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{SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT -1 1 "\n" }{TEXT 256 17 "RIEMAN
N PACKAGE  " }}{PARA 260 "" 0 "" {TEXT -1 39 "Tools to manipulate tens
or components, " }}{PARA 261 "" 0 "" {TEXT -1 42 "Applications to Gene
ral 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 - portugal@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 267 "" 0 "" {TEXT -1 18 "Example
 with  the " }{TEXT 257 11 "Kerr metric" }{TEXT -1 45 " in vierbein ba
sis in  spherical coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 306 "    This worksheet describes the calcula
tions of many tensors of the gravitational theory in the Kerr riemanni
an manifold  using vierbein basis. Also, this example shows that a sui
ted choice of the simplifying function  and applying specific simplifi
cation command is crucial to obtain  readable  results. " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 128 "     The first step is to give the path \+
of the directory where the package have been saved in order to load th
e Riemann package:" }}{PARA 0 "" 0 "" {TEXT -1 19 "     For DOS users:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# libname:=libname,`c/riemann`:
    # for example" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "    \n     F
or unix users:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# libname:=libnam
e,`/local/Riemann`: # for example" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
4 "    " }}{PARA 0 "" 0 "" {TEXT -1 25 "     To load the package:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read Riemann; # or with(Riem
ann);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7M%&acompG%%amapG%/antisymmet
rizeG%&applyG%*bintnamesG%%calcG%&clearG%*clearbintG%*clearcompG%,clea
rtensorG%'codiffG%%compG%)complistG%,coordinatesG%-definetensorG%,ente
rtensorG%&evaltG%%initG%*ivierbeinG%)lptensorG%(ltensorG%'metricG%(met
ricVG%$offG%#onG%%pathG%'petrovG%,printtensorG%+readmetricG%+readtenso
rG%-readvierbeinG%+savemetricG%+savetensorG%-savevierbeinG%%showG%(sim
pfcnG%%sumtG%)switchesG%+symmetrizeG%&tdiffG%%tmapG%)vierbeinG%,vierto
coordG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "     The user can work \+
in any dimension.  In this example it is used the default dimension 4.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Dimension;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 113 "     The user must begin by defining two funda
mentals objects: the coordinates and the metric tensor in vierbein." }
}{PARA 0 "" 0 "" {TEXT -1 31 "     To define the coordinates:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "coordinates(t,r, theta, phi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~coordinates~are:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&%\"XG\"\"\")%!G%\"1GF&%\"tG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*&%\"XG\"\"\"%!G\"\"#%\"rG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"XG\"\"\"%!G\"\"$%&thetaG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%\"XG\"\"\"%!G\"\"%%$phiG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 271 "\n     There are two ways to enter the metric componen
ts: the interactive and the direct ways. In the present example it is \+
used the interactive way. There are  cases which are more usefull to e
ntry  the metric by the direct way. More details can be found in help \+
on line." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "metricV();" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%I1->~metric~in~orthonormal~vierbein~basisG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%J2->~metric~in~complex~null~vierbein~
basisG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G3->~metric~in~semi-null~vi
erbein~basisG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*4->~otherG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%1ENTER~THE~OPTIONG" }}}{EXCHG {PARA 
0 "option 1,2,3 or 4 ?" 0 "" {MPLTEXT 1 0 3 " 2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%UThe~components~of~the~metric~in~the~rigid~frame~are:G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%$etaG\"\"\"&%!G6#%\"iGF&&F(6#%
\"jGF&-%'MATRIXG6#7&7&\"\"!F&F3F37&F&F3F3F37&F3F3F3!\"\"7&F3F3F6F3" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "     To entry the vierbein compo
