa+144y2a2+b2+4xba+4x2a2D_ya2,4aD_z,cosazyD_xa+12cosaz2ax+bD_ya3/2+2ysinazD_z2ax+b,cosazD_xa+2sinazD_z2ax+b,sinazyD_xa12sinaz2ax+bD_ya3/2+2ycosazD_z2ax+b,sinazD_xa+2cosazD_z2ax+b,D_y,a=0,b=0,c=c,a=a,b=b,c=14b2a,a=0,b=b,c=c,a=a,b=b,c=c,a=a,b=b,c=14b2a

(2.21)

 

Exceptional Case 1: a=0,b=0,c=c

M > 

CKV1

xzD_xc+yzD_yc+12cz2y2x2D_zc2,xyD_xc12cz2y2+x2D_yc+yzD_zc,xD_xc+yD_yc+zD_zc,12cz2x2+y2D_xc+xyD_yc+xzD_zc,zD_y+yD_zc,D_zc,zD_x+xD_zc,yD_x+xD_y,D_y,D_x

(2.22)
M > 

nopsCKV11+nopsCKV12

10

(2.23)

Exceptional Case 2: a=a,b=b,c=14b2a

M > 

CKV2

122ax+bD_xa+yD_y,y2ax+bD_xa+144y2a2+b2+4xba+4x2a2D_ya2,cosazD_xa+2sinazD_z2ax+b,sinazD_xa+2cosazD_z2ax+b,D_za,D_y