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Re: Non-inertial: Part I

On Fri, 26 Apr 1996, Rauber, Joel Phys wrote:

...
Marlow seems to use as an operational definition of a force,"those readings
that pressure sensors give you". (For purposes of this discussion I assume
we may consider the object in question to be a small cube with mass which
has pressure sensors located on each of its 6 faces.) If this is one's
operational definition of forces then I agree with what Marlow says in
quotes I and II below. Particularly quote II regarding the analysis of
forces acting on the cube which is sitting on the table (cube instead
person) as analyzed in general relativity in a free-fall frame (free-fall
frames being the analogous frame of an inertial frame in Newtonian
mechanics). I.e. there indeed is an unbalanced force on the cube as its
world line deviates from the geodesic with an acceleration of +32 ft/s^s.

However, I don't think this is a good way to operationally define forces
acting on our cube, because it leads to certain conundrums; (if I'm wrong
here correct me, its not completely thought out).
...

Pressure gauges are not the only operational way of defining forces.
Old-fashioned spring extension scales (stretch scales) are quite useful;
besides springs, more generally, the strain or distortion produced in
standard materials (fibers, Etc.) can be used.

...
a) If we decide to analyze the cube on the table using Newtonian Mechanics,
in a Newtonian inertial frame, we are forced to say gravity is a fictional
force and my force diagram for the cube will have only one force vector on
it and therefore there is an unbalanced force. ...

Agreed, in the following sense: MOST of what people before Einstein
believed was gravitation (including even great Newton) turns out to be
simply accelerations produced by the reference frame we are using. After
you have transformed away all the accelerations masquerading as forces,
and this was midwife's role played by all of Einstein's thought experiments
with falling elevator's and such, the hard, solid residue left is true
gravitation -- the stresses and strains produced by tidal effects, e. g.,
the greater compression in your feet versus that in your head because your
feet are in a region of stronger spacetime curvature than your head, Etc.
This is why 20 century physics regards gravitation as weakest of all
interactions. Unless you are in a region of much greater curvature than
our region of the solar system, the effects are not very great relative,
say, to the electromagnetic effects of daily life (the force vector in
your description above). Ignoring the small gravitational effects, what
you say above is true.


... This contradicts the 2nd law, since there is zero acceleration.

I don't see this. It doesn't seem to contradict the 2nd law at all.
Any correct statement of the second law must start with "Relative to an
inertial frame the net force acting on an object equals ... ." Without
the initial proviso, none of the laws of Newton are valid or applicable.
Thus, when we detect an unbalanced force, as in your above example, but
zero acceleration, the logical conclusion from the 2nd law is that we
are not using an inertial frame for our acceleration measurements. This
was in fact the conclusion drawn by Einstein in formulating the current
best theory we have of gravitation, and this alone should be a strong
argument against giving in to the alluring temptation to invent
fictitious forces. They serve to conceal reality from us, and keep
us from understanding what's going on. If Einstein had given in to that
temptation we would not have the incredibly accurate theory of gravitation
we now have. In other posts we have seen that introduction of fictitious
forces leads people to miscompute the mass of the Sun. The fact that
300 years ago Newton could see no other way to go than to introduce a
fictitious action-at-a-distance force is no excuse for us today, once
the matter has been cleared up in 20th century physics. (Very presciently,
Newton himself deplored the very concept of action-at-a-distance, but he
could see no other way at that time. I think he would be very happy
today to see that you really don't need it.)

This ought to confuse a student who is using force diagrams and the
pressure sensor definition of a force.

Who knows, it might even confuse her enough to search for a better view
of reality and discover something greater even than general relativity!

And
therefore we shouldn't use these force diagrams in studying statics, even in
inertial frames!


I simply have to disagree completely with this statement for all the reasons
already put forward in this and other postings in this thread.

I should point out that free-fall frames are not inertial frames in
Newtonian physics.
...

They weren't able to be recognized as such by Newton, but they are certainly
recognized as such in 20th century Newtonian physics. I use that term to
refer to the very accurate Newtonian approximatian to GR that works so well
in the weak field of the solar system. By the way, it was pointed out in
someones else's post that the notion of force doesn't seem to survive
well in relativity; I would qualify this by noting that force doesn't
survive well in relativity as a vector notion, but it gets subsumed very
successfully into components of the stress-energy tensor, which involves the
rates of change of momentum relative to both time and space coordinates, and
this reduces to standard force components in the Newtonian approximation.

