Re: Does gravity do work on the freely falling body?

On Apr 22, 7:51 pm, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
Phil wrote:
Let's say one takes the
position that freely-falling motion is unaccelerated inertial motion.
How does that position stack up to "all the evidence" that gravity
accelerates the object which is inertial?

That is a reasonable question to ask (unlike most of this thread).

In GR, indeed we do consider freely-falling motion to be inertial, and
therefore unaccelerated, in the sense that the 4-acceleration of such an
object is identically zero. Note that I qualified what I meant by
"acceleration", because in GR that word has several meanings; some apply
to what I said, some don't.

In GR one must be careful of coordinate-dependent quantities, because
artifacts of the coordinate system can confuse the discussion. It's
usually MUCH better to discuss invariants (quantities independent of
coordinate choice).

4-acceleration is a 4-vector, and is completely independent
of coordinates. That's why it is so useful, and why I used
it above.

In coordinates fixed to my office, a falling object has nonzero
acceleration. Note that this statement is explicitly coordinate
dependent. When I consider the 4-acceleration of the object, it is zero
(neglecting air resistance and other minuscule effects). The
un-qualified word "acceleration" could refer to either of these, and the
resulting confusion is the basis of your question.

Tom, I appreciate your response. I am not sure where you are confused
regarding the relationship of coordinate acceleration. In regard to
gravity, it is also a reasonable question to ask, how does gravity
effect the motion of objects. We seemingly have two choices. Either
gravity is a force impressed from the surface --- OR --- gravity is
the acceleration of inertial frames. And here I qualify that to mean
that the 4-acceleration of the object IS ZERO in a system of co-
accelerating inertial frames. So do keep in mind that I am in
agreement of your application of the concept in the restricted way you
are using it.

I am suggesting, that if one takes in sufficient evidence it is
possible to determine whether it is the inertial frames which are
accelerated OR the surface frame.

For example, you made mention of the physical law F/m=a. Is this
generally true in the surface frame? I think its an important
question and their are experiments one can conduct (independent of the
local inertial frames) which can answer this question unambiguously.


When you say "gravity accelerates the object" you mean a coordinate
acceleration using coordinates that are fixed to the earth (or similar).

That's close enough for now.

This COORDINATE acceleration of the object is really an artifact of the
coordinates, as any object at rest in them has nonzero 4-acceleration.

Right. But then that is an artifact of a system of co-accelerating
inertial frames. Keep in mind I am agreeing these object are moving
by their inertia. I am qualifying that by saying that because of
their inertia they are accelerated by gravity. Because the system of
inertial frames and the surface accelerate with respect to each other
it must be acknowledged that the coordinate relationship of
acceleration is ambiguous (locally).


Those coordinates are not inertial coordinates, so naturally any object
moving inertially is accelerated relative to them. In GR, we say that an
object "moves inertially" when its 4-acceleration is zero [%].

[%] Technically this holds only for "test particles" --
objects that have no internal structure and are so small
that their effect on the geometry can be completely neglected.
For example, this marble (1 cm glass sphere) can be considered
to be a test particle here on earth.

In summary, in GR the idea that freely-falling motion is unaccelerated
inertial motion is completely consistent with "all the evidence that
gravity accelerates the object". One must apply the appropriate meanings
to those words, that's all.

For a local system of inertial objects, one conceives the 4-
acceleration of the local objects is ZERO when and only when one
removes the artifact of their co-acceleration (that artifact is their
acceleration property wrt a field centered non-rotating inertial
coordinate system). This is basically the same conceptually as
"removing the gravity". Then all that remains is the remnant tidal
acceleration (due to field non-uniformity).

In this regard is is not accurate to say simply that the idea that
freely falling motion is unaccelerated inertial motion is completely
consistent with gravity accelerates an object.

For me the accurate description is "The idea that freely falling
motion is accelerated inertial motion is completely consistent with
all the evidence that objects freely-falling in gravity within
sufficiently restricted domain are moving inertially with respect to
one another.

It's a question of which is the dog and which is the tail. Is it the
co-acceleration which makes the local domain inertial for freely
falling objects ... OR ... is it the inertial relationship between
freely falling objects in a local domain which makes them accelerate
with respect to a gravitationally centered non-rotating frame of
reference. The property of the objects to be inertial wrt to one
other is limited in domain. Another way of saying it is that their
motion is not a continuous function without the context of their
acceleration wrt to the gravity centered inertial frame.

It wasn't as much about
Work as it is about what work is.

In GR we do not normally attempt to discuss work, because it is
exceedingly complicated and not very useful [#]. That's primarily
because work is a type of energy, and energy is ALWAYS coordinate
dependent; re-read my caveat about coordinate-dependent quantities
above. In short, once one studies GR in any detail, it becomes QUITE
CLEAR that whenever possible, one wants to discuss invariant quantities,
not coordinate-dependent ones.

