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{SECT 0 {EXCHG {PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT 258 16 "RIEMANN PACKAGE " }}{PARA 260 "" 0 "" {TEXT -1 39 "Tools
 to manipulate tensor components, " }}{PARA 261 "" 0 "" {TEXT -1 42 "A
pplications to General Relativity Theory," }}{PARA 262 "" 0 "" {TEXT 
-1 33 "Some symbolic manipulation tools." }}{PARA 264 "" 0 "" {TEXT 
-1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 43 "Renato Portugal/CBPF - portug
al@cat.cbpf.br" }}{PARA 265 "" 0 "" {TEXT -1 46 "Sandra L.  Sautu/CBPF
 - sautu@lca1.drp.cbpf.br" }}{PARA 266 "" 0 "" {TEXT -1 24 "version 2 \+
- Aug 18, 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" 
{TEXT 256 43 "Examples in Spherically Symmetric Spacetime" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 351 "     This workshe
et describes the calculations of many geometrical quantities of the Sc
hwarzschild spacetime. Some quantities are built-in tensor and are cal
culated automatically while others are calculated through expressions \+
defined interactivelly. Many functions of the package are describe her
e. More details can be obtained from the help on line." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "     To load the Ri
emann package first give the path of the directory where the package h
ave been saved. For DOS users:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#
 libname:=libname,`c:/riemann`:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
127 "This command can be put in the file maple.ini in the directory c:
\\mapleV4\\lib. In this case it will be processed automatically." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "For unix \+
users:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "libname:=libname,`/local/
Riemann`:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "This command can be \+
put in the file .mapleinit in the home directory to be processed autom
atically." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "     To load the package:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "with(Riemann); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7M%&acompG%
%amapG%/antisymmetrizeG%&applyG%*bintnamesG%%calcG%&clearG%*clearbintG
%*clearcompG%,cleartensorG%'codiffG%%compG%)complistG%,coordinatesG%-d
efinetensorG%,entertensorG%&evaltG%%initG%*ivierbeinG%)lptensorG%(lten
sorG%'metricG%(metricVG%$offG%#onG%%pathG%'petrovG%,printtensorG%+read
metricG%+readtensorG%-readvierbeinG%+savemetricG%+savetensorG%-savevie
rbeinG%%showG%(simpfcnG%%sumtG%)switchesG%+symmetrizeG%&tdiffG%%tmapG%
)vierbeinG%,viertocoordG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 126 "     To calculate any built-in tensor in
 coordinate basis one must first give the coordinates and the metric o
f the spacetime:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "coordinates(t,r
,theta,phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~coordinates~are:
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"XG\"\"\")%!G%\"1GF&%\"tG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"XG\"\"\"%!G\"\"#%\"rG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&%\"XG\"\"\"%!G\"\"$%&thetaG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&%\"XG\"\"\"%!G\"\"%%$phiG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "metric(-a(r),1/a(r),r^2,r^2*sin(theta)^2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BThe~components~of~the~metric~are
:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\"&%!G6#%\"tGF&&F(F)
F&,$-%\"aG6#%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"
\"&%!G6#%\"rGF&&F(F)F&*$-%\"aGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/*(%\"gG\"\"\"&%!G6#%&thetaGF&&F(F)F&*$%\"rG\"\"#" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*(%\"gG\"\"\"&%!G6#%$phiGF&&F(F)F&*&%\"rG\"\"#-%$si
nG6#%&thetaGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 198 "     The built-in tensors are: Riemann, Ricci, Ricc
i scalar, Einstein, Weyl, Riemann square, Weyl square. The inverse of \+
the metric, its determinant and the Christoffel symbols are also imple
mented." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
87 "      For example, the components of the inverse metric are printe
d using the function " }{HYPERLNK 17 "show" 2 "show" "" }{TEXT -1 1 ":
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "show(g[i,j]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%\"tGF&)F(F)F&,$*$-%\"aG6#%\"rG!\"
\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%\"rGF&)F(F)
F&-%\"aG6#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%&th