nents:\n\n     In this example is convenient to define a variable A in
 order to simplify the typing of the vierbein components." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "     The user can
 entry the vierbein components or the inverse components, as he prefer
s.\n     The definition is:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "prin
ttensor(Theta[Alpha]=omega[Alpha,-alpha]*dx[alpha]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*&%&ThetaG\"\"\")%!G%&AlphaGF&*,%&omegaGF&)F(F)F&&F
(6#%&alphaGF&%#dxGF&)F(F/F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "w
here the first index of  vierbein component (omega) is a vierbein inde
x and the second one is a coordinate index.\n\n     In this example we
 entry the inverse vierbein:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "A:=r^2+a^2*cos(theta)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
AG,&*$%\"rG\"\"#\"\"\"*&%\"aGF(-%$cosG6#%&thetaGF(F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "ivierbein();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%>Convention~for~omega~indices:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%_oFirst~index~is~a~vierbein~index~and~second~one~is~a~
coordinate~indexG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6'%5ENTER~THE~COMPONENTSG*(%&omegaG\"\"\"&%!G6
#%\"tGF&)F(F*F&*(F%F&&F(F)F&)F(%\"rGF&*(F%F&&F(F)F&)F(%&thetaGF&*(F%F&
&F(F)F&)F(%$phiGF&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 54 "(r^2+a^
2)/(r^2-2*M*r+a^2) , 1, 0,  a/(r^2-2*M*r+a^2) ;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'%5ENTER~THE~COMPONENTSG*(%&omegaG\"\"\"&%!G6#%\"rGF&)F(
%\"tGF&*(F%F&&F(F)F&)F(F*F&*(F%F&&F(F)F&)F(%&thetaGF&*(F%F&&F(F)F&)F(%
$phiGF&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 55 "1/2*(r^2+a^2)/A, -
1/2*(r^2-2*M*r+a^2)/A , 0,  1/2*a/A ;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6'%5ENTER~THE~COMPONENTSG*(%&omegaG\"\"\"&%!G6#%&thetaGF&)F(%\"tGF&*(
F%F&&F(F)F&)F(%\"rGF&*(F%F&&F(F)F&)F(F*F&*(F%F&&F(F)F&)F(%$phiGF&" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 135 "1/2*I*a*sin(theta)*(r-I*a*cos
(theta))/A*2^(1/2) , 0, 1/2*(r-I*a*cos(theta))*2^(1/2)/A,   1/2*I*csc(
theta)*(r-I*a*cos(theta))/A*2^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6'%5ENTER~THE~COMPONENTSG*(%&omegaG\"\"\"&%!G6#%$phiGF&)F(%\"tGF&*(F%F
&&F(F)F&)F(%\"rGF&*(F%F&&F(F)F&)F(%&thetaGF&*(F%F&&F(F)F&)F(F*F&" }}}
{EXCHG {PARA 11 "" 1 "" {MPLTEXT 1 0 130 "1/2*a*sin(theta)*(a*cos(thet
a)-I*r)*2^(1/2)/A, 0,  1/2*(r+I*a*cos(theta))*2^(1/2)/A,  1/2*csc(thet
a)*(a*cos(theta)-I*r)*2^(1/2)/A;" }{XPPMATH 20 "6#%LThe~components~of~
the~inverse~vierbein~are:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%LThe~co
mponents~of~the~inverse~vierbein~are:G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*(%&omegaG\"\"\"&%!G6#%\"tGF&)F(F*F&*&,&*$%\"rG\"\"#F&*$%\"aGF0
F&F&,(F.F&*&%\"MGF&F/F&!\"#F1F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*(%&omegaG\"\"\"&%!G6#%\"tGF&)F(%\"rGF&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%\"tGF&)F(%$phiGF&*&%\"aGF&,(*$%
\"rG\"\"#F&*&%\"MGF&F1F&!\"#*$F.F2F&!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%\"rGF&)F(%\"tGF&,$*&,&*$F*\"\"#
F&*$%\"aGF1F&F&,&F0F&*&F3F1-%$cosG6#%&thetaGF1F&!\"\"#F&F1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%\"rGF&)F(F*F&,$*&,(*
$F*\"\"#F&*&%\"MGF&F*F&!\"#*$%\"aGF0F&F&,&F/F&*&F5F0-%$cosG6#%&thetaGF
0F&!\"\"#F<F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G
6#%\"rGF&)F(%$phiGF&,$*&%\"aGF&,&*$F*\"\"#F&*&F/F2-%$cosG6#%&thetaGF2F
&!\"\"#F&F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#
%&thetaGF&)F(%\"tGF&,$*.%\"IGF&%\"aGF&-%$sinGF)F&,&%\"rGF&*(F/F&F0F&-%
$cosGF)F&!\"\"F&,&*$F4\"\"#F&*&F0F;F6F;F&F8F;#F&F;F=" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%&thetaGF&)F(F*F&,$*(,&%\"rGF
&*(%\"IGF&%\"aGF&-%$cosGF)F&!\"\"F&\"\"##F&F6,&*$F/F6F&*&F2F6F3F6F&F5F
7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%&thetaGF&
)F(%$phiGF&,$*,%\"IGF&-%$cscGF)F&,&%\"rGF&*(F/F&%\"aGF&-%$cosGF)F&!\"
\"F&,&*$F3\"\"#F&*&F5F;F6F;F&F8F;#F&F;F=" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*(%&omegaG\"\"\"&%!G6#%$phiGF&)F(%\"tGF&,$*,%\"aGF&-%$sinG6#%&t
hetaGF&,&*&F/F&-%$cosGF2F&F&*&%\"IGF&%\"rGF&!\"\"F&\"\"##F&F<,&*$F:F<F
&*&F/F<F6F<F&F;F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"
