1 2 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 % Twisting type N solutions of GR % 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6 7% The problem is to analyse an ansatz for a particular type of vacuum 8% solution to Einstein's equations for general relativity. The analysis was 9% described by Finley and Price (Proc Aspects of GR and Math Phys 10% (Plebanski Festschrift), Mexico City June 1993). The equations resulting 11% from the ansatz are: 12 13% F - F*gamma = 0 14% 3 3 15% 16% F *x + 2*F *x + x *F - x *Delta*F = 0 17% 2 2 1 2 1 2 1 2 2 1 18% 19% 2*F *x + 2*F *x + 2*F *x + 2*F *x + x *F = 0 20% 2 3 2 3 2 2 3 3 3 2 2 3 3 2 2 3 2 2 3 3 21% 22% Delta =0 Delta neq 0 23% 3 1 24% 25% gamma =0 gamma neq 0 26% 2 1 27 28% where the unknowns are {F,x,gamma,Delta} and the indices refer to 29% derivatives with respect to an anholonomic basis. The highest order is 4, 30% but the 4th order jet bundle is too large for practical computation, so 31% it is necessary to construct partial prolongations. There is a single 32% known solution, due to Hauser, which is verified at the end. 33 34on evallhseqp,edssloppy,edsverbose; 35off arbvars,edsdebug; 36 37pform {F,x,Delta,gamma,v,y,u}=0; 38pform v(i)=0,omega(i)=1; 39indexrange {i,j,k,l}={1,2,3}; 40 41% Construct J1({v,y,u},{x}) and transform coordinates. Use ordering 42% statement to get v eliminated in favour of x where possible. 43% NB Coordinate change cc1 is invertible only when x(-1) neq 0. 44 45J1 := contact(1,{v,y,u},{x}); 46korder x(-1),x(-2),v(-3); 47cc1 := {x(-v) = x(-1), 48 x(-y) = x(-2), 49 x(-u) = -x(-1)*v(-3)}; 50J1 := restrict(pullback(J1,cc1),{x(-1) neq 0}); 51 52% Set up anholonomic cobasis 53 54bc1 := {omega(1) = d v - v(-3)*d u, 55 omega(2) = d y, 56 omega(3) = d u}; 57J1 := transform(J1,bc1); 58 59% Prolong to J421: 4th order in x, 2nd in F and 1st in rest 60 61J2 := prolong J1$ 62J20 := J2 cross {F}$ 63J31 := prolong J20$ 64J310 := J31 cross {Delta,gamma}$ 65J421 := prolong J310$ 66cc4 := first pullback_maps; 67 68% Apply first order de and restrictions 69 70de1 := {Delta(-3) = 0, 71 gamma(-2) = 0, 72 Delta(-1) neq 0, 73 gamma(-1) neq 0}; 74 75J421 := pullback(J421,de1)$ 76 77% Main de in original coordinates 78 79de2 := {F(-3,-3) - gamma*F, 80 x(-1)*F(-2,-2) + 2*x(-1,-2)*F(-2) 81 + (x(-1,-2,-2) - x(-1)*Delta)*F, 82 x(-2,-3)*(F(-2,-3)+F(-3,-2)) + x(-2,-2,-3)*F(-3) 83 + x(-2,-3,-3)*F(-2) + (1/2)*x(-2,-2,-3,-3)*F}; 84 85% This is not expressed in terms of current coordinates. 86% Missing coordinates are seen from 1-form variables in following 87 88d de2 xmod cobasis J421; 89 90% The necessary equation is contained in the last prolongation 91 92pullback(d de2,cc4) xmod cobasis J421; 93 94% Apply main de 95 96pb1 := first solve(pullback(de2,cc4),{F(-3,-3),F(-2,-2),F(-2,-3)}); 97Y421 := pullback(J421,pb1)$ 98 99% Check involution 100 101on ranpos; 102characters Y421; 103dim_grassmann_variety Y421; 104 105% 15+2*7 = 29 > 28: Y421 not involutive, so prolong 106 107Y532 := prolong Y421$ 108 109characters Y532; 110dim_grassmann_variety Y532; 111 112% 22+2*6 = 34: just need to check for integrability conditions 113 114torsion Y532; 115 116% Y532 involutive. Dimensions? 117 118dim Y532; 119length one_forms Y532; 120 121% The following puts in part of Hauser's solution and ends up with an ODE 122% system (all characters 0), so no more solutions, as described by Finley 123% at MG6. 