Re: Evil Twin Paradox
- From: Dono <sa_ge@xxxxxxxxxxx>
- Date: Tue, 07 Aug 2007 21:09:52 -0700
On Aug 7, 8:51 pm, David <dsepp...@xxxxxxxxxxxxx> wrote:
On Mon, 06 Aug 2007 11:10:14 -0700, Dono <sa...@xxxxxxxxxxx> wrote:
On Aug 6, 6:52 am, David <dsepp...@xxxxxxxxxxxxx> wrote:
This problem is a variation of the barn-pole paradox. It is a simple
application of the length contraction concept, but the results don't
seem to make sense. I would appreciate it if someone would point out
the error.
Evil Twin Paradox
An evil twin and a good twin are intially in the same reference frame
which I'll call the rest frame. The feuding twins are separated by an
extremely long steel wall that extends along the x-axis. One twin is
on the postivie y side of the steel wall and the other twin is on the
negative y side of the steel wall. Each twin has a weapon that can
shoot a 100 millimeter projectile in the y direction perpendicular to
the x-axis. The steel wall prevents the twins from shooting at each
other, but there is a section of this wall that has a long gap
extending along the x-direction. The length of this gap is L=10
light-years.
The good twin happens to be at rest at the mid-point of this 10
light-year gap. The evil twin learns of this and decides to leave the
rest frame and travel at velocity V parallel to the x-axis and wall,
and fire his weapon (which is always pointed perpendicular to the
x-axis as measured in his moving reference frame) at the good twin as
he passes the good twin who is at the mid-point of this 10 light-year
wide gap in the steel wall. The good twin learns of the evil twin's
plan well before the evil twin takes off. So the good twin prepares
his weapon so that his 100 mm projectile can hit the evil twin as he
passes the good twin's position as the evil twin moves with velocity V
just parallel to the x-axis.
Both twins know the exact initial positions, and time that the evil
twin starts moving parallel to the x-axis, and they each know the
exact relative velocity V. But the evil twin has an ultra-fast
propulsion system that makes his velocity V relative to the rest frame
almost equal to c. The good twin has zero velocity with respect to
the rest frame (and the steel wall).
When I apply Einstein's length contraction formula
L' = L * SQRT(1-(V**2/c**2))
in this situation, since V is abitrarily close to c, L' gets
arbitrarily close to zero. In other words, this 10 light-year gap in
the steel wall as measured in the moving frame of the evil twin will
be smaller than the size of a proton if V is almost equal to c. If
the moving twin attempts to fire his 100 millimeter projectile through
this proton sized gap (he fires in the y-direction as the wall and
good twin pass), the 100 millimeter projectile will hit the steel wall
and cannot pass through this proton sized gap.
The good twin is firing his 100 millimeter projectile in the y
direction through the 10 light-year wide gap so his projectile can
easily hit the evil twin as he passes.
If the steel wall were not there, either twin could shoot the other
twin as they passed each other. Or if the steel wall prior to the gap
was removed, then the evil twin could shoot the passing good twin. Or
if the steel wall following the gap was removed, the evil twin could
hit the good twin. But if both sections of the steel wall are
present, then using this length contraction formula, the evil twin's
100 millimeter projectile cannot pass through the proton sized gap.
That makes no sense to me.
Thanks ahead for your physics insights (and also for the usual barrage
of comments about what an idiot I am).
David Seppala
Ok,
We'll do this a little differently, Seppala.
Here is the data for your problem:
1. the gap width is W and it is centered on the Oy axis in the good
twin frame (S)
2. the good twin is at location (0,-a)
3. evil twin is flying along the line y=b at the speed +V (very close
to light speed) in frame S
4. the evil twin shoots a bullet of width w and speed v along his Oy'
axis (in frame S')
What are the conditions necessary and sufficient for the evil twin to
hit the good twin. Please use SR and use symbolic computation.
If the steel wall wasn't there, the evil twin is simply watching the
good twin travel along the x-axis, or to be precise, along a line
parallel and very close to the x-axis (in the negative direction). The
evil twin is firing a projectile in the y direction perpendicular to
the x axis. Without the wall there the condition to hit the passing
twin is to simply time the firing of the projectile such that when it
crosses the x axis, the good twin happens to be at those same x
coordinates.
So write the equations. I gave you the parameters, don't give me a
story.
When we add the steel wall to the problem, the length contraction
equation L' = L * SQRT(1-V**2/c**2) results in the 10 light-year gap
for v very close to c being less than the width of a proton as
measured in the moving frame (the evil twin frame). A 100 millimeter
projectile traveling perpendicular to the x-axis will hit all points
along the x axis simultaneously over a 100 millimeter span. That
projectile cannot go through a gap less than the width of a proton.
David
You are giving me stories because you are applying the "cartoon" form
of SR. Your result is wrong, try learning SR, apply the complete
equations and you will find where you made the mistake.
.
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