Simultaneity in Special Relativity

by

Ardeshir Mehta

Wednesday, October 10, 2001



  1. Special Relativity claims that events which are simultaneous from the inertial frame of reference (or IFR) of one observer are definitely not simultaneous from the IFR of another observer moving rectilinearly at constant velocity relative to the first observer.

  2.  
  3. This means that if two events, which we shall call E1 and E2, occur simultaneously in the IFR of an observer whom we shall call Adam, then events E1and E2 could not possibly occur simultaneously in the IFR of another observer — whom we shall call Eve — if Eve is moving rectilinearly at a constant velocity v relative to Adam.

  4.  
  5. This in turn means that if according to Adam’s watch, event E1 occurred precisely at a specific instant in time t, and if as indicated by Adam’s watch event E2 also occurred precisely at the same specific instant in time t — as it must if event E2 is to be simultaneous in Adam’s IFR with event E1 — then event E1, as indicated by Eve’s watch, must have occurred at a specific instant in time t' which is different from t as indicated by Adam’s watch; and as indicated by Eve’s watch, event E2 must have occurred at another specific instant in time t" which is different from both, the instant t as indicated by Adam’s watch and from the instant t' as indicated by Eve’s watch. 

  6.  
  7. (For if the instants t' and t" as indicated by Eve’s watch were exactly the same, then events E1 and E2 would have occurred simultaneously in Eve’s IFR too!)

  8.  
  9. Thus by sentences 3. and 4. above, t' is definitely not equal to t".

  10.  
  11. The above is however incompatible with the Lorentz transformation equations, which are essential for Special Relativity. 

  12. According to the Lorentz transformation equations, the instant t' as indicated by Eve’s watch must be related to the instant t indicated by Adam’s watch by the formula t' = <gamma>[t-(xv/c2)] where <gamma> = 1/[1-(v2/c2)]0.5, and x is the distance, as measured by Adam, between Adam's watch and Eve's watch.

  13.  
  14. And according to the Lorentz transformation equations, the instant t" as indicated by Eve’s watch must be related to the instant t indicated by Adam’s watch by the formula t" = <gamma>[t-(xv/c2)] where <gamma> = 1/[1-(v2/c2)]0.5, and x is the distance as measured by Adam between Adam's watch and Eve's watch.

  15.  
  16. At any given instant t, as indicated by Adam's watch, there can be only one distance x, as measured by Adam, between Adam's watch and Eve's watch.

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  18. Thus the value of x must be exactly the same in both 7. and 8. above.

  19.  
  20. Since the values of the terms on the right hand sides of the equations in 7. and 8. above are exactly identical, t' must be exactly equal to t" and cannot possibly be different from it.

  21.  
  22. Thus by sentence 9. above, t' = t" — which contradicts sentence 5. above, according to which t' is not equal to t" ... and which therefore proves that the Special Theory of Relativity must be self-contradictory.

 

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