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Timothy van der Linden writes:
> ReplyTo : Steve and Kevin
> ReplyFrom: Timothy
> >I say that "relativistic mass increase" is a misnomer and that
> >you are better off treating mass as invariant.
> Both methods are valid to use. The difference is more a physical matter than
> a mathematical. I was used to working with the "wrong" formulas and Steve
> was used working with the "right" formulas.
> The so called "wrong" formulas look a bit more like the classic formulas so
> they may be easier to understand.
> Both methods are being taught at universities and both are valid.
I think we'd both agree that, for example, p(v) = m * v / sqrt(1
- v^2). What I have learned to be suspicious of is interpreting
this as (m/sqrt(1 - v^2)) * v, as if the mass somehow increases
with velocity. What I find to be a more intuitive and less
misleading interpretation is that mass is invariant (m^2 = E^2 -
p^2) but that energy and momentum can increase without limit as
the moving object accelerates. When doing more elaborate
kinematics problems it's also easier to just keep track of the
conserved E and p components, then sort out the resultant masses
of the reaction products.
Often in math there is more than one way to the right answer. I
happen to find that concentrating on conservation of energy and
momentum and keeping track of invariant quantities makes it
easier to get the right answer and to prove that it's right. If
you can do it the "wrong" way, get the right answer, and prove
that it's right, then fine.
> I've always looked at is as follows: When you move faster and faster, part
> of the energy is transformed into mass, the other part is used to get the
> extra momentum.
I used to look at it this way, but Taylor and Wheeler talked me
out of it (see chapter 8 of _Spacetime Physics_ for a lengthy,
careful discussion on "Use and Abuse of the Concept of Mass").
The problem here is that relative velocity or acceleration do not
cause any fundamental changes in the structure of the moving
object. Where is this extra mass? If it's really stashed on the
ship somewhere then the people on the ship could measure it. But
they don't feel the ship getting heavier or see any increase in
the mass of the ship in their frame.
You also seem to be falling into the same trap that Kelly did
earlier, in not treating energy and momentum as separate
components. Most of the counterintuitive results of relativistic
kinematics problems come from failing to understand that the
conserved quantity in a reaction is a _vector_ quantity, and that
the magnitude of that vector is calculated using Lorentz rather
than Euclidean geometry.
Taylor and Wheeler's wisdom on the subject is that the definition
of mass is sqrt(E^2 - p^2); then every observer sees the same
mass for the same object, no matter what their relative motion.
The quick treatment in many physics texts is that mass is sort of
like E, but they do a lot of tapdancing to keep everything
consistent. This appears to be motivated by trying explain the
famous "E = m * c^2", which turns out to be a lot less profound
an observation than some of the other implications of
relativistic physics. Taylor and Wheeler look at "E = m * c^2"
as a mere matter of unit conversion; it's only really true if you
are at rest relative to the mass in question, and they consider
the use of Lorentz geometry for spacetime and invariance of
spacetime interval as more fundamental and revealing concepts.
> Now I only wonder, does such a fast moving particle excert greater
> gravitation on a non-moving observer?
This I can't answer with certainty. Offhand, I'd say "no." If
the particle's mass doesn't change, then how could its
> Steve if the answer is yes, how do you explain that not using "relativistic
> mass increase"?
If, on the other hand, a moving object did exert greater
gravitation than a stationary object of the same mass, I'd
probably be looking for a relation to a quantity that did change,
like the object's total energy.