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   This is a note I'd like to convey here regarding a matter that I   
   (and a few of my friends) have been giving thought to recently.   
      
   Hawking has been on much on my mind over the past 2 weeks. On a   
   sudden lark, I posed as a problem to a (former) s.p.r. moderator   
   to uncover some of the interesting details regarding the 7-dimensional   
   geometry with coordinates (t,u,v,w,x,y,z) ... and corresponding   
   differential operators (T,U,V,W,X,Y,Z) ... possessing the following   
   vector fields as symmetries:   
                   Jx = zY-yZ, Kx = wtX+x(vT-U),   
   		Jy = xZ-zX, Ky = wtY+y(vT-U),   
   		Jz = yX-xY, Kz = wtZ+z(vT-U).   
   The fields have the following remarkable property: though they are   
   non-linear, and have Lie brackets with variable structure coefficients,   
   the coefficients are all invariants, so that the fields reduce to   
   Lie algebras on each of its orbits. But different Lie algebras on   
   different orbits; thus making the geometry a "signature changing"   
   geometry with different "Lie domains", on whose boundaries lie   
   "signature-changing" interfaces whose Lie algebras are contractions   
   of the Lie algebras in the domains that are immediately adjacent.   
      
   In particular, since [Kx,Ky] = [wtX,y(vT-U)] + [x(vT-U),wtY] =   
   vw(xY-yX) = -vw Jx, then the 3 Lie algebras are so(3,1) on orbits   
   where vw > 0, se(3) where vw = 0, so(4) where vw < 0. The vw > 0   
   so(3,1) domain passes through a vw = 0 se(3) interface over to the   
   vw < 0 so(4) domain.   
      
   The invariants are v, w, s def= t + uv, rr def= xx + yy + zz + tuw   
   + uuvw, where rr < 0 is possible if w != 0, so r can be imaginary.   
   The first 2 invariants, v and w are the two mentioned above that   
   control signature. The last 2 invariants are inhomogeneous, but are   
   homogeneous with respect to a reduced set of coordinates -- either   
   (t,u,x,y,z) or (t,v,w,x,y,z). So one can go further and add in   
   translation generators for the corresponding subsets of the fields   
   (T,U,V,W,X,Y,Z). Alternatively, the two inhomogeneous invariants   
   combine to yield the homogeneous invariant: wss - vrr = wtt -   
   v(xx+yy+zz). This shows that the actions of the Lie algebras on the   
   coordinates are Lorentz/Poinca\'e for so(3,1) when vw > 0, Euclid-4D   
   for so(4) when vw < 0 and either Galilei, Carroll or Static for vw   
   == 0 (the cases v = 0, w != 0; v != 0, w = 0 or v = 0, w = 0   
   respectively). This puts the focus on the coordinate set (t,u,x,y,z)   
   instead of (t,v,w,x,y,z). Since U is already used in one of the   
   operators, then (t,u,x,y,z) should be the set chosen, while v,w   
   remain as just parameters.   
      
   So, adding in the translation generators   
   	Px = -X, Py = -Y, Pz = -Z   
   one immediately finds a generator M for "mass":   
   	M = [Kx,Px] = [Ky,Py] = [Kz,Pz] = vT-U,   
   The other combinations of K's and P's producing 0 brackets. In both   
   relativity and non-relativistic theory, a generator H for Kinetic   
   energy would yield the brackets [Kx,H] = Px, [Ky,H] = Py, [Kz,H] = Pz,   
   which goes naturally with the identification   
           H = -T.   
   The invariant mass m would be the Galilei limit lim_{v->0} M:   
   	m = -U   
   and appears in the v = 0 case as a central charge. For the so(3,1)   
   or so(4) cases, one could replace M by E def= M/v, identifying it   
   as the "total energy".   
      
   When translation generators are added in, the invariants have to   
   re-stated in terms of coordinate differences or differentials. In   
   differential form the inhomogeneous invariants I previously mentioned   
   would be rewritten as:   
           ds = dt + v du and dr dr = dx dx + dy dy + dz dz + 2 w dt   
           du + w w du du   
   	w(dr dr) - v (ds ds) = w(dx dx + dy dy + dz dz) - v(dt dt).   
      
   It is also possible too get [anti-]deSitter into the mix. As a side   
   effect, the 5 groups I previously mentioned now become 5*3 - 1 =   
   14; the 10-1 additions being:   
   [anti-]de{Sitter,Galilei,Euclid,Carroll,Static} with deStatic =   
   anti-deStatic. The 2 extra deEuclids are, of course, 4D hyperbolic   
   and 4D hyperspherical, and the 2 deGalileis are already known as   
   [anti-]Newton-Hooke.   
      
   Such a geometry can be endowed with any of these 14 kinematic groups   
   by adding another coordinate, q, and adopting, in place of the   
   translation generators, X,Y,Z,T,U, the deformed versions   
   aX,aY,aZ,aT-bU,cU, where   
           a = root(1 + q(wss-vrr)), c = root(1 + qwss), b = qrr/(a+c);   
   noting that wss-vrr = wtt-v(xx+yy+zz) is independent of u.   
      
   The bracket for the boost generator Kx and translation generator   
   Px deform from [Kx,Px] = vT-U to v(aT-bU) - cU = a(vT-U), since c   
   = a+bv; the central charge U deforms to cU and remains a central   
   charge for non-zero q.   
      
   And now, the note, itself:   
   For the reasons about to be cited, the very existence of the Cosmic   
   Microwave Background (CMB) should be regarded as direct proof that   
   the universe must be:   
           *       non-Riemannian,   
   	*	signature changing, with an initial null hypersurface,   
   	*	geodesically complete near its initial hypersurface,   
   	*	causally connected across spacelike separations, and   
   	*	must have an early inflationary period, as well as   
           *       a non-trivial self-consistency requirement that may   
           suppress backwards propagation along the past light cone!   
      
   The reasons are as follows. The CMB marks the point of last scattering,   
   the point where outer space first became transparent, before which   
   the universe was radiation dominant. A big bang metric that is   
   radiation dominant in early times must *always* be signature changing   
   with an initial null hypersurface at t = 0 and must always be smooth   
   (i.e. C-infinity and even analytic) across t = 0; even though it   
   reduces to rank 1 at t = 0. For t < 0 it is Euclidean. The converse   
   may be true as well, if assuming the weak energy condition and   
   excluding Minkowski geometry (smooth signature changing metric if   
   and only if early radiation dominant).   
      
   For metrics in the family   
   	dt dt - A(t) B(r) (dx dx + dy dy + dz dz)   
   where A(t) -> 0 as t -> 0, a C-infinity growth condition implies a   
   power-law near t = 0 of the form A(t) ~ t^K for some integer K =   
   0,1,2,3,... Excluding the cases of Minkowski/Hyperspherical/Hyperbolic   
   (K = 0), or the case K = 2, which lies at the upper limit of what's   
   permitted by the Weak Energy condition, this implies K = 1, which   
   corresponds to the case of a universe with early radiation dominance   
   with a null hypersurface at t = 0.   
      
   Such a geometry must also have the following properties: (1) it is   
   geodesically complete!, (2) the past-directed null and timelike   
   geodesics are almost all tangent to the t = 0 surface and reverse   
   direction to become future-pointing before that, the sole exceptions   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)