kerrinvariants.mws

>    restart:

>    grtw():

>    qload(newkerr);

`Default spacetime` = newkerr

`For the newkerr spacetime:`

Coordinates

x(up)

`x `^a = vector([r, u, phi, t])

`Line element`

` ds`^2 = (r^2+u^2)/(r^2-2*m*r+a^2)*` d`*r^`2 `+(r^2+u^2)/(a^2-u^2)*` d`*u^`2 `+(a^2-u^2)/a^2*(r^2+a^2+2*(a^2-u^2)*m*r/(r^2+u^2))*` d`*phi^`2 `-4*(a^2-u^2)/a*m*r/(r^2+u^2)*` d`*phi^` `*`d `*t^` `+(-1+2...

Constraints = [u = a*cos(theta)]

`The Kerr metric in Boyer-Lindquist type coordinates (u=a*cos(theta)).`

>    grdef(`WstarSq:=C{a b c d}*Cstar{^a ^b ^c ^d}`); 

Created definition for WstarSq

>    grcalc(WeylSq,WstarSq);

`CPU Time ` = .221

>    grmap(_,subs,r=x*a,`x`); 

Applying routine subs to WeylSq

Applying routine subs to WstarSq

>    gralter(_,13,expandsqrt,6,7); 

Component simplification of a GRTensorII object:

Applying routine `Apply constraints repeatedly` to object WeylSq

Applying routine `Apply constraints repeatedly` to object WstarSq

Applying routine expandsqrt to object WeylSq

Applying routine expandsqrt to object WstarSq

Applying routine expand to object WeylSq

Applying routine expand to object WstarSq

Applying routine factor to object WeylSq

Applying routine factor to object WstarSq

`CPU Time ` = .50e-1

>    Ww:=grcomponent(WeylSq,[])*a^6/(48*m^2):

>    Wwstar:=grcomponent(WstarSq,[])*a^6/(96*m^2):

>    W:=(x-x1)*(x+x1)*(x-x2)*(x+x2)*(x-x3)*(x+x3)/(x^2+x1^2)^6;

W := (x-x1)*(x+x1)*(x-x2)*(x+x2)*(x-x3)*(x+x3)/(x^2+x1^2)^6

>    Wstar:=-3*x*x1*(x-x4)*(x+x4)*(x-x5)*(x+x5)/(x^2+x1^2)^6;

Wstar := -3*x*x1*(x-x4)*(x+x4)*(x-x5)*(x+x5)/(x^2+x1^2)^6

>    x1:=cos(theta):

>    x2:=(2+sqrt(3))*cos(theta):

>    x3:=(2-sqrt(3))*cos(theta):

>    x4:=sqrt(3)*cos(theta):

>    x5:=cos(theta)/sqrt(3):

Verification of equations (1) and (2)

>    factor(simplify(Ww-W));

0

>    factor(simplify(Wwstar-Wstar));

0

The loci W=0 (red) Wstar=0 (blue)

>    with(plots):

>    plot1:=implicitplot(Ww=0,x=0..4,theta=0..Pi,color=red,numpoints=40000):

>    plot2:=implicitplot(Wwstar=0,x=0..4,theta=0..Pi,color=blue,numpoints=40000):

>    display({plot1,plot2});

[Maple Plot]

>