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