From: toadastronomer@gmail.com   
      
   On Monday, June 21, 2021 at 4:01:07 AM UTC-4, mark horn wrote:   
   > 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]]   
      
   21-JUN-2021   
      
   Thanks so much.   
   I'll chalk up another one to the unreasonable effectiveness   
   of my ignorance to lead me astray.   
      
   thanks again, m   
      
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