&%!G6#%$phiGF&)F(%&thetaGF&,$*(,&%\"rGF&*(%\"IGF&%\"aGF&-%$cosG6#F,F&F
&F&\"\"##F&F7,&*$F0F7F&*&F3F7F4F7F&!\"\"F8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%$phiGF&)F(F*F&,$**-%$cscG6#%&th
etaGF&,&*&%\"aGF&-%$cosGF0F&F&*&%\"IGF&%\"rGF&!\"\"F&\"\"##F&F;,&*$F9F
;F&*&F4F;F5F;F&F:F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 107 "    The user can save the metric, the vierbein
 and the coordinates in a specified file: (see ?savevierbein)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "# path(`/user/...`);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# savevierbein(exampleKerrV); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "# readvierbein(exampleKerrV)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
102 "     Now, the user is ready to calculate the components of the te
nsors of  General Relativity theory: " }}{PARA 0 "" 0 "" {TEXT -1 173 
"Line element,\nCommutation Coefficients,\nRicci Coefficients,\nRiemma
n tensor,\nRicci tensor,\nRicci Scalar,\nWeyl tensor,\nEinstein tensor
,\nRiemann Square Scalar,\nInvariants,\netc." }}{PARA 0 "" 0 "" {TEXT 
-1 5 "....." }}{PARA 0 "" 0 "" {TEXT -1 55 "The user can define new te
nsors using the above ones.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 178 "     The covariant (contravariant) indic
es of any tensor are indicated by  negative (positive) names. They mus
t be enclosed by square brackets and follow the  name of the tensor." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "     A
TTENTION! Due to the complexity of  Kerr metric, it is NECESSARY TO SI
MPLIFY the components that have been calculated in order to obtain rea
dable outputs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 419 "     In Riemann package, a simplifying function is appli
ed  to every component that is being calculated. This function can be \+
changed through the simpfcn command. The default function is normal. I
n order to obtain the simplest results for the Kerr metric, we will ch
oose x->factor(simplify(normal(x),trig)) as the simplifying function. \+
Note that this choise was done having in mind simple results and not f
ast results ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "    To change the simplifying funtion:\n" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "simpfcn(x->factor(simplify(normal(x),trig)));" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "\n     The vierbein components a
re:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "show(omega[i,-j]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"tGF&&F(6#F)F&,$*(-%$
cscG6#%&thetaGF&,(*$%\"rG\"\"#F&*&%\"MGF&F4F&!\"#*$%\"aGF5F&F&,(*&F.F&
F4F5!\"\"*&F.F&F:F5F=*&F:F5-%$sinGF0F&F&F=#F=F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"tGF&&F(6#%\"rGF&#F&\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"tGF&&F(6#%$phi
GF&,$**,(*$%\"rG\"\"#F&*&%\"MGF&F1F&!\"#*$%\"aGF2F&F&F7F&-%$sinG6#%&th
etaGF&,(*&-%$cscGF:F&F1F2!\"\"*&F>F&F7F2F@*&F7F2F8F&F&F@#F&F2" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"rGF&&F(6#%\"tG
F&,$*(,&*$F)\"\"#F&*&%\"aGF1-%$cosG6#%&thetaGF1F&F&-%$cscGF6F&,(*&F8F&
F)F1!\"\"*&F8F&F3F1F<*&F3F1-%$sinGF6F&F&F<F<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"rGF&&F(6#F)F&,$*&,(*$F)\"\"#F&*
&%\"MGF&F)F&!\"#*$%\"aGF0F&!\"\",&F/F&*&F5F0-%$cosG6#%&thetaGF0F&F&F6
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"rGF&&F(6#%
$phiGF&**,&*$F)\"\"#F&*&%\"aGF0-%$cosG6#%&thetaGF0F&F&F2F&-%$sinGF5F&,
(*&-%$cscGF5F&F)F0!\"\"*&F;F&F2F0F=*&F2F0F7F&F&F=" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%&thetaGF&&F(6#%\"tGF&,$*.%\"aGF&,
&*$%\"rG\"\"#F&*&F/F3-%$cosG6#F)F3F&F&,&F2F&*(%\"IGF&F/F&F5F&F&F&F3#F&
F3,&F2!\"\"F9F&F=,.**-%$cscGF7F&F2F3F/F&F5F&F=*(F:F&F@F&F2\"\"$F&*(F@F
&F/FCF5F&F=**F:F&F@F&F/F3F2F&F&**F:F&F/F3-%$sinGF7F&F2F&F=*(F/FCFGF&F5
F&F&F=F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%&the
taGF&&F(6#F)F&,$*(,&*$%\"rG\"\"#F&*&%\"aGF1-%$cosGF+F1F&F&F1#F&F1,&F0!
\"\"*(%\"IGF&F3F&F4F&F&F8#F8F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*(%
&omegaG\"\"\")%!G%&thetaGF&&F(6#%$phiGF&,$*.,&*$%\"rG\"\"#F&*$%\"aGF2F
&F&,&F0F&*&F4F2-%$cosG6#F)F2F&F&,&F1F&*(%\"IGF&F4F&F7F&F&F&F2#F&F2,&F1
!\"\"F;F&F?,.**-%$cscGF9F&F1F2F4F&F7F&F?*(F<F&FBF&F1\"\"$F&*(FBF&F4FEF
7F&F?**F<F&FBF&F4F2F1F&F&**F<F&F4F2-%$sinGF9F&F1F&F?*(F4FEFIF&F7F&F&F?
