Re: Proper time differences

Le 08/07/2024 à 16:45, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :

  From various sources I gather,
 dt = "gamma" d"tau".
   Where t is the coordinate time in the rest frame, "gamma"
  is the Lorentz gamma factor and "tau" is the proper time.
   Now, if "gamma" is constant, I think we can replace the "d"
  by "D" (triangle which is flat at its bottom), i.e., we can
  use finite difference instead of infinitesimal ones:
 Dt = "gamma" D"tau".
   I believe 0<="gamma"<=1, so, for an example, we can assume   "gamma" to be 0.5:
 Dt = 0.5 D"tau",
   which means just,
 D"tau" = 2 Dt.
   So, that would mean: For a moving thing the proper time
  difference D"tau" (I assume: between two fixed events) is
  /larger/ than the coordinate time difference.
   But since falling muons live longer, the proper time distance
  should be /smaller/, not larger!
   What's wrong here? TIA!

You made a mistake in the wording.
The correct equation is To=Tr.gamma
but gamma is 1/sqrt(1-v²/c²).
And not sqrt(1-v²/c²).
Offer you gamma=0.5
It's impossible.
R.H.