Commit ff3952c7 by grtensor

First pass of working worksheets and associated metrics

parent a3cb697d
Ndim_ := 3:
x1_ := r:
x2_ := phi:
x3_ := t:
complex_ := {}:
g11_ := 1/(r^2/sigma^2-m+Q^2*ln(r)):
g22_ := r^2:
g33_ := -r^2/sigma^2+m-Q^2*ln(r):
Ndim_ := 3 :
x1_ := x :
x2_ := y :
x3_ := z :
g11_ := c(x,y,z) :
g22_ := d(x,y,z) :
g33_ := e(x,y,z) :
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
g11_ := diff(R(r,t),r)^2/(1+f(r)) :
g22_ := R(r,t)^2 :
g33_ := R(r,t)^2*sin(theta)^2 :
g44_ := -1 :
constraint_ := [diff(diff(R(r,t),r),t) = (2*diff(m(r),r)/R(r,t)-2*m(r)*diff(R(r,t),r)/R(r,t)^2+
diff(f(r),r))/(2*sqrt(2*m(r)/R(r,t)+f(r))), diff(R(r,t),t) = sqrt(2*m(r)/R(r,t)+f(r)),
diff(diff(R(r,t),t),t) = -m(r)/R(r,t)^2, diff(diff(diff(R(r,t),t),r),t) =
-diff(m(r),r)/R(r,t)^2+2*m(r)*diff(R(r,t),r)/R(r,t)^3] :
Info_:=`The Tolman dust solution (Proc. Nat. Acad. Sci. 20, 169,1934)`:
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
g11_ := diff(R(r,t),r)^2/(1 + f(r)) :
g22_ := R(r,t)^2 :
g33_ := R(r,t)^2*sin(theta)^2 :
g44_ := -1 :
constraint_ := [
diff(R(r,t), r, t, t) = 2*M(r)*diff(R(r,t), r)/R(r,t)^3
- diff(M(r), r)/R(r,t)^2 ,
diff(R(r,t), r, t) = ( 2*diff(M(r), r)/R(r,t)
- 2*M(r)*diff(R(r,t), r)/R(r,t)^2
+ diff(f(r), r) )
/ ( 2*diff(R(r,t), t) ),
diff(R(r,t), t, t) = - M(r)/R(r,t)^2,
diff(R(r,t), t)^2 = 2*M(r)/R(r,t) + f(r)
]:
Info_:=`Tolman metric with constraints by C.W.Hellaby`:
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
sig_ := 2 :
g11_ := diff(R(r,t),r)^2/(1+f(r)) :
g22_ := R(r,t)^2 :
g33_ := R(r,t)^2*sin(theta)^2 :
g44_ := -1 :
constraint_ := [diff(diff(R(r,t),r),t) = (2*diff(m(r),r)/R(r,t)-2*m(r)*diff(R(r,t),r)/R(r,t)^2+
diff(f(r),r)+2*Lambda*R(r,t)*diff(R(r,t),r)/3)/(2*sqrt(2*m(r)/R(r,t)+f(r)+Lambda*R(r,t)^2/3)), diff(R(r,t),t) = sqrt(2*m(r)/R(r,t)+f(r)+Lambda*R(r,t)^2/3),
diff(diff(R(r,t),t),t) = -m(r)/R(r,t)^2-Lambda*R(r,t)/3, diff(diff(diff(R(r,t),t),r),t) =
-diff(m(r),r)/R(r,t)^2+2*m(r)*diff(R(r,t),r)/R(r,t)^3-Lambda*diff(R(r,t),r)/3] :
Info_:=`The Tolman dust solution with cosmological constant `:
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
sig_ := 2 :
g11_ := diff(R(r,t),r)^2 :
g22_ := R(r,t)^2 :
g33_ := R(r,t)^2*sin(theta)^2 :
g44_ := -1 :
constraint_ := [R(r,t)=((6*m(r)/Lambda)*sinh((3*Lambda)^(1/2)*(t[c](r)-t)/2)^2)^(1/3)] :
Info_:=`The Tolman dust solution with cosmological constant `:
Ndim_ := 5:
x1_ := t:
x2_ := r:
x3_ := theta:
x4_ := phi:
x5_ := y:
complex_ := {}:
g11_ := -exp(F(r))^2*A(t,y)^2:
