From: toadastronomer@gmail.com   
      
   20-JUN-2021   
   hello -   
      
   I find a maximum value for the Lorentz gamma factor,   
   gamma = ((1-((v)^2/c^2))^(1/2))^-1 = 54794158.005943767726,   
   for a relative velocity v = 299792457.99999997 m/s.   
   For an electron with mass m_e = 510998.91 ev/c^2 and momentum p_e=m_ev   
   the max velocity is v_e = p_e/m_e = 299792457.9999999404 m/s.   
   Plugging v_e into the gamma equation yields the same gamma max.   
   Computing a higher velocity past the eighth decimal place does   
   not change the gamma value either; until it blows up as gamma = inf.   
      
   Is there a good turn of phrase to explain this limit?   
      
   Cheers,   
   mj horn   
      
   [[Mod. note -- I think "floating-point rounding errors" is the phrase   
   you're looking for.  If v/c is very close to 1, then the formula for   
   gamma tends to be very sensitive to rounding errors, causing the sorts   
   of anomolous behavior you noticed.   
      
   The computation can be reorganized to be less sensitive to rounding   
   errors, but the easy solution is to just use brute force, i.e., use   
   higher precision in the computation.  For example, software systems   
   such as Sage, Maple, and Mathematica can all easily do computations   
   in higher precision than standard C "double" (which typically gives   
   about 16-digit accuracy).  For example, in Sage:   
      
   sage: gamma(v_over_c) = 1/sqrt(1 - v_over_c^2)   
   sage: gamma(1 - 1/(10**20))   
   100000000000000000000/199999999999999999999*sqrt(199999999999999999999)   
   sage: n(gamma(1 - 1/(10**20)), digits=50)   
   7.0710678118654752440261212905781540809584467771981e9   
   sage:   
      
   As to what relevance this has for *physics*: the current record for the   
   highest-energy cosmic ray has a gamma factor of over 10**20, corresponding   
   to v/c of over 1 - 10**-40.   
   -- jt]]   
      
   --- SoupGate-Win32 v1.05   
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