On Saturday, July 22, 2017 at 11:00:24 AM UTC-5, Tom Roberts wrote:   
   > On 7/21/17 7/21/17   7:29 PM, I wrote:   
   >> Physical quantities form a TYPED ALGEBRA.   
   >>   
   >> The system of types form an Abelian group with the identity 1   
   >> standing for the type of dimensionless quantities   
   >>   
   >> A type judgement e: T means quantity e has type T, which in dimensional   
   >> analysis means [e] = T.   
   >>   
   >> Addition and subtraction are subject to type-restriction: e + f and   
   >> e - f are only defined if e:T and f:T, in which case (e +/- f): T.   
   >   
   > OK, except there is also the concept of compatible types: cm and inches (in)   
   are not equal (same type), but are compatible:   
      
   You're mixing concepts. 2 and 3 are "int" in C, and are not equal, but   
   they are the same type. For the units, the types are   
      
   in: Length, cm: Length.   
      
   They're the same type and (as you tried to note), are related by in/cm =   
   2.54.   
      
   Now, a more interesting question is whether the c that appears in   
      p = mv/root(1 - (v/c)^2)   
   is the same type as the one that appears in   
      ds^2 = dt^2 - c^2 (dx^2 + dy^2 + dz^2).   
      
   Within the setting of the a more general framework that embodies ALL of   
   the Kinematic groups in the Bacry Levy-Leblond classification   
   (particularly, the Carroll "c = 0" group and Static "c = 0 & c =   
   infinity" group), this becomes a non-trivial question.   
      
   And so... the seguey into the first of the two earlier mentioned case   
   studies...   
      
   First case study:   
   Consider the following enveloping framework for kinematic symmetry groups (the   
   BLL or Bacry Levy-Leblond) framework,   
   given by the following transformation laws:   
   	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 + kappa epsilon x J - upsilon M - lambda tau K   
   	delta(H) = -beta upsilon.P - lambda epsilon.K   
   	delta(M) = -gamma upsilon.P - kappa epsilon.K   
   with the structure coefficients related by   
   	gamma = alpha beta, kappa = alpha lambda, and   
   	zeta = beta lambda   
   also added in for convenience but otherwise unused. The quantities are the   
   infinitesimal forms of:   
   	omega, upsilon, epsilon:   
   		(vector) rotation, boost and spatial translation   
   	tau: (scalar) time translation   
   while:   
   	J, K, P:   
   		(vector forms of) angular momentum,   
   		moving mass moment, linear momentum   
   	H, M: (scalar) kinetic energy and "total" mass.   
   This corresponds to the Poisson bracket   
   	delta(_) = { _, Lambda }   
   where   
   	Lambda = omega.J + upsilon.K + epsilon.P - tau H + psi (M - alpha H) with the   
   invariants   
   	mu = M - alpha H: invariant mass   
   	nu = beta P^2 - 2 M H + alpha H^2: mass shell   
      
   All cases with (alpha,beta,lambda) of the same signs are equivalent: so   
   that reduces the count to 27.   
      
   The cases (alpha,beta,lambda) and (-alpha,-beta,-lambda) are equivalent,   
   so the 27 cases reduce to 14; 13 equivalent pairs + 1 single case (the   
   'static' group).   
      
   Assuming the type:   
   	delta: 1   
   and adopting the dimensions:   
   	upsilon: V, epsilon: L, tau: T, M: M, J: H   
   this leads to the equation H = MLV and to the following typing:   
   	J: MLV, K: ML, P: MV, H: MLV/T, M: M   
   	omega: 1, upsilon: V, epsilon: L, tau: T, psi:   
   	alpha: T/LV, beta: L/VT, gamma: 1/V^2   
   	kappa; 1/L^2, lambda: V/TL, zeta; 1/T^2   
   	mu: M, nu: M^2 L^3/VT   
      
   The kinematic groups with lambda non-zero are suitable for handling   
   homogeneous curved geometries.   
      
   For flat geometries, one has lambda = 0 and the count reduces to 5:   
   	beta = 0, alpha = 0	-	static (c = 0 and c = infinity)   
   	beta = 0, alpha != 0	-	Carroll (c = 0)   
   	beta != 0, alpha = 0	-	Bargmann/Galilei (c = infinity)   
   	gamma > 0		-	Lorentzian   
   	gamma < 0		-	Euclidean   
      
   In this geometry, V has to be made a separate dimension from L and T!   
   They have to be kept separate because of the possibility of beta being   
   0!  In the case where beta != 0, beta can be normalized to 1, and only   
   then can one recover V = L/T.   
      
   The transformation laws for (P,H,M) then reduce to:   
   	delta(P) = omega x P - upsilon M   
   	delta(H) = -beta upsilon.P   
   	delta(M) = -gamma upsilon.P   
   as well as:   
   	delta(mu) = 0   
      
   A geometry emerges with the correspondence:   
   	P <-> del = d/dr, H <-> -d/dt, mu <-> d/du   
   resulting in the transform laws:   
   	delta(dr) = omega x dr - beta upsilon dt   
   	delta(dt) = -alpha upsilon.dr   
   	delta(du) = upsilon.dr   
   with invariants:   
   	ds = dt + alpha du   
   	dr^2 + 2 beta dt du + gamma du^2   
      
   In this case, for Lorentzian geometries, the role of the invariant speed   
   can be determined by replacing u by s:   
   dr^2 - beta/alpha dt^2 + beta/alpha ds^2.   
      
   The quantity beta/alpha has the dimensions (L/T)^2 with the invariant   
   speed being c = (beta/alpha)^{1/2}   
      
   This is distinct from the electromagnetic wave speed 1/gamma^{1/2},   
   differing from it by a factor of |beta|.   
      
   The beta = 0 geometries are suited for event horizons and Hubble or "big   
   rip" horizons, while the alpha = 0 geometries are suited for   
   Newton-Cartan spacetime and for the cosmological "big bang" t = 0   
   horizons.   
      
   Second case study:   
   This point, particularly, needs to be underscored with the so-called   
   "constancy of light speed" axiom, given what was said in the earlier   
   article about the ill-definedness of the notions of "constant" and   
   "variable" with respect to dimensioned quantities.   
      
   The correct statement that characterizes special relativity should   
   therefore be made that makes no mention of the constancy or variability   
   of anything -- namely, that special relativity has a Lorentzian   
   chronogeometry! Or, in the case of general relativity: locally   
   Lorentzian.   
      
   The (E,B) fields combine into the 2-form   
   	F	= (E_x dx + E_y dy + E_z dz) ^ dt   
   		+ B^x dy^dz + B^y dz^dx + B^z dx^dy   
   and the (D,H) fields, which combine into the 2-form   
   	G	= D^x dy^dz + D^y dz^dx + D^z dx^dy   
   		- (H_x dx + H_y dy + H_z dz) ^ dt   
   are always defined, as the Lagrangian derivatives:   
   		delta L = -(delta F) ^ G.   
   For the field components, this reduces to:   
   		delta L = 1/2 ((delta E).D - (delta B).H) dt^dx^dy^dz   
      
   As dimensioned quantities, they have the types   
   		G: Q = charge,   
   		F: P = H/Q = unit of gauge = magnetic charge/flux   
   where (H = unit of action)   
   		(dx,dy,dz): L = length, dt: T = time   
   		(E_x,E_y,E_z): P/(LT), (B^x,B^y,B^Z): P/L^2   
   		(D^x,D^y,D^z): Q/L^2, (H_x,H_y,H_z):  Q/(LT)   
   The Maxwell-Lorentz-Einstein Lagrangian can be written as a 4-form:   
   	L = K root(-g) g^{mr} g^{ns} F_{mn} F_{rs} dt^dx^dy^dz   
      
   So, now consider this field in the space-time geometry given by the metric:   
   	ds^2 = dt^2 - k t (dx^2 + dy^2 + dz^2),   
   where k > 0 is a dimensioned constant (i.e. dk/dt = 0, del k = 0)   
   		k: T/LL.   
      
   This is:   
   	Radiation-dominant metric   
      
   	Signature change across t = 0 to t < 0:   
   	NOT Lorentzian t < 0 but Euclidean.   
      
   	Null surface at t = 0,   
      
   	as per Mansouri, inflation is a *predicted* consequence,   
   	not something "written in by hand"   
   	as part of an otherwise-unresolved "fine tuning problem".   
      
   	Geodesically complete (despite the singularity at t = 0!)   
      
   Here, with t = x^0, (x,y,z) = (x^1,x^2,x^3), we have   
   	g_00 = g^00 = 1,   
   	g_0j = 0 = g_i0, g^0j = 0 = g^i0, for i, j = 1, 2, 3   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)