From: Lee Corbin (lcorbin@tsoft.com)
Date: Sat Apr 19 2003 - 16:48:54 MDT
I have come up with a new paradox in Special Relativity.
I was explaining a typical thought experiment to an
acquaintance in order to convey why mass increase
occurs. You have two bowlers
on the different spaceships who have their bowling
alleys aligned and perpendicular to their relative
motion. If you believe in time dilation (for which
there is a very standard TE), then it happens that
when one bowler "measures" the mass of the other
bowlers ball (as it collides with his), he must conclude
that that slow moving ball on the other ship is pretty
massive.
Anyway, it occurred to me that this TE was all about
*inertial* mass, and that I didn't have a TE that was
about gravitational mass. So here it is. Fortunately,
it can be explained without any math, but I do include
some math at the bottom of this email. The trouble is
that its conclusion is apparently opposite to the
predictions of SR!
Okay, so you are approaching a solar system at speed v
in your rocket ship, and you see a planet orbiting a
star. You are traveling perpendicularly to the plane
of the orbit of the planet. So imagine that the planet
is orbiting the star like the tip of a minute hand is
orbiting the center of a clock on the wall, and you are
walking towards the clock straight on.
Now because physical processes happen in that solar
system in your frame of reference more slowly than they
do in the that solar system's frame, you observe that
the planet is moving around its star not very fast. In
fact, judging from the usual mass of that kind of star,
its amazing how slowly that planet is revolving about it.
So using some typical formulas, you calculate the mass
of the star, and find it to be rather picayune. But
hold on! Special relativity predicts that your measurements
of the star should make it *more* massive, not less. Paradox!
Anyone see a resolution to this?
(Since Einstein had troubles incorporating gravity into special
relativity, I'm hopeful that this is a very meaty paradox
indeed, and shows the need for GR!)
Lee
P.S. In order to calculate the mass of the star, note that
the acceleration of the planet can be obtained from Newton's
gravitational formula and also from the formula for centrifugal
force. When we set these equal, we obtain v^2/r = GM/r^2,
or r = GM/v^2. The paradox then consists of reckoning that
since the radius of the orbit is unaffected by your motion
(it's orthogonal to it, after all), it's a constant: so
the mass of the star is proportional to the square of the
velocity of the planet around it. Since the velocity in
your frame is so puny, the mass is puny, but it should be
magnified, not reduced, according to SR! (Sorry about the
variable v: here it means the orbital speed of the planet,
not your relative velocity to the whole system as usual,
which should be referred to by another variable, say nu.)
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