#F?F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%$phiGF&
&F(6#%\"tGF&,$**\"\"##F&F/%\"aGF&,&*$%\"rGF/F&*&F1F/-%$cosG6#%&thetaGF
/F&F&,.**-%$cscGF8F&F4F/F1F&F6F&!\"\"*(%\"IGF&F<F&F4\"\"$F&*(F<F&F1FAF
6F&F>**F@F&F<F&F1F/F4F&F&**F@F&F1F/-%$sinGF8F&F4F&F>*(F1FAFEF&F6F&F&F>
F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%$phiGF&&F(
6#%&thetaGF&,$**%\"IGF&,&*$%\"rG\"\"#F&*&%\"aGF3-%$cosGF+F3F&F&F3#F&F3
,&*&F5F&F6F&F&*&F/F&F2F&!\"\"F<#F<F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*(%&omegaG\"\"\")%!G%$phiGF&&F(6#F)F&,$**\"\"##F&F.,&*$%\"rGF.F&*&%
\"aGF.-%$cosG6#%&thetaGF.F&F&,&F1F&*$F4F.F&F&,.**-%$cscGF7F&F2F.F4F&F5
F&!\"\"*(%\"IGF&F=F&F2\"\"$F&*(F=F&F4FBF5F&F?**FAF&F=F&F4F.F2F&F&**FAF
&F4F.-%$sinGF7F&F2F&F?*(F4FBFFF&F5F&F&F?#F?F." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 135 "\n\n     Note that these results can be simplified fu
rther. One can take only one component to learn about the procedure to
 simplify it :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "show(omega[1,-1])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"tGF&&F(6#
F)F&,$*(-%$cscG6#%&thetaGF&,(*$%\"rG\"\"#F&*&%\"MGF&F4F&!\"#*$%\"aGF5F
&F&,(*&F.F&F4F5!\"\"*&F.F&F:F5F=*&F:F5-%$sinGF0F&F&F=#F=F5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tmap(simplify,omega[1,-1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*$%\"rG\"\"#\"\"\"*&%\"MGF)F'F)!
\"#*$%\"aGF(F)F),&F&F)*&F.F(-%$cosG6#%&thetaGF(F)!\"\"#F)F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "when the convenient simplification
 command is found , it can be  applied to all  components:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "amap(simplify,omega[i,-j]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"tGF&&F(6#F)F&,$*&,(*$%\"rG
\"\"#F&*&%\"MGF&F0F&!\"#*$%\"aGF1F&F&,&F/F&*&F6F1-%$cosG6#%&thetaGF1F&
!\"\"#F&F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"
tGF&&F(6#%\"rGF&#F&\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omega
G\"\"\")%!G%\"tGF&&F(6#%$phiGF&,$*(%\"aGF&,.*$%\"rG\"\"#!\"\"*&F2F3-%$
cosG6#%&thetaGF3F&*&%\"MGF&F2F&F3*(F;F&F2F&F6F3!\"#*$F/F3F4*&F/F3F6F3F
&F&,&F1F&F?F&F4#F&F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"
\"\")%!G%\"rGF&&F(6#%\"tGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&
omegaG\"\"\")%!G%\"rGF&&F(6#F)F&,$*&,(*$F)\"\"#F&*&%\"MGF&F)F&!\"#*$%
\"aGF0F&!\"\",&F/F&*&F5F0-%$cosG6#%&thetaGF0F&F&F6" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%\"rGF&&F(6#%$phiGF&*&,&*$-%$cos
G6#%&thetaG\"\"#F&!\"\"F&F&%\"aGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*(%&omegaG\"\"\")%!G%&thetaGF&&F(6#%\"tGF&,$*.%\"IGF&,&%\"rGF&*(F/F&%
\"aGF&-%$cosG6#F)F&F&F&F3F&\"\"##F&F7-%$sinGF6F&,&*$F1F7F&*&F3F7F4F7F&
!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%&thet
aGF&&F(6#F)F&,$*(,&*$%\"rG\"\"#F&*&%\"aGF1-%$cosGF+F1F&F&F1#F&F1,&F0!
\"\"*(%\"IGF&F3F&F4F&F&F8#F8F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%
&omegaG\"\"\")%!G%&thetaGF&&F(6#%$phiGF&,$*,%\"IGF&,**(F/F&-%$cosG6#F)
F&%\"aG\"\"$F&**F/F&%\"rG\"\"#F5F&F2F&F&*&F5F9F8F&F&*$F8F6F&F&F9#F&F9-
%$sinGF4F&,&*$F8F9F&*&F5F9F2F9F&!\"\"#FBF9" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%$phiGF&&F(6#%\"tGF&,$**%\"aGF&\"
\"##F&F0-%$sinG6#%&thetaGF&,&*&F/F&-%$cosGF4F&F&*&%\"IGF&%\"rGF&!\"\"F
=#F=F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\")%!G%$phiGF
&&F(6#%&thetaGF&,$**%\"IGF&,&*$%\"rG\"\"#F&*&%\"aGF3-%$cosGF+F3F&F&F3#
F&F3,&*&F5F&F6F&!\"\"*&F/F&F2F&F&F;F8" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/*(%&omegaG\"\"\")%!G%$phiGF&&F(6#F)F&,$**,&*$%\"rG\"\"#F&*$%\"aGF1
F&F&F1#F&F1-%$sinG6#%&thetaGF&,&*&F3F&-%$cosGF7F&!\"\"*&%\"IGF&F0F&F&F
=#F=F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "\n     The user can als
