From: Hal Finney (hal@finney.org)
Date: Sat Apr 19 2003 - 18:35:54 MDT
Lee Corbin writes:
> 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.
Actually, in modern treatments of Special Relativity, relativistic mass
increase is no longer considered to be the best way of understanding
the phenomena. My son is in college and just studied SR last term,
and he confirms this. Teachers downplay the notion of "relativistic
mass" because it leads to so many confusing situations like the one
you describe. Instead, they use the idea of "relativistic momentum"
which is equal to rest mass, times velocity, times the "gamma" parameter,
1/sqrt(1-v^2/c^2). By sticking with momentum and energy, you don't have
to imagine that mass increases.
The old way took that same formula, p = v * m * gamma (where m is rest
mass), and grouped the last two terms to produce the "relativistic mass",
which went to infinity as v approached c. But as I said that produced a
lot of confusion. Among the problems would be the question of whether
a sufficiently fast moving particle would become a black hole when its
"relativistic mass" got high enough. Basically anything involving
gravitation causes problems.
> 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!
Despite what I said above, I will give a solution in terms of relativistic
mass. The answer is that gravitational attraction depends on rest mass
and not relativistic mass. Of course, that is an over-simplification because
you need General Relativity to analyze gravitation. But for small fields
you can do OK with Special Relativity, and in that case you need to do it
this way.
By definition, the rest mass of the planet and star are constant and
independent of velocity; therefore the gravitational attraction is
unchanged. Yet the planet's inertial mass has increased because of its
velocity (remember I am using "relativistic mass" here). Therefore it
is much heavier and will respond much more slowly to the same force,
hence you see it revolving around the star very slowly.
Or you can solve it without relativistic mass, by going back to the
formula above, p = m * v * gamma, and understanding that p and v are
vectors while m and gamma are scalars. Gamma is large in this situation,
and it boosts the momentum components in the orbital plane as well as the
big component perpendicular to the orbit. So we have a higher momentum
than would be calculated by Newtonian physics, hence a given force will
produce a smaller change in momentum, where we use F = dp/dt.
Hal
This archive was generated by hypermail 2.1.5 : Sat Apr 19 2003 - 18:46:19 MDT