Simultaneity
in Special
Relativity
by
Ardeshir
Mehta
Wednesday, October 10,
2001

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.

This means that if two
events, which we shall
call E_{1} and E_{2},
occur simultaneously
in the IFR of an observer whom we shall call Adam,
then events E_{1}and
E_{2} 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.

This in turn means that
if according to Adam’s
watch, event E_{1} occurred
precisely at a specific instant in
time t, and if as indicated by Adam’s watch
event E_{2 }also
occurred precisely at the same specific
instant in time
t — as it must
if event E_{2} is to be simultaneous
in Adam’s IFR with
event E_{1} — then event E_{1},
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 E_{2}
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.

(For if the instants t'
and t"
as indicated by Eve’s watch were exactly the same,
then events
E_{1} and E_{2} would
have occurred simultaneously
in Eve’s IFR too!)

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

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

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/c^{2})]
where <gamma> = 1/[1(v^{2}/c^{2})]^{0.5},
and x is the distance, as measured by Adam,
between Adam's watch
and Eve's watch.

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/c^{2})]
where <gamma> = 1/[1(v^{2}/c^{2})]^{0.5},
and x is the distance as measured by Adam
between Adam's watch and
Eve's watch.

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.

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

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.

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 selfcontradictory.
Any comments? email
me.
