Sun Aug 18 18:06:18 2002 run on Windows
*** ^ redefined
+++ depends redefined
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Twisting type N solutions of GR %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The problem is to analyse an ansatz for a particular type of vacuum
% solution to Einstein's equations for general relativity. The analysis was
% described by Finley and Price (Proc Aspects of GR and Math Phys
% (Plebanski Festschrift), Mexico City June 1993). The equations resulting
% from the ansatz are:
% F - F*gamma = 0
% 3 3
%
% F *x + 2*F *x + x *F - x *Delta*F = 0
% 2 2 1 2 1 2 1 2 2 1
%
% 2*F *x + 2*F *x + 2*F *x + 2*F *x + x *F = 0
% 2 3 2 3 2 2 3 3 3 2 2 3 3 2 2 3 2 2 3 3
%
% Delta =0 Delta neq 0
% 3 1
%
% gamma =0 gamma neq 0
% 2 1
% where the unknowns are {F,x,gamma,Delta} and the indices refer to
% derivatives with respect to an anholonomic basis. The highest order is 4,
% but the 4th order jet bundle is too large for practical computation, so
% it is necessary to construct partial prolongations. There is a single
% known solution, due to Hauser, which is verified at the end.
on evallhseqp,edssloppy,edsverbose;
off arbvars,edsdebug;
pform {F,x,Delta,gamma,v,y,u}=0;
pform v(i)=0,omega(i)=1;
indexrange {i,j,k,l}={1,2,3};
% Construct J1({v,y,u},{x}) and transform coordinates. Use ordering
% statement to get v eliminated in favour of x where possible.
% NB Coordinate change cc1 is invertible only when x(-1) neq 0.
J1 := contact(1,{v,y,u},{x});
j1 := EDS({d x - x *d u - x *d v - x *d y},d u^d v^d y)
u v y
korder x(-1),x(-2),v(-3);
cc1 := {x(-v) = x(-1),
x(-y) = x(-2),
x(-u) = -x(-1)*v(-3)};
cc1 := {x =x ,
v 1
x =x ,
y 2
x = - x *v }
u 1 3
J1 := restrict(pullback(J1,cc1),{x(-1) neq 0});
j1 := EDS({d x + v *x *d u - x *d v - x *d y},d u^d v^d y)
3 1 1 2
% Set up anholonomic cobasis
bc1 := {omega(1) = d v - v(-3)*d u,
omega(2) = d y,
omega(3) = d u};
1 2 3
bc1 := {omega = - v *d u + d v,omega =d y,omega =d u}
3
J1 := transform(J1,bc1);
1 2 1 2 3
j1 := EDS({d x - x *omega - x *omega },omega ^omega ^omega )
1 2
% Prolong to J421: 4th order in x, 2nd in F and 1st in rest
J2 := prolong J1$
Prolongation using new equations:
- x
2 3
v =---------
3 2 x
1
- x
1 3
v =---------
3 1 x
1
x =x
2 1 1 2
x neq 0
1
J20 := J2 cross {F}$
J31 := prolong J20$
Prolongation using new equations:
2*x *x - x *x
1 3 2 3 1 2 3 3
v =-------------------------
3 3 2 2
(x )
1
2
- x *x + 2*(x )
1 3 3 1 1 3
v =--------------------------
3 3 1 2
(x )
1
- x *x + x *x
1 2 2 3 1 2 2 3
x =--------------------------
2 3 2 x
1
x *x - x *x
1 2 3 1 1 2 1 3
x =-----------------------
2 3 1 x
1
x =x
2 2 1 1 2 2
- x *x + x *x
1 1 2 3 1 2 3 1
x =--------------------------
1 3 2 x
1
x *x - x *x
1 1 3 1 1 1 1 3
x =-----------------------
1 3 1 x
1
x =x
1 2 1 1 1 2
x neq 0
1
J310 := J31 cross {Delta,gamma}$
J421 := prolong J310$
Prolongation using new equations:
- f *x + f *x
1 2 3 2 3 1
f =----------------------
3 2 x
1
f *x - f *x
1 3 1 1 1 3
f =-------------------
3 1 x
1
f =f
2 1 1 2
2 2
3*x *x *x - 6*(x ) *x + 3*x *x *x - (x ) *x
1 3 3 1 2 3 1 3 2 3 1 3 1 2 3 3 1 2 3 3 3
v =-----------------------------------------------------------------------
3 3 3 2 3
(x )
1
2 3
- x *(x ) + 6*x *x *x - 6*(x )
1 3 3 3 1 1 3 3 1 3 1 1 3
v =--------------------------------------------------
3 3 3 1 3
(x )
1
x
2 3 3 2
2
- 2*x *x *x + 2*x *x *x - x *x *x + (x ) *x
1 2 3 1 2 3 1 2 1 3 2 3 1 2 1 2 3 3 1 2 2 3 3
=--------------------------------------------------------------------------
2
(x )
1
2 2
x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
1 2 3 3 1 1 2 3 1 3 1 1 2 1 3 3 1 1 2 1 3
x =---------------------------------------------------------------------
2 3 3 1 2
(x )
1
- x *x + x *x
1 2 2 2 3 1 2 2 2 3
x =------------------------------
2 2 3 2 x
1
x *x - x *x
1 2 2 3 1 1 2 2 1 3
x =---------------------------
2 2 3 1 x
1
x =x
2 2 2 1 1 2 2 2
x
1 3 3 2
2
- 2*x *x *x + 2*x *x *x - x *x *x + x *(x )
1 1 3 1 2 3 1 1 1 3 2 3 1 1 1 2 3 3 1 2 3 3 1
=--------------------------------------------------------------------------
2
(x )
1
2 2
x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
1 1 3 3 1 1 1 3 1 3 1 1 1 1 3 3 1 1 1 1 3
x =---------------------------------------------------------------------
1 3 3 1 2
(x )
1
- x *x + x *x
1 1 2 2 3 1 2 2 3 1
x =------------------------------
1 2 3 2 x
1
x *x - x *x
1 1 2 3 1 1 1 2 1 3
x =---------------------------
1 2 3 1 x
1
x =x
1 2 2 1 1 1 2 2
- x *x + x *x
1 1 1 2 3 1 1 2 3 1
x =------------------------------
1 1 3 2 x
1
x *x - x *x
1 1 1 3 1 1 1 1 1 3
x =---------------------------
1 1 3 1 x
1
x =x
1 1 2 1 1 1 1 2
x neq 0
1
cc4 := first pullback_maps;
x *f - f *x
1 2 3 1 2 3
cc4 := {f =-------------------,
3 2 x
1
x *f - f *x
1 1 3 1 1 3
f =-------------------,