etaGF&)F(F)F&*$%\"rG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"
\"\")%!G%$phiGF&)F(F)F&*&%\"rG!\"#-%$sinG6#%&thetaGF-" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The cont
ravariant indices are  indicated by positive names and the covariant o
nes by negative names. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 33 "The determinant of the metric is:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "show(detg[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
%%detgG,$*&%\"rG\"\"%-%$sinG6#%&thetaG\"\"#!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "All tensors must h
ave square brackets (including scalars). The line element is:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "show(ds2[]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$ds2G,**&-%\"aG6#%\"rG\"\"\"-%\"dG6#%\"tG\"\"#!\"\"*&
F'F1-F-F)F0F+*&F*F0-F-6#%&thetaGF0F+*(F*F0-%$sinGF6F0-F-6#%$phiGF0F+" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 
"The Christoffel symbols are:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sh
ow(Gamma[i,-j,-k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&GammaG\"\"
\")%!G%\"rGF&&F(6#%\"tGF&&F(F+F&,$*&-%\"aG6#F)F&-%%diffG6$F0F)F&#F&\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%\"tGF&&F(
6#F)F&&F(6#%\"rGF&,$*&-%\"aGF-!\"\"-%%diffG6$F1F.F&#F&\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%\"rGF&&F(6#F)F&&F(F+F&
,$*&-%\"aGF+!\"\"-%%diffG6$F/F)F&#F1\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%&thetaGF&&F(6#%\"rGF&&F(6#F)F&*$F
,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%$phiGF
&&F(6#%\"rGF&&F(6#F)F&*$F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**
%&GammaG\"\"\")%!G%\"rGF&&F(6#%&thetaGF&&F(F+F&,$*&-%\"aG6#F)F&F)F&!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%$phiGF&&F(
6#%&thetaGF&&F(6#F)F&*&-%$sinGF+!\"\"-%$cosGF+F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%\"rGF&&F(6#%$phiGF&&F(F+F&,$*(-%
\"aG6#F)F&F)F&-%$sinG6#%&thetaG\"\"#!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&GammaG\"\"\")%!G%&thetaGF&&F(6#%$phiGF&&F(F+F&,$*&
-%$sinG6#F)F&-%$cosGF2F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 58 "The totally covariant component of th
e Riemann tensor are:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "show(R[-i,
-j,-k,-l]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"
tGF&&F(6#%\"rGF&&F(F)F&&F(F,F&,$-%%diffG6$-F26$-%\"aGF,F-F-#!\"\"\"\"#
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"tGF&&F(6#%&
thetaGF&&F(F)F&&F(F,F&,$*(-%%diffG6$-%\"aG6#%\"rGF8F&F5F&F8F&#!\"\"\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"tGF&&F(6
#%$phiGF&&F(F)F&&F(F,F&,$**-%%diffG6$-%\"aG6#%\"rGF8F&F5F&F8F&-%$sinG6
#%&thetaG\"\"##!\"\"F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"
\"&%!G6#%\"rGF&&F(6#%&thetaGF&&F(F)F&&F(F,F&,$*(-%\"aGF)!\"\"-%%diffG6
$F2F*F&F*F&#F&\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&
%!G6#%\"rGF&&F(6#%$phiGF&&F(F)F&&F(F,F&,$**-%\"aGF)!\"\"-%%diffG6$F2F*
F&F*F&-%$sinG6#%&thetaG\"\"##F&F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*,%\"RG\"\"\"&%!G6#%&thetaGF&&F(6#%$phiGF&&F(F)F&&F(F,F&,&*(%\"rG\"\"#
-%\"aG6#F2F&-%$sinGF)F3F&*&F2F3F7F3!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "One can print mixed compo
nents:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "show(R[i,-j,-k,-l]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"tGF&&F(6#%\"rGF&&
F(6#F)F&&F(F+F&,$*&-%\"aGF+!\"\"-%%diffG6$-F66$F2F,F,F&#F&\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"rGF&&F(6#%\"tGF&&
F(F+F&&F(6#F)F&,$*&-%\"aGF/F&-%%diffG6$-F56$F2F)F)F&#F&\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"tGF&&F(6#%&thetaGF&&F(6
#F)F&&F(F+F&,$*&-%%diffG6$-%\"aG6#%\"rGF8F&F8F&#F&\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%&thetaGF&&F(6#%\"tGF&&F(F+F&&
F(6#F)F&,$*(%\"rG!\"\"-%%diffG6$-%\"aG6#F2F2F&F7F&#F&\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"tGF&&F(6#%$phiGF&&F(6#F
)F&&F(F+F&,$*(-%%diffG6$-%\"aG6#%\"rGF8F&F8F&-%$sinG6#%&thetaG\"\"##F&
F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%$phiGF&&F(6#%
\"tGF&&F(F+F&&F(6#F)F&,$*(%\"rG!\"\"-%%diffG6$-%\"aG6#F2F2F&F7F&#F&\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"rGF&&F(6#%
&thetaGF&&F(6#F)F&&F(F+F&,$*&-%%diffG6$-%\"aGF.F)F&F)F&#F&\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%&thetaGF&&F(6#%\"rG
F&&F(F+F&&F(6#F)F&,$*(F,!\"\"-%\"aGF+F2-%%diffG6$F3F,F&#F2\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"rGF&&F(6#%$phiGF&
&F(6#F)F&&F(F+F&,$*(-%%diffG6$-%\"aGF.F)F&F)F&-%$sinG6#%&thetaG\"\"##F
