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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Demonstration 1(br):Defin
ition of the Bel-Robinson Tensor and check of identities in the Kerr-N
ewman metric." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R6)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%02~February~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayll~LakeG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~the~authors.G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~available~from:~http
://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4e:/G
rtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(n
ewkn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&newkn
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~newkn~spacetime: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&%\"rG%\"uG%$phiG%\"tG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%-Line~elementG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*$)
%$~dsG\"\"#\"\"\",,*&*(,&*$)%\"rGF'F(F(*$)%\"uGF'F(F(F(%#~dGF()F/%#2~G
F(F(,*F-F(*(F'F(%\"mGF(F/F(!\"\"*$)%\"aGF'F(F(*$)%\"QGF'F(F(F9F(*&*(F,
F(F3F()F2F5F(F(,&F:F(F0F9F9F(*&**FCF(,(F-F(F:F(*&*&FCF(,&*&F8F(F/F(F'F
=F9F(F(F,F9F(F(F3F()%$phiGF5F(F(*$F;F(F9F(*&*0F'F(FCF(FIF(F3F()FL%\"~G
F(%#d~GF()%\"tGFQF(F(*&F<F(F,F(F9F9*(,&F9F(*&FIF(F,F9F(F(F3F()FTF5F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,ConstraintsG7#/%\"uG*&%\"aG\"\"
\"-%$cosG6#%&thetaGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oKerr~Newma
n~Solution~in~Boyer-Lindquist~coordinates~(u=a*cos(theta))G" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "grdef(`T\{(c d e f)\}:=C\{a \+
c d b\}*C\{^a e f^b\}+Cstar\{a c d b\}*Cstar\{^a e f^b\}`);" }}{PARA 
6 "" 1 "" {TEXT -1 38 "Created definition for C(up,dn,dn,up) " }}
{PARA 6 "" 1 "" {TEXT -1 42 "Created definition for Cstar(up,dn,dn,up)
 " }}{PARA 6 "" 1 "" {TEXT -1 37 "Created definition for T(dn,dn,dn,dn
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "grdef(`TT\{c d\}:=T\{^
a a c d\}`);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Created definition for T(
up,dn,dn,dn) " }}{PARA 6 "" 1 "" {TEXT -1 32 "Created definition for T
T(dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "grdef(`TC\{b c \+
d\}:=T\{^a b c d ;a\}`);" }}{PARA 6 "" 1 "" {TEXT -1 43 "Created a def
inition for T(up,dn,dn,dn,cdn)" }}{PARA 6 "" 1 "" {TEXT -1 35 "Created
 definition for TC(dn,dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "grcalc(T(dn,dn,dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CP
U~Time~G$\"%#)=!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gral
ter(_,expand,factor);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simpli
fication of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}
{PARA 6 "" 1 "" {TEXT -1 48 "Applying routine expand to object T(dn,dn
,dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 48 "Applying routine factor to obj
ect T(dn,dn,dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"
$5$!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Check that $T^\{a\}_\{
~acd\}=0$." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grcalc(TT(dn,dn));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$?\"!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9For~the~newkn~spacetime:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%*TT(dn,dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#TTG6
$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#TTG6#%\"aG\"\"\"&%!G