3 1 x
1
f =f ,
2 1 1 2
2
v =( - (x ) *x + 3*x *x *x + 3*x *x *x
3 3 3 2 1 2 3 3 3 1 1 3 3 2 3 1 1 3 2 3 3
2 3
- 6*(x ) *x )/(x ) ,
1 3 2 3 1
2 3
- (x ) *x + 6*x *x *x - 6*(x )
1 1 3 3 3 1 1 3 3 1 3 1 3
v =--------------------------------------------------,
3 3 3 1 3
(x )
1
2
x =((x ) *x - 2*x *x *x - x *x *x
2 3 3 2 1 2 2 3 3 1 1 2 3 2 3 1 1 2 2 3 3
2
+ 2*x *x *x )/(x ) ,
1 2 1 3 2 3 1
x
2 3 3 1
2 2
(x ) *x - 2*x *x *x - x *x *x + 2*x *(x )
1 1 2 3 3 1 1 2 3 1 3 1 1 2 1 3 3 1 2 1 3
=---------------------------------------------------------------------,
2
(x )
1
x *x - x *x
1 2 2 2 3 1 2 2 2 3
x =---------------------------,
2 2 3 2 x
1
x *x - x *x
1 1 2 2 3 1 2 2 1 3
x =---------------------------,
2 2 3 1 x
1
x =x ,
2 2 2 1 1 2 2 2
2
x =((x ) *x - 2*x *x *x - x *x *x
1 3 3 2 1 1 2 3 3 1 1 1 3 2 3 1 1 1 2 3 3
2
+ 2*x *x *x )/(x ) ,
1 1 1 3 2 3 1
x
1 3 3 1
2 2
(x ) *x - 2*x *x *x - x *x *x + 2*x *(x )
1 1 1 3 3 1 1 1 3 1 3 1 1 1 1 3 3 1 1 1 3
=---------------------------------------------------------------------,
2
(x )
1
x *x - x *x
1 1 2 2 3 1 1 2 2 3
x =---------------------------,
1 2 3 2 x
1
x *x - x *x
1 1 1 2 3 1 1 2 1 3
x =---------------------------,
1 2 3 1 x
1
x =x ,
1 2 2 1 1 1 2 2
x *x - x *x
1 1 1 2 3 1 1 1 2 3
x =---------------------------,
1 1 3 2 x
1
x *x - x *x
1 1 1 1 3 1 1 1 1 3
x =---------------------------,
1 1 3 1 x
1
x =x ,
1 1 2 1 1 1 1 2
x neq 0}
1
% Apply first order de and restrictions
de1 := {Delta(-3) = 0,
gamma(-2) = 0,
Delta(-1) neq 0,
gamma(-1) neq 0};
de1 := {delta =0,
3
gamma =0,
2
delta neq 0,
1
gamma neq 0}
1
J421 := pullback(J421,de1)$
% Main de in original coordinates
de2 := {F(-3,-3) - gamma*F,
x(-1)*F(-2,-2) + 2*x(-1,-2)*F(-2)
+ (x(-1,-2,-2) - x(-1)*Delta)*F,
x(-2,-3)*(F(-2,-3)+F(-3,-2)) + x(-2,-2,-3)*F(-3)
+ x(-2,-3,-3)*F(-2) + (1/2)*x(-2,-2,-3,-3)*F};
de2 := {f - f*gamma,
3 3
f *x + 2*f *x + x *f - x *delta*f,
2 2 1 2 1 2 1 2 2 1
2*f *x + 2*f *x + 2*f *x + 2*f *x + x *f
2 3 2 3 2 2 3 3 3 2 2 3 3 2 2 3 2 2 3 3
--------------------------------------------------------------------}
2
% This is not expressed in terms of current coordinates.
% Missing coordinates are seen from 1-form variables in following
d de2 xmod cobasis J421;
{d f *x }
3 2 2 3
% The necessary equation is contained in the last prolongation
pullback(d de2,cc4) xmod cobasis J421;
{}
% Apply main de
pb1 := first solve(pullback(de2,cc4),{F(-3,-3),F(-2,-2),F(-2,-3)});
pb1 := {f =f*gamma,
3 3
- 2*f *x - x *f + x *delta*f
2 1 2 1 2 2 1
f =--------------------------------------,
2 2 x
1
2
2*f *(x ) - 2*f *x *x - 2*f *x *x - x *x *f
1 2 3 2 1 2 3 3 3 1 2 2 3 1 2 2 3 3
f =----------------------------------------------------------------}
2 3 4*x *x
1 2 3
Y421 := pullback(J421,pb1)$
% Check involution
on ranpos;
characters Y421;
{15,7,0}
dim_grassmann_variety Y421;
28
% 15+2*7 = 29 > 28: Y421 not involutive, so prolong
Y532 := prolong Y421$
Prolongation using new equations:
- gamma *x
1 2 3
gamma =----------------
3 2 x
1
gamma *x - gamma *x
1 3 1 1 1 3
gamma =---------------------------
3 1 x
1
gamma =0
1 2
delta *x
1 2 3
delta =-------------
2 3 x
1
delta =delta
2 1 1 2
delta *x
1 1 3
delta =-------------
1 3 x
1
2 2
f =(2*f *x *x + f *x *x - 2*f *(x ) + f *(x ) *gamma
1 3 3 1 3 1 3 1 1 1 3 3 1 1 1 3 1 1
2 2
+ gamma *(x ) *f)/(x )
1 1 1
3 2 2
f =( - 2*f *x *(x ) + 4*f *x *x *(x ) - 2*f *(x ) *x *x
1 3 2 1 1 1 2 3 1 2 1 3 1 2 3 1 2 1 2 3 3 2 3
2 3 2
- 2*f *(x ) *x *x - 2*f *x *(x ) + 2*f *x *x *(x )
1 3 1 2 2 3 2 3 1 1 1 2 3 1 1 2 3 1 2 3
2
- 2*f *x *x *(x ) + 2*f *x *x *x *x
1 1 2 1 3 2 3 1 1 3 1 2 2 3 2 3
2 2
- f *(x ) *x *x - 2*f *x *(x ) *x
1 1 2 2 3 3 2 3 2 1 2 3 3 1 2 3
2
+ 4*f *x *x *x *x + 2*f *x *(x ) *x
2 1 2 3 1 3 1 2 3 2 1 2 3 1 2 3 3
2
+ 2*f *x *x *x *x - 4*f *x *(x ) *x
2 1 2 1 3 3 1 2 3 2 1 2 1 3 2 3
2
- 2*f *x *x *x *x - 2*f *x *(x ) *x
2 1 2 1 3 1 2 3 3 3 1 2 2 3 1 2 3
2
+ 2*f *x *x *x *x + 2*f *x *(x ) *x
3 1 2 2 1 3 1 2 3 3 1 2 3 1 2 2 3
2
- 2*f *x *x *x *x + x *(x ) *x *f
3 1 2 1 3 1 2 2 3 1 2 3 1 2 2 3 3
2 2 2
- x *x *x *x *f - (x ) *x *x *f)/(4*(x ) *(x ) )
1 2 1 3 1 2 2 3 3 1 2 2 3 3 1 2 3 1 2 3
f *x - f *x
1 1 3 1 1 1 1 3
f =-----------------------
1 3 1 x
1
3 2 2
f =(2*f *x *(x ) + 4*f *x *x *(x ) - 2*f *(x ) *x *x
1 2 3 1 1 1 2 3 1 2 1 3 1 2 3 1 2 1 2 3 3 2 3
2 3 2
- 2*f *(x ) *x *x - 2*f *x *(x ) + 2*f *x *x *(x )
1 3 1 2 2 3 2 3 1 1 1 2 3 1 1 2 3 1 2 3
2
- 2*f *x *x *(x ) + 2*f *x *x *x *x
1 1 2 1 3 2 3 1 1 3 1 2 2 3 2 3
2 2
- f *(x ) *x *x - 2*f *x *(x ) *x
1 1 2 2 3 3 2 3 2 1 2 3 3 1 2 3
2
+ 4*f *x *x *x *x + 2*f *x *(x ) *x
2 1 2 3 1 3 1 2 3 2 1 2 3 1 2 3 3
2
+ 2*f *x *x *x *x - 4*f *x *(x ) *x
2 1 2 1 3 3 1 2 3 2 1 2 1 3 2 3
2
- 2*f *x *x *x *x - 2*f *x *(x ) *x
2 1 2 1 3 1 2 3 3 3 1 2 2 3 1 2 3
2
+ 2*f *x *x *x *x + 2*f *x *(x ) *x
