On Friday, May 15, 2020 at 4:06:46 PM UTC-5, Nicolaas Vroom wrote:   
   > Today we understand group theory (which Einstein did not in 1905). We   
   > can use group theory to show that there can be only three transformation   
   > groups among inertial frames, and that just one of them, the Lorentz   
   > group [#], agrees with all the experiments.   
      
   14, not 3. There are 14 kinematic groups consistent with a general set   
   of assumptions - a 3-parameter family of groups distinguished by the   
   sign of each parameter, which I'll call here (alpha,beta,kappa), such   
   that the (-alpha,-beta,-kappa) group is isomorphic to the   
   (alpha,beta,kappa) group.   
      
   They are:   
   Static: (alpha,beta,kappa) = (0,0,0)   
   Galilei: (alpha,kappa) = (0,0), beta != 0   
   Carroll: (beta,kappa) = (0,0), alpha != 0   
   Para-Galilei: (alpha,beta) = (0,0), kappa != 0   
   Newton-Hooke(+/-): alpha = 0, zeta > 0 (+) or zeta < 0 (-)   
   Para-X: beta = 0, lambda > 0 (X = Poincare') or lambda < 0 (X = Euclid)   
   X: kappa = 0, gamma > 0 (X = Poincare') or gamma < 0 (X = Euclid-4D)   
   deSitter(+/-): gamma > 0; lambda > 0 (+) or lambda < 0 (-)   
   Hyper-X: gamma < 0; zeta > 0 (X = bolic), zeta < 0 (X = spherical)   
      
   where gamma = alpha beta and lambda = alpha kappa and zeta = beta kappa.   
      
   The Bacry Levi-Leblond (BLL) 1968 classification.   
      
   Roughly speaking: alpha = 0 corresponds to c = infinity, beta = 0   
   corresponds to c = 0, with beta/alpha = c^2, when alpha beta > 0. The 3   
   groups where alpha beta < 0 correspond to geometries with signature 4+0   
   and were excluded in the 1968 paper ... though they are relevant for   
   geometries with signature-changing metrics.   
      
   Both the c = 0 and c = infinity cases can occur together: alpha = 0 and   
   beta = 0 (the Static and Carroll groups). Static and Carroll have the   
   same central extension, even though one is trivial; Galilei has the   
   Bargmann group as its central extension.   
      
   If the symmetries are denoted:   
   * Spatial translations P = (P1,P2,P3)   
   * Spatial rotations J = (J1,J2,J3)   
   * Time translations H   
   * Boosts K = (K1,K2,K3)   
   then the groups are those with the Lie brackets:   
   * [Ja,Vb] = Vc, where V is (J,K,P), (a,b,c)=(1,2,3), (2,3,1) or (3,1,2)   
   * [J,H] = 0   
   * [Ka,Kb] = -gamma Jc; [Pa,Pb] = lambda Jc, with (a,b,c) as above   
   * [Ka,H] = beta Pa, [Pa,H] = kappa Ka   
   * [Ka,Pb] = alpha delta_{a,b} H where delta_{a,b} = 1 if a = b, 0 else   
   where gamma = alpha beta, lambda = alpha kappa. The combination zeta =   
   beta kappa is also useful, but does not appear as a structure   
   coefficient.   
      
   This set is derived from the assumptions:   
   * isotropy to fix all the J brackets as above   
   * consistency with the discrete transforms (J,K,P,H) -> (J,aK,bP,abH)   
   for time reversal (a,b)=(-,+) and parity reversal (a,b)=(+,-), which   
   fixes the forms of the brackets,   
   * Jacobi identities (from which follow gamma = alpha beta, lambda =   
   alpha kappa).   
   A slightly more general classification can be derived by seeking out all   
   possible deformations of the (alpha,beta,kappa) = (0,0,0) case.   
      
   The dimensions of the various quantities (using M,L,T,V respectively for   
   mass, length, time, speed) are:   
      
   alpha: T/VL, beta: L/TV, kappa: V/LT   
   gamma: 1/VV, lambda: 1/LL, zeta: 1/TT   
   J: MLV, K: ML, P: MV, H: MLV/T   
      
   The V dimension has to be treated as independent of L/T for consistency   
   with beta -> 0, though it can be normalized to L/T for non-zero beta by   
   setting beta = +1 or beta = -1.   
      
   All of the kinematic groups have a central extension of the form:   
      
   [Ka,Pb] = M delta_{ab}, where M def= mu + alpha H and mu is the central   
   charge.   
      