&F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%$phiGF&&F(6#
%\"rGF&&F(F+F&&F(6#F)F&,$*(F,!\"\"-%\"aGF+F2-%%diffG6$F3F,F&#F2\"\"#" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%&thetaGF&&F(6#%$p
hiGF&&F(6#F)F&&F(F+F&*&-%$sinGF.\"\"#,&-%\"aG6#%\"rGF&!\"\"F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%$phiGF&&F(6#%&theta
GF&&F(F+F&&F(6#F)F&,&-%\"aG6#%\"rG!\"\"F&F&" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "or only one component:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "show(R[t,r,t,r]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\")%!G%\"tGF&)F(%\"rGF&)F(F)F&)F(F+F
&,$-%%diffG6$-F06$-%\"aG6#F+F+F+#!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The covariant component
s of the Ricci tensor are:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "show(
R[-i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#%\"t
GF&&F(F)F&,$*(-%\"aG6#%\"rGF&,&*&-%%diffG6$-F56$F.F1F1F&F1F&F&F7\"\"#F
&F1!\"\"#F&F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#%
\"rGF&&F(F)F&,$*(,&*&-%%diffG6$-F16$-%\"aGF)F*F*F&F*F&F&F3\"\"#F&F*!\"
\"F5F8#F8F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#%&t
hetaGF&&F(F)F&,(*&-%%diffG6$-%\"aG6#%\"rGF4F&F4F&!\"\"F1F5F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#%$phiGF&&F(F)F&,(*
(-%%diffG6$-%\"aG6#%\"rGF4F&F4F&-%$sinG6#%&thetaG\"\"#!\"\"*&F5F9F1F&F
:*$F5F9F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "The Ricci scalar is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "show(R[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"RG,$*&,**&-%%dif
fG6$-F*6$-%\"aG6#%\"rGF1F1\"\"\"F1\"\"#F2*&F,F2F1F2\"\"%F.F3!\"#F2F2F1
F6!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 56 "The contravariant components of the Einstein tensor are:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "show(G[i,j]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*(%\"GG\"\"\")%!G%\"tGF&)F(F)F&,$*(,(*&-%%diffG6$-%
\"aG6#%\"rGF5F&F5F&F&F2F&!\"\"F&F&F2F6F5!\"#F6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"GG\"\"\")%!G%\"rGF&)F(F)F&*(-%\"aG6#F)F&,(*&-%%di
ffG6$F,F)F&F)F&F&F,F&!\"\"F&F&F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*(%\"GG\"\"\")%!G%&thetaGF&)F(F)F&,$*&,&*&-%%diffG6$-F06$-%\"aG6#%
\"rGF7F7F&F7F&F&F2\"\"#F&F7!\"$#F&F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*(%\"GG\"\"\")%!G%$phiGF&)F(F)F&,$*(,&*&-%%diffG6$-F06$-%\"aG6#%\"r
GF7F7F&F7F&F&F2\"\"#F&F7!\"$-%$sinG6#%&thetaG!\"##F&F8" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "     Let
 us use the functions of the package to build the Einstein equation in
 vacuum to find the function a(r). The function complist gives the lis
t of the components of a tensor:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"complist(G[-i,-j]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,,$*(-%\"aG6#
%\"rG\"\"\",(*&-%%diffG6$F&F)F*F)F*F*F&F*!\"\"F*F*F)!\"#F0\"\"!F2F2*(F
+F*F&F0F)F1F2F2,&F,F**&-F.6$F-F)F*F)\"\"##F*F8F2,&*(F-F*F)F*-%$sinG6#%
&thetaGF8F**(F<F8F6F*F)F8F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 24 "Let us cancel the zeros:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "subs(0=NULL,\");" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7&,$*(-%\"aG6#%\"rG\"\"\",(*&-%%diffG6$F&F)F*F)F*F*F&F*!\"\"F*F*
F)!\"#F0*(F+F*F&F0F)F1,&F,F**&-F.6$F-F)F*F)\"\"##F*F7,&*(F-F*F)F*-%$si
nG6#%&thetaGF7F**(F;F7F5F*F)F7F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "and solve the differential equatio
ns:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "map(dsolve,\",a(r));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/-%\"aG6#%\"rG*&F(!\"\",&F(\"\"\"%$_
C1GF,F,F$/F%,&F-F,*&%$_C2GF,F(F*F,F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The most restrictive solution \+
is the first one. Let us put the integration constant _C1 equal to -2*
m:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(_C1=-2*m,\"[1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"aG6#%\"rG*&F'!\"\",&F'\"\"\"%\"mG
!\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(\");" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Le
t us confirm now that the Ricci component are zero:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "show(R[-i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*(%\"RG\"\"\"&%!G6#%\"tGF&&F(F)F&,$*(%\"rG!\"#,&F.F&%\"mGF/F&,(*&,&*
&F.!\"$F0F&\"\"#*$F.F/F/F&F.F&F&*&F.F/F0F&F/*$F.!\"\"F7F&#F&F7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#%\"rGF&&F(F)F&,$*&
,(*&,&*&F*!\"$,&F*F&%\"mG!\"#F&\"\"#*$F*F5F5F&F*F&F&*&F*F5F3F&F5*$F*!
