And another thing, Nick,
especially for April Fool's Day.
Once the ideas of relativity began to sink in, it became clear that it
was very easy to confuse oneself and others. In John Bell's book,
"Speakable and Unspeakable in Quantum Mechanics", there is a chapter
showing that even the professionals get confused and have to sit down
and think very carefully and calmly about apparently simple problems.
It's a platform problem, because you have different observers
observing different things. If you use expressions which combine some
of A's measurements with others of B's, you can get totally mixed up.
The way to avoid this is only to make statements which are
transparently true in all coordinate systems. The buzzword is
'explicit Lorentz covariance'. If you have an expression that is
explicitly Lorentz covariant, then you can be sure that if a statement
involving it is true in one coordinate system, then it is true in all
coordinate systems.
In the equation
p = mv
the momentum p is not too bad, because it turns out that it is the
spatial part of a 4-vector, the energy-momentum vector. When you throw
in the energy E, the complete set transforms according to a definite,
relatively simple, set of rules. However, the quantity v, the
velocity, is the derivative of a position with respect to time t. But
whose time: A's, B's, or that of a third observer C? It does have a
law of transformation, but it's a rather complicated one.
One of the hallmarks of this complication is that space and time enter
into v in different ways: space is in the numerator and time is in the
denominator.
That, in fact, is the problem with your traditional definition of mass
m. It privileges a specific coordinate system, and when you use it in
everyday conversation it is all to easy to think that you are talking
about something with absolute significance. It actually carries with
it the hidden implication of a specific preferred coordinate system.
You can use it unambiguously if you are careful, but it became clear
over the years it always carries with it this risk of confusion. Since
it isn't absolutely necessary, it is best to ban it completely. The
only quantity connected to 'mass' which is safe to continue using is
the 'rest mass', which can therefore now be shortened to 'mass'.
You may have noticed a subtle difference between the way I wrote the
equation involving m and the way Mati, Greg and others wrote it. I
wrote
E^2 - p^2*c^2 = m^2*c^4
whereas they wrote
E^2 = m^2*c^4 + p^2*c^2
I don't like their way because it treats the space- and
time-components of the energy-momentum vector differently, and
suggests that energy E is somehow different from and more important
than momentum p. Not a lot, but still differently. I like my way
because it is possible to see that it is equivalent to the tensor
equation
g^ij*p_i*p_j = m^2*c^2
which is explicitly Lorentz covariant and treats space and time
scrupulously on the same footing.
L'obsession d'un viellard gâteux sans doute, certainement poisson
d'avril.
Cheers,
Zigoteau.
.
|
