[Phys-L] foundations of physics: Galilean relativity, including KE

[John Denker <jsd@av8n.com>]

On 09/29/2015 12:52 PM, Jeffrey Schnick wrote:
> In interactions, what matters is the kinetic energy of the system in
> the reference frame in which the center of mass is at rest.

That's not what the laws of physics say.

All the laws I've ever seen, when properly formulated,
uphold Galileo's principle of relativity.  That means
they all get the same answer, no matter what frame of
reference -- if any -- you choose.

As a starting point, consider this:  In grade school
you learned to add vectors geometrically, tip to tail.
You can do this without reference to any coordinate 
system and without reference to any vector basis.
Meanwhile, if-and-when you choose a basis, the vector
sum is the same geometric object, independent of your
choice.

That's true unto itself, and also true as a metaphor
for more complicated situations:  The laws of physics
produce the same physical result, no matter what frame
of reference -- if any -- you choose.

To say the same thing the other way:  You are under 
no obligation to choose a reference frame at all, and 
certainly no obligation to choose the frame comoving 
with the CM.  If-and-when you find it convenient to 
choose the CM frame, you're free to do that, but other
folks remain free to choose differently.

There exist situations where the CM frame would be
exceedingly inconvenient.  For example, consider a
charged particle interacting with an electric field.
If you write the field as a bivector (as you should),
the Lorentz force law is expressed as:

          ∂p
	 ----  =  q u · F                 [1]
          ∂τ

where p is the particle's 4-momentum, u is its 4-velocity,
q is its charge, F is the electromagnetic field bivector,
and τ is the proper time.  This expression is manifestly
independent of whatever reference frame -- if any -- you
choose.

Optionally, you may find it convenient to choose the 
frame in which the EM field is purely an electric field,
in which case the equation reduces to:

          ∂p
	 ----  =  q E                    [2]
          ∂τ

Please notice that this is not the CM frame.  The mass
of the electric field is not specified and not relevant
to this problem.  It would be quite incorrect to say 
that the CM frame is "what matters".

Examples like this, where the CM frame is not convenient,
are a dime a dozen.


In many familiar situations, the laws of physics require
there to be a /metric/.  This allows us to form the dot
product of two vectors, and thereby define notions of 
length and angle.  It must be emphasized that the metric
is independent of the choice of vector basis, which in 
turn is independent of the choice of coordinate system.
For example, the distance from the village school to the
post office is independent of whether you draw the map
using polar coordinates or rectangular coordinates.  It
is independent of whether you orient the map with north
at the top or east at the top (the latter being the 
original meaning of ORIENTation).

Homework:  Prove that the usual Euclidean metric is
invariant with respect to a rotation in the xy plane.
Hint:  Feel free to use Cartesian coordinates in the
proof, but keep in mind that the result is independent
of whatever coordinate system -- if any -- you choose.

Homework:  Prove that the Minkowski metric (as used e.g.
in equation [1]) is invariant with respect to a rotation 
in the xy plane, and (!) also invariant with respect to
a rotation in the xt plane, i.e. invariant with respect 
to a change in velocity.


> Observers in all reference frames would all agree on what happens in
> the interaction

You can, if you wish, introduce a concept of "KE relative
to the center of mass" for some specified system.  Call it
the "proper internal KE" if you wish.  All observers will
agree as to its numerical value.  However, to call it 
*"the"* kinetic energy would be an astonishing abuse of 
the terminology.  If you want to be understood, don't go 
there.  In physics, there is a well-established notion of 
"the" KE, and that ain't it.

Forsooth, there is a nifty theorem that says the overall
kinetic energy can be decomposed into the internal KE
(relative to the center of mass) /plus/ the macroscopic 
KE of the CM itself (relative to some chosen reference
frame).

Homework:  Formulate a precise statement of this idea,
then prove it.