3 1 2 2 1 3 1 2 3 3 1 2 3 1 2 2 3
2
- 2*f *x *x *x *x + x *(x ) *x *f
3 1 2 1 3 1 2 2 3 1 2 3 1 2 2 3 3
2 2 2
- x *x *x *x *f - (x ) *x *x *f)/(4*(x ) *(x ) )
1 2 1 3 1 2 2 3 3 1 2 2 3 3 1 2 3 1 2 3
2 2
f =(delta *(x ) *f - 2*f *x *x - f *x *x + f *(x ) *delta
1 2 2 1 1 1 2 1 2 1 1 1 2 2 1 1 1
- 2*f *x *x + 2*f *x *x - x *x *f + x *x *f)/
2 1 1 2 1 2 1 1 1 2 1 1 2 2 1 1 1 1 2 2
2
(x )
1
f =f
1 2 1 1 1 2
2
v =(4*x *(x ) *x - 24*x *x *x *x
3 3 3 3 2 1 3 3 3 1 2 3 1 3 3 1 3 1 2 3
2 3 2
+ 6*x *(x ) *x + 24*(x ) *x - 12*(x ) *x *x
1 3 3 1 2 3 3 1 3 2 3 1 3 1 2 3 3
2 3 4
+ 4*x *(x ) *x - (x ) *x )/(x )
1 3 1 2 3 3 3 1 2 3 3 3 3 1
3 2 2 2
v =( - x *(x ) + 8*x *x *(x ) + 6*(x ) *(x )
3 3 3 3 1 1 3 3 3 3 1 1 3 3 3 1 3 1 1 3 3 1
2 4 4
- 36*x *(x ) *x + 24*(x ) )/(x )
1 3 3 1 3 1 1 3 1
2 3 3
x =( - 12*f *(x ) *(x ) + 12*f *x *x *(x )
2 3 3 3 2 1 3 1 2 3 1 1 3 1 2 3
2 2 3
- 6*f *(x ) *x *(x ) - 4*f *(x ) *x *x
1 1 2 3 3 2 3 2 1 2 3 3 3 2 3
3 2 3 2
+ 6*f *(x ) *(x ) - 8*f *(x ) *(x ) *gamma
2 1 2 3 3 2 1 2 3
3 3
- 6*f *(x ) *x *x + 6*f *(x ) *x *x
3 1 2 2 3 3 2 3 3 1 2 2 3 2 3 3
2 2 2
- 6*x *(x ) *(x ) *f + 12*x *x *x *(x ) *f
1 2 3 3 1 2 3 1 2 3 1 3 1 2 3
2 2
- 6*x *(x ) *x *x *f + 6*x *x *x *(x ) *f
1 2 3 1 2 3 3 2 3 1 2 1 3 3 1 2 3
2 2
- 12*x *(x ) *(x ) *f + 6*x *x *x *x *x *f
1 2 1 3 2 3 1 2 1 3 1 2 3 3 2 3
2 3
- 2*x *(x ) *x *x *f + 3*(x ) *x *x *f
1 2 1 2 3 3 3 2 3 1 2 2 3 3 2 3 3
3 3
- 4*(x ) *x *x *f*gamma)/(2*(x ) *x *f)
1 2 2 3 2 3 1 2 3
3 2 2
x =(x *(x ) - 3*x *x *(x ) - 3*x *x *(x )
2 3 3 3 1 1 2 3 3 3 1 1 2 3 3 1 3 1 1 2 3 1 3 3 1
2 2
+ 6*x *(x ) *x - x *x *(x ) + 6*x *x *x *x
1 2 3 1 3 1 1 2 1 3 3 3 1 1 2 1 3 3 1 3 1
3 3
- 6*x *(x ) )/(x )
1 2 1 3 1
3 3 2
x =( - 12*f *x *(x ) + 12*f *x *(x ) - 6*f *x *x *(x )
2 2 3 3 3 1 3 1 2 3 1 1 3 2 3 1 1 2 3 3 2 3
2 2 2
- 4*f *(x ) *x *x + 6*f *(x ) *(x )
2 1 2 3 3 3 2 3 2 1 2 3 3
2 2 2
- 8*f *(x ) *(x ) *gamma - 6*f *(x ) *x *x
2 1 2 3 3 1 2 2 3 3 2 3
2 2
+ 6*f *(x ) *x *x + 3*(x ) *x *x *f
3 1 2 2 3 2 3 3 1 2 2 3 3 2 3 3
2 2
- 4*(x ) *x *x *f*gamma)/(2*(x ) *x *f)
1 2 2 3 2 3 1 2 3
3 2
x =(12*f *x *(x ) + 6*f *x *x *(x )
2 2 3 3 2 1 2 1 2 3 1 1 2 2 3 2 3
2 2
+ 24*f *x *x *(x ) - 24*f *x *x *(x )
2 1 2 3 1 2 3 2 1 2 1 3 2 3
2 2
- 6*f *(x ) *x *x + 6*f *(x ) *x *x
2 1 2 2 3 3 2 3 2 1 2 2 3 2 3 3
2
+ 12*f *x *x *(x ) - 12*f *x *x *x *x
3 1 2 2 1 2 3 3 1 2 1 2 2 3 2 3
2 2 2
- 4*f *(x ) *x *x + 6*f *(x ) *(x )
3 1 2 2 2 3 2 3 3 1 2 2 3
2 2 2
- 8*f *(x ) *(x ) *delta + 8*x *x *(x ) *f
3 1 2 3 1 2 2 3 1 2 3
2
- 8*x *x *(x ) *f + 4*x *x *x *x *f
1 2 2 1 3 2 3 1 2 2 1 2 3 3 2 3
2
- 6*x *x *x *x *f + 3*(x ) *x *x *f
1 2 1 2 2 3 3 2 3 1 2 2 3 3 2 2 3
2 2
- 4*(x ) *x *x *delta*f)/(2*(x ) *x *f)
1 2 3 3 2 3 1 2 3
3 2
x =(12*f *x *(x ) + 6*f *x *x *(x )
2 2 2 3 3 1 2 1 2 3 1 1 2 2 3 2 3
2 2
+ 24*f *x *x *(x ) - 24*f *x *x *(x )
2 1 2 3 1 2 3 2 1 2 1 3 2 3
2 2
- 6*f *(x ) *x *x + 6*f *(x ) *x *x
2 1 2 2 3 3 2 3 2 1 2 2 3 2 3 3
2
+ 12*f *x *x *(x ) - 12*f *x *x *x *x
3 1 2 2 1 2 3 3 1 2 1 2 2 3 2 3
2 2 2
- 4*f *(x ) *x *x + 6*f *(x ) *(x )
3 1 2 2 2 3 2 3 3 1 2 2 3
2 2 2
- 8*f *(x ) *(x ) *delta + 12*x *x *(x ) *f
3 1 2 3 1 2 2 3 1 2 3
2
- 12*x *x *(x ) *f + 6*x *x *x *x *f
1 2 2 1 3 2 3 1 2 2 1 2 3 3 2 3
2
- 6*x *x *x *x *f + 3*(x ) *x *x *f
1 2 1 2 2 3 3 2 3 1 2 2 3 3 2 2 3
2 2
- 4*(x ) *x *x *delta*f)/(2*(x ) *x *f)
1 2 3 3 2 3 1 2 3
- x *x + x *x
1 2 2 2 2 3 1 2 2 2 2 3
x =----------------------------------
2 2 2 3 2 x
1
x *x - x *x
1 2 2 2 3 1 1 2 2 2 1 3
x =-------------------------------
2 2 2 3 1 x
1
x =x
2 2 2 2 1 1 2 2 2 2
2
x =( - 3*x *(x ) *x + 6*x *x *x *x
1 3 3 3 2 1 1 3 3 1 2 3 1 1 3 1 3 1 2 3
2
- 3*x *(x ) *x + 3*x *x *x *x
1 1 3 1 2 3 3 1 1 1 3 3 1 2 3
2 2
- 6*x *(x ) *x + 3*x *x *x *x - x *(x ) *x
1 1 1 3 2 3 1 1 1 3 1 2 3 3 1 1 1 2 3 3 3
3 3
+ x *(x ) )/(x )
1 2 3 3 3 1 1
3 2 2
x =(x *(x ) - 3*x *x *(x ) - 3*x *x *(x )
1 3 3 3 1 1 1 3 3 3 1 1 1 3 3 1 3 1 1 1 3 1 3 3 1
2 2
+ 6*x *(x ) *x - x *x *(x ) + 6*x *x *x *x
1 1 3 1 3 1 1 1 1 3 3 3 1 1 1 1 3 3 1 3 1
3 3
- 6*x *(x ) )/(x )
1 1 1 3 1
x =( - 2*x *x *x + 2*x *x *x - x *x *x
1 2 3 3 2 1 1 2 3 1 2 3 1 1 2 1 3 2 3 1 1 2 1 2 3 3
2
+ 2*x *x *x + x *x *x - 2*x *(x )
1 2 2 3 1 3 1 1 2 2 1 3 3 1 1 2 2 1 3
2 2
+ (x ) *x )/(x )
1 2 2 3 3 1 1
x
1 2 3 3 1
2 2
x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
1 1 2 3 3 1 1 1 2 3 1 3 1 1 1 2 1 3 3 1 1 1 2 1 3
=-----------------------------------------------------------------------------
2
(x )
1
2
x =(2*x *x *x + x *x *x - 2*x *(x )
1 2 2 3 3 1 2 2 3 1 3 1 1 2 2 1 3 3 1 1 2 2 1 3
2 2
+ (x ) *x )/(x )
1 2 2 3 3 1 1
- x *x + x *x
1 1 2 2 2 3 1 2 2 2 3 1
x =----------------------------------
1 2 2 3 2 x
1
x *x - x *x
1 1 2 2 3 1 1 1 2 2 1 3
x =-------------------------------
1 2 2 3 1 x
1
x =x
1 2 2 2 1 1 1 2 2 2