o save any calculated tensor : (see ?savetensor)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "#savetensor(omega[i,-j],tempomegaKerr); # for example
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "#readtensor(tempomegaKe
rr);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "#show(omega[i,-j]);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "\n   In the following, we sho
w how to calculate many geometrical objects:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Metric Tensor in coordinates:" 
}}{PARA 0 "" 0 "" {TEXT -1 29 "Totally covariant components:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "show(g[-i,-j]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"gG\"\"\"&%!G6#%\"tGF&&F(F)F&*&,(*$%\"rG\"\"#F&*&%
\"MGF&F/F&!\"#*&%\"aGF0-%$cosG6#%&thetaGF0F&F&,&F.F&F4F&!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\"&%!G6#%\"tGF&&F(6#%$phiG
F&,$*.%\"MGF&%\"rGF&%\"aGF&,&-%$cosG6#%&thetaGF&!\"\"F&F&,&F4F&F&F&F&,
&*$F1\"\"#F&*&F2F<F4F<F&F8!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%
\"gG\"\"\"&%!G6#%\"rGF&&F(F)F&,$*&,(*$F*\"\"#F&*&%\"MGF&F*F&!\"#*$%\"a
GF0F&!\"\",&F/F&*&F5F0-%$cosG6#%&thetaGF0F&F&F6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"gG\"\"\"&%!G6#%&thetaGF&&F(F)F&,&*&%\"aG\"\"#-%$c
osGF)F/!\"\"*$%\"rGF/F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"
\"\"&%!G6#%$phiGF&&F(F)F&**,&-%$cosG6#%&thetaGF&!\"\"F&F&,&F.F&F&F&F&,
.*(%\"aG\"\"#F.F7%\"rGF7F&**F8F&F6F7%\"MGF&F.F7!\"#*&F6\"\"%F.F7F&*$F8
F=F&*(F8F&F6F7F:F&F7*&F8F7F6F7F&F&,&*$F8F7F&*&F6F7F.F7F&F2" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 32 "\nTotally contravariant omponent:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "show(g[i,j]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%\"tGF&)F(F)F&*(,.*(%\"aG\"\"#-%$cosG
6#%&thetaGF/%\"rGF/F&**F4F&F.F/%\"MGF&F0F/!\"#*&F.\"\"%F0F/F&*$F4F9F&*
(F4F&F.F/F6F&F/*&F4F/F.F/F&F&,(*$F4F/F&*&F6F&F4F&F7*$F.F/F&!\"\",&F>F&
*&F.F/F0F/F&FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%
\"tGF&)F(%$phiGF&,$*,%\"MGF&%\"rGF&%\"aGF&,(*$F/\"\"#F&*&F.F&F/F&!\"#*
$F0F3F&!\"\",&F2F&*&F0F3-%$cosG6#%&thetaGF3F&F7F3" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%\"rGF&)F(F)F&,$*&,(*$F)\"\"#F&*&%\"M
GF&F)F&!\"#*$%\"aGF/F&F&,&F.F&*&F4F/-%$cosG6#%&thetaGF/F&!\"\"F;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%&thetaGF&)F(F)F&,$*
$,&*$%\"rG\"\"#F&*&%\"aGF0-%$cosG6#F)F0F&!\"\"F6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%$phiGF&)F(F)F&*,,(*$%\"rG\"\"#F&*&%
\"MGF&F.F&!\"#*&%\"aGF/-%$cosG6#%&thetaGF/F&F&,&F5F&!\"\"F&F:,&F5F&F&F
&F:,&F-F&F3F&F:,(F-F&F0F2*$F4F/F&F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 18 "\nMixed components:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "show(
g[i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%\"tGF&
&F(6#F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%\"rG
F&&F(6#F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%!G%&t
hetaGF&&F(6#F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"gG\"\"\")%
!G%$phiGF&&F(6#F)F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "\nThe de
terminant of the metric:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "show(de
tg[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%detgG*(,&*$%\"rG\"\"#\"\"
\"*&%\"aGF)-%$cosG6#%&thetaGF)F*F),&F-F*!\"\"F*F*,&F-F*F*F*F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "\nThe Commutation Coefficients:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "show(C[-i,-j,-k]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%\"rGF&&F(6#%\"tGF&&F(F)F&,$*&
,**(F*F&%\"aG\"\"#-%$cosG6#%&thetaGF4F&*&%\"MGF&F*F4F&*(F:F&F3F4F5F4!
