Artifact 0f47eea5913cf913b3f3d87c295b97c7a3141816dfa7e9324692be251e25762a:
- Executable file
r38/packages/eds/eds.rlg
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 75749) [annotate] [blame] [check-ins using] [more...]
Tue Feb 10 12:28:07 2004 run on Linux *** ^ 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 2 3 3 s := {u=(k1*x + 3*k1*x*y*z - 6*k10*y*z - 6*k11 - 6*k12*z - k2*x *z - k2*x*y*z 2 3 2 - 6*k3*x*y*z - 6*k4*x*y - 3*k5*x *z - k5*y*z - 6*k6*x*z - 3*k7*x 2 - 3*k7*y*z - 6*k8*x - 6*k9*y)/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: 5850 ms, plus GC time: 170 ms