Problem with velocity composition in SR - math/guru help
- From: David <dseppala@xxxxxxxxxxxxx>
- Date: Mon, 30 Oct 2006 15:15:57 GMT
I don't understand the velocity composition of SR. I've setup two
simple examples to illustrate the problem I'm having. Can anyone
clarify how Einstein's notions of space and time apply to these very
simple situations?
Example 1.
An inertial reference frame is traveling with velocity V relative
to another inertial reference frame. I'll call the first frame the
moving frame and the second frame the rest frame. Let V be very close
to the speed of light. In the moving frame, let there be an object
that is moving with velocity Vm relative to V ( in the same direction
as viewed in the rest frame). If Newtonian addition of velocities
were used the moving object would exceed the speed of light as
measured in the rest frame. With Einstein's velocity composition
formula, the two velocities V and Vm result in a velocity as measured
in the rest frame that is less than the speed of light. Okay, I see
how to apply the equation to this case. But now, I try the same thing
with Example 2.
Example 2.
At the equator of a planet, the surface has a velocity V that is
very close to the speed of light (just as V was in Example 1). A
person puts up a pole on the equator. Because of the planet is a
sphere, the tip of this pole is moving at Vm relative to V (due to its
increased radius from the center of the planet). If Newtonian
addition of velocities were used the moving object would exceed the
speed of light as measured in the rest frame of the center of the
planet. However, I don't see how to apply Einstein's velocity
composition formula to this case. The problem occurs because both the
point on the surface and the tip of the pole each make one revolution
in the same amount of time T as measured in the rest frame of the
center of the planet. To compute the instantaneous velocity of the
point on the equator surface is moving (as measured by the rest frame
of the center of the planet), don't I just divide the circumference by
T? If I try the same thing with the tip of the pole, I find that the
circumference there divided by T exceeds the speed of light. If I say
that the tip is moving slower than c, then both the surface point and
the tip of the pole must have a different time T of rotation (as
measured in the rest frame of the center of the planet). That is
impossible unless we say something physically happens to the tip of
the pole - but then the same problem would have to appear in Example
1. Since V and Vm are the same in both examples (with the exception
of a slight curvature in the path), I don't see how to apply the same
logic to both cases. I can't make the distance traveled by the tip of
the pole during each revolution equal the radius times two times pi
unless I let the tip of the pole exceed the speed of light.
Can anyone explain how to apply Einstein's concepts in a consistent
way to these two examples.
Thanks,
David Seppala
.
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