\"\"*&F3F4F*F&F<F&,&*$F*F4F&*&F3F4F5F4F&!\"#F<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%&thetaGF&&F(6#%\"tGF&&F(6#%\"rGF&,
$*.F0F&%\"aGF&\"\"##F&F4-%$sinGF)F&,&*$F0F4F&*&F3F4-%$cosGF)F4F&!\"\",
&*&F3F&F;F&F=*&%\"IGF&F0F&F&F=F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
*%\"CG\"\"\"&%!G6#%$phiGF&&F(6#%\"tGF&&F(6#%\"rGF&,$*.%\"aGF&\"\"##F&F
4-%$sinG6#%&thetaGF&F0F&,&*&F3F&-%$cosGF8F&!\"\"*&%\"IGF&F0F&F&F&,&*$F
0F4F&*&F3F4F<F4F&!\"#F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%\"CG\"
\"\"&%!G6#%$phiGF&&F(6#%\"tGF&&F(6#%&thetaGF&*(,(*$%\"rG\"\"#!\"\"**%
\"IGF&F4F&%\"aGF&-%$cosGF/F&F5*&F9F5F:F5F&F&,&F3F&F<F&F6,&F4F6*(F8F&F9
F&F:F&F&F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%&th
etaGF&&F(6#%\"tGF&&F(6#%$phiGF&,$*(,**(%\"IGF&%\"aG\"\"$-%$cosGF)F7F&*
(%\"rGF&F6\"\"#F8F<F7**F5F&F;F<F6F&F8F&!\"$*$F;F7!\"\"F&,&*$F;F<F&*&F6
F<F8F<F&F@,(FBF&FCF@**F5F&F;F&F6F&F8F&F<F@F@" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%\"tGF&&F(6#%\"rGF&&F(6#%&thetaGF&*
.%\"aG\"\"#-%$cosGF/F&-%$sinGF/F&,&F-!\"\"*(%\"IGF&F2F&F4F&F&F&F3#F&F3
,&*$F-F3F&*&F2F3F4F3F&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%\"CG
\"\"\"&%!G6#%$phiGF&&F(6#%\"rGF&&F(6#%&thetaGF&,$**,(*$F-\"\"#F&*&%\"M
GF&F-F&!\"#*$%\"aGF5F&F&,(F4!\"\"**%\"IGF&F-F&F:F&-%$cosGF/F&F5*&F:F5F
?F5F&F&,&F4F&FAF&F8,&F-F<*(F>F&F:F&F?F&F&F<#F<F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%\"tGF&&F(6#%\"rGF&&F(6#%$phiGF&,$*
.%\"aG\"\"#-%$cosG6#%&thetaGF&-%$sinGF7F&,&*$F-F4F&*&F3F4F5F4F&!\"#,&F
-F&*(%\"IGF&F3F&F5F&F&F&F4#F&F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/**%\"CG\"\"\"&%!G6#%&thetaGF&&F(6#%\"rGF&&F(6#%$phiGF&,$**,(*$F-\"
\"#F&*&%\"MGF&F-F&!\"#*$%\"aGF5F&F&,**(%\"IGF&F:\"\"$-%$cosGF)F>F&*(F-
F&F:F5F?F5F>**F=F&F-F5F:F&F?F&!\"$*$F-F>!\"\"F&,&F4F&*&F:F5F?F5F&F8,(F
4F&FGFE**F=F&F-F&F:F&F?F&F5FE#F&F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/**%\"CG\"\"\"&%!G6#%\"tGF&&F(6#%&thetaGF&&F(6#%$phiGF&,$**%\"IGF&%\"a
GF&-%$cosGF,F&,&*$%\"rG\"\"#F&*&F4F:F5F:F&!\"\"!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%\"rGF&&F(6#%&thetaGF&&F(6#%$phiG
F&,$*,%\"IGF&,(*$F*\"\"#F&*&%\"MGF&F*F&!\"#*$%\"aGF6F&F&F;F&-%$cosGF,F
&,&F5F&*&F;F6F<F6F&F9!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%\"CG
\"\"\"&%!G6#%&thetaGF&&F(F)F&&F(6#%$phiGF&,$*.,&*&%\"rGF&-%$cosGF)F&F&
*&%\"IGF&%\"aGF&F&F&\"\"##F&F9-%$sinGF)F&,&F4F&!\"\"F&F>,&F4F&F&F&F>,(
*$F3F9F&*&F8F9F4F9F>**F7F&F3F&F8F&F4F&F9F>#F>F9" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%\"CG\"\"\"&%!G6#%$phiGF&&F(6#%&thetaGF&&F(F)F&,$**,
.*(%\"aG\"\"#%\"rGF&-%$cosGF,\"\"$!\"\"*(%\"IGF&F6F4F3F8F&**F;F&F6F4F3
F&F5F4F4*(F6F&F3F4F5F&F4*&F6F&F5F8F&*(F;F&F3F&F5F4F9F&F4#F&F4,&*$F5F4F
&*&F3F4F6F4F&!\"#-%$sinGF,F9#F9F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The Ricci Coefficients:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "show(gamma[-a,-b,-c]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"tGF&&F(6#%\"rGF&&F(F,F&,$*&
,**(F-F&%\"aG\"\"#-%$cosG6#%&thetaGF4F&*&%\"MGF&F-F4F&*(F:F&F3F4F5F4!