\"\"F6F&F3F:#F:F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!
G6#%&thetaGF&&F(F)F&,(*&,&*&%\"rG!\"#,&F0F&%\"mGF1F&!\"\"*$F0F4F&F&F0F
&F4*&F0F4F2F&F4F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&
%!G6#%$phiGF&&F(F)F&,(*(,&*&%\"rG!\"#,&F0F&%\"mGF1F&!\"\"*$F0F4F&F&F0F
&-%$sinG6#%&thetaG\"\"#F4*(F6F:F0F4F2F&F4*$F6F:F&" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "To simplify each c
omponent:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "tmap(normal, _);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"\"\"&%!G6#%\"iGF&&F(6
#%\"jGF&\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 62 "The character _ refers to the tensor used in the last \+
command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
204 "The components of the Ricci tensor has not been changed since we \+
have not used any function that assigns the results to the components \+
(see ?amap, ?calc, ?acomp). So we can obtain back the initial values:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a(r):='a(r)';" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>-%\"aG6#%\"rGF$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "show(R[-i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
(%\"RG\"\"\"&%!G6#%\"tGF&&F(F)F&,$*(-%\"aG6#%\"rGF&,&*&-%%diffG6$-F56$
F.F1F1F&F1F&F&F7\"\"#F&F1!\"\"#F&F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*(%\"RG\"\"\"&%!G6#%\"rGF&&F(F)F&,$*(,&*&-%%diffG6$-F16$-%\"aGF)F*F*
F&F*F&F&F3\"\"#F&F*!\"\"F5F8#F8F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*(%\"RG\"\"\"&%!G6#%&thetaGF&&F(F)F&,(*&-%%diffG6$-%\"aG6#%\"rGF4F&F4F
&!\"\"F1F5F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#
%$phiGF&&F(F)F&,(*(-%%diffG6$-%\"aG6#%\"rGF4F&F4F&-%$sinG6#%&thetaG\"
\"#!\"\"*&F5F9F1F&F:*$F5F9F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 38 "The components of the Weyl tensor are:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "show(C[i,j,k,l]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\")%!G%\"tGF&)F(%\"rGF&)F(F)F&)F(F+F
&,$*&,**&-%%diffG6$-F36$-%\"aG6#F+F+F+F&F+\"\"#F&*&F5F&F+F&!\"#F7F:F<F
&F&F+F<#!\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\")%
!G%\"tGF&)F(%&thetaGF&)F(F)F&)F(F+F&,$*(%\"rG!\"%-%\"aG6#F0!\"\",**&-%
%diffG6$-F96$F2F0F0F&F0\"\"#F&*&F;F&F0F&!\"#F2F=F?F&F&#F&\"#7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\")%!G%\"tGF&)F(%$phiGF&)F
(F)F&)F(F+F&,$**%\"rG!\"%-%$sinG6#%&thetaG!\"#-%\"aG6#F0!\"\",**&-%%di
ffG6$-F>6$F7F0F0F&F0\"\"#F&*&F@F&F0F&F6F7FBF6F&F&#F&\"#7" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\")%!G%\"rGF&)F(%&thetaGF&)F(F)F&)F(
F+F&,$*(F)!\"%-%\"aG6#F)F&,**&-%%diffG6$-F76$F1F)F)F&F)\"\"#F&*&F9F&F)
F&!\"#F1F;F=F&F&#!\"\"\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"CG
\"\"\")%!G%\"rGF&)F(%$phiGF&)F(F)F&)F(F+F&,$**F)!\"%-%$sinG6#%&thetaG!