x =( - 2*x *x *x + 2*x *x *x - x *x *x
1 1 3 3 2 1 1 1 3 1 2 3 1 1 1 1 3 2 3 1 1 1 1 2 3 3
2 2
+ x *(x ) )/(x )
1 1 2 3 3 1 1
x
1 1 3 3 1
2 2
x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
1 1 1 3 3 1 1 1 1 3 1 3 1 1 1 1 1 3 3 1 1 1 1 1 3
=-----------------------------------------------------------------------------
2
(x )
1
- x *x + x *x
1 1 1 2 2 3 1 1 2 2 3 1
x =----------------------------------
1 1 2 3 2 x
1
x *x - x *x
1 1 1 2 3 1 1 1 1 2 1 3
x =-------------------------------
1 1 2 3 1 x
1
x =x
1 1 2 2 1 1 1 1 2 2
- x *x + x *x
1 1 1 1 2 3 1 1 1 2 3 1
x =----------------------------------
1 1 1 3 2 x
1
x *x - x *x
1 1 1 1 3 1 1 1 1 1 1 3
x =-------------------------------
1 1 1 3 1 x
1
x =x
1 1 1 2 1 1 1 1 1 2
x neq 0
1
x neq 0
2 3
f neq 0
characters Y532;
{22,6,0}
dim_grassmann_variety Y532;
34
% 22+2*6 = 34: just need to check for integrability conditions
torsion Y532;
{}
% Y532 involutive. Dimensions?
dim Y532;
79
length one_forms Y532;
48
% The following puts in part of Hauser's solution and ends up with an ODE
% system (all characters 0), so no more solutions, as described by Finley
% at MG6.
hauser := {x=-v+(1/2)*(y+u)**2,delta=3/(8x),gamma=3/(8v)};
2 2
u + 2*u*y - 2*v + y
hauser := {x=-----------------------,
2
3
delta=-----,
8*x
3
gamma=-----}
8*v
H532 := pullback(Y532,hauser)$
New 0-form conditions detected
2
- 8*gamma *v - 3*v
3 3
-----------------------
2
8*v
2
- 8*gamma *v - 3
1
--------------------
2
8*v
3*(v - u - y)
3
----------------------------------------------------------------------------
4 3 2 2 2 3 2 2 4
2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
4 3 2 2 2
( - 2*delta *u - 8*delta *u *y + 8*delta *u *v - 12*delta *u *y
2 2 2 2
3 2 2 4
+ 16*delta *u*v*y - 8*delta *u*y - 8*delta *v + 8*delta *v*y - 2*delta *y
2 2 2 2 2
4 3 2 2 2 3 2
- 3*u - 3*y)/(2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v
2 4
- 4*v*y + y ))
4 3 2 2 2
( - 2*delta *u - 8*delta *u *y + 8*delta *u *v - 12*delta *u *y
1 1 1 1
3 2 2 4
+ 16*delta *u*v*y - 8*delta *u*y - 8*delta *v + 8*delta *v*y - 2*delta *y
1 1 1 1 1
4 3 2 2 2 3 2 2
+ 3)/(2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y
4
+ y ))
- v + u + y
3
- x + u + y
2
- (x + 1)
1
lift ws;
Solving 0-forms
New equations:
- 3*(u + y)
gamma =--------------
3 2
8*v
- 3
gamma =------
1 2
8*v
delta
2
- 3*(u + y)
=----------------------------------------------------------------------------
4 3 2 2 2 3 2 2 4
2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
delta
1
3
=----------------------------------------------------------------------------
4 3 2 2 2 3 2 2 4
2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
v =u + y
3
x =u + y
2
x =-1
1
New 0-form conditions detected
3 2 2
- 8*gamma *v + 6*u + 12*u*y - 3*v + 6*y
3 3
-----------------------------------------------
3
8*v
3*(x - 1)
2 3
--------------
2
8*v
3
- 8*gamma *v + 3*x *v + 6*u + 6*y
1 3 1 3
-----------------------------------------
3
8*v
3
- 4*gamma *v + 3*u + 3*y
1 3
------------------------------
3
4*v
3
- 4*gamma *v + 3
1 1
----------------------
3
4*v
3*(x - 1)
2 3
----------------------------------------------------------------------------
4 3 2 2 2 3 2 2 4
2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
8 7 6 6 2
( - 2*delta *u - 16*delta *u *y + 16*delta *u *v - 56*delta *u *y
2 2 2 2 2 2 2 2
5 5 3 4 2
+ 96*delta *u *v*y - 112*delta *u *y - 48*delta *u *v
2 2 2 2 2 2
4 2 4 4 3 2
+ 240*delta *u *v*y - 140*delta *u *y - 192*delta *u *v *y
2 2 2 2 2 2
3 3 3 5 2 3
+ 320*delta *u *v*y - 112*delta *u *y + 64*delta *u *v
2 2 2 2 2 2
2 2 2 2 4 2 6
- 288*delta *u *v *y + 240*delta *u *v*y - 56*delta *u *y
2 2 2 2 2 2
3 2 3 5
+ 128*delta *u*v *y - 192*delta *u*v *y + 96*delta *u*v*y
2 2 2 2 2 2
7 4 3 2 2 4
- 16*delta *u*y - 32*delta *v + 64*delta *v *y - 48*delta *v *y
2 2 2 2 2 2 2 2
6 8 4 3 2 2 2
+ 16*delta *v*y - 2*delta *y + 9*u + 36*u *y - 12*u *v + 54*u *y
2 2 2 2
3 2 2 4 8 7 6
- 24*u*v*y + 36*u*y - 12*v - 12*v*y + 9*y )/(2*(u + 8*u *y - 8*u *v
6 2 5 5 3 4 2 4 2 4 4
+ 28*u *y - 48*u *v*y + 56*u *y + 24*u *v - 120*u *v*y + 70*u *y
3 2 3 3 3 5 2 3 2 2 2
+ 96*u *v *y - 160*u *v*y + 56*u *y - 32*u *v + 144*u *v *y
2 4 2 6 3 2 3 5 7
- 120*u *v*y + 28*u *y - 64*u*v *y + 96*u*v *y - 48*u*v*y + 8*u*y
4 3 2 2 4 6 8
+ 16*v - 32*v *y + 24*v *y - 8*v*y + y ))
8 7 6 6 2
( - delta *u - 8*delta *u *y + 8*delta *u *v - 28*delta *u *y
1 2 1 2 1 2 1 2
5 5 3 4 2
+ 48*delta *u *v*y - 56*delta *u *y - 24*delta *u *v
1 2 1 2 1 2
4 2 4 4 3 2
+ 120*delta *u *v*y - 70*delta *u *y - 96*delta *u *v *y
1 2 1 2 1 2
3 3 3 5 2 3
+ 160*delta *u *v*y - 56*delta *u *y + 32*delta *u *v
1 2 1 2 1 2
2 2 2 2 4 2 6
- 144*delta *u *v *y + 120*delta *u *v*y - 28*delta *u *y
1 2 1 2 1 2
3 2 3 5
+ 64*delta *u*v *y - 96*delta *u*v *y + 48*delta *u*v*y
1 2 1 2 1 2