\"\"*&F3F4F-F&F<F&,&*$F-F4F&*&F3F4F5F4F&!\"#F<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"tGF&&F(6#%\"rGF&&F(6#%&thetaG
F&,$*.,&F-!\"\"*(%\"IGF&%\"aGF&-%$cosGF/F&F&F&F7F&\"\"##F&F:-%$sinGF/F
&,&*$F-F:F&*&F7F:F8F:F&F4,&*&F7F&F8F&F4*&F6F&F-F&F&F4F;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"tGF&&F(6#%\"rGF&&F(6#%
$phiGF&,$*,%\"aGF&\"\"##F&F4-%$sinG6#%&thetaGF&,(*(F3F&F-F&-%$cosGF8F&
F4*&%\"IGF&F-F4!\"\"*(F?F&F3F4F<F4F&F&,&*$F-F4F&*&F3F4F<F4F&!\"#F5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"tGF&&F(6#%&t
hetaGF&&F(6#%\"rGF&,$*.,&F0F&*(%\"IGF&%\"aGF&-%$cosGF,F&F&F&F6F&\"\"##
F&F9-%$sinGF,F&,&*$F0F9F&*&F6F9F7F9F&!\"\",&*&F6F&F7F&F@*&F5F&F0F&F&F@
#F@F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"tGF
&&F(6#%&thetaGF&&F(6#%$phiGF&,$*$,&%\"rG!\"\"*(%\"IGF&%\"aGF&-%$cosGF,
F&F&F5F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"
tGF&&F(6#%$phiGF&&F(6#%\"rGF&,$*,%\"IGF&%\"aGF&\"\"##F&F5-%$sinG6#%&th
etaGF&,&*$F0F5F&*&F4F5-%$cosGF9F5F&!\"\"#F@F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"tGF&&F(6#%$phiGF&&F(6#%&theta
GF&*$,&%\"rGF&*(%\"IGF&%\"aGF&-%$cosGF/F&F&!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"rGF&&F(6#%&thetaGF&&F(6#%\"tG
F&,$*.,&F*!\"\"*(%\"IGF&%\"aGF&-%$cosGF,F&F&F&F7F&\"\"##F&F:-%$sinGF,F
&,&*$F*F:F&*&F7F:F8F:F&F4,&*&F7F&F8F&F4*&F6F&F*F&F&F4#F4F:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"rGF&&F(6#%&thetaGF
&&F(6#%$phiGF&,$*(,(*$F*\"\"#F&*&%\"MGF&F*F&!\"#*$%\"aGF5F&F&,&F4F&*&F
:F5-%$cosGF,F5F&!\"\",&F*F&*(%\"IGF&F:F&F=F&F&F?#F?F5" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"rGF&&F(6#%$phiGF&&F(6#%
\"tGF&,$*,%\"aGF&\"\"##F&F4-%$sinG6#%&thetaGF&,(*(F3F&F*F&-%$cosGF8F&F
4*&%\"IGF&F*F4!\"\"*(F?F&F3F4F<F4F&F&,&*$F*F4F&*&F3F4F<F4F&!\"##F@F4" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%\"rGF&&F(6#%
$phiGF&&F(6#%&thetaGF&,$*(,(*$F*\"\"#F&*&%\"MGF&F*F&!\"#*$%\"aGF5F&F&,
&F4F&*&F:F5-%$cosGF/F5F&!\"\",&F*F?*(%\"IGF&F:F&F=F&F&F?#F&F5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%&thetaGF&&F(6#
%$phiGF&&F(6#%\"rGF&,$*.%\"aGF&-%$cosGF)F&,(*$F0\"\"#F&*&%\"MGF&F0F&!
\"#*$F3F8F&F&,&*&F3F&F4F&!\"\"*&%\"IGF&F0F&F&F&,&F7F&*&F3F8F4F8F&F;,&F
0F&*(FAF&F3F&F4F&F&F?F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/**%&gammaG
\"\"\"&%!G6#%&thetaGF&&F(6#%$phiGF&&F(F)F&,$*.,&*&%\"rGF&-%$cosGF)F&F&
*&%\"IGF&%\"aGF&F&F&\"\"##F&F9-%$sinGF)F&,&F4F&!\"\"F&F>,&F4F&F&F&F>,(
*$F3F9F&*&F8F9F4F9F>**F7F&F3F&F8F&F4F&F9F>#F>F9" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/**%&gammaG\"\"\"&%!G6#%&thetaGF&&F(6#%$phiGF&&F(F,F&,$
**,.*(%\"aG\"\"#%\"rGF&-%$cosGF)\"\"$F&*(%\"IGF&F6F4F3F8!\"\"**F:F&F6F
4F3F&F5F4!\"#*(F6F&F3F4F5F&F=*&F6F&F5F8F;*(F:F&F3F&F5F4F&F&F4#F&F4-%$s
inGF)F;,&*$F5F4F&*&F3F4F6F4F&F=FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Riemann Tensor:" }}{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&,$
**%\"MGF&F-F&,&*&%\"aG\"\"#-%$cosG6#%&thetaGF6\"\"$*$F-F6!\"\"F&,&F<F&
F4F&!\"$F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"t
GF&&F(6#%\"rGF&&F(6#%&thetaGF&&F(6#%$phiGF&,$*0,&*$F-\"\"#\"\"$*&%\"aG
F8-%$cosGF/F8!\"\"F&,,*&%\"IGF&F-\"\"%F&*(F-F9F;F&F<F&!\"%**FAF&F-F8F;
F8F<F8!\"'*(F-F&F;F9F<F9FB*(FAF&F;FBF<FBF&F&%\"MGF&F;F&F<F&,&F7F&F:F&!