   Both M and mu have the dimension M. The central extension is only   
   non-trivial when alpha = 0; it is "trivial" otherwise; i.e. the group,   
   for non-zero alpha, splits into (J,K,P,H,mu) -> (J,K,P,M) x (mu). In   
   this case, E = M/alpha can be used in place of M. This corresponds to   
   "total energy" in Relativity, while M corresponds to "relativistic   
   mass".   
      
   For uniformity, however, it's necessary to use (J,K,P,H,M) or (J,K,P,H,mu);   
   since E is ill-defined for alpha = 0.   
      
   Two of the groups coincide under central extensions, so the number then   
   drops to 13.   
      
   The transformations in infinitesimal form are   
   * rotations: omega   
   * boosts: upsilon   
   * spatial translations: epsilon   
   * time translations: tau   
   * extra translation: psi   
   which have dimensions:   
      
   omega: 1, upsilon: V, epsilon: L, tau: T, psi: LV   
      
   Their actions on (J,K,P,H,M,mu) are all given by   
      
   delta X = [X, J.omega + K.upsilon + P.epsilon - H tau + mu psi]   
      
   leading to the following infinitesimal transformations:   
      
   delta J = omega x J + upsilon x K + epsilon x P   
   delta K = omega x K - gamma upsilon x J + epsilon M - beta tau P   
   delta P = omega x P - upsilon M + lambda epsilon x J - kappa tau K   
   delta H = -beta upsilon.P - kappa epsilon.K   
   delta M = -gamma upsilon.P - lambda epsilon.K   
   delta mu = 0   
   where ()x() denotes vector cross-product and ().() vector dot product.   
      
   They can be integrated to finite form to see how each transforms under the   
   respective symmetry.   
      
   Rotations: (J,K,P,H,M,mu) -> (RJ,RK,RP,RH,RM,R mu)   
   where R is a rotation operator given by   
      
   RV = V + sin(theta) n x V + (1 - cos(theta)) n x (n x V),   
   where the unit vector n is the axis of rotation, theta the rotation angle   
      
   Time Translations by time-shift s:   
   (J,H,M,mu) -> (J,H,M,mu)   
   (K,P) -> (K - beta s P)/r, (P - kappa s K)/r)   
   where r = root(1 + zeta s^2) and zeta s^2 > -1   
      
   For zeta < 0, the transform has an horizon at |s| = root(-1/zeta); time   
   is hyperbolic. For zeta > 0, time is circular and the transform is   
   actually a rotation and you can take either the + or - sign of the   
   square root, root (1 + zeta s^2).   
      
   Spatial Translations by shift-vector a:   
   (mu,J0,K1,P0) -> (mu,J0,K1,P0)   
   (J1,P1) -> ((J1 + a x P1)/r, (P1 + lambda a x J1)/r)   
   (K0,M -> ((K0 + a M)/r, (M - lambda a.K0)/r)   
   H -> H - kappa a.K0/r + M/r kappa a^2/(1 + r)   
   where r = root(1 - lambda a^2) and lambda a^2 < 1.   
      
   The components (J0,K0,P0) are those parallel to epsilon, while   
   (J1,K1,P1) are perpendicular to epsilon.   
   J = J0 + J1, K = K0 + K1, P = P0 + P1   
   epsilon.V1 = 0 = epsilon x V0, where V is (J,K,P).   
      
   This has an horizon at a = 1/root(lambda) if lambda > 0; the spatial   
   dimensions are hyperbolic if lambda > 0, circular if lambda < 0 and flat   
   if lambda = 0. For lambda < 0, both signs of the root can be used and   
   the transformations are ordinary sinusoidal rotations.   
      
   Boosts by a velocity vector v:   
   (J0,K0,P1,mu) -> (J0,K0,P1,mu)   
   (J1,K1) -> ((J1 + v x K1)/r, (K1 - gamma v x J1)/r)   
   (P0,M) -> ((P0 - v M)/r, (M - gamma v.P0)/r)   
   H -> H - beta v.P/r + M/r beta v^2/(1 + r)   
   where r = root(1 - gamma v^2) and gamma v^2 < 1.   
      
   This has an horizon at v = 1/root(gamma) if gamma > 0 and reduces to a   
   circular rotation if gamma < 0. For gamma = 0, it reduces to the form:   
      
      
   [continued in next message]   
      
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