7 4 3 2 2 4
- 8*delta *u*y - 16*delta *v + 32*delta *v *y - 24*delta *v *y
1 2 1 2 1 2 1 2
6 8 3 2 2
+ 8*delta *v*y - delta *y - 6*u - 18*u *y + 12*u*v - 18*u*y + 12*v*y
1 2 1 2
3 8 7 6 6 2 5 5 3 4 2
- 6*y )/(u + 8*u *y - 8*u *v + 28*u *y - 48*u *v*y + 56*u *y + 24*u *v
4 2 4 4 3 2 3 3 3 5
- 120*u *v*y + 70*u *y + 96*u *v *y - 160*u *v*y + 56*u *y
2 3 2 2 2 2 4 2 6 3
- 32*u *v + 144*u *v *y - 120*u *v*y + 28*u *y - 64*u*v *y
2 3 5 7 4 3 2 2 4
+ 96*u*v *y - 48*u*v*y + 8*u*y + 16*v - 32*v *y + 24*v *y
6 8
- 8*v*y + y )
3*x
1 3
----------------------------------------------------------------------------
4 3 2 2 2 3 2 2 4
2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
6 5 4 4 2
( - delta *u - 6*delta *u *y + 6*delta *u *v - 15*delta *u *y
1 2 1 2 1 2 1 2
3 3 3 2 2
+ 24*delta *u *v*y - 20*delta *u *y - 12*delta *u *v
1 2 1 2 1 2
2 2 2 4 2
+ 36*delta *u *v*y - 15*delta *u *y - 24*delta *u*v *y
1 2 1 2 1 2
3 5 3 2 2
+ 24*delta *u*v*y - 6*delta *u*y + 8*delta *v - 12*delta *v *y
1 2 1 2 1 2 1 2
4 6 6 5 4 4 2
+ 6*delta *v*y - delta *y - 6*u - 6*y)/(u + 6*u *y - 6*u *v + 15*u *y
1 2 1 2
3 3 3 2 2 2 2 2 4 2
- 24*u *v*y + 20*u *y + 12*u *v - 36*u *v*y + 15*u *y + 24*u*v *y
3 5 3 2 2 4 6
- 24*u*v*y + 6*u*y - 8*v + 12*v *y - 6*v*y + y )
6 5 4 4 2
( - delta *u - 6*delta *u *y + 6*delta *u *v - 15*delta *u *y
1 1 1 1 1 1 1 1
3 3 3 2 2
+ 24*delta *u *v*y - 20*delta *u *y - 12*delta *u *v
1 1 1 1 1 1
2 2 2 4 2
+ 36*delta *u *v*y - 15*delta *u *y - 24*delta *u*v *y
1 1 1 1 1 1
3 5 3 2 2
+ 24*delta *u*v*y - 6*delta *u*y + 8*delta *v - 12*delta *v *y
1 1 1 1 1 1 1 1
4 6 6 5 4 4 2
+ 6*delta *v*y - delta *y + 6)/(u + 6*u *y - 6*u *v + 15*u *y
1 1 1 1
3 3 3 2 2 2 2 2 4 2
- 24*u *v*y + 20*u *y + 12*u *v - 36*u *v*y + 15*u *y + 24*u*v *y
3 5 3 2 2 4 6
- 24*u*v*y + 6*u*y - 8*v + 12*v *y - 6*v*y + y )
- v + 1
3 3
- x + 1
2 3
- x + 1
2 2
- x
1 3
- x
1 2
- x
1 1
Solving 0-forms
New equations:
2 2
3*(2*u + 4*u*y - v + 2*y )
gamma =-----------------------------
3 3 3
8*v
3*(u + y)
gamma =-----------
1 3 3
4*v
3
gamma =------
1 1 3
4*v
4 3 2 2 2 3 2
delta =(3*(3*u + 12*u *y - 4*u *v + 18*u *y - 8*u*v*y + 12*u*y - 4*v
2 2
2 4 8 7 6 6 2 5
- 4*v*y + 3*y ))/(2*(u + 8*u *y - 8*u *v + 28*u *y - 48*u *v*y
5 3 4 2 4 2 4 4 3 2
+ 56*u *y + 24*u *v - 120*u *v*y + 70*u *y + 96*u *v *y
3 3 3 5 2 3 2 2 2 2 4
- 160*u *v*y + 56*u *y - 32*u *v + 144*u *v *y - 120*u *v*y
2 6 3 2 3 5 7 4
+ 28*u *y - 64*u*v *y + 96*u*v *y - 48*u*v*y + 8*u*y + 16*v
3 2 2 4 6 8
- 32*v *y + 24*v *y - 8*v*y + y ))
3 2 2 3 8 7
delta =(6*( - u - 3*u *y + 2*u*v - 3*u*y + 2*v*y - y ))/(u + 8*u *y
1 2
6 6 2 5 5 3 4 2 4 2
- 8*u *v + 28*u *y - 48*u *v*y + 56*u *y + 24*u *v - 120*u *v*y
4 4 3 2 3 3 3 5 2 3
+ 70*u *y + 96*u *v *y - 160*u *v*y + 56*u *y - 32*u *v
2 2 2 2 4 2 6 3 2 3
+ 144*u *v *y - 120*u *v*y + 28*u *y - 64*u*v *y + 96*u*v *y
5 7 4 3 2 2 4 6 8
- 48*u*v*y + 8*u*y + 16*v - 32*v *y + 24*v *y - 8*v*y + y )
6 5 4 4 2 3 3 3 2 2
delta =6/(u + 6*u *y - 6*u *v + 15*u *y - 24*u *v*y + 20*u *y + 12*u *v
1 1
2 2 2 4 2 3 5 3
- 36*u *v*y + 15*u *y + 24*u*v *y - 24*u*v*y + 6*u*y - 8*v
2 2 4 6
+ 12*v *y - 6*v*y + y )
v =1
3 3
x =1
2 3
x =1
2 2
x =0
1 3
x =0
1 2
x =0
1 1
New 0-form conditions detected
- v
3 3 3
- x
2 3 3
- x
2 2 3
- x
2 2 2
- x
1 3 3
- x
1 2 3
- x
1 2 2
- x
1 1 3
- x
1 1 2
- x
1 1 1
Solving 0-forms
New equations:
v =0
3 3 3
x =0
2 3 3
x =0
2 2 3
x =0
2 2 2
x =0
1 3 3
x =0
1 2 3
x =0
1 2 2
x =0
1 1 3
x =0
1 1 2
x =0
1 1 1
New 0-form conditions detected
- v
3 3 3 3
- x
2 3 3 3
- x
2 2 3 3
- x
2 2 2 3
- x
2 2 2 2
- x
1 3 3 3
- x
1 2 3 3
- x
1 2 2 3
- x
1 2 2 2
- x
1 1 3 3
- x
1 1 2 3
- x
1 1 2 2
- x
1 1 1 3
- x
1 1 1 2
- x
1 1 1 1
Solving 0-forms
New equations:
v =0
3 3 3 3
x =0
2 3 3 3
x =0
2 2 3 3
x =0
2 2 2 3
x =0
2 2 2 2
x =0
1 3 3 3
x =0
1 2 3 3
x =0
1 2 2 3
x =0
1 2 2 2
x =0
1 1 3 3
x =0
1 1 2 3
x =0
1 1 2 2
x =0
1 1 1 3
x =0
1 1 1 2
x =0
1 1 1 1
New 0-form conditions detected
- v
3 3 3 3 3
- x
2 3 3 3 3
3*( - 4*f *v + f )
1 3 2
----------------------
2*f*v
2 2
3*(2*f *u + 4*f *u*y - 4*f *v + 2*f *y + f )
1 2 1 2 1 2 1 2 3
--------------------------------------------------------
2 2
f*(u + 2*u*y - 2*v + y )
- x
2 2 2 2 3
- x
2 2 2 2 2
- x
1 3 3 3 3
- x
1 2 3 3 3
- x
2 2 3 3 1
- x
1 2 2 2 3
- x
1 2 2 2 2
- x
1 1 3 3 3
- x
1 1 2 3 3
- x
1 1 2 2 3
- x
1 1 2 2 2
- x
1 1 1 3 3
- x
1 1 1 2 3
- x
1 1 1 2 2
- x
1 1 1 1 3
- x
1 1 1 1 2
- x
1 1 1 1 1
Solving 0-forms
New equations:
v =0
3 3 3 3 3
x =0
2 3 3 3 3
x =0
2 2 3 3 1
x =0
2 2 2 2 3
x =0
2 2 2 2 2
x =0
1 3 3 3 3
x =0
1 2 3 3 3
x =0
1 2 2 2 3
x =0
1 2 2 2 2
x =0
1 1 3 3 3
x =0
1 1 2 3 3
x =0
1 1 2 2 3
x =0
1 1 2 2 2
x =0
1 1 1 3 3
x =0
1 1 1 2 3
x =0
1 1 1 2 2
x =0
1 1 1 1 3
x =0
1 1 1 1 2
x =0
1 1 1 1 1
f
2
f =-----
1 3 4*v
- f
3
f =---------------------------
1 2 2 2