\"$,,*$F-FBF&**FAF&F;F&F<F&F-F9FB*(F;F8F<F8F-F8FF**FAF&F;F9F<F9F-F&FD*
&F;FBF<FBF&F>F8" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6
#%\"tGF&&F(6#%&thetaGF&&F(6#%\"rGF&&F(6#%$phiGF&**%\"MGF&,0*$F0\"\"'F&
**%\"IGF&%\"aG\"\"&-%$cosGF,F<F0F&F8*(F0\"\"%F;\"\"#F=FA!#:**F:F&F0\"
\"$F;FDF=FD!#?**F:F&F0F<F;F&F=F&F8*(F;F@F=F@F0FA\"#:*&F;F8F=F8!\"\"F&,
&*$F0FAF&*&F;FAF=FAF&!\"$,**(F:F&F;FDF=FDFJ*(F0F&F;FAF=FAFN**F:F&F0FAF
;F&F=F&FD*$F0FDF&FJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&
%!G6#%\"tGF&&F(6#%$phiGF&&F(6#%\"rGF&&F(6#%&thetaGF&*(,&*&%\"aGF&-%$co
sGF2F&!\"\"*&%\"IGF&F0F&F&F&%\"MGF&,,*&F<F&F0\"\"%F&*(F0\"\"$F7F&F8F&!
\"%**F<F&F0\"\"#F7FEF8FE!\"'*(F0F&F7FBF8FBF@*(F<F&F7F@F8F@F&F:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%&thetaGF&&F(6#%$p
hiGF&&F(F)F&&F(F,F&,$**%\"MGF&%\"rGF&,&*&%\"aG\"\"#-%$cosGF)F7\"\"$*$F
3F7!\"\"F&,&F;F&F5F&!\"$F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "The Ricci Tensor:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "show(R[-i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
(%\"RG\"\"\"&%!G6#%\"iGF&&F(6#%\"jGF&\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Ricci Scalar:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "show(R[]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"RG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "The Einstein Tensor:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "show(G[-i,-j]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*(%\"GG\"\"\"&%!G6#%\"iGF&&F(6#%\"jGF&\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The Weyl tensor:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "show(C[-i,-j,-k,-l]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\"&%!G6#%\"tGF&&F(6#%\"rGF&&F(F)F&&F
(F,F&,$**%\"MGF&F-F&,&*&%\"aG\"\"#-%$cosG6#%&thetaGF6\"\"$*$F-F6!\"\"F
&,&F<F&F4F&!\"$F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\"&%!
G6#%\"tGF&&F(6#%\"rGF&&F(6#%&thetaGF&&F(6#%$phiGF&,$*0,&*$F-\"\"#!\"$*
&%\"aGF8-%$cosGF/F8F&F&,,*&%\"IGF&F-\"\"%F&*(F-\"\"$F;F&F<F&!\"%**F@F&
F-F8F;F8F<F8!\"'*(F-F&F;FCF<FCFA*(F@F&F;FAF<FAF&F&%\"MGF&F;F&F<F&,&F7F
&F:F&F9,,*$F-FAF&**F@F&F;F&F<F&F-FCFA*(F;F8F<F8F-F8FF**F@F&F;FCF<FCF-F
&FD*&F;FAF<FAF&!\"\"!\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*,%\"CG\"
\"\"&%!G6#%\"tGF&&F(6#%&thetaGF&&F(6#%\"rGF&&F(6#%$phiGF&**%\"MGF&,0*$
F0\"\"'!\"\"**%\"IGF&%\"aG\"\"&-%$cosGF,F=F0F&!\"'*(F0\"\"%F<\"\"#F>FC
\"#:**F;F&F0\"\"$F<FFF>FF\"#?**F;F&F0F=F<F&F>F&F@*(F<FBF>FBF0FC!#:*&F<
F8F>F8F&F&,&*$F0FCF&*&F<FCF>FCF&!\"$,**(F;F&F<FFF>FFF&*(F0F&F<FCF>FCFF
**F;F&F0FCF<F&F>F&FO*$F0FFF9F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%
\"CG\"\"\"&%!G6#%\"tGF&&F(6#%$phiGF&&F(6#%\"rGF&&F(6#%&thetaGF&,$*(,&*
&%\"aGF&-%$cosGF2F&F&*&%\"IGF&F0F&!\"\"F&%\"MGF&,,*&F<F&F0\"\"%F&*(F0
\"\"$F8F&F9F&!\"%**F<F&F0\"\"#F8FFF9FF!\"'*(F0F&F8FCF9FCFA*(F<F&F8FAF9
FAF&F=F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"CG\"\"\"&%!G6#%&thet
aGF&&F(6#%$phiGF&&F(F)F&&F(F,F&,$**%\"MGF&%\"rGF&,&*&%\"aG\"\"#-%$cosG
F)F7\"\"$*$F3F7!\"\"F&,&F;F&F5F&!\"$F7" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 203 "      More details about
 the functions of Riemann Package and their applications, see the work
sheets Riemann1.mws (general tutorial) and Riemann2.mws (example of Sp
herically Symmetric metric). See also ?" }{HYPERLNK 17 "Riemann" 2 "ri
emann" "" }{TEXT -1 1 "." }}}}{MARK "0 0 1" 17 }{VIEWOPTS 1 1 0 1 1 
1803 }