==================================================================== Just exactly what is the "Lorentz group"? ________________________________ I want to tell you about a clutch of closely related groups of geometrical transformations --- Lie groups --- some familiar, some less so. Each is essentially defined by preserving a quadratic form on m variables, which acts like a metric to specify the topology in some suitable coordinate system. Now it is well-known (trans: I think it's true, but can't find the reference --- anybody?) that the signature of this form --- the numbers (p, n, z) of respectively positive, negative, zero coefficients in the diagonal form --- is an invariant of the group (modulo p,n transposition anyway). Signature provides a concise method for distinguishing groups which are NOT isomorphic. A less technical means to achieve the same end is comparing their topological dimension: unfortunately, all these groups happen to boast the same dimension m(m-1)/2 . The coincidence has in the past contributed to much confusion, a scarcely credible example of which will later be unveiled. Such transformations are traditionally constructed as compositions of elementary "reflectors" leaving invariant prime (maximal nontrivial) subspaces of the underlying space. However the groups so generated are neither connected nor compact, and engineering or physical applications usually restrict attention to their "kinematic" subgroups: connected to the identity, and generated by "rotators" composed of two reflectors. Compactification for computational purposes incorporates non-invertible projections as limits, which may then be identified with members and subspaces of the space. *Exhibit* EucKin(m-1) , proper Euclidean group in real (m-1)-space. Eschewing improper reflections in hyperplanes, this is generated instead by rotations around coline axes, together with translations. Continuous rigid motions arise as infinitesimal limits of compositions of rotators. Each isometry decomposes uniquely in general into at most [m/2] orthogonal commuting rotators: eg. when m = 5 a 3-space helical screw with freedom 6 factorises into rotation composed with translation along the same axis line. Translations constitute an (m-1)-dimensional normal subgroup. *Signature* (m-1, 0, 1) . *Exhibit* MobKin(m-2) , kinematic Moebius group in (m-2)-space: conformal, conserving points and spheres. Again, the usual geometric inversions are disconnected reflectors, and instead we compose rotators, of three flavours: elliptic (including EucKin(m-2) rotations), parabolic (including EucKin(m-2) translations), hyperbolic (including Euclidean dilations). Hyperbolic rotators ("boosts") are easiest to visualise, with eigenspace comprising two real, distinct "limit points": a test sphere initially dilates from source point and finally shrinks into sink point, meantime remaining tangent to some arbitrary (m-2)-set of guiding (m-3)-spheres meeting both points (so fixed by the motion). *Signature* (m-1, 1, 0) . *Exhibit* LagKin(m-2) , kinematic Laguerre group in (m-2)-space: equilong (preserving distance between tangencies), conserving primes and "oriented" spheres (alternatively, half-spaces internal or external to spheres). Besides being disconnected, reflectors are dismally unintuitive; rotators include EucKin(m-2) rotations and translations, exclude similarities (this convention is regrettably not universal). A (hyperbolic) rotator has a pair of (real) "limit primes": a test sphere initially detaches from source prime, finally merges into sink prime, meantime remaining tangent to fixed m-2 spheres, themselves tangent to both primes. A noteworthy parabolic rotator is the "offset", under which the radius of every sphere increases by the same extent; offsets constitute a 1-dimensional normal subgroup. *Signature* (m-2, 1, 1) . *Exhibit* LieKin(m-3) , kinematic Lie-sphere group in (m-3)-space: conserving contact (oriented tangency), acting on & conserving oriented spheres (not separately lines or points). Includes MobKin(m-3) and LagKin(m-3) , which in turn both include EucKin(m-3) . Rotators are as before, except that their eigenspaces now comprise pairs of arbitrary spheres (conjugate complex, coincident, or real). Despite its failure to preserve angles, physicists are reputed to to describe this group as "conformal". *Signature* (m-2, 2, 0) . Note MobKin(m-2) is transitive on points, and separately on other spheres; in contrast LagKin(m-2) leaves fixed the single prime at inversive infinity, where lurk eigenspaces of translations, offsets etc. MobKin(m-2) has no small normal subgroups, essentially because every (say) elliptic rotator is conjugate to every other elliptic rotator equipped with the same (angular) extent; whereas LagKin(m-2) has its one-dimensional normal subgroup of offsets. So those two groups are distinct on these grounds alone. It may also be worth emphasising the distinction between continuous Moebius dilation: hyperbolic, with distance from some fixed centre point an exponential function of extent; and continuous Laguerre offset: parabolic, with all radii a linear function of extent, but no detectable finite axis. Summing up then: if you believe in their signatures (or even if you don't), despite their sharing the same dimension, for m > 3 *** all four groups above are non-isomorphic in pairs! *** Now take say m = 5 ... ==================================================================== Why have I been bending your ear and inflicting this obscure guff on you? https://en.wikipedia.org/wiki/Lorentz_group http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html refer to various luminaries who apparently accept unquestioningly that the (kinematic) Lorentz group is isomorphic to what I have called MobKin(3) above. Whereas an impressive list of authorities --- including Cartan, Poincare, Klein, Blaschke, Coolidge, Bateman, Pottmann --- are alleged at https://en.wikipedia.org/wiki/Spherical_wave_transformation to assert equally firmly that the same group is isomorphic to LagKin(3) instead. [ Implicit in these and similar surveys seems to be an assumption that the z component of the signature may be casually disregarded, the remaining p,n somehow guaranteeing isomorphism; although the quadratic forms are not positive definite, and do not entail classical metrics. ] At this stage in my life, I am resigned to the prospect of never personally acquiring a more than rudimentary understanding of relativistic mechanics. However it is most disconcerting to discover that the grasp of their subject commanded by a substantial proportion of cognoscenti, to whom I might have entrusted the task of allaying my ignorance, appears evidently feebler than my own. Come on, people --- 127 years after Michelson-Morley experimented and Fitzgerald-Lorentz contracted, surely we should be able to get our act together sufficiently to engineer agreement about whether their bloomin' group either (1) has degenerate signature, and/or some one-dimensional normal subgroup, and/or some fixed plane; or else (2) has not? It isn't rocket science, is it ... On second thoughts, it IS rocket science --- Heaven help us all ... Fred Lunnon, Maynooth 02/09/14 ====================================================================