2*(u + 2*u*y - 2*v + y )
New 0-form conditions detected
- 4*f *v - 2*f *u - 2*f *y + 3*f
1 2 2
-----------------------------------
2
8*v
2 2 2
- 8*f *u *v - 16*f *u*v*y + 16*f *v - 8*f *v*y + 3*f
1 1 1 1 1 1 1 1
-----------------------------------------------------------------
2 2
16*v*(u + 2*u*y - 2*v + y )
2 2 2 3 2 2 2
( - 8*f *u *v - 16*f *u*v *y + 16*f *v - 8*f *v *y - 2*f *u
1 1 3 1 1 3 1 1 3 1 1 3 2
2 2 2 2
- 4*f *u*y + 4*f *v - 2*f *y - f *v)/(8*v *(u + 2*u*y - 2*v + y ))
2 2 2 3
2 2 2
8*f *u *v + 16*f *u*v*y - 16*f *v + 8*f *v*y - 3*f
1 1 1 1 1 1 1 1
--------------------------------------------------------------
2 2
16*v*(u + 2*u*y - 2*v + y )
2 2
- 2*f *u - 4*f *u*y + 4*f *v - 2*f *y + 2*f *u + 2*f *y - 3*f
1 1 1 1 3 3
----------------------------------------------------------------------------
4 3 2 2 2 3 2 2 4
2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
4 3 2 2 2 2
( - 8*f *u *v - 32*f *u *v*y + 32*f *u *v - 48*f *u *v*y
1 1 2 1 1 2 1 1 2 1 1 2
2 3 3 2 2
+ 64*f *u*v *y - 32*f *u*v*y - 32*f *v + 32*f *v *y
1 1 2 1 1 2 1 1 2 1 1 2
4 2 2
- 8*f *v*y - f *u - 2*f *u*y + 2*f *v - f *y - 8*f *v)/(8*v
1 1 2 2 2 2 2 3
4 3 2 2 2 3 2 2 4
*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y ))
Solving 0-forms
New equations:
2 2 2 2
f =(3*(2*f *u *v + 4*f *u*v*y - 4*f *v + 2*f *v*y - 2*f*u - 4*f*u*y
1 1 3 1 1 1 1
2 2
+ 3*f*v - 2*f*y ))/(16*v
3 2 2 3
*(u + 3*u *y - 2*u*v + 3*u*y - 2*v*y + y ))
2 2 2 2
f =(3*( - 4*f *u *v - 8*f *u*v*y + 8*f *v - 4*f *v*y - f*u - 2*f*u*y
1 1 2 1 1 1 1
2 5 4 3 3 2 2
- 6*f*v - f*y ))/(16*v*(u + 5*u *y - 4*u *v + 10*u *y - 12*u *v*y
2 3 2 2 4 2 3 5
+ 10*u *y + 4*u*v - 12*u*v*y + 5*u*y + 4*v *y - 4*v*y + y ))
3*f
f =-----------------------------
1 1 2 2
8*v*(u + 2*u*y - 2*v + y )
2 2
2*f *u + 4*f *u*y - 4*f *v + 2*f *y + 3*f
1 1 1 1
f =---------------------------------------------
3 2*(u + y)
- 4*f *v + 3*f
1
f =-----------------
2 2*(u + y)
New 0-form conditions detected
4 2 3 2 2 3 2 2 2
( - 8*f *u *v - 32*f *u *v *y + 32*f *u *v - 48*f *u *v *y
1 1 1 1 1 1 1 1 1 1 1 1
3 2 3 4 3 2
+ 64*f *u*v *y - 32*f *u*v *y - 32*f *v + 32*f *v *y
1 1 1 1 1 1 1 1 1 1 1 1
2 4 2 2 2 2
- 8*f *v *y + 3*f *u *v + 6*f *u*v*y - 6*f *v + 3*f *v*y - 3*f*u
1 1 1 1 1 1 1
2 2
- 6*f*u*y + 12*f*v - 3*f*y )/(8*v
4 3 2 2 2 3 2 2 4
*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y ))
Solving 0-forms
New equations:
f
1 1 1
2 2 2 2 2
3*(f *u *v + 2*f *u*v*y - 2*f *v + f *v*y - f*u - 2*f*u*y + 4*f*v - f*y )
1 1 1 1
=-------------------------------------------------------------------------------
2 4 3 2 2 2 3 2 2 4
8*v *(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
4*f *v - 3*f
1 1 2
EDS({d f - f *omega + --------------*omega
1 2*(u + y)
2 2
- 2*f *u - 4*f *u*y + 4*f *v - 2*f *y - 3*f
1 1 1 1 3
+ ------------------------------------------------*omega ,
2*(u + y)
3*f 1
d f - -----------------------------*omega
1 2 2
8*v*(u + 2*u*y - 2*v + y )
2 2
2*f *u + 4*f *u*y - 4*f *v + 2*f *y + 3*f
1 1 1 1 2
+ -----------------------------------------------*omega
3 2 2 3
4*(u + 3*u *y - 2*u*v + 3*u*y - 2*v*y + y )
4*f *v - 3*f
1 3 1 2 3
+ --------------*omega },omega ^omega ^omega )
8*v*(u + y)
characters ws;
{0,0,0}
clear v(i),omega(i);
clear F,x,Delta,gamma,v,y,u,omega;
off ranpos;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Isometric embeddings of Ricci-flat R(4) in ISO(10) %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Determine the Cartan characters of a Ricci-flat embedding of R(4) into
% the orthonormal frame bundle ISO(10) over flat R(6). Reference:
% Estabrook & Wahlquist, Class Quant Grav 10(1993)1851
% Indices
indexrange {p,q,r,s}={1,2,3,4,5,6,7,8,9,10},
{i,j,k,l}={1,2,3,4},{a,b,c,d}={5,6,7,8,9,10};
% Metric for R10
pform g(p,q)=0;
g(p,q) := 0$
g(-p,-q) := 0$
g(-p,-p) := g(p,p) := 1$
% Hodge map for R4
pform epsilon(i,j,k,l)=0;
index_symmetries epsilon(i,j,k,l):antisymmetric;
epsilon(1,2,3,4) := 1;
1 2 3 4
epsilon := 1
% Coframe for ISO(10)
% NB index_symmetries must come after o(p,-q) := ... (EXCALC bug)
pform e(r)=1,o(r,s)=1;
korder index_expand {e(r)};
e(-p) := g(-p,-q)*e(q)$
o(p,-q) := o(p,r)*g(-r,-q)$
index_symmetries o(p,q):antisymmetric;
% Structure equations
flat_no_torsion := {d e(p) => -o(p,-q)^e(q),
d o(p,q) => -o(p,-r)^o(r,q)};
p p q
flat_no_torsion := {d e => - o ^e ,
q
p q p r q
d o => - o ^o }
r
% Coframing structure
ISO := coframing({e(p),o(p,q)},flat_no_torsion)$
dim ISO;
55
% 4d curvature 2-forms
pform F(i,j)=2;
index_symmetries F(i,j):antisymmetric;
F(-i,-j) := -g(-i,-k)*o(k,-a)^o(a,-j);
1 10 2 10 1 5 2 5 1 6 2 6 1 7 2 7 1 8 2 8 1 9 2 9
f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
1 2
1 10 3 10 1 5 3 5 1 6 3 6 1 7 3 7 1 8 3 8 1 9 3 9
f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
1 3
