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Re: Work/Energy theorem ? (very long)



I'm getting into this discussion against my better judgement (and am doing so
only because I don't want to fight the urge right now.) I don't have
anything (much) against what Bob Sciamanda has said in this discussion, I
just need to vent some things which have been building for a longer while and
Bob merely triggered them.

Bob Sciamanda wrote:
To say that a gravitational field and an accelerating frame will
produce equivalent effects is certainly a "numerical equality" (to
intentionally harken back to the flavor of a previous discussion).
The question (as with Bernoulli and the W.E theorem) is "what physical
(conceptual) sense (model) are we to make of it?"

The equivalence principle IS a physical and conceptual explanation. I
sometimes get annoyed with the idea that a "conceptual" or "physical"
model or explanation is somehow divorced from a mathematical one. The
mathematical equations of physics are the *best* (both in the sense of
precision and in the sense of the definition of the concepts involved)
models we have to explain physical effects. We (maybe, at least, I)
understand the physical meaning of the concepts *via* the mathematics, *not*
the other way around. If I want to understand what momentum is, I look at
its mathematical role as the generator of a transformation inducing
infinitesimal spatial displacements of the state of a physical system. If I
want to know how fictitious forces behave, I look mathematically how they are
generated by a transformation from an inertial coordinate system to an
accelerated one. If I want to know what the physical meaning of a quantum
wave function is (when written in a basis which diagonalizes a particular
observable such as position), I look at how its square magnitude yields the
probability distribution for the results of a measurement of the observable
in question. If I want to understand the conceptual meaning of entropy, I
look at its mathematical definition and realize that it is a quantitative
measure of the amount of information that must be supplied regarding a
macroscopic system (over and above that information contained in its
macroscopic description) to uniquely specify its microscopic state. If I
want to understand the physical meaning of a uniform gravitational field, I
invoke the Equivalence Principle. Come to think of it, in fact, if I want to
understand the physical meaning of space and time themselves (as opposed to
the psychological feelings of them) I look to the mathematics of the
spacetime manifold described by GR. Here I see that space and time
coordinates are merely labels used to coordinate that manifold, but the
proper time interval calculated along some spacetime trajectory gives the
actual physical time measured by a tiny clock carried along that path.
Similarly, I can use the mathematics of GR to find out and understand the
physical proper distance between two simultaneous spatially separated events
along a spacelike geodesic (even if to do so may be a complicated task).

The mathematics does *more* than merely give us calculational algorithms for
finding numerical values for things (physical quantities); it tells us the
very meanings of the physical concepts themselves via their roles in the
mathematical formulations. Physical conceptual understanding is dependent on
the theoretical framework which explains the phenomena in question. That is
why our modern conceptual ideas of things like momentum, energy, gravitation,
etc. are different from those of Aristotle, or even Newton, for that matter.
Our concepts are different because our physical theories are different, and
those theories live in the abstract world of mathematics where the concepts
themselves are incarnated with an abstract mathematical substance.

So, if you want to know "what physical (conceptual) sense (model) are we to
make of it?" (as with Bernoulli and the W.E theorem), then I say look at the
mathematics of the theory, and it will tell you.

There, I feel better now.

Bob goes on:
I don't think that your train rider is making "perfect sense" out of this.
You have merely given him an alternative calculational algorithm, but he
is at a loss to explain why it works, because these gravitational fields
appear as a magic result of the engineer's actions! I think he can make
better physical (and equivalent calculational) sense by acknowledging his
acceleration!

What's wrong with having the gravitational fields appear and disappear in
tandem with a pilot's actions? When the Apollo astronauts went to the moon
and back the gravitational field that they experienced throughout the whole
trip was certainly influenced by the actions of the pilot. In fact for a
couple of days on the moon the gravitational field that they experienced
was 1/6 (in magnitude) that of what they had experienced before and after the
trip. While in transit the gravitational field was relatively intense during
the various rocket motor burns, and was essentialy zero for most of the rest
of the trip.

I am certainly out of my field here, but it seems to me that it was
just such questions which frustrated Einstein as he sought to incorporate
E&M forces into a "unified theory". G.E seems to work fine for purely
gravitational (astronomical) situations; it is a theory of the
gravitational "force"; IMHO its relevance in more general situations
needs work!

I may get in trouble here (as I have in the past) for posting from memory
without looking things up, but as I recall, the failure of Einstein's unified
field theory, and Kalusa's gravito-electro-magnetic theory before it, was not
due to anything discussed so far in this thread. Rather it was due to an
inability to incorporate other nonelectromagnetic, i.e. strong and weak,
interactions into the theory, and the inability to have a consistently
quantized theory. E&M by itself gave correct predictions of experimental
phenomena only when it was quantized to give QED. The early unified theories
that combined E&M with gravity were necessarily *classical* theories which
could not accomodate quantization of neither the EM field nor of the various
matter fields, and had no place at all for the strong and weak interactions.

