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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "Demonstration 0 (horizon)
:Geometry of the Kerr horizon. The Gauss curvature, area and Euler cha
racteristic of the Kerr horizon are evaluated for $r=R$ where $R=m \\p
m (m^2-a^2)^\{1/2\}$ in Boyer-Lindquist coordinates at constant t." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "grtw():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTenso
rII~Version~1.79~(R6)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%02~February
~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrav
e,~Denis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D
Copyright~1994-2001~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%inLatest~version~available~from:~http://grtensor.phy.queensu.ca/G" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F(theta):=(R^2+a^2*cos(th
eta)^2)^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"FG6#%&thetaG*$-
%%sqrtG6#,&*$)%\"RG\"\"#\"\"\"F1*&)%\"aGF0F1)-%$cosGF&F0F1F1F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "G(theta):=(R^2+a^2)*sin(thet
a)/F(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"GG6#%&thetaG*&*&,
&*$)%\"RG\"\"#\"\"\"F/*$)%\"aGF.F/F/F/-%$sinGF&F/F/*$-%%sqrtG6#,&F+F/*
&F1F/)-%$cosGF&F.F/F/F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "qload(twod):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~space
timeG%%twodG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8For~the~twod~spaceti
me:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
)%#x~G%\"aG-%'vectorG6#7$%&thetaG%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"
\"#\"\"\",&*(,&*$)%\"RGF'F(F(*&)%\"aGF'F()-%$cosG6#%&thetaGF'F(F(F(%#~
dGF()F6%#2~GF(F(*&**),&F,F(*$F0F(F(F'F()-%$sinGF5F'F(F7F()%$phiGF9F(F(
F+!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "grcalc(Riccisc
alar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#S!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gralter(_,1,7);" }}{PARA 6 "
" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" 
}}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applyin
g routine simplify to object Ricciscalar" }}{PARA 6 "" 1 "" {TEXT -1 
45 "Applying routine factor to object Ricciscalar" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"#g!\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 45 "The Gauss curvature (K)  is the Ricciscalar/2" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 33 "K:=grcomponent(Ricciscalar,[])/2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"KG,$*&*&,&*$)%\"RG\"\"#\"\"\"F-*$)%\"aGF
,F-F-F-,&*&F/F-)-%$cosG6#%&thetaGF,F-\"\"$F)!\"\"F-F-*$),&F)F-F2F-F8F-
F9F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Defining $X=R/a$, we have
 $L=a^2K$" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L:=a^2*factor(subs(R^2
=a^2*X^2,K));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,$*&*&,&*$)%\"X
G\"\"#\"\"\"F-F-F-F-,&*$)-%$cosG6#%&thetaGF,F-\"\"$F)!\"\"F-F-*$),&F)F
-F/F-F5F-F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "For the area of \+
the horizon we have" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "int(int(F(th
eta)*G(theta),theta=0..Pi),phi=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*&)%\"RG\"\"#\"\"\"%#PiGF(\"\"%*(F*F()%\"aGF'F(F)F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "For the Euler characteristic" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "int(int(K*F(theta)*G(theta),theta=0
..Pi),phi=0..2*Pi)/(2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 2" 0 }
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