Re: Simple four velocity question

PD wrote:
On Apr 2, 10:59 pm, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
_ALL_ measured quantities are
invariant,

I believe you are confusing the measurement with the quantity being
*measured*.

I say the same to you.


Yes, of course, if I make a particular measurement in
frame A, then that measurement *in frame A* is invariant,

You are confusing frames with measurements. Discuss the MEASUREMENT INSTRUMENT, not the frame.


Every valid "law" that you consider to be "real but involving
frame-dependent quantities" is, at base, the PROJECTION of some related
law expressed in invariant form, PROJECTED onto the coordinates that
make your version use frame-dependent quantities.

Yes. And that *projected* quantity may well be represented in physical
laws. This is why I referred to conservation of mechanical energy. It
is quite clear that the kinetic energy that appears in that
conservation law is a *projection* onto that reference frame.
Nevertheless, it is remarkable and notable that this *projected* value
seems to play a role in a useful and broad physical law.

Not really. The "broad physical law" is the conservation of 4-momentum. That is more general and of much broader applicability than mere energy conservation. Just because energy conservation came first historically does not mean it is more basic, fundamental, or important. Indeed, in this case it's clear that we learned something essential, and the conservation of 4-momentum is clearly more basic, fundamental, and important.


To give a more explicit example, you referred to Mandelstam variables.
We could also call s the magnitude of the system's 4-vector momenergy
(or invariant mass). While it is certainly true that this invariant
plays exhibits itself in fundamental laws, it is *also* true that in a
given inertial reference frame, the *components* of that momenergy are
*also* conserved, independent of each other. I see no reason to say
that the conservation law about the Mandelstam variable (which has the
added feature that the value is the same in all inertial reference
frames) is somehow more of a physical law than the conservation law
that applies to the components (which do not share that feature).

Again, the fundamental, underlying law is the conservation of 4-momentum. You can choose to project it onto certain coordinates, but I see no way in which those PROJECTIONS are "more fundamental" or "more important" or "more physical" than the underlying law to which they refer.


The invariant
relationships are clearly more fundamental (in some abstract sense) than
their projections onto certain coordinates, even though the invariant
form might have come historically long after the frame-dependent
version. No matter, as we know this NOW.

Yes, I agree that they are more fundamental. I just don't agree that
makes them more "real", or that this makes the others less "real".

Again that wishy-washy word. Remember that I reserve it for quantities that can be valid models of physical phenomena. As nature clearly uses no coordinates, no coordinate-dependent quantity can be a valid physical model (nature cannot depend on arbitrary human choices like coordinates).

My usage makes it easy to distinguish "centrifugal force"
from real forces. Your usage makes that impossible.


Tom Roberts
.

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