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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 102 "This worksheet is desig
ned to determine the connections, and curvatures for the flat Minkowsk
i metric " }}{PARA 257 "" 0 "" {TEXT -1 52 "          expressed in obl
ate spheroidal coordinates" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "S:=sqrt(sigma^2+a^2*(cos(theta))^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG*$,&*$)%&sigmaG\"\"#\"\"\"F+*&)%
\"aGF*F+)-%$cosG6#%&thetaGF*F+F+#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Delta:=sqrt(sigma^2+a^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&DeltaG*$,&*$)%&sigmaG\"\"#\"\"\"F+*$)%\"aGF*F+F+#F+F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%06~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#%4
C:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mak
eg(OblSpher);" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 46 "Makeg 2.0: GRTensor metr
ic/basis entry utility" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 
1 "" {TEXT -1 41 "To quit makeg, type 'exit' at any prompt." }}{PARA 
6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 44 "Do you wish to
 enter a 1) metric [g(dn,dn)]," }}{PARA 6 "" 1 "" {TEXT -1 44 "       \+
                2) line element [ds]," }}{PARA 6 "" 1 "" {TEXT -1 63 "
                       3) non-holonomic basis [e(1)...e(n)], or" }}
{PARA 6 "" 1 "" {TEXT -1 49 "                       4) NP tetrad [l,n,
m,mbar]?" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "makeg>" 
0 "" {MPLTEXT 1 0 2 "3;" }}{PARA 6 "" 1 "" {TEXT -1 50 "Enter coordina
tes as a LIST (eg. [t,r,theta,phi]):" }}}{EXCHG {PARA 0 "makeg>" 0 "" 
{MPLTEXT 1 0 20 "[sigma,theta,phi,t];" }}{PARA 6 "" 1 "" {TEXT -1 48 "
Would you like to enter 1) covariant components," }}{PARA 6 "" 1 "" 
{TEXT -1 55 "                        2) contravariant components, or" 
}}{PARA 6 "" 1 "" {TEXT -1 32 "                        3) both." }}}
{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 6 "" 1 "" 
{TEXT -1 111 "Enter the covariant components of basis vector '1' as a \+
LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):" }}}{EXCHG 
{PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 19 "(S/Delta)*d[sigma];" }}{PARA 6 
"" 1 "" {TEXT -1 111 "Enter the covariant components of basis vector '
2' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):" }}}
{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 11 "S*d[theta];" }}{PARA 6 "
" 1 "" {TEXT -1 111 "Enter the covariant components of basis vector '3
' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):" }}}
{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 24 "Delta*sin(theta)*d[phi];
" }}{PARA 6 "" 1 "" {TEXT -1 111 "Enter the covariant components of ba
sis vector '4' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d
[y]):" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 5 "d[t];" }}{PARA 
6 "" 1 "" {TEXT -1 43 "Is the basis inner product  1) Diagonal, or" }}
{PARA 6 "" 1 "" {TEXT -1 41 "                            2) Symmetric?
" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 6 "" 1 "
" {TEXT -1 15 "Enter eta[1,1]:" }}}{EXCHG {PARA 0 "makeg>" 0 "" 
{MPLTEXT 1 0 2 "1;" }}{PARA 6 "" 1 "" {TEXT -1 15 "Enter eta[2,2]:" }}
}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 6 "" 1 "" 
{TEXT -1 15 "Enter eta[3,3]:" }}}{EXCHG {PARA 0 "makeg>" 0 "" 
{MPLTEXT 1 0 2 "1;" }}{PARA 6 "" 1 "" {TEXT -1 15 "Enter eta[4,4]:" }}
}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 3 "-1;" }}{PARA 6 "" 1 "" 
{TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 67 "If there are any complex \+
valued coordinates, constants or functions" }}{PARA 6 "" 1 "" {TEXT 
-1 66 "for this spacetime, please enter them as a SET ( eg. \{ z, psi \+
\} )." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 33 "
Complex quantities [default=\{\}]: " }}}{EXCHG {PARA 0 "makeg>" 0 "" 
{MPLTEXT 1 0 1 ";" }}{PARA 6 "" 1 "" {TEXT -1 2 "\{\}" }}{PARA 6 "" 1 
"" {TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%AThe~values~you~h
ave~entered~are:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,CoordinatesG7&
%&sigmaG%&thetaG%$phiG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Basis
~1-forms:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&omegaG6#\"\"\"-%'vec
torG6#7&*&,&*$)%&sigmaG\"\"#F'F'*&)%\"aGF1F')-%$cosG6#%&thetaGF1F'F'#F
'F1,&F.F'*$F3F'F'#!\"\"F1\"\"!F?F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/&%&omegaG6#\"\"#-%'vectorG6#7&\"\"!*$,&*$)%&sigmaGF'\"\"\"F2*&)%\"aGF
'F2)-%$cosG6#%&thetaGF'F2F2#F2F'F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/&%&omegaG6#\"\"$-%'vectorG6#7&\"\"!F,*&,&*$)%&sigmaG\"\"#\"\"\"F3*$