g22_ := exp(G(r))^2*B(t,y)^2:
g33_ := exp(G(r))^2*B(t,y)^2*H(t,y)^2*r^2:
g44_ := exp(G(r))^2*B(t,y)^2*H(t,y)^2*r^2*sin(theta)^2:
g55_ := exp(K(r))^2*C(t,y)^2:
Ndim_ := 4:
x1_ := r:
x2_ := u:
x3_ := phi:
x4_ := t:
complex_ := {}:
g11_ := 1/((1-2*m/r)^gamma)*((r^2-2*m*r)/(r^2-2*m*r+m*u^2))^delta:
g22_ := 1/((1-2*m/r)^gamma)*(r^2-2*m*r)^epsilon/((r^2-2*m*r+m*u^2)^delta)/(m-u^2):
g33_ := 1/((1-2*m/r)^gamma)*u^2/m*r^2-2/((1-2*m/r)^gamma)*u^2*r:
g44_ := -(1-2*m/r)^gamma:
Info_ := ` gamma metric, u = m^(1/2)sin(theta) `:
constraint_ := [delta = gamma^2-1, epsilon = gamma^2]:
Ndim_ := 4 :
x1_ := x :
x2_ := y :
x3_ := z :
x4_ := t :
g11_ := a^2 :
g22_ := -1/2*a^2*exp(x)^2 :
g24_ := -a^2*exp(x)*c :
g33_ := a^2 :
g44_ := -a^2*c^2 :
Info_ := ` Godel's universe `:
Ndim_ := 3 :
x1_ := t :
x2_ := theta :
x3_ := r :
sig_ := 1 :
g11_ := -cos(2*alpha(r)) :
g12_ := -r*sin(2*alpha(r)) :
g22_ := r^2*cos(2*alpha(r)) :
g33_ := 1 :
Info_:=`Combed hedgehog metric (Williams G.R.G. 23, 181)`:
Ndim_ := 4 :
x1_ := u :
x2_ := w :
x3_ := theta :
x4_ := phi :
g11_ := 1/2*w^2/m/r(u,w) :
g12_ := 1 :
g33_ := r(u,w)^2 :
g44_ := r(u,w)^2*sin(theta)^2 :
constraint_ := [r(u,w) = 2*m+1/4*u*w/m] :
Info_:=`Israel coordinates (Phys. Rev. 143,1016)`:
Ndim_ := 3 :
x1_ := r :
x2_ := theta :
x3_ := t :
g11_ := b^2/r^2 :
g22_ := b^2 :
g33_ := -(1+a*ln(r)-a*ln(b))^2 :
Info_ := ` The back hole that went away `:
Ndim_ := 4:
x1_ := Theta:
x2_ := Phi:
x3_ := Psi:
x4_ := T:
sig_ := 2:
complex_ := {}:
eta11_ := 1:
eta22_ := 1:
eta33_ := 1:
eta44_ := -1:
bd11_ := -exp(a(T))*(1-Psi^2)^(1/2)/(1-Theta^2)^(1/2):
bd12_ := exp(a(T))*Psi*(1-Theta^2)^(1/2):
bd21_ := -exp(b(T))*Psi/(1-Theta^2)^(1/2):
bd22_ := -exp(b(T))*(1-Theta^2)^(1/2)*(1-Psi^2)^(1/2):
bd32_ := Theta*exp(c(T)):
bd33_ := exp(c(T))/(1-Psi^2)^(1/2):
bd44_ := exp(a(T)+b(T)+c(T)):
Info_ := `Mixmaster metric (e.g. MTW Box 30.1, Theta = cos(theta), Psi = sin(psi))`:
Ndim_ := 4 :
x1_ := x :
x2_ := y :
x3_ := z :
x4_ := t :
sig_ := 2 :
g11_ := L^2/(x^2+y^2+z^2) :
g22_ := L^2/(x^2+y^2+z^2) :
g33_ := L^2/(x^2+y^2+z^2) :
g44_ := -P(x,y,z,t)^2 :
constraint_ := [P(x,y,z,t) = a(t)*cos(ln((x^2+y^2+z^2)^(1/2)/L))+b(t)*sin(ln((x^2+y^2+z^2)^(1/2)/L))] :
Ndim_ := 4 :
x1_ := x :
x2_ := y :
x3_ := z :
x4_ := t :
eta11_ := -1 :
eta22_ := -1 :
eta33_ := -1 :
eta44_ := 1 :
bd11_ := L/(x^2+y^2+z^2)^(1/2) :
bd22_ := L/(x^2+y^2+z^2)^(1/2) :
bd33_ := L/(x^2+y^2+z^2)^(1/2) :
bd44_ := P(x,y,z,t) :
constraint_ := [P(x,y,z,t) = a(t)*cos(ln((x^2+y^2+z^2)^(1/2)/L))+b(t)*sin(ln((x^2+y^2+z^2)^(1/2)/L))] :