6#%\"bGF)%8All~components~are~zeroG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 52 "Check that $T^\{abcd\}_\{~~~~;a\}=0$ for vacuum ($Q=0$)." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Q:=0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"QG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "grcalc
(TC(dn,dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"%U5
!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gralter(_,expand,fa
ctor);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a G
RTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 46 "Applying routine expand to object TC(dn,dn,dn)" }}{PARA 
6 "" 1 "" {TEXT -1 46 "Applying routine factor to object TC(dn,dn,dn)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#5!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9For~the~newkn~spacetime:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%-TC(dn,dn,dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%-T
C(dn,dn,dn)G%8All~components~are~zeroG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "Display $T_\{abcd\}$ for the Kerr metric" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "grmap(T(dn,dn,dn,dn),subs,u=a*cos(theta),`x`);" 
}}{PARA 6 "" 1 "" {TEXT -1 39 "Applying routine subs to T(dn,dn,dn,dn)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%9For~the~newkn~spacetime:G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%/T(dn,dn,dn,dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/&%\"TG6#**%\"rG\"\"\"%\"rGF)%\"rGF)%\"rGF),$*&,&*&)%\"mG\"\"#F))%
\"rGF3F)F)*()%\"aGF3F))-%$cosG6#%&thetaGF3F)F1F)F)F)*&),&*$F4F)F)*&F7F
)F9F)F)F3F)),(FAF)*(F3F)F2F)F5F)!\"\"*$F7F)F)F3F)FF\"\"'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/&%\"TG6#**%\"rG\"\"\"%\"rGF)%\"uGF)%\"uGF),$*&,
&*&)%\"mG\"\"#F))%\"rGF3F)F)*()%\"aGF3F))-%$cosG6#%&thetaGF3F)F1F)F)F)
**),&*$F4F)F)*&F7F)F9F)F)F3F),&F8F)*&F8F)F:F)!\"\"F),&F8F)FDF)F),(FAF)
*(F3F)F2F)F5F)FE*$F7F)F)F)FE!\"%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&
%\"TG6#**%\"rG\"\"\"%\"rGF)%$phiGF)%$phiGF),$*&**,&*&)%\"mG\"\"#F))%\"
rGF4F)F)*()%\"aGF4F))-%$cosG6#%&thetaGF4F)F2F)F)F),&F9F)*&F9F)F;F)!\"
\"F),&F9F)F@F)F),0*$)F9\"\"%F)\"\"$*(\"\"&F)F8F)F5F)F)*&FEF)F:F)FA**F4
F)F8F)F3F)F6F)FA*,F4F)F8F)F:F)F3F)F6F)F)*(F5F)F8F)F:F)FA*&F4F))F6FFF)F
)F)F)*(,(*$F5F)F)*(F4F)F3F)F6F)FA*$F8F)F)F)F8F)),&FRF)*&F8F)F:F)F)FFF)
FA!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"TG6#**%\"rG\"\"\"%\"rGF
)%$phiGF)%\"tGF),$*&**,&*&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cos
G6#%&thetaGF4F)F2F)F)F),&F9F)*&F9F)F;F)!\"\"F),&F9F)F@F)F),(*$F8F)\"\"
$*(F4F)F3F)F6F)FA*&FEF)F5F)F)F)F)*(F9F)),&*$F5F)F)*&F8F)F:F)F)\"\"%F),
(FKF)*(F4F)F3F)F6F)FAFDF)F)FAF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%
\"TG6#**%\"rG\"\"\"%\"rGF)%\"tGF)%\"tGF),$*&*&,&*&)%\"mG\"\"#F))%\"rGF
4F)F)*()%\"aGF4F))-%$cosG6#%&thetaGF4F)F2F)F)F),**&F3F)F6F)!\"#*$F5F)F
)*&\"\"$F)F8F)F)*(F4F)F8F)F:F)!\"\"F)F)*&,(FBF)*(F4F)F3F)F6F)FF*$F8F)F
)F)),&FBF)*&F8F)F:F)F)\"\"%F)FFFA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
&%\"TG6#**%\"uG\"\"\"%\"uGF)%\"uGF)%\"uGF),$*&,&*&)%\"mG\"\"#F))%\"rGF
3F)F)*()%\"aGF3F))-%$cosG6#%&thetaGF3F)F1F)F)F)*(),&*$F4F)F)*&F7F)F9F)
F)F3F)),&F8F)*&F8F)F:F)!\"\"F3F)),&F8F)FEF)F3F)FF\"\"'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/&%\"TG6#**%\"uG\"\"\"%\"uGF)%$phiGF)%$phiGF),$*&*
&,&*&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cosG6#%&thetaGF4F)F2F)F)
F),0*(F8F)F3F)F6F)!\"%*,\"\"%F)F8F)F:F)F3F)F6F)F)*$)F6FCF)F)*(FCF)F8F)
F5F)F)**F4F)F5F)F8F)F:F)!\"\"*(F4F))F9FCF)F:F)FH*&\"\"$F)FJF)F)F)F)*&F