)%\"aGF2F3F3#F3F2-%$sinG6#%&thetaGF3F," }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/&%&omegaG6#\"\"%-%'vectorG6#7&\"\"!F,F,\"\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%@Inner~product~of~basis~vectors:G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/%$etaG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+
F+F*F+7&F+F+F+!\"\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "You may choose to \+
0) Use the metric WITHOUT saving it," }}{PARA 6 "" 1 "" {TEXT -1 46 " \+
                 1) Save the metric as it is," }}{PARA 6 "" 1 "" 
{TEXT -1 45 "                  2) Re-enter a basis vector," }}{PARA 6 
"" 1 "" {TEXT -1 48 "                  3) Re-enter the inner product,
" }}{PARA 6 "" 1 "" {TEXT -1 44 "                  4) Add/change const
raints," }}{PARA 6 "" 1 "" {TEXT -1 48 "                   5) Add a te
xt description, or" }}{PARA 6 "" 1 "" {TEXT -1 61 "                  6
) Abandon this metric and return to Maple." }}{PARA 6 "" 1 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 6 
"" 1 "" {TEXT -1 58 "Information written to: `C:/Grtii(6)/Metrics/OblS
pher.mpl`" }}{PARA 6 "" 1 "" {TEXT -1 57 "Do you wish to use this spac
etime in the current session?" }}{PARA 6 "" 1 "" {TEXT -1 29 "(1=yes [
default], other=no): " }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 
"1;" }}{PARA 6 "" 1 "" {TEXT -1 22 "Initializing: OblSpher" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%)OblSpherG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%<For~the~OblSpher~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-%'vect
orG6#7&%&sigmaG%&thetaG%$phiG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%4Basis~inner~productG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$etaG6$%$b
upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$etaG%$(a)G\"\"\")%!G%$(
b)GF(-%'matrixG6#7&7&F(\"\"!F1F17&F1F(F1F17&F1F1F(F17&F1F1F1!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Basis~(covariant~components)G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w1G6#%#dnG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%'omega1G6#%\"aG-%'vectorG6#7&*&,&*$)%&sigmaG\"\"#\"
\"\"F2*&)F'F1F2)-%$cosG6#%&thetaGF1F2F2#F2F1,&F.F2*$F4F2F2#!\"\"F1\"\"
!F?F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w2G6#%#dnG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/&%'omega2G6#%\"aG-%'vectorG6#7&\"\"!*$,&*$)%&sigm
aG\"\"#\"\"\"F3*&)F'F2F3)-%$cosG6#%&thetaGF2F3F3#F3F2F,F," }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%#w3G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/&%'omega3G6#%\"aG-%'vectorG6#7&\"\"!F,*&,&*$)%&sigmaG\"\"#\"\"\"F3*
$)F'F2F3F3#F3F2-%$sinG6#%&thetaGF3F," }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%#w4G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega4G6#%\"aG-
%'vectorG6#7&\"\"!F,F,\"\"\"" }}{PARA 6 "" 1 "" {TEXT -1 18 "makeg() c
ompleted." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grcalc(rot(bdn
,bdn,bdn));" }}{PARA 6 "" 1 "" {TEXT -1 38 "Created a definition for e
(bdn,dn,pdn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$q\"!
\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "grdisplay(rot(bdn,bd
n,bdn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~OblSpher~spaceti
me:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6Rotation~CoefficientsG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&gammaG6#*(-%!G6#%\"1G\"\"\"-F)6#%
\"2GF,-F)6#%\"1GF,,$**,&*$)%&sigmaG\"\"#F,F,*&)%\"aGF9F,)-%$cosG6#%&th
etaGF9F,F,#!\"$F9F<F9F>F,-%$sinGF@F,!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%&gammaG6#*(-%!G6#%\"1G\"\"\"-F)6#%\"2GF,-F)6#%\"2GF,
,$*(,&*$)%&sigmaG\"\"#F,F,*&)%\"aGF9F,)-%$cosG6#%&thetaGF9F,F,#!\"$F9F
8F,,&F6F,*$F;F,F,#F,F9!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&gam
maG6#*(-%!G6#%\"1G\"\"\"-F)6#%\"3GF,-F)6#%\"3GF,,$*(,&*$)%&sigmaG\"\"#
F,F,*&)%\"aGF9F,)-%$cosG6#%&thetaGF9F,F,#!\"\"F9,&F6F,*$F;F,F,FBF8F,FC
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&gammaG6#*(-%!G6#%\"2G\"\"\"-F)
6#%\"3GF,-F)6#%\"3GF,,$*(-%$cosG6#%&thetaGF,,&*$)%&sigmaG\"\"#F,F,*&)%
\"aGF=F,)F5F=F,F,#!\"\"F=-%$sinGF7FCFC" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "grcalc(R(bdn,bdn,bdn,bdn));" }}{PARA 6 "" 1 "" {TEXT 
-1 40 "Created definition for rot(bdn,bup,bdn) " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"$+\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "grdisplay(R(bdn,bdn,bdn,bdn));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"RG6#**-%!G6#%\"1G\"\"\"-F)6#%\"2GF,-F)6#%\"1GF,-F)
6#%\"2GF,*(%\"aG\"\"#,.*&)F7F8F,)-%$cosG6#%&thetaG\"\"%F,F,*(F;F,)F=F8
F,)-%$sinGF?F8F,F,*&F;F,FCF,!\"\"*&)%&sigmaGF8F,FCF,FH*&FDF,FJF,FH*$FJ
F,F,F,,&FMF,FGF,!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"RG6#**-%!