Ndim_ := 4 :
x1_ := r :
x2_ := u :
x3_ := phi :
x4_ := t :
sig_:= 2:
g11_ := (r^2+u^2)/(r^2-2*m*r+a^2) :
g22_ := (r^2+u^2)/(a^2-u^2) :
g33_ := (a^2-u^2)/a^2*(r^2+a^2+2*(a^2-u^2)*m*r/(r^2+u^2)) :
g34_ := -2*(a^2-u^2)/a*m*r/(r^2+u^2) :
g44_ := -1+2*m*r/(r^2+u^2) :
Info_:=`The Kerr metric in Boyer-Lindquist type coordinates (u=a*cos(theta)).`:
constraint_ := [u=a*cos(theta)]:
Ndim_ := 4:
x1_ := r:
x2_ := u:
x3_ := phi:
x4_ := t:
sig_ := 2:
complex_ := {}:
g11_ := (r^2+u^2)/(r^2-2*m*r+a^2+Q^2):
g22_ := (r^2+u^2)/(a^2-u^2):
g33_ := ((a^2-u^2)/a^2)*(r^2+a^2+(a^2-u^2)*(2*m*r-Q^2)/(r^2+u^2)):
g34_ := -((a^2-u^2)/a)*(2*m*r-Q^2)/(r^2+u^2):
g43_ := -((a^2-u^2)/a)*(2*m*r-Q^2)/(r^2+u^2):
g44_ := -(1-(2*m*r - Q^2)/(r^2+u^2)):
constraint_ := [u=a*cos(theta)]:
Info_:=`Kerr Newman Solution in Boyer-Lindquist coordinates (u=a*cos(theta))`:
Ndim_ := 4:
x1_ := r:
x2_ := psi:
x3_ := theta:
x4_ := phi:
complex_ := {}:
g11_ := 1/alpha(r):
g22_ := 1/4*r^2*beta(r):
g24_ := 1/4*r^2*beta(r)*cos(theta):
g33_ := 1/4*r^2*alpha(r)*cos(psi)^2+1/4*r^2/beta(r)*sin(psi)^2:
g34_ := 1/4*r^2*alpha(r)*cos(psi)*sin(psi)*sin(theta)-1/4*r^2/beta(r)*cos(psi)*sin(psi)*sin(theta):
g44_ := 1/4*r^2*alpha(r)*sin(psi)^2*sin(theta)^2+1/4*r^2/beta(r)*cos(psi)^2*sin(theta)^2+1/4*r^2*beta(r)*cos(theta)^2:
Ndim_ := 4 :
x1_ := t :
x2_ := r :
x3_ := theta :
x4_ := phi :
eta12_ := 1 :
eta34_ := -1 :
bd11_ := 1 :
bd12_ := -(r^2+a^2*cos(theta)^2)/(r^2-2*M*r+a^2+Q^2) :
bd14_ := -a*sin(theta)^2 :
bd21_ := 1/2*(r^2-2*M*r+a^2+Q^2)/(r^2+a^2*cos(theta)^2) :
bd22_ := 1/2 :
bd24_ := -1/2*a*(r^2-2*M*r+a^2+Q^2)*sin(theta)^2/(r^2+a^2*cos(theta)^2) :
bd31_ := 1/2*(I*sin(theta)*r+sin(theta)*a*cos(theta))*a*2^(1/2)/(r^2+a^2*cos(theta)^2) :
bd33_ := -1/2*(r-I*a*cos(theta))*2^(1/2) :
bd34_ := -1/2*(I*sin(theta)*r+sin(theta)*a*cos(theta))*(r^2+a^2)*2^(1/2)/(r^2+a^2*cos(theta)^2) :
bd41_ := 1/2*(-I*sin(theta)*r+sin(theta)*a*cos(theta))*a*2^(1/2)/(r^2+a^2*cos(theta)^2) :
bd43_ := -1/2*(r+I*a*cos(theta))*2^(1/2) :
bd44_ := -1/2*(-I*sin(theta)*r+sin(theta)*a*cos(theta))*(r^2+a^2)*2^(1/2)/(r^2+a^2*cos(theta)^2) :
Info_:=`Covariant NPtetrad for Kerr-Newman metric(Boyer-Lindquist coordinates (J. Math. Phys. 8 265))`:
Ndim_ := 4:
x1_ := r:
x2_ := theta:
x3_ := phi:
x4_ := t:
sig_:= 2:
g11_ := 1/(1-2*m/r-Lambda*r^2/3+e^2/r^2):
g22_ := r^2:
g33_ := r^2*sin(theta)^2:
g44_ := -(1-2*m/r-Lambda*r^2/3+e^2/r^2):
Info_:=`The Reissner-Nordstrom-de Sitter metric`:
Ndim_ := 4:
x1_ := r:
x2_ := theta:
x3_ := phi:
x4_ := t:
sig_:= 2:
g11_ := 1/(1-2*m/r-Lambda*r^2/3+Q^2/r^2):
g22_ := r^2:
g33_ := r^2*sin(theta)^2:
g44_ := -(1-2*m/r-Lambda*r^2/3+Q^2/r^2):
Info_:=`The Reissner-Nordstrom-de Sitter metric`:
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
sig_ := 2:
g11_ := exp(lambda(r,t)) :
g22_ := exp(mu(r,t)) :
g33_ := exp(mu(r,t))*sin(theta)^2 :
g44_ := -exp(nu(r,t)) :
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
g11_ := 1/A(r) :
g22_ := r^2 :
g33_ := r^2*sin(theta)^2 :
g44_ := -A(r) :
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
g11_ := exp(lambda(r/t)) :
g22_ := r^2 :
g33_ := r^2*sin(theta)^2 :
g44_ := -exp(nu(r/t)) :
Info_ := ` Self-similar metric "curvature form" `:
Ndim_ := 4 :
x1_ := r :
x2_ := theta :
x3_ := phi :
x4_ := t :
g11_ := exp(Lambda(r/t)) :
g22_ := R(r/t)^2*t^2 :
g33_ := R(r/t)^2*t^2*sin(theta)^2 :
g44_ := -exp(Delta(r/t)) :
Info_ := ` Self-similar metric "comoving form" `:
Ndim_ := 4 :
x1_ := v :
x2_ := r :
x3_ := theta :
x4_ := phi :
g11_ := -c(r/v)^2+2*c(r/v)^2*h(r/v)/r*v :
g12_ := c(r/v) :
g33_ := r^2 :
g44_ := r^2*sin(theta)^2 :
Info_ := ` Self-similar metric "Bondi form" `:
Ndim_ := 4 :
x1_ := x :
x2_ := y :
x3_ := z :
x4_ := t :
g11_ := 1/V(x,y,z,t)^2 :
g22_ := 1/V(x,y,z,t)^2 :
g33_ := 1/V(x,y,z,t)^2 :
g44_ := -9*diff(V(x,y,z,t),t)^2/Theta(t)^2/V(x,y,z,t)^2 :
constraint_ := [diff(V(x,y,z,t),t,z,x) =0,diff(V(x,y,z,t),y,t,z)=0,diff(V(x,y,z,t),y,x,t)=0,diff(V(x,y,z,t),z,z,t)=diff(V(x,y,z,t),x,x,t),diff(V(x,y,z,t),y,y,t)=diff(V(x,y,z,t),x,x,t),
diff(V(x,y,z,t),x,y)=0,diff(V(x,y,z,t),x,z)=0,diff(V(x,y,z,t),y,z)=0,diff(V(x,y,z,t),y,y)=diff(V(x,y,z,t),x,x),diff(V(x,y,z,t),z,z)=diff(V(x,y,z,t),x,x)] :
Ndim_ := 4 :
x1_ := u :
x2_ := v :
x3_ := t :
x4_ := phi :
g11_ := U(u,v)/((1+u)*u) :
g22_ := U(u,v)/((1-v)*v) :
g33_ := -f(u,v) :
g34_ := f(u,v)*omega(u,v) :
g43_ := f(u,v)*omega(u,v) :
g44_ := m^2*p^2*u*v/(4*f(u,v))-f(u,v)*omega(u,v)^2 :
constraint_ := [omega(u,v)=2*m*q*c(u,v)*v/a(u,v),U(u,v)=b(u,v)*m^2/(16*p^2*(u+v)^3),f(u,v)=a(u,v)/b(u,v)]:
Ndim_ := 2 :
x1_ := theta :
x2_ := phi :
g11_ := F(theta)^2 :
g22_ := G(theta)^2 :
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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 211 184 "Demonstration 1 (kawai)
: Here we take a look at \"the black hole that went away\" . Note tha
t we avoid use of $r_o$to designate a constant in the input since r i
s used as a coordinate." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(grtensor);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "Q-GRTensor~III6\"" }}{PARA 11 "" 1 ""