One reply to this conundrum, is that I have it exactly correct and that is
why Newtonian physics doesn't match reality and why we need the General
Theory of Relativity.


Agreed, if you actually need the greater accuracy of general relativity,
but otherwise Newtonian physics does match reality with great accuracy, as
long as you don't misstate Newtonian laws: each of Newton's laws involves
the proviso "Relative to an inertial frame,..." or it is simply not being
correctly stated or applied.

...I'd respond, yes; but Newtonian physics should be a self-consistent
theory, ...

It is.

even if it doesn't match experiment

But it does, to great accuracy!

... and I think the paradox above can also
be viewed as a self consistancy problem with how one defines forces, within
Newtonian Physics. (More with this thought after I hear the responses.)


The paradox was precisely removed when Einstein cleared the air with his
thought experiments involving falling elevators that showed that MOST of
what people had been confusing with gravitation were mere coordinate
accelerations. In other words, he precisely showed that any effect that
could be transformed away by changing your frame of reference must not be
a true interaction, but instead must be simply an acceleration produced by
your choice of reference frame. In that one stroke he cleared up what
inertial reference frames should be (in which Newton's laws work) and what
actually constitutes the interaction of gravitation -- the residue that
is left when you transform to a true inertial frame. If he had not done
this he would not have been able to see that gravitation is most
accurately described as geometric curvature of spacetime. We lose all
this both for ourselves and our students when we regress to the stage of
inventing fictitious interactions.


b) A more serious conundrum in my opinion.

Put a charge on the cube and levitate it between large parallel plate
capacitors (again in a uniform gravitational field as in the above example).
Now the pressure sensors
read no force present (they all have the same reading).

Pressure sensors are obviously not appropriate tools in all circumstances.
You must use all available means for testing for forces in determining
whether or not you have an inertial frame. For example, the simple spring
stretch scale mentioned previously will do the job. Attached to various
sides of your capacitor plates (with and without the charged cube inside),
you find that the spring scale must exert a real outward force (as shown
by the reading on the scale) on one of the sides of the capacitor plate
apparatus -- the side we'll, for convenience, refer to as the "top" --
if the whole apparatus is to be prevented from accelerating relative to
your lab. Trying the same thing on the opposite side (the "bottom") results
in the spring being compressed rather than stretched if you are going
to prevent the apparatus from accelerating. Searching further by all
known means (except very, very sensitive stress detectors, which would
detect the slight effects of gravitation), you can find no other forces
acting on your apparatus. If you are not going to invent some far-fetched
fictitious force, the most reasonable conclusion you can come to is
that your lab is not an inertial frame, since in an inertial frame
(e. g., one of Einstein's falling elevator's) you would not have to use
any force to keep something from accelerating. In other words, your
lab is behaving exactly like a turntable, where mysterious accelerations
appear without any detectable forces.

... And this now
contradicts the General relativity analysis of quote II.

Sorry, but I'm afraid it CONFIRMS it rather than contradicts it.

In a free fall
frame there is the +32ft/s^2 acceleration (the acceleration which deviates
the world line of the cube from being a geodesic, i.e. the free-fall path of
an identical uncharged cube) , but the pressure sensors indicate no force
present; GR says there is a force present, because the worldline is
deviating from the geodesic.


Simply try other force detectors rather than pressure sensors.


Conclusion: The pressure sensor method only can detect contact forces and is
not useful for non-contact forces, and therefore is inadequate as an
operational definition of force.


I never said that pressure sensor methods were the only means of detection
of forces (possibly better, interactions), and in previous posts have
specifically mentioned fiber torsion, spring stretching, Etc.

In general, any method of making precise the pushes and pulls experienced
in real life will do. Just keep these aspects of experience distinct from
the purely kinematical concepts of position, velocity and acceleration.


A. R. Marlow E-MAIL: marlow@beta.loyno.edu
Department of Physics PHONE: (504) 865 3647 (Office)
Loyola University 865 2245 (Home)
New Orleans, LA 70118 FAX: (504) 865 2453