8F)),&*$F5F)F)*&F8F)F:F)F)FCF)FHF4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/&%\"TG6#**%\"uG\"\"\"%\"uGF)%$phiGF)%\"tGF),$*&*&,&*&)%\"mG\"\"#F))%
\"rGF4F)F)*()%\"aGF4F))-%$cosG6#%&thetaGF4F)F2F)F)F),(*&F3F)F6F)!\"%*&
\"\"$F)F5F)F)*&FCF)F8F)F)F)F)*&),&*$F5F)F)*&F8F)F:F)F)\"\"%F)F9F)!\"\"
!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"TG6#**%\"uG\"\"\"%\"uGF)%
\"tGF)%\"tGF),$*&*&,&*&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cosG6#
%&thetaGF4F)F2F)F)F),**&F3F)F6F)!\"%*&F4F)F5F)F)*&F8F)F:F)!\"\"*&\"\"$
F)F8F)F)F)F)*(,&F9F)*&F9F)F;F)FDF),&F9F)FIF)F)),&*$F5F)F)FCF)\"\"%F)FD
F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"TG6#**%$phiG\"\"\"%$phiGF)%
$phiGF)%$phiGF),$*&**,&*&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cosG
6#%&thetaGF4F)F2F)F)F)),&F9F)*&F9F)F;F)!\"\"F4F)),&F9F)FAF)F4F),P*$)F9
\"\")F)\"\"'**\"\"%F))F6FIF)F8F)F:F)FB*$)F6FHF)F)**\"#;F))F9FIF)F5F)F:
F)FB*(\"#=F)FQF)F5F)F)*(\"#>F))F9FKF))F6FKF)F)*(FHF)FLF)F8F)F)*&FGF))F
;FKF)F)*(FIF)FGF)F:F)FB*(FWF)FVF)FZF)F)**\"#7F)FQF)F3F)F6F)FB*,FHF)F2F
)F5F)FVF)F:F)FB*,FPF)FQF)F6F)F3F)F:F)F)*,\"#CF)FVF))F6\"\"$F)F3F)F:F)F
)*,FKF)FQF)F6F)FZF)F3F)FB*,FHF)F3F))F6\"\"&F)F8F)F:F)F)*,FKF)F3F)F]oF)
FVF)FZF)FB*,FKF)F2F)F5F)FVF)FZF)F)**\"#?F)FVF)F]oF)F3F)FB**FHF)FaoF)F8
F)F3F)FB**FKF)FVF)F2F)F5F)F)**F4F)FQF)F5F)FZF)F)**\"#9F)FVF)FWF)F:F)FB
F)F)*&),&*$F5F)F)*&F8F)F:F)F)FIF)FVF)FBFI" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/&%\"TG6#**%$phiG\"\"\"%$phiGF)%$phiGF)%\"tGF),$*&**,&*
&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cosG6#%&thetaGF4F)F2F)F)F)),
&F9F)*&F9F)F;F)!\"\"F4F)),&F9F)FAF)F4F),>*$)F9\"\"'F)FH**\"#7F))F9\"\"
%F)F3F)F6F)FB*(\"\"$F)FGF)F:F)FB*(\"#:F)FKF)F5F)F)**\"#;F)F8F))F6FNF)F
3F)FB**FHF)FKF)F5F)F:F)FB*(FJF)F8F))F6FLF)F)**FLF)F8F)F2F)F5F)F)*,\"\"
)F)FKF)F:F)F3F)F6F)F)*(FLF))F6\"\"&F)F3F)FB*,FYF)FSF)F8F)F:F)F3F)F)*&F
NF))F6FHF)F)*,FLF)F8F)F:F)F2F)F5F)FB**FNF)FVF)F8F)F:F)FBF)F)*&),&*$F5F
)F)*&F8F)F:F)F)FHF))F9FNF)FB!\"'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&
%\"TG6#**%$phiG\"\"\"%$phiGF)%\"tGF)%\"tGF),$*&**,&*&)%\"mG\"\"#F))%\"
rGF4F)F)*()%\"aGF4F))-%$cosG6#%&thetaGF4F)F2F)F)F),&F9F)*&F9F)F;F)!\"
\"F),&F9F)F@F)F),D*(F5F))F9\"\"%F))F;FFF)F4*(FFF))F6\"\"&F)F3F)FA*(F4F
))F9\"\"'F)FGF)F)**\"#9F))F6FFF)F8F)F:F)FA**\"#7F)F8F)F2F)F5F)F)**\"#O
F)F8F))F6\"\"$F)F3F)FA**\"#KF)FEF)F5F)F:F)FA*,\"#GF)FUF)F8F)F:F)F3F)F)
*,FRF)F8F)F:F)F2F)F5F)FA*,FFF)FEF)FGF)F3F)F6F)FA*&F4F))F6FMF)F)*(\"#?F
)F8F)FPF)F)*(FTF)FEF)F5F)F)*(\"#=F)FLF)F:F)FA*&F]oF)FLF)F)**FTF)FEF)F3
F)F6F)FA*,FTF)FEF)F:F)F3F)F6F)F)F)F)*&),&*$F5F)F)*&F8F)F:F)F)FMF)F8F)F
AF4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"TG6#**%$phiG\"\"\"%\"tGF)%
\"tGF)%\"tGF),$*&**,&*&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cosG6#
%&thetaGF4F)F2F)F)F),&F9F)*&F9F)F;F)!\"\"F),&F9F)F@F)F),4*$)F9\"\"%F)
\"\"'*(\"\"*F)F8F)F5F)F)**\"#7F)F8F)F3F)F6F)FA*(\"\"$F)FEF)F:F)FA*(FFF
)F2F)F5F)F)*,FFF)F8F)F:F)F3F)F6F)F)*(\"\")F)F3F))F6FMF)FA**FMF)F5F)F8F
)F:F)FA*&FMF))F6FFF)F)F)F)*&),&*$F5F)F)*&F8F)F:F)F)FGF)F9F)FA!\"'" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"TG6#**%\"tG\"\"\"%\"tGF)%\"tGF)%
\"tGF),$*&*&,&*&)%\"mG\"\"#F))%\"rGF4F)F)*()%\"aGF4F))-%$cosG6#%&theta
GF4F)F2F)F)F),6F1\"\"%*(F@F)F3F))F6\"\"$F)!\"\"*,\"\")F)F8F)F:F)F3F)F6
F)F)**\"#7F)F8F)F3F)F6F)FD*$)F6F@F)F)*(\"\"'F)F8F)F5F)F)**F@F)F5F)F8F)
F:F)FD*&)F9F@F))F;F@F)F)*&FLF)FOF)F)*(FLF)FOF)F:F)FDF)F)*$),&*$F5F)F)*
&F8F)F:F)F)FLF)FDFL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
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