General Relativity is not so much a theory of "gravitational 'force'" as it
is one of differential local accelerations in a given frame produced by a
curvature of spacetime which itself is produced by the stress-energy-momentum
of the matter present in the spacetime. What situations are more general
than astronomical ones? If you mean microscopic particle physics, then it
(GR) does need work if we are to understand how quantized gravitational
interactions (a la gravitons) work during the initial Planck time of the
Big Bang. Otherwise it doesn't need work. GR works just fine, as is, --even
in a particle physics context-- to provide a classical background metric for
spacetime in which to do either quantized particle physics or classical
Newtonian physics. In cases where the relevant speeds are small compared to
c and the gravitational potential is small compared to c^2, then GR just
boils down to Newtonian physics anyway.

This is a field in which I am anxious to learn! If you have physical
answers (not just alternative calculational algorithms) please teach me!

Again, the mathematics of modern physical theories provides the "physical
answers", and are not "(not just alternative calculational algorithms)". If
you want to learn the physical meaning of things in GR then study GR.

John Mallinckrodt *so appropriately* wrote:
The principle of equivalence does not fool around! It is not at all
content simply to say that uniform gravitational fields give the same
numerical results as you get in an accelerating frame. It says that there
is no way--*no* way--to distinguish the effects of a uniform gravitational
field from those of being in an accelerated frame; it says that they are,
in fact, the *same* thing. Einstein made hay by taking the principle
seriously. I think we have to as well.
....
It is *not* merely a mathematical game when we do physics from
the moving frame of the car and it is no more of a mathematical game when
we do physics from the "accelerating frame" of the train.

These nicely underscore my point that the mathematics of physical theory is
far more than "just alternative calculational algorithms". Rather it is
the embodiement of our whole understanding about the matter including the
physical and conceptual meanings, as well as its calculational aspects.

Bob goes on to write:
John,
You keep talking around my question. If you can "do physics" from the
"accelerating frame" then you must identify gravitational SOURCES which are
synchronized to the engineer's actions. DO it!
(It can be made to work - it all depends on how "wierd" you are willing to
get!) Let's have your model.

Here is where things get subtle. The "gravitational sources" do not directly
produce any gravitational forces. Rather they produce a local curvature of
spacetime which is manifested as a tidal differential acceleration in the
relative motion of freely falling neighboring test particles. In a frame in
which one of the freely falling particles is at rest then the other one is
seen to exhibit a slow relative acceleration wrt the first one. Also in that
frame the gravitational field vanishes at the location of the first test
particle (as does the gravitational force acting on it) but not at the
location of the second one. It is this tidal differential acceleration which
betrays the presence of nearby, "true gravitational sources", (i.e. stress-
energy-momentum). Otherwise we would just have a uniform gravitational
field in a flat spacetime which would really just be a case of a gravity-free
region described in an accelerated frame, and the frame in which the test
particles are at rest (or, equivalently, uniform motion) is(are) the inertial
frame(s) for that gravity-free system.

Since gravitational forces (but not tidal force gradients) are merely
artifacts of a local noninertial coordinate system we see that it is no
wonder that the engineer's actions affect the observed gravitational
forces experienced by the passengers on the train. IOW the inertial
(fictitious) forces experienced in any non-inertial frame are gravitational
forces just as much as any other gravitational force produced by a planet or
a star is also a fictitious force *to the extent that the tidal field
gradient can be ignored*, since, after all, it is the field *gradient* that
gives away the spacetime curvature produced by the gravitating matter
sources. It is the effect of the curvature of spacetime that prevents the
earth's surface from simultaneously expanding outward in all directions at
9.8 m/s^2 resulting in an globally inflating earth, while in any local region
centered at some point on the earth's surface the surface is indeed
accelerating upward at 9.8 m/s^2 wrt a local inertial (i.e. free falling)
reference frame. If you really want a full-blown GR description of the
decelerating train problem it can be done using a Schwarzschild metric (if we
neglect the earth's rotation, and aspherical mass anisotropies, otherwise we
would need a more complicated metric such as a Kerr metric) where the train
is moving on a path with a fixed radial coordinate and varying angular
coordinates. To calculate the gravitational field experienced onboard the
train at some point on the train's world curve, one transforms coordinates to
a non-uniformly rotating coordinate system whose fixed angular coordinates go
around the earth with the locally accelerating train. In that coordinate
system (in which the train is at rest) the coefficients of the Levi-Civita
connection are calculated at each of the different times of the train's trip,
and those coefficients give the time-dependent gravitational field for the
passengers in the train.

David Bowman
dbowman@gtc.georgetown.ky.us