\"#-%\"aG6#F)F&,**&-%%diffG6$-F<6$F6F)F)F&F)\"\"#F&*&F>F&F)F&F5F6F@F5F
&F&#!\"\"\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\")%!G%&t
hetaGF&)F(%$phiGF&)F(F)F&)F(F+F&,$*(%\"rG!\"'-%$sinG6#F)!\"#,**&-%%dif
fG6$-F96$-%\"aG6#F0F0F0F&F0\"\"#F&*&F;F&F0F&F5F=F@F5F&F&#F&\"\"'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Th
e Riemann square scalar is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "show
(RieSq[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&RieSqG*&%\"rG!\"%,,*&
-%%diffG6$-F+6$-%\"aG6#F&F&F&\"\"#F&\"\"%\"\"\"*&F-F2F&F2F3*$F/F2F3F/!
\")F3F4F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 26 "The Weyl square scalar is:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "show(WeylSq[]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'WeylSqG
,$*&,**&-%%diffG6$-F*6$-%\"aG6#%\"rGF1F1\"\"\"F1\"\"#F2*&F,F2F1F2!\"#F
.F3F5F2F3F1!\"%#F2\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 133 "     Let us show now how to calculate ne
w objects using the functions of the package. For example, the square \+
of the Einstein tensor:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalt(G[
i,j]*G[-i,-j]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&,(*&-%%diffG6$-
%\"aG6#%\"rGF-\"\"\"F-F.F.F*F.!\"\"F.\"\"#F-!\"%F0*(,&*&-F(6$F'F-F.F-F
.F.F'F0F.F-!\"$,&F&F.*&F5F.F-F0#F.F0F.F:**F3F.F-F7-%$sinG6#%&thetaG!\"
#,&*(F'F.F-F.F<F0F.*(F<F0F5F.F-F0F:F.F:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "normal(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,2
*&-%%diffG6$-%\"aG6#%\"rGF-\"\"#F-F.\"\")*(F'\"\"\"F*F1F-F1F/*&F'F1F-F
1!\")*$F*F.\"\"%F*F3F5F1*&-F(6$F'F-F.F-F5F1*(F-\"\"$F7F1F'F1F5F1F-!\"%
#F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalt(\" - RicciSq
[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We can calculate new t
ensors:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalt(G[i,j]-3*g[i,j]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"\"\")%!G%\"tGF&)F
(F)F&,&*(,(*&-%%diffG6$-%\"aG6#%\"rGF5F&F5F&F&F2F&!\"\"F&F&F2F6F5!\"#F
6*$F2F6\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"\"\")
%!G%\"rGF&)F(F)F&,&*(-%\"aG6#F)F&,(*&-%%diffG6$F-F)F&F)F&F&F-F&!\"\"F&
F&F)!\"#F&F-!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"
\"\")%!G%&thetaGF&)F(F)F&,&*&,&*&-%%diffG6$-F06$-%\"aG6#%\"rGF7F7F&F7F
&F&F2\"\"#F&F7!\"$#F&F8*$F7!\"#F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*(%*componentG\"\"\")%!G%$phiGF&)F(F)F&,&*(,&*&-%%diffG6$-F06$-%\"aG6#
%\"rGF7F7F&F7F&F&F2\"\"#F&F7!\"$-%$sinG6#%&thetaG!\"##F&F8*&F7F>F:F>F9
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "tmap(normal,_);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"\"\")%!G%\"tGF&)F(F)F
&,$*(,**&-%%diffG6$-%\"aG6#%\"rGF5F&F5F&F&F2F&!\"\"F&*$F5\"\"#!\"$F&F2
F6F5!\"#F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"\"\")%!
