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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 66 " \+
Riemann[vierbein] - give values to the components of the vierbein" }}
{PARA 0 "" 0 "" {TEXT -1 94 "                    Riemann[ivierbein] - \+
give values to the components of the inverse vierbein" }}{PARA 0 "" 0 
"usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 15 "\n   vierbein( )
" }}{PARA 0 "" 0 "" {TEXT -1 42 "    vierbein(w_11, w_12, w_13, ... , \+
w_nn)" }}{PARA 0 "" 0 "" {TEXT -1 18 "    ivierbein( )  " }}{PARA 0 "
" 0 "" {TEXT -1 43 "    ivierbein(a_11, a_12, a_13, ... , a_nn)" }}
{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 47 "\n    w_ij - t
he (i,j) component of the vierbein" }}{PARA 0 "" 0 "" {TEXT -1 54 "   \+
 a_ij - the (i,j) component of the inverse vierbein" }}}{SECT 0 {PARA 
0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT 
-1 267 "The call vierbein( ) allows the user to enter the components o
f the vierbein interactively. The first index is the vierbein index (c
ontravariant) and the second one is the coordinate index (covariant). \+
These components will be referred to as omega[i,-j] subsequently." }}
{PARA 15 "" 0 "" {TEXT -1 253 "The call ivierbein( ) allows to enter t
he components of the inverse vierbein. The first index is the viervein
 index (covariant) and the second one is the coordinate index (contrav
ariant). These components will be referred to as omega[-i,j] subsequen
tly." }}{PARA 15 "" 0 "" {TEXT -1 164 "One can enter all components of
 the vierbein as arguments the function vierbein (ivierbein). In this \+
case the number of arguments must be equal to Dimension square." }}
{PARA 15 "" 0 "" {TEXT -1 198 "The function vierbein (iveirbein) calcu
lates the (inverse) vierbein that are called by omega[-i, j] (omega[i,
-j]). The user should simplify these components before proceeding the \+
calculations. (ser " }{HYPERLNK 17 "amap" 2 "amap" "" }{TEXT -1 4 " or
 " }{HYPERLNK 17 "simpfcn" 2 "simpfcn" "" }{TEXT -1 1 ")" }}{PARA 15 "
" 0 "" {TEXT -1 86 "The coordinates and the metric in vierbein and mus
t be given before the vierbein (see " }{HYPERLNK 17 "coordinates" 2 "c
oordinates" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "metricV" 2 "metricV
" "" }{TEXT -1 2 ")." }}{PARA 15 "" 0 "" {TEXT -1 119 "The default nam
e for vierbein is omega. This name can be changed by assigning another
 one to the variable VierbeinName." }}{PARA 15 "" 0 "" {TEXT -1 73 "Th
e first index is a vierbein index and the second is a coordinate index
." }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Riemann):" }}{PARA 6 "" 1 "" 
{TEXT -1 23 "Share Library:  Riemann" }}{PARA 6 "" 1 "" {TEXT -1 44 "A
uthors: Portugal, Renato, Sautu, Sandra L.." }}{PARA 6 "" 1 "" {TEXT 
-1 199 "Description:  With this package, one can perform the tensoralg
ebra, like to add, multiply or contract tensors.One can define tensors
 with symmetries and can apply Maple functions to their components." }
}{PARA 6 "" 1 "" {TEXT -1 265 "One can use this package to calculate t
he maintensors used in General Relativity Theory, likeRiemann, Ricci, \+
Traceless Ricci, Einstein, Weyltensors or the 16 Carminatti-McLenaghan
 invariants.The calculations can be performed in coordinatebasis or in
 vierbein basis." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Dimensi
on := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*DimensionG\"\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "In the following, we calulate the
 vierbein components of Riemann and Ricci tensors for the Schwarzschil
d metric: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "coordinates(t
,r,theta,phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5The~coordinates~ar
e: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\"\"$\"\"\"%&the
taG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"XG\"\"\")%!G\"\"%\"\"\"%$
phiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "metricV(1,-1,-1,-1)
;  # the metric in tetrads" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%UThe~co