2 10 3 10 2 5 3 5 2 6 3 6 2 7 3 7 2 8 3 8 2 9 3 9
f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
2 3
1 10 4 10 1 5 4 5 1 6 4 6 1 7 4 7 1 8 4 8 1 9 4 9
f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
1 4
2 10 4 10 2 5 4 5 2 6 4 6 2 7 4 7 2 8 4 8 2 9 4 9
f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
2 4
3 10 4 10 3 5 4 5 3 6 4 6 3 7 4 7 3 8 4 8 3 9 4 9
f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
3 4
% EDS for vacuum GR (Ricci-flat) in 4d
GR0 := eds({e(a),epsilon(i,j,k,l)*F(-j,-k)^e(-l)},
{e(i)},
ISO)$
% Find an integral element, and linearise
Z := integral_element GR0$
45 free variables
39 free variables
29 free variables
21 free variables
GRZ := linearise(GR0,Z)$
% This actually tells us the characters already:
% {45-39,39-29,29-21,21} = {6,10,8,21}
% Get the characters and dimension at Z
characters GRZ;
Cauchy characteristics detected from characters
{6,10,8,21}
dim_grassmann_variety GRZ;
134
% 6+2*10+3*8+4*21 = 134, so involutive
clear e(r),o(r,s),g(p,q),epsilon(i,j,k,l),F(i,j);
clear e,o,g,epsilon,F,Z;
indexrange 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%
% Janet's PDE system %
%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is something of a standard test problem in analysing integrability
% conditions. Although it looks very innocent, it must be prolonged five
% times from the second jet bundle before reaching involution. The initial
% equations are just
%
% u =w, u =u *y + v
% y y z z x x
load sets;
off varopt;
pform {x,y,z,u,v,w}=0$
janet := contact(2,{x,y,z},{u,v,w})$
janet := pullback(janet,{u(-y,-y)=w,u(-z,-z)=y*u(-x,-x)+v})$
% Prolong to involution
involutive janet;
0
involution janet;
Prolongation using new equations:
u =u *y + u + v
y z z x x y x x y
u =w
y y z z
u =u *y + v
x z z x x x x
u =w
x y y x
Reduction using new equations:
- v - w *y + w
y y x x z z
u =-------------------------
x x y 2
Reduction using new equations:
w =v + w *y + 3*w
y z z y y y x x y x x
Prolongation using new equations:
w =v + w *y + 3*w
y z z z y y y z x x y z x x z
w =v + w *y + 4*w
y y z z y y y y x x y y x x y
w =v + w *y + 3*w
x y z z x y y y x x x y x x x
2
2*u - v *y + 2*v - w *y + w *y
x x x x y y x y x x x x z z
u =-----------------------------------------------------
x y z z 2
u =w
x y y z x z
u =u *y + v
x x z z x x x x x x
- v - w *y + w
y y z x x z z z z
u =-------------------------------
x x y z 2
- v - w *y + w
x y y x x x x z z
u =-------------------------------
x x x y 2
Reduction using new equations:
w
z z z z
2
=2*u - v *y + 2*v + v - w *y + 2*w *y
x x x x x x y y x x y y y z z x x x x x x z z
EDS({d u - u *d x - u *d y - u *d z,
x y z
d v - v *d x - v *d y - v *d z,
x y z
d w - w *d x - w *d y - w *d z,
x y z
d u - u *d x - u *d y - u *d z,
x x x x y x z
d u - u *d x - w*d y - u *d z,
y x y y z
d u - u *d x - u *d y - (u *y + v)*d z,
z x z y z x x
d v - v *d x - v *d y - v *d z,
x x x x y x z
d v - v *d x - v *d y - v *d z,
y x y y y y z
d v - v *d x - v *d y - v *d z,
z x z y z z z
d w - w *d x - w *d y - w *d z,
x x x x y x z
d w - w *d x - w *d y - w *d z,
y x y y y y z
d w - w *d x - w *d y - w *d z,
z x z y z z z
v + w *y - w
y y x x z z
d u - u *d x + ----------------------*d y - u *d z,
x x x x x 2 x x z
v + w *y - w
y y x x z z
d u + ----------------------*d x - w *d y - u *d z,
x y 2 x x y z
d u - u *d x - u *d y - (u *y + v )*d z,
x z x x z x y z x x x x
d u - u *d x - w *d y
y z x y z z
2
- 2*u + v *y - 2*v + w *y - w *y
x x y y y x x z z
+ ----------------------------------------------*d z,
2
d v - v *d x - v *d y - v *d z,
x x x x x x x y x x z
d v - v *d x - v *d y - v *d z,
x y x x y x y y x y z
d v - v *d x - v *d y - v *d z,
x z x x z x y z x z z
d v - v *d x - v *d y - v *d z,
y y x y y y y y y y z
d v - v *d x - v *d y - v *d z,
y z x y z y y z y z z
d v - v *d x - v *d y - v *d z,
z z x z z y z z z z z
d w - w *d x - w *d y - w *d z,
x x x x x x x y x x z
d w - w *d x - w *d y - w *d z,
x y x x y x y y x y z
d w - w *d x - w *d y - w *d z,
x z x x z x y z x z z
d w - w *d x - w *d y - w *d z,
y y x y y y y y y y z
d w - w *d x - w *d y + ( - v - w *y - 3*w )*d z,
y z x y z y y z y y y x x y x x
d w - w *d x + ( - v - w *y - 3*w )*d y - w *d z,
z z x z z y y y x x y x x z z z
v + w *y - w
x y y x x x x z z
d u - u *d x + ----------------------------*d y - u *d z,
x x x x x x x 2 x x x z
v + w *y - w
y y z x x z z z z
d u - u *d x + ----------------------------*d y
x x z x x x z 2
- (u *y + v )*d z,
x x x x x x
v + w *y - w
y y z x x z z z z
d u + ----------------------------*d x - w *d y
x y z 2 x z
2
- 2*u + v *y - 2*v + w *y - w *y
x x x x y y x y x x x x z z
+ --------------------------------------------------------*d z,
2
d v - v *d x - v *d y - v *d z,
x x x x x x x x x x y x x x z