124 125hauser := {x=-v+(1/2)*(y+u)**2,delta=3/(8x),gamma=3/(8v)}; 126H532 := pullback(Y532,hauser)$ 127lift ws; 128characters ws; 129 130clear v(i),omega(i); 131clear F,x,Delta,gamma,v,y,u,omega; 132off ranpos; 133 134 135 136 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 137 % Isometric embeddings of Ricci-flat R(4) in ISO(10) % 138 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 139 140% Determine the Cartan characters of a Ricci-flat embedding of R(4) into 141% the orthonormal frame bundle ISO(10) over flat R(6). Reference: 142% Estabrook & Wahlquist, Class Quant Grav 10(1993)1851 143 144% Indices 145 146indexrange {p,q,r,s}={1,2,3,4,5,6,7,8,9,10}, 147 {i,j,k,l}={1,2,3,4},{a,b,c,d}={5,6,7,8,9,10}; 148 149% Metric for R10 150 151pform g(p,q)=0; 152g(p,q) := 0$ g(-p,-q) := 0$ g(-p,-p) := g(p,p) := 1$ 153 154% Hodge map for R4 155 156pform epsilon(i,j,k,l)=0; 157index_symmetries epsilon(i,j,k,l):antisymmetric; 158epsilon(1,2,3,4) := 1; 159 160% Coframe for ISO(10) 161% NB index_symmetries must come after o(p,-q) := ... (EXCALC bug) 162 163pform e(r)=1,o(r,s)=1; 164korder index_expand {e(r)}; 165e(-p) := g(-p,-q)*e(q)$ 166o(p,-q) := o(p,r)*g(-r,-q)$ 167index_symmetries o(p,q):antisymmetric; 168 169% Structure equations 170 171flat_no_torsion := {d e(p) => -o(p,-q)^e(q), 172 d o(p,q) => -o(p,-r)^o(r,q)}; 173 174% Coframing structure 175 176ISO := coframing({e(p),o(p,q)},flat_no_torsion)$ 177dim ISO; 178 179% 4d curvature 2-forms 180 181pform F(i,j)=2; 182index_symmetries F(i,j):antisymmetric; 183F(-i,-j) := -g(-i,-k)*o(k,-a)^o(a,-j); 184 185% EDS for vacuum GR (Ricci-flat) in 4d 186 187GR0 := eds({e(a),epsilon(i,j,k,l)*F(-j,-k)^e(-l)}, 188 {e(i)}, 189 ISO)$ 190 191% Find an integral element, and linearise 192 193Z := integral_element GR0$ 194GRZ := linearise(GR0,Z)$ 195 196% This actually tells us the characters already: 197% {45-39,39-29,29-21,21} = {6,10,8,21} 198 199% Get the characters and dimension at Z 200 201characters GRZ; 202dim_grassmann_variety GRZ; 203 204% 6+2*10+3*8+4*21 = 134, so involutive 205 206clear e(r),o(r,s),g(p,q),epsilon(i,j,k,l),F(i,j); 207clear e,o,g,epsilon,F,Z; 208indexrange 0; 209 210 %%%%%%%%%%%%%%%%%%%%%%%%%% 211 % Janet's PDE system % 212 %%%%%%%%%%%%%%%%%%%%%%%%%% 213 214% This is something of a standard test problem in analysing integrability 215% conditions. Although it looks very innocent, it must be prolonged five 216% times from the second jet bundle before reaching involution. The initial 217% equations are just 218% 219% u =w, u =u *y + v 220% y y z z x x 221 222load sets; 223off varopt; 224pform {x,y,z,u,v,w}=0$ 225 226janet := contact(2,{x,y,z},{u,v,w})$ 227janet := pullback(janet,{u(-y,-y)=w,u(-z,-z)=y*u(-x,-x)+v})$ 228 229% Prolong to involution 230 231involutive janet; 232involution janet; 233involutive ws; 234 235% Solve the homogeneous system, for which the 236% involutive prolongation is completely integrable 237 238fdomain u=u(x,y,z),v=v(x,y,z),w=w(x,y,z); 239 240janet := {@(u,y,y)=0,@(u,z,z)=y*@(u,x,x)}; 241janet := involution pde2eds janet$ 242 243% Check if completely integrable 244if frobenius janet then write "yes" else write "no"; 245length one_forms janet; 246 247% So there are 12 constants in the solution: there should be 12 invariants 248 249length(C := invariants janet); 250solve(for i:=1:length C collect 251 part(C,i) = mkid(k,i),coordinates janet \ {x,y,z})$ 252S := select(lhs ~q = u,first ws); 253 254% Check solution 255mkdepend dependencies; 256sub(S,{@(u,y,y),@(u,z,z)-y*@(u,x,x)}); 257 258clear u(i,j),v(i,j),w(i,j),u(i),v(i),w(i); 259clear x,y,z,u,v,w,C,S; 260 261end; 262