Demonstration 0 (horizon):Geometry of the Kerr horizon. The Gauss curvature, area and Euler characteristic of the Kerr horizon are evaluated for $r=R$ where $R=m \pm (m^2-a^2)^{1/2}$ in Boyer-Lindquist coordinates at constant t.

> restart:

> grtw():

> F(theta):=(R^2+a^2*cos(theta)^2)^(1/2);

> G(theta):=(R^2+a^2)*sin(theta)/F(theta);

> qload(twod):

> grcalc(Ricciscalar);

> gralter(_,1,7);

Component simplification of a GRTensorII object:

Applying routine simplify to object Ricciscalar

Applying routine factor to object Ricciscalar

The Gauss curvature (K) is the Ricciscalar/2

> K:=grcomponent(Ricciscalar,[])/2;

Defining $X=R/a$, we have $L=a^2K$

> L:=a^2*factor(subs(R^2=a^2*X^2,K));

For the area of the horizon we have

> int(int(F(theta)*G(theta),theta=0..Pi),phi=0..2*Pi);

For the Euler characteristic

> int(int(K*F(theta)*G(theta),theta=0..Pi),phi=0..2*Pi)/(2*Pi);

>