d v - v *d x - v *d y - v *d z,
x x y x x x y x x y y x x y z
d v - v *d x - v *d y - v *d z,
x x z x x x z x x y z x x z z
d v - v *d x - v *d y - v *d z,
x y y x x y y x y y y x y y z
d v - v *d x - v *d y - v *d z,
x y z x x y z x y y z x y z z
d v - v *d x - v *d y - v *d z,
x z z x x z z x y z z x z z z
d v - v *d x - v *d y - v *d z,
y y y x y y y y y y y y y y z
d v - v *d x - v *d y - v *d z,
y y z x y y z y y y z y y z z
d v - v *d x - v *d y - v *d z,
y z z x y z z y y z z y z z z
d v - v *d x - v *d y - v *d z,
z z z x z z z y z z z z z z z
d w - w *d x - w *d y - w *d z,
x x x x x x x x x x y x x x z
d w - w *d x - w *d y - w *d z,
x x y x x x y x x y y x x y z
d w - w *d x - w *d y - w *d z,
x x z x x x z x x y z x x z z
d w - w *d x - w *d y - w *d z,
x y y x x y y x y y y x y y z
d w - w *d x - w *d y
x y z x x y z x y y z
+ ( - v - w *y - 3*w )*d z,
x y y y x x x y x x x
d w - w *d x + ( - v - w *y - 3*w )*d y
x z z x x z z x y y y x x x y x x x
- w *d z,
x z z z
d w - w *d x - w *d y - w *d z,
y y y x y y y y y y y y y y z
d w - w *d x - w *d y
y y z x y y z y y y z
+ ( - v - w *y - 4*w )*d z,
y y y y x x y y x x y
d w - w *d x + ( - v - w *y - 3*w )*d y + (
z z z x z z z y y y z x x y z x x z
2
- 2*u + v *y - 2*v - v + w *y
x x x x x x y y x x y y y z z x x x x
- 2*w *y)*d z,
x x z z
d u ^d x + d u ^d z
x x x x x x x z
- v - w *y + w
x x y y x x x x x x z z
+ -------------------------------------*d x^d y
2
v + w *y - w
x y y z x x x z x z z z
+ ----------------------------------*d y^d z,
2
1
d u ^d z + ---*d u ^d x
x x x x y x x x z
- v - w *y + w v
x y y z x x x z x z z z x x x
+ -------------------------------------*d x^d y + --------*d x^d z
2*y y
v + w *y - w
x x y y x x x x x x z z
+ ----------------------------------*d y^d z,
2
y 1
d u ^d z - ---*d v ^d z + ---*d v ^d y
x x x x 2 x x y y 2 y y y z
2
1 y y
+ ---*d v ^d z - ----*d w ^d z + ---*d w ^d y
2 y y z z 2 x x x x 2 x x y z
3*w
1 x x x z
+ y*d w ^d z + ---*d w ^d x + ------------*d x^d y
x x z z 2 x z z z 2
v - 2*w *y - w
x x y y x x x x x x z z
+ v *d x^d z + ------------------------------------*d y^d z,
x x x y 2
d v ^d x + d v ^d y + d v ^d z,
x x x x x x x y x x x z
d v ^d x + d v ^d y + d v ^d z,
x x x y x x y y x x y z
d v ^d x + d v ^d y + d v ^d z,
x x x z x x y z x x z z
d v ^d x + d v ^d y + d v ^d z,
x x y y x y y y x y y z
d v ^d x + d v ^d y + d v ^d z,
x x y z x y y z x y z z
d v ^d x + d v ^d y + d v ^d z,
x x z z x y z z x z z z
d v ^d x + d v ^d y + d v ^d z,
x y y y y y y y y y y z
d v ^d y + y*d w ^d y + d w ^d x + d w ^d z
x y y y x x x y x x z z x z z z
+ 3*w *d x^d y - 3*w *d y^d z,
x x x x x x x z
d v ^d z + y*d w ^d z + d w ^d x + d w ^d y
x y y y x x x y x x y z x y y z
+ 3*w *d x^d z + 4*w *d y^d z,
x x x x x x x y
d v ^d x + d v ^d y + d v ^d z,
x y y z y y y z y y z z
d v ^d x + d v ^d y + d v ^d z,
x y z z y y z z y z z z
d v ^d x + d v ^d y + d v ^d z,
x z z z y z z z z z z z
d v ^d z + y*d w ^d z + d w ^d x + d w ^d y
y y y y x x y y x y y z y y y z
+ 4*w *d x^d z + 5*w *d y^d z,
x x x y x x y y
d w ^d x + d w ^d y + d w ^d z,
x x x x x x x y x x x z
d w ^d x + d w ^d y + d w ^d z,
x x x y x x y y x x y z
d w ^d x + d w ^d y + d w ^d z,
x x x z x x y z x x z z
d w ^d x + d w ^d y + d w ^d z,
x x y y x y y y x y y z
d w ^d x + d w ^d y + d w ^d z},d x^d y^d z)
x y y y y y y y y y y z
involutive ws;
1
% Solve the homogeneous system, for which the
% involutive prolongation is completely integrable
fdomain u=u(x,y,z),v=v(x,y,z),w=w(x,y,z);
janet := {@(u,y,y)=0,@(u,z,z)=y*@(u,x,x)};
janet := {@ u=0,@ u=@ u*y}
y y z z x x
janet := involution pde2eds janet$
Prolongation using new equations:
u =u *y + u
y z z x x y x x
u =0
y y z
u =u *y
x z z x x x
u =0
x y y
Reduction using new equations:
u =0
x x y
Prolongation using new equations:
u =u
x y z z x x x
u =0
x y y z
u =u *y
x x z z x x x x
u =0
x x y z
u =0
x x x y
Reduction using new equations:
u =0
x x x x
Prolongation using new equations:
u =0
x x x z z
u =0
x x x y z
u =0
x x x x z
% Check if completely integrable
if frobenius janet then write "yes" else write "no";
yes
length one_forms janet;
12
% So there are 12 constants in the solution: there should be 12 invariants
length(C := invariants janet);
12
solve(for i:=1:length C collect
part(C,i) = mkid(k,i),coordinates janet \ {x,y,z})$
S := select(lhs ~q = u,first ws);
3 3
s := {u=( - k1*x *z - k1*x*y*z - 6*k10*x*z - 6*k11*y*z - 6*k12*y - 6*k2*x*y
3 2 2 2
- 6*k3*x*y*z + k4*x + 3*k4*x*y*z - 6*k5 - 3*k6*x - 3*k6*y*z
2 3
- 6*k7*x - 6*k8*z - 3*k9*x *z - k9*y*z )/6}
% Check solution
mkdepend dependencies;
sub(S,{@(u,y,y),@(u,z,z)-y*@(u,x,x)});
{0,0}
clear u(i,j),v(i,j),w(i,j),u(i),v(i),w(i);
clear x,y,z,u,v,w,C,S;
end;
Time for test: 242603 ms, plus GC time: 3724 ms