G%\"rGF&)F(F)F&*(-%\"aG6#F)F&,**&-%%diffG6$F,F)F&F)F&F&F,F&!\"\"F&*$F)
\"\"#!\"$F&F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"
\"\")%!G%&thetaGF&)F(F)F&,$*&,(*&-%%diffG6$-F06$-%\"aG6#%\"rGF7F7F&F7F
&F&F2\"\"#F7!\"'F&F7!\"$#F&F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*
componentG\"\"\")%!G%$phiGF&)F(F)F&,$*(,(*&-%%diffG6$-F06$-%\"aG6#%\"r
GF7F7F&F7F&F&F2\"\"#F7!\"'F&F7!\"$-%$sinG6#%&thetaG!\"##F&F8" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "We
 can extend the built-in tensors through the " }{HYPERLNK 17 "definete
nsor" 2 "definetensor" "" }{TEXT -1 9 " command:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "definetensor(TclR[-i,-j]=R[-i,-j]-1/4*g[-i,-j]*R[],sy
mm[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\"&%!G6#%\"
iGF&&F(6#%\"jGF&,&*(%\"RGF&&F(F)F&&F(F,F&F&**%\"gGF&&F(F)F&&F(F,F&F0F&
#!\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 252 "Now, the tensor TclR has been added to the list of the
 built-in tensors.  The totally covariant components will be calculate
d through the definition and the contravariant components will be obta
ined by raising the indices using the metric. For example:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "show(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*(%%TclRG\"\"\"&%!G6#%\"tGF&&F(F)F&,$*(-%\"aG6#%\"rGF&,(*&-%%diffG6$-
F56$F.F1F1F&F1\"\"#!\"\"F.F9!\"#F&F&F1F;#F:\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%%TclRG\"\"\"&%!G6#%\"rGF&&F(F)F&,$*(,(*&-%%diffG6$-
F16$-%\"aGF)F*F*F&F*\"\"#!\"\"F5F7!\"#F&F&F5F8F*F9#F&\"\"%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\"&%!G6#%&thetaGF&&F(F)F&,(-%
\"aG6#%\"rG#!\"\"\"\"##F&F3F&*&-%%diffG6$-F76$F-F0F0F&F0F3#F&\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\"&%!G6#%$phiGF&&F(F)F&,
(*&-%$sinG6#%&thetaG\"\"#-%\"aG6#%\"rGF&#!\"\"F2*$F.F2#F&F2*(F.F2-%%di
ffG6$-F=6$F3F6F6F&F6F2#F&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "show(TclR[i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%%TclR
G\"\"\")%!G%\"tGF&&F(6#F)F&,$*&,(*&-%%diffG6$-F16$-%\"aG6#%\"rGF8F8F&F
8\"\"#!\"\"F5F9!\"#F&F&F8F;#F&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*(%%TclRG\"\"\")%!G%\"rGF&&F(6#F)F&,$*&,(*&-%%diffG6$-F16$-%\"aGF+F)
F)F&F)\"\"#!\"\"F5F7!\"#F&F&F)F9#F&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*(%%TclRG\"\"\")%!G%&thetaGF&&F(6#F)F&,$*&,(*&-%%diffG6$-F16$-%
\"aG6#%\"rGF8F8F&F8\"\"#!\"\"F5F9!\"#F&F&F8F;#F:\"\"%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\")%!G%$phiGF&&F(6#F)F&,$*&,(*&-%%di
ffG6$-F16$-%\"aG6#%\"rGF8F8F&F8\"\"#!\"\"F5F9!\"#F&F&F8F;#F:\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "show(TclR[i,j]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\")%!G%\"tGF&)F(F)F&,$*(,(*&-%
%diffG6$-F06$-%\"aG6#%\"rGF7F7F&F7\"\"#!\"\"F4F8!\"#F&F&F4F9F7F:#F9\"
\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\")%!G%\"rGF&)F(F
)F&,$*(-%\"aG6#F)F&,(*&-%%diffG6$-F36$F-F)F)F&F)\"\"#!\"\"F-F7!\"#F&F&
F)F9#F&\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%%TclRG\"\"\")%!G%&
thetaGF&)F(F)F&,$*&%\"rG!\"%,(*&-%%diffG6$-F26$-%\"aG6#F-F-F-F&F-\"\"#
!\"\"F6F9!\"#F&F&#F:\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%%TclR
G\"\"\")%!G%$phiGF&)F(F)F&,$*(%\"rG!\"%-%$sinG6#%&thetaG!\"#,(*&-%%dif
fG6$-F76$-%\"aG6#F-F-F-F&F-\"\"#!\"\"F;F>F3F&F&#F?\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalt(TclR[i,-i]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 236 "Thi
s tensor is the traceless Ricci tensor, and it is  present in the defa
ult list of tensor in Riemann package with the name S (see ?bintnames)
, then we can compare the last results with the components of S tensor
,  they must be equal:." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalt(Tc