{XPPMATH 20 "QZCopyright~2016,~Peter~Musgrave,~Denis~Pollney,~Kayll~La
ke6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q-grtensor.org6\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "79I%AsymG6\"I.KillingCoordsGF$I$SymGF$I*autoAliasG
F$I(gralterGF$I'grcalcGF$I,grcalcalterGF$I,grcomponentGF$I(grdebugGF$I
&grdefGF$I*grdisplayGF$I'grinitGF$I'grloadGF$I*grloaddefGF$I&grmapGF$I
*groptionsGF$I*grsavedefGF$I,grtestinputGF$I,grtransformGF$I'kdeltaGF$
I&makegGF$I)nptetradGF$I&qloadGF$" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "qload(kawai);" }}{PARA 6 "" 1 "" {TEXT 212 39 "Calcul
ated ds for kawai (0.001000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I2
Default~spacetimeG6\"I&kawaiGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I9For
~the~kawai~spacetime:G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I,Coordinat
esG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"xG6\"6#I#upGF$" }}{PARA 11
"" 1 "" {XPPMATH 20 "/)I#x~G6\"I\"aGF%=F%6#;\"\"\"\"\"$E\\[l$F*I\"rGF
%\"\"#I&thetaGF%F+I\"tGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "I-Line~elem
entG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/*$I$~dsG6\"\"\"#,(**I\"bGF%F
&I\"rGF%!\"#I#~dGF%\"\"\")F*I#2~GF%F-F-*(F)F&F,F-)I&thetaGF%F/F-F-*(,(
F-F-*&I\"aGF%F--I#lnG6$%*protectedGI(_syslibGF%6#F*F-F-*&F6F--F86#F)F-
!\"\"F&F,F-)I\"tGF%F/F-F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "IE~~~~The~b
ack~hole~that~went~away~~~~G6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 23 "grcalc(R(dn,dn,dn,dn));" }}{PARA 6 "" 1 "" {TEXT 212 49 "Calcu
lated g(dn,dn,pdn) for kawai (0.001000 sec.)" }}{PARA 6 "" 1 "" {TEXT
212 50 "Calculated Chr(dn,dn,dn) for kawai (0.001000 sec.)" }{TEXT
212 42 "\nCalculated detg for kawai (0.004000 sec.)" }{TEXT 212 8 "\np
recalc" }{TEXT 212 46 "\nCalculated g(up,up) for kawai (0.006000 sec.)
" }{TEXT 212 52 "\nCalculated R(dn,dn,dn,dn) for kawai (0.001000 sec.)
" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I*CPU~Time~G6\"$\"#7!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 ""
1 "" {XPPMATH 20 "I9For~the~kawai~spacetime:G6\"" }}{PARA 11 "" 1 ""
{XPPMATH 20 "I2Covariant~RiemannG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
/I/R(dn,dn,dn,dn)G6\"I8All~components~are~zeroGF$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
33 1 1 }
\ No newline at end of file
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