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[Phys-L] invariances



Hi --

The reason I am making such a fuss about moving clocks not running slower
is that there are some deep and important physics principles involved.

I have been making a point of treating boosts and rotations on the same
footing. For example:

proper length: invariant w.r.t rotations invariant w.r.t boosts

proper time: invariant w.r.t rotations invariant w.r.t boosts

Two people have tried to use velocity or momentum as a counterexample. Far
from being a counterexample, it simply illustrates the same point, or rather
the other side of the same coin:

momentum: not invariant w.r.t rotations not invariant w.r.t boosts

So if you are trying to prove that boosts must be treated differently from
rotations, the velocity/momentum example will certainly not get the job done.

The deeper point involves a technique for doing physics, which involves writing
things in frame-independent language. Anything worth writing down can be
written in frame-independent form. If ever you see anything that appears
to be frame-dependent, you should be verrrry suspicious. You should seek a
better way of writing it.

For example, the idea of conservation of 4-momentum is frame-independent:

conservation
of 4-momentum: invariant w.r.t rotations invariant w.r.t boosts

To say the same thing in more detail, when there is a reaction of the form
reactants --> products
all observers agree that the 4-momentum of the reactants equals the 4-momentum
of the products. That is
P_joe(reactants) = P_joe(products)
and
P_moe(reactants) = P_moe(products)
even though in general P_moe() will be wildly different from P_joe().

Concentrating on the P_moe minus P_joe difference is a bad technique, a
bad habit, a huge waste of time. Don't do it. Don't teach it. Instead,
express the laws in a manifestly frame-invariant form. You don't need
to know anything about Moe or Joe to know that 4-momentum is conserved.

=========

Here is an example of how to take something that is frame-dependent and
rewrite in a way that is frame-independent.

In the "clocks" thread, nobody has yet exhibited an example of something
that is invariant w.r.t rotations but not invariant w.r.t boosts. But it
is possible. Let V be some 4-vector. Project out "the" t-component.

V_t : invariant w.r.t rotations not invariant w.r.t boosts

However this leaves us with the nagging question of _why_ you would want to
do such a thing. That is a terrible approach, not least because it doesn't
specify whose t axis is to be used.

The recommended approach would be to let u be the 4-velocity of whatever
physical system V is interacting with. Then we can form the dot product
u · V which is frame-invariant in the same sense that all the laws of
physics are frame-invariant. Moe and Joe will generally disagree about the
components of u, and they will generally disagree about the components of V,
but they will agree as to the value of u · V.

u · V : invariant w.r.t rotations invariant w.r.t boosts

Remember what Charlie Peck said: "The purpose of this class is not to teach
you how to do Lorentz transformations. The purpose is to teach you how to
avoid doing Lorentz transformations."

If you write everything in manifestly invariant form, you won't have to do
nearly so many Lorentz transformations. A nice pedagogical example is the
antiproton production energy calculation, as was discussed on this list a
couple times in the last couple years, e.g.
http://lists.nau.edu/cgi-bin/wa?A2=ind0304&L=phys-l&P=R1581

=================================

Additional insight as to the relationship between boosts and rotations comes
from the structure of the Lorentz group.

The rotation group is a subgroup of the Lorentz group. Interestingly, boosts
do not form a subgroup. Boosts are not closed ... by which I mean it is easy
to arrange a succession of boosts that combine to make a pure rotation (with no
net boost). An example of this the Thomas precession.
http://www.google.com/search?q=boosts+thomas-precession

You don't need Clifford algebra to see the intimate connection between boosts
and rotations. For an elementary audience you just write rotations as

[ x' ] [ cos sin ] [ x ]
[ ] = [ ] [ ] [1]
[ y' ] [ -sin cos ] [ y ]

and then write boosts as

[ t' ] [ cosh sinh ] [ t ]
[ ] = [ ] [ ] [2]
[ x' ] [ sinh cosh ] [ x ]

which has the same meaning as the usual "Lorentz transformation" equations
but is very much easier to remember.

Do you really want to argue that equation [2] is not analogous to equation [1]?

I'm aware that the two equations are not completely identical, but I daresay
they are not completely unrelated, either. The analogy is readily discernible,
and if you have even a little bit of skill you can put the analogy to good use.
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