lR[-i,-j] - S[-i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*compone
ntG\"\"\"&%!G6#%\"iGF&&F(6#%\"jGF&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "evalt(TclR[i,-j] - S[i,-j]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%*componentG\"\"\")%!G%\"iGF&&F(6#%\"jGF&\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalt(TclR[i,j] - S[i,j]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*componentG\"\"\")%!G%\"iGF&)F(%
\"jGF&\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 
"" {TEXT -1 239 "There is a function to calculate the covariant deriva
tive of a tensor. The result is a symbolic expression that can be used
 as argument of the function evalt, calc, definetensor, etc. For examp
le, the covariant derivative of the metric is:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "codiff(g[i,j],X[k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,(-&%&tdiffG6#%\"kG6#*(%\"gG\"\"\")%!G%\"iGF,)F.%\"jGF,F,*0%&GammaG
F,)F.F/F,&F.6#%#_xGF,&F.F'F,F+F,)F.F7F,)F.F1F,F,*0F3F,)F.F1F,&F.F6F,&F
.F'F,F+F,)F.F/F,)F.F7F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 37 "To show that the components are zero:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalt(\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%*componentG\"\"\")%!G%\"iGF&)F(%\"jGF&&F(6#%\"kGF&
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 74 "The expression to the covariant derivative of the mixed R
iemann tensor is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "codiff(R[i,-j,
-k,-l],X[m]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-&%&tdiffG6#%\"mG6#
*,%\"RG\"\"\")%!G%\"iGF,&F.6#%\"jGF,&F.6#%\"kGF,&F.6#%\"lGF,F,*4%&Gamm
aGF,)F.F/F,&F.6#%#_vGF,&F.F'F,F+F,)F.F>F,&F.F1F,&F.F4F,&F.F7F,F,*4F:F,
)F.F>F,&F.F1F,&F.F'F,F+F,)F.F/F,&F.F=F,&F.F4F,&F.F7F,!\"\"*4F:F,)F.F>F
,&F.F4F,&F.F'F,F+F,)F.F/F,&F.F1F,&F.F=F,&F.F7F,FL*4F:F,)F.F>F,&F.F7F,&
F.F'F,F+F,)F.F/F,&F.F1F,&F.F4F,&F.F=F,FL" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The expression to the cov
ariant derivative of the Ricci tensor is:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "codiff(R[-mu,-nu],X[lambda]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(-&%&tdiffG6#%'lambdaG6#*(%\"RG\"\"\"&%!G6#%#muGF,&F.6
#%#nuGF,F,*0%&GammaGF,)F.%'_gammaGF,&F.F/F,&F.F'F,F+F,&F.6#F7F,&F.F2F,
!\"\"*0F5F,)F.F7F,&F.F2F,&F.F'F,F+F,&F.F/F,&F.F;F,F=" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Let us calcula
te the following invariant (where DR is the covariant derivative of th
e Ricci tensor):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "definetensor(DR
[-mu,-nu,-lambda]=codiff(R[-mu,-nu],X[lambda]),symm[1,2]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/**%#DRG\"\"\"&%!G6#%#muGF&&F(6#%#nuGF&&F(6#
%'lambdaGF&,(-&%&tdiffGF/6#*(%\"RGF&&F(F)F&&F(F,F&F&*0%&GammaGF&)F(%%_
etaGF&&F(F)F&&F(F/F&F7F&&F(6#F=F&&F(F,F&!\"\"*0F;F&)F(F=F&&F(F,F&&F(F/
F&F7F&&F(F)F&&F(FAF&FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sh
ow(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%#DRG\"\"\"&%!G6#%\"tGF&&
F(F)F&&F(6#%\"rGF&,$*(-%\"aGF-F&,(*&-%%diffG6$-F66$-F66$F1F.F.F.F&F.\"
\"#F&*&F8F&F.F&F<F:!\"#F&F.F>#F&F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/**%#DRG\"\"\"&%!G6#%\"rGF&&F(F)F&&F(F)F&,$*(-%\"aGF)!\"\",(*&-%%diffG
6$-F56$-F56$F/F*F*F*F&F*\"\"#F&*&F7F&F*F&F;F9!\"#F&F*F=#F1F;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/**%#DRG\"\"\"&%!G6#%\"rGF&&F(6#%&thetaGF&&F
(F,F&,$*&,(-%\"aGF)!\"#\"\"#F&*&-%%diffG6$-F86$F2F*F*F&F*F5F&F&F*!\"\"
#F<F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%#DRG\"\"\"&%!G6#%\"rGF&&F
(6#%$phiGF&&F(F,F&,$*(-%$sinG6#%&thetaG\"\"#,(-%\"aGF)!\"#F5F&*&-%%dif