mponents~of~the~metric~in~the~rigid~frame~are:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*(%$etaG\"\"\"&%!G6#%\"iGF&&F(6#%\"jGF&-%'MATRIXG6#7&7
&F&\"\"!F3F37&F3!\"\"F3F37&F3F3F5F37&F3F3F3F5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "vierbein();  # the Schwarzschild solution" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Convention~for~omega~indices:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%_oFirst~index~is~a~vierbein~index~and
~second~one~is~a~coordinate~indexG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%\"~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%5ENTER~THE~COMPONENTSG*(%&om
egaG\"\"\")%!G%\"tGF&&F(6#F)F&*(F%F&)F(F)F&&F(6#%\"rGF&*(F%F&)F(F)F&&F
(6#%&thetaGF&*(F%F&)F(F)F&&F(6#%$phiGF&" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 22 "(1-2*m/r)^(1/2),0,0,0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6&*$-%%sqrtG6#,&\"\"\"F(*&%\"mG\"\"\"%\"rG!\"\"!\"#F+\"\"!F/F/" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "0,(1-2*m/r)^(-1/2),0,0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!*&\"\"\"F%*$-%%sqrtG6#,&\"\"\"F+*
&%\"mGF%%\"rG!\"\"!\"#F%F/F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "0,0,r,0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#%\"rGF#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "0,0,0,r*sin(theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#F#*&%\"rG\"\"\"-%$sinG6#%&theta
GF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The components of the inve
rve vierbein are:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "show
(omega[-i,j]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%
!G6#%\"tGF&)F(F*F&*&\"\"\"F-*$-%%sqrtG6#,&F&F&*&%\"mGF-%\"rG!\"\"!\"#F
-F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%\"rGF&)
F(F*F&*$-%%sqrtG6#,&F&F&*&%\"mG\"\"\"F*!\"\"!\"#F3" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%&thetaGF&)F(F*F&*&\"\"\"F-%\"
rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%&omegaG\"\"\"&%!G6#%$ph
iGF&)F(F*F&*&\"\"\"F-*&%\"rG\"\"\"-%$sinG6#%&thetaG\"\"\"!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Riemann tensor (covariant componen
ts):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " show(R[-i,-j,-k,-l
]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"tGF&&F(
6#%\"rGF&&F(F)F&&F(F,F&,$*&%\"mG\"\"\"*$)F-\"\"$F3!\"\"!\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"tGF&&F(6#%&thetaGF&&F
(F)F&&F(F,F&*&%\"mG\"\"\"*$)%\"rG\"\"$F2!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%\"tGF&&F(6#%$phiGF&&F(F)F&&F(F,F&*
&%\"mG\"\"\"*$)%\"rG\"\"$F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
,%\"RG\"\"\"&%!G6#%\"rGF&&F(6#%&thetaGF&&F(F)F&&F(F,F&,$*&%\"mG\"\"\"*
$)F*\"\"$F3!\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"
\"&%!G6#%\"rGF&&F(6#%$phiGF&&F(F)F&&F(F,F&,$*&%\"mG\"\"\"*$)F*\"\"$F3!
\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*,%\"RG\"\"\"&%!G6#%&the
taGF&&F(6#%$phiGF&&F(F)F&&F(F,F&,$*&%\"mG\"\"\"*$)%\"rG\"\"$F3!\"\"\"
\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Ricci tensor:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "show(R[-i,-j]); " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*(%\"RG\"\"\"&%!G6#%\"iGF&&F(6#%\"jGF&\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 0 "" 0 "
" {TEXT 26 9 "See Also:" }{TEXT -1 2 "  " }{HYPERLNK 17 "coordinates" 
2 "coordinates" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "metricV" 2 "metricV
" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "simpfcn" 2 "simpfcn" "" }{TEXT 
-1 2 ", " }{HYPERLNK 17 "definetensor" 2 "definetensor" "" }{TEXT -1 
2 ", " }{HYPERLNK 17 "evalt" 2 "evalt" "" }{TEXT -1 2 ", " }{HYPERLNK 
17 "viertocoord" 2 "viertocoord" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "cl
earbint" 2 "clearbint" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "codiff" 2 "c
odiff" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "metric" 2 "metric" "" }
{TEXT -1 2 ", " }{HYPERLNK 17 "readvierbein" 2 "readvierbein" "" }
{TEXT -1 2 ", " }{HYPERLNK 17 "savevierbein" 2 "savevierbein" "" }
{TEXT -1 2 ".." }}}}{MARK "2 17 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }