From: PointedEars@web.de   
      
   Ross Finlayson wrote:   
   > The idea that everything physics is always parameterized   
   > by time or 't' is often formalized "the Lagrangian",   
      
   No, the parametrization by time is a concept in Lagrangian _mechanics_ which   
   is based on the _principle of stationary ("least") action_.  The action is   
   defined as   
      
     S[x(t)] = ∫ dt L[x(t), dx(t)/dt, t],   
      
   where x may be a vector (field), and L is the Lagrangian (function).   
   [Both S and L are *functionals*: they depend on a function, x(t);   
   hence the customary notation with rectangular brackets.]   
      
   In special relativity, one finds from the Minkowski metric   
      
     ds^2 = c^2 dτ² = c^2 dt^2 - dx^2 - dy^2 - dz^2   
                    = c^2 dt^2 [1 - (dx/dt)^2 - (dy/dt)^2 - (dz/dt)^2]   
                    = c^2 dt^2 (1 - V^2/c^2)   
      
   that   
      
     S[x] = -m c ∫ ds = -m c ∫ dt c √(1 - V^2/c^2)   
                      = ∫ dt [-m c^2 √(1 - V^2/c^2)],   
      
   where the prefactor -m c is introduced so as to produce a quantity with   
   dimensions of action (energy × time, cf. ℎ and ℏ) and the correct   
   canonical   
   momentum [*], and in the integrand one with dimensions of energy; so the   
   relativistic non-interacting Lagrangian is   
      
     L = -m c^2 √(1 - V^2/c^2) = -m c^2 √[1 - (dX/dt)^2/c^2].   
      
   It turns out that this leads to the correct energy--momentum relation,   
   as I pointed out earlier.   
      
     [*] For example, the canonical 3-momentum is, from the Euler--Lagrange   
         equations   
      
           0 = d/dt ∂L/∂(dX/dt) - ∂L/∂X = d/dt ∂L/∂V - ∂L/∂X =   
   d/dt ∂L/∂V   
      
           P = ∂L/∂V   
             = -m c^2 ∂/∂V √(1 - V^2/c^2)   
             = -m c^2/[2 √(1 - V^2/c^2)] ∂/∂V (1 - V^2/c^2)   
             = -m c^2/[2 √(1 - V^2/c^2)] (-2 V/c^2)   
             = m V/√(1 - v^2/c^2)   
             = γ(v) m V.   
      
     [It is interesting to note that this way the relativistic/exact 3-momentum   
      for a massive particle can be derived purely from the Minkowski metric,   
      without a Lorentz transformation (but the Minkowski metric is Lorentz-   
      invariant, somewhat by design [I showed before that you do not even   
      need to assume Lorentz invariance to derive it, just a constant speed   
      with which information propagates in space)].   
      
   Since from the above follows that ds = c dτ, one can also write   
      
     S[x(τ)] = -m c ∫ dτ c = -m c^2 ∫ dτ.   
      
   The physical paths of free motion, which (one can prove) are spacetime   
   geodesics, are those where the action S[x(t)] is minimal (stationary in   
   general).  From the form above one can see that those are the trajectories W   
   along which the elapsed proper time ∆τ = ∫_W dτ is maximal, which is   
   another   
   way of describing "time dilation" when there is relative motion, and finally   
   explaining the "twin paradox" as nothing more than a consequence of   
   different elapsed proper times along different worldlines.   
      
   One can also see here that mass arises naturally from assuming the principle   
   of stationary action.   
      
   > sort of like "the Machian" is a usual notion of far-field.   
      
   No, nonsense.   
      
   > [pseudo-scientific word salad]   
      
   You are a hopeless case.   
      
   --   
   PointedEars   
      
   Twitter: @PointedEars2   
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