fG6$-F<6$F7F*F*F&F*F5F&F&F*!\"\"#F@F5" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/**%#DRG\"\"\"&%!G6#%&thetaGF&&F(F)F&&F(6#%\"rGF&,$*&,(-%\"aGF-!\"#
\"\"#F&*&-%%diffG6$-F86$F2F.F.F&F.F5F&F&F.!\"\"F<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%#DRG\"\"\"&%!G6#%$phiGF&&F(F)F&&F(6#%\"rGF&,$*(-%$s
inG6#%&thetaG\"\"#,(-%\"aGF-!\"#F5F&*&-%%diffG6$-F<6$F7F.F.F&F.F5F&F&F
.!\"\"F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalt(DR[i,j,k]
*DR[-i,-j,-k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,(*&-%%diffG6$-
F(6$-F(6$-%\"aG6#%\"rGF1F1F1\"\"\"F1\"\"#F2*&F*F2F1F2F3F,!\"#F3F1!\"%F
.F2#F2F3*(F1!\"'F.F2,(F.F5F3F2*&F*F2F1F3F2F3\"\"$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "normal(\");" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#,$*(-%\"aG6#%\"rG\"\"\",8*&F(\"\"'-%%diffG6$-F.6$-F.6$F%F(F(F(\"\"#
F)*(F(\"\"&F-F)F0F)\"\"%*(F(F7F-F)F2F)!\"%*&F0F4F(F7\"#5*(F(\"\"$F0F)F
2F)!\")*&F2F4F(F4F7*$F%F4\"#CF%!#[*(F(F4F%F)F0F)!#CFAF)*&F0F)F(F4FAF)F
(!\"'#F)F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 36 "The Carminati-McLenaghan invariants:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 17 "invariants(show);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"RG6\",$*&,**&-%%diffG6$-F,6$-%\"aG6#%\"rGF3F3\"\"
\"F3\"\"#F4*&F.F4F3F4\"\"%F0F5!\"#F4F4F3F8!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%#r1G6\",$*&,(-%\"aG6#%\"rG!\"#\"\"#\"\"\"*&-%%diffG6
$-F36$F*F-F-F0F-F/F0F/F-!\"%#F0\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/&%#r2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#r3G6\",$*&,(-%
\"aG6#%\"rG!\"#\"\"#\"\"\"*&-%%diffG6$-F36$F*F-F-F0F-F/F0\"\"%F-!\")#F
0\"%C5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$w1rG6\",$*&,**&-%%diffG6
$-F,6$-%\"aG6#%\"rGF3F3\"\"\"F3\"\"#F4*&F.F4F3F4!\"#F0F5F7F4F5F3!\"%#F
4\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$w1IG6\"\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/&%$w2rG6\",$*&,**&-%%diffG6$-F,6$-%\"aG6#%\"rG
F3F3\"\"\"F3\"\"#F4*&F.F4F3F4!\"#F0F5F7F4\"\"$F3!\"'#!\"\"\"$)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$w2IG6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%$m1rG6\",$*(,(-%\"aG6#%\"rG!\"#\"\"#\"\"\"*&-%%diffG
6$-F36$F*F-F-F0F-F/F0F/,*F1F0*&F5F0F-F0F.F*F/F.F0F0F-!\"'#!\"\"\"#'*" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$m1IG6\"\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/&%$m2rG6\",$*(,**&-%%diffG6$-F,6$-%\"aG6#%\"rGF3F3\"
\"\"F3\"\"#F4*&F.F4F3F4!\"#F0F5F7F4F5,(F0F7F5F4F*F4F5F3!\")#F4\"$w&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$m2IG6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%#m3G6\",$*(,**&-%%diffG6$-F,6$-%\"aG6#%\"rGF3F3\"\"
\"F3\"\"#F4*&F.F4F3F4!\"#F0F5F7F4F5,(F0F7F5F4F*F4F5F3!\")#F4\"$w&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#m4G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%$m5rG6\",$*(,**&-%%diffG6$-F,6$-%\"aG6#%\"rGF3F3\"\"
\"F3\"\"#F4*&F.F4F3F4!\"#F0F5F7F4\"\"$F3!#5,(F0F7F5F4F*F4F5#!\"\"\"%cM
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$m5IG6\"\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "To print only o
ne invariant:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "show(m5r[]
);  # remember the bracketes [] for scalars" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$m5rG,$*(,**&-%%diffG6$-F*6$-%\"aG6#%\"rGF1F1\"\"\"F1
\"\"#F2*&F,F2F1F2!\"#F.F3F5F2\"\"$F1!#5,(F.F5F3F2F(F2F3#!\"\"\"%cM" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 180 "
     More details about the functions of the Riemann package and their
 applications, see the worksheets Riemann1.mws (tutorial) and Riemann3
.mws (Kerr in null vierbein). See also ?" }{HYPERLNK 17 "Riemann" 2 "r
iemann" "" }{TEXT -1 1 "." }}}}{MARK "0 10 0" 7 }{VIEWOPTS 1 1 0 1 1 
1803 }