It is quite easy to discuss work in gravity but it is IMPOSSIBLE to
make it coordinate independent. So what does that mean? It means
that gravitational work IS NOT coordinate independent. One can't just
choose any frame willy-nilly. There is a source of gravitational
acceleration. The work done on a freely falling object would be
related functionally to its inertial frame.

I understand its not a useful concept for GR. But is it an important
concept for gravity? I can assure you, that if gravity does work,
then it is important for the very reason you mentioned above. It must
be accounted for in energy conservation and this is where I have found
it a (conceptually) very interesting concept to contemplate and
model.

This is true even in Newtonian mechanics. Compare solving
F=ma to solving the equations of Lagrangian mechanics....
The Lagrangian equations are independent of coordinates,
while F=ma is not.

Consider a lab on the surface of Earth. Assume it to be accelerating
consistent with the EP. Does this acceleration effect any change to
the motion of the lab? I mean does it cause an accelerating
displacement with respect to anything other than a system of local
inertial frames? I think this is a very important question for
answering the question of whether the lab is accelerating or the
inertial frames. Also does the physical law a=dv/dt hold true in the
conceptualization that the surface is accelerating. This can be
unambiguously answered simply by seeing if distant starlight is blue-
shifted as a consequence of the lab's acceleration. I can tell you
that any blueshift one may observe of starlight is small and IS NOT
functionally related to the acceleration of the surface (the frame
assuming the EP fails a=dv/dt in the direction of gravitational
acceleration). Yet on the other hand the freely falling frames detect
a redshift of the distant starlight (meaning that assuming a=dv/dt is
true in all directions if acceleration of the inertial frames is
assumed). Measuring the doppler shift of starlight is unambiguous and
totally COORDINATE INDEPENDENT (though it is SPATIALLY DEPENDENT).
The only way to resolve this evidence is to arrive at the awareness
that it is the inertial frames which are accelerating because the
inertial frames can't have the stars co-accelerating with the
surface. So then what about the surface frame? It is accelerating
wrt the inertial frames and "feels like" it is accelerating. But it
is not having an accelerating displacement ... rather ... it is merely
stressed. The surface does no work on the lab, objects in the lab are
compressed into stress like one might compress a spring in a vise.


[#] One aspect of the complexity is related to the fact
that work is inherently an integral, and integration in a
curved manifold has thorns....

You were correct IMHO in your
reference to evidence supporting the notion that Gravity does work on
the freely falling body. There are many other ways to cite evidence
in this regard also.

Sure. But it's all COORDINATE-DEPENDENT evidence, using NON-inertial
coordinates. See above for my caveat about coordinate-dependent quantities.

PMB's selection of evidence was, yes, but I mentioned above an
unambiguous coordinate independent evidence. Arguments can also be
based on the tidal stress of inertial objects which is also coordinate
independent.

Use co-moving locally-inertial coordinates, and it's obvious that no
work at all is being done on the object (it of course remains at rest in
such coordinates).

So if one choose a "special set" of accelerating coordinate systems
one arrive at a coordinate dependent conclusion that no work is done
on the freely falling object.


So simply asking whether or not "work is done" is not very useful....

Remember that selecting coordinates is an ARBITRARY human
choice, and such choices cannot possibly have any effect on
physical phenomena.

But nature doesn't always give us the choice. In that regard, nature
isn't arbitrary. An object is the source of gravity. Effects are
related to coordinate system of that source. Only when we abstract
them away are we able to make effects independent of this coordinate
system.

Even GR can not do that entirely if one wants to incorporate any
curvature into local freefall domain. That curvature will have a
relationship to the source centered inertial frame. The effects of
that curvature will also be manifest as accelerations vectored to the
center of gravitation. If it isn't ... it will not properly predict
effects.

Since I can make "work" appear or disappear
merely by thinking differently about the situation, it should
be clear that this "work" is unrelated to any physical
phenomena --

Absolutely false. One can't make work appear or disappear. He can
choose the wrong coordinate system or select a class of coordinate
systems where locally he can make work seem to disappear. Can you see
what is wrong with the argument. You say a concept should be
independent of coordinates and in same breath choose a class of
coordinates "to make" a concept disappear.

I mentioned earlier that tidal stress in the inertial body falling
freely in gravity is evidence of work. Stress is clearly present
within the tidally stressed object. Further if that object is a
meterstick, guess what? It no longer has a connection to space-time
with regard to its length (its longer than 1 meter of space-time).
This has always confused me because if GR were purely geometrical
wouldn't a meter-stick just curve with space-time always being the
same length? Curvature as it is applied in GR seems more closely
related to force or acceleration than geometry when one gives
consideration to the effects of curvature. Why is it that the tidally
stressed meter-stick loses its grip with space-time and no longer is
extended one meter of space-time?
.

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