G6#%\"1G\"\"\"-F)6#%\"3GF,-F)6#%\"2GF,-F)6#%\"3GF,,$*0%&sigmaGF,-%$cos
G6#%&thetaGF,%\"aG\"\"#,(*$)F9F>F,F,*$)-%$sinGF;F>F,F,F,!\"\"F,,&*$)F8
F>F,F,*&)F=F>F,FAF,F,!\"#,&FHF,*$FKF,F,#FFF>FDFFFF" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "gralter(R(bdn,bdn,bdn,bdn),simplify,trig);
" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTenso
rII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT 
-1 54 "Applying routine simplify to object R(bdn,bdn,bdn,bdn)" }}
{PARA 6 "" 1 "" {TEXT -1 62 "Applying routine `simplify[trig]` to obje
ct R(bdn,bdn,bdn,bdn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G
$\"$h&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "grdisplay(R(b
dn,bdn,bdn,bdn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3R(bdn,bdn,bdn,
bdn)G%8All~components~are~zeroG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
232 "The plots just below first show graphs of curves, in the y,z-plan
e, for instance, of rho versus z (in cylindrical coordinates), for var
ious fixed values of sigma:  a, 2a, 3a, 4a, and 5a, where everything i
s measured in units of a.  " }}{PARA 0 "" 0 "" {TEXT -1 76 "          \+
    The 2-dimensional ones are ellipses, with foci at +1 and -1.  " }}
{PARA 0 "" 0 "" {TEXT -1 110 "If we had also plotted the situation for
 sigma = 0, we would have just the horizontal axis between  +1 and -1.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "Then
 they show a 3-dimensional plot for a given, single value of sigma; in
 this case sigma=1.  It is of course an oblate spheroid." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Rho:=sqrt(sigma^2+a^2)*sin(theta); \+
Z:=sigma*cos(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$RhoG*&,&*$)
%&sigmaG\"\"#\"\"\"F+*$)%\"aGF*F+F+#F+F*-%$sinG6#%&thetaGF+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"ZG*&%&sigmaG\"\"\"-%$cosG6#%&thetaGF'" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "plot([seq([subs(a=1,sigma
=n,Rho),subs(sigma=n,Z),theta=0..Pi],n=1..5),seq([subs(a=1,sigma=n,-Rh
o),subs(sigma=n,Z),theta=0..Pi],n=1..5)],scaling=`constrained`,color=[
red,green,yellow,blue,cyan,red,green,yellow,blue,cyan], title=`y,z-pla
ne; constant sigma`);" }}{PARA 13 "" 1 "" {GLPLOT2D 363 363 363 
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oidG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 1 1 1.000000 
51.000000 -37.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 58 "The graphs below show graphs for constant values of theta." }}
{PARA 0 "" 0 "" {TEXT -1 98 " You can see, on the 2-dimensional one be
low, in the y,z-plane, that they are pairs of hyperbolae." }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "plot([seq([subs
(a=1,theta=2*n,Rho),subs(theta=2*n,Z),sigma=-5..5],n=1..5),seq([subs(a
=1,theta=2*n,-Rho),subs(theta=2*n,Z),sigma=-5..5],n=1..5)],scaling=`co
nstrained`,color=[red,green,yellow,blue,cyan,red,green,yellow,blue,cya
n],title=`constant theta; y,z-plane`);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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%:constant~theta;~y,z-planeG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$%(DEF
AULTGFfbs" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cu
rve 8" "Curve 9" "Curve 10" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "B
elow I also give two different perspectives of the surfaces of constan
t theta, which are hyperboloids of one sheet; the generalization of " 
}}{PARA 0 "" 0 "" {TEXT -1 128 "    the pair of cones that one would h
ave gotten for such surfaces if theta had been a member of a set of sp
herical coordinates." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 171 "plot3d([subs(a=1,theta=5,Rho*cos(phi)),subs(a=1,thet
a=5,Rho*sin(phi)),subs(a=1,theta=1.5*Pi,Z)],sigma=-5..5,phi=0..2*Pi,ti
tle=`constant theta; hyperboloids of one sheet`);" }}{PARA 13 "" 1 "" 
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" {MPLTEXT 1 0 181 "plot3d([subs(a=1,theta=1.5*Pi,Rho*cos(phi)),subs(a
=1,theta=1.5*Pi,Rho*sin(phi)),subs(a=1,theta=1.5*Pi,Z)],sigma=-5..5,ph
i=0..2*Pi,title=`constant theta; hyperboloids of one sheet`);" }}
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