[Phys-L] conservation of momentum

[John Denker <jsd@av8n.com>]

Executive summary:
 -- Momentum is conserved.
 -- Momentum is always conserved.
 -- Momentum is strictly conserved.
 -- Momentum is locally conserved.  That means no matter where you
  look, momentum is conserved right here, right now.

 ++ You have to account for all forms of momentum.

On 08/30/2016 07:48 AM, Moses Fayngold wrote:

> "action-reaction" concept is most simple and fruitful only in 
> classical and static situation, in which case the notion of their 
> starting and ending together is meaningless.

The action/reaction concept is synonymous with conservation of
momentum.  If it doesn't appear synonymous, you're doing it wrong.

> In more realistic situations involving dynamics this notion does not 
> work or is, at best, not straightforward.

It does work.  It always works.

> Consider, for instance, an electron-positron pair production from 
> collision of 2 neutral particles in the field of a distant proton.

That's a complicated setup.  It's neither simple nor straightforward,
for reasons having nothing to do with conservation.

> Each member of the newly-born pair immediately feels the field of
> the proton and experiences the corresponding action,

OK....

> but the proton will start feeling their field (and respective
> reaction) much later.

In the early going, the electric field of the two leptons, as
observed from far away, is zero.  That's because they have opposite
charges, at initially the same position.  The reaction on the proton
is initially zero, as it should be.

Later on, as they move apart, there is a changing electromagnetic
field.  This must be analyzed using tools appropriate to the task.
In particular, there is RADIATION, and there is MOMENTUM IN THE
FIELDS.  This is small, as it should be.  It is just sufficient
to account for the small reaction on the proton.

It is a fundamental fallacy to focus attention to one small effect
while neglecting other equally-small effects.

If you do it right, momentum is conserved.  Always.  Everywhere.
Strictly.  Locally.



On 08/30/2016 08:29 PM, LaMontagne, Bob wrote:

> > If the information travels at the speed of light, then aren't the
> > "action" and "reaction" simultaneous in Space-Time?

Conservation does not depend on any notion of "travel".  It does not
care about travel at the speed of light, or any other speed.

Relativity is a challenge to any notion of /simultaneity at a distance/
but that is a non-problem for conservation, since the relevant distance
is zero.  Momentum is conserved right here, right now.

To say the same thing in slightly different words:  The notion of
"same time at the same position" is manifestly independent of the
choice of reference frame.
  In contrast, the notion of same time at different positions would
  require looking at the spacetime diagram, but for the purposes of
  conservation that's not our problem.  We do not need to ask the
  question, much less answer it.

=====

As a separate matter, suppose we have two different events (A and B)
in spacetime, separated by a lightlike interval (i.e. null interval).
The interval is a vector in spacetime  The components of this vector
are nonzero.
   The norm of this vector is zero.  In Euclidean space that would
   imply that each component is zero, but in spacetime it doesn't.

The /proper time/ separating the two events is zero, but that's not
what anybody means by simultaneity.  Proper time would be interesting
in a comoving frame ... but there cannot possibly be a frame moving
with the speed of light.  (If you tried it, all the laws of physics
would look like "zero divided by zero" and would be quite uninformative.)

Instead, consider the spacetime vector (B-A) i.e. the interval from
A to B.  In some chosen frame F it has components [t, x]_@F.
  -- Simultaneity in frame F means t_@F = 0.
  -- In some other frame G, B will be later than A, i.e. t_@G > 0.
  -- In some other frame H, B will be earlier than A, i.e. t_@H < 0.

All frames will agree that the proper time is τ = 0, but that's the
answer to a different question.  It is irrelevant to questions of
simultaneity at a distance.  It's doubly irrelevant to conservation,
because conservation doesn't involve distance.  Momentum is conserved
right here, right now.