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Re: "acceleration due to gravity" (rhetorical?)



Regarding Michael Edmiston's comments and questions about the Equivalence
Principle:
...
One way to state the principle of equivalence is to say something like: (1)
"There is no way to distinguish, on a local scale, between gravitational and
inertial forces."

OK.

Another way to state the principle of equivalence is to say something like:
(2) "There is no way to distinguish between a reference frame in a
homogeneous gravitational field and a reference frame undergoing uniform
acceleration."

This second version is incorrect.

Some people seem to be saying the word "local" in version (1) somehow
invalidates the idea that acceleration and gravity are truly equivalent or
truly indistinguishable.

Yes. The reason for the emphasis on the word "local" is precisely
because in general a universe with real gravitation is *not*
indistinguishable from one without it but merely described from an
accelerated coordinate system. The effects of an accelerated reference
frame and those of gravity are only asymptotically indistinguishable in
the limit of a local region of spacetime of vanishingly small size. They
*are* distinguishable on the large scale. To claim otherwise is sort of
like saying the surface of a sphere is indistinguishable from a plane.
It simply is not true in general. *But* in the limit of a sufficiently
small neighborhood patch on the surface of a sphere its geometric
properties *do* become indistinguishable from those of a flat plane.

Indeed, if I am taken to a lab where the
gravitational field is not homogenous, I should be able to discern the
accelerations I measure are caused by gravity rather than any uniform
acceleration of the lab. I don't see why that invalidates anything, but it
does lead me to prefer statement (2).

The problem is that the very *concept* of a *uniform* acceleration of the
lab becomes subtlely problematic when the lab has been accelerating for
such a long time interval that its speed has changed by an appreciable
fraction of c, and when the size of the "lab" is so big that the light
travel time across it becomes comparable to the time it takes for one
part of that lab to accelerate up to a speed comparable to c (from some
initial fiducial "rest" state). There is no way to have an infinite lab
'uniformly' rigidly accelerate for all time in flat Minkowski space
without generating some serious coordinate singularites somewhere and/or
somewhen. And there are many accelerating coordinate systems that
*locally* are equivalent to each other in some small region of spacetime
(and which have the same nonrelativistic limit), but which are really
globally inequivalent and behave very differently from each other in
their outer reaches. But *none* of them act like a universe with a
'uniform' real gravitational field (which is *also* a problematic
concept).

BTW, there are also multiple ways to define a 'uniform' gravitational
field that mutually contradict each other in their distant (in spacetime)
outer reaches from some local region of interest, all differing on the
details of just how a vague 'unifomity' concept is implemented in a
spacetime metric that locally is like that of flat special relativity.
The quantity (Levi-Civita connection) that plays the role of the
gravitational field in GR is a 3-index object that has 40 distinct
components (in 4-D spacetime), but in Newtonian physics the gravitational
field or the acceleration field of a noninertial coordinate system have
only 3 components. Clearly, there are multiple inequivalent ways to
implement the notions of a "uniform/homogeneous Newtonian gravitational
field" and a "uniformly accelerating coordinate system" into GR that in
the Newtonian limit behave like you would want them to. The particular
implementations that that cause a particular nonlinear and derivative
form of the Levi-Civita connection, called the Riemann curvature tensor,
(which is the GR generalization of the idea of a tidal field or field
gradient) to globally vanish in all its components are all just different
(possibly) acclerated coordinate systems for a flat space without any
real gravity present. Those particular implementations that have the
Riemann tensor not globally vanish are *not* just acclerated versions of
a nongravitational universe, but actually describe real gravitational
effects (e.g tidal effects for example).

The weird thing is that it is the "uniformly" accelerating coordinate
systems in flat nongravitational Minkowski spacetime are ones that
require the (coordinate) singularities in their periphery. But there
are "uniform" gravitational field universes that have no such
singularities, but just describe an appropriately curved spacetime that
is *not* globally equivalent to a nongravitational universe in an
accelerated frame.

For an example of an accelerated reference frame (in a flat
universe) which locally has objects dropped near its origin to "fall"
with an (initial from rest) acceleration -g along the z-axis, and for
an example of an (inequivalent to the first) universe that has a
homogeneous gravitational field -g along the z-axis (which has a
curved spacetime) see my post in the PHYS-L archives dated 5 SEP 99
whose subject is "Re: Symmetry in Lorentz transformation equations
(long)" which can be found at:

http://mailgate.nau.edu/cgi-bin/wa?A2=ind9909&L=phys-l&O=A&P=8149

I think the heart of the matter I am trying to get at is: If there is any
difference between a homogeneous gravitational field and a reference frame
in uniform acceleration, what is this difference?

*Yes*. See above and see the archived post.

If there isn't any, then
there also isn't any difference between F=mg and F=ma. If this is true then
why can't we call "g" an acceleration and why can't we call "a" a gravity
field? What could we notice that would make it more appropriate to choose
one word over the other?

Since your premise is false the questions are moot.

It seems another way to say this is to ask gravitational mass versus
inertial mass questions. If these masses are indeed the same, that seems to
say the same thing as (2).

No. That is a related but different question.

Is the problem some people have with what I am saying simply the fact that
we do not know of any homogeneous gravitational fields?

Not really, but just defining the very concept of a globally
'homogeneous' gravitational field must be done with much care. When it
is done one discovers that a spacetime that supports such a thing is
typically *not* flat, and is inequivalent to a nongravitational universe
viewed from an accelerated frame (which happens typically to be not as
"smooth", "regular", "homogeneous", etc. in terms of its coordinates for
the accelerating frame as in the previous case of the 'uniform'
gravitational field).

If so, is it valid
to claim (2) is not true.

The "if so" here is not the case. But the conclusion that "(2) is not
true" *is* true anyway.

Or is it valid to say that F=mg and F=ma are
always distinguishable?

I'm fearful of the word "always", but in general they *are*
distinguishable assuming one has sufficient apparatus and room in
spacetime to work.

I'm asking questions here. I may be implying answers but I am not sure
about them. Here's another question: It's common in electrostatic
discussions to talk about an infinite plane of charge. When we imagine such
a thing, we say the electric field it creates is homogenous. Couldn't we
also imagine an infinite sheet of mass that would produce a uniform
gravitational field?

This is a good question. I'm not sure. I suppose it might be possible
assuming much care is taken in setting the gedanken situation up. I have
not calculated the actual components of the Ricci tensor for the
homogeneous field in the 2nd (gravitating) example in the earlier
5 SEP 99 post to check if that field actually exists in an empty vacuum,
or if it is filled with some space-filling distribution of matter. If it
*does* describe a vacuum, then presumably we could splice that metric for
z > 0 to a mirror image version of it for z < 0 in the x-y plane so that
the singularity across the x-y plane was due to an infinite sheet of
matter there with no matter anywhere else.

Suppose we created an infinite sheet of mass in an attempt to create a
homogeneous gravitational field. You decide how this sheet is constructed.
Maybe it is a sheet of atoms; maybe it is a two-dimensional array of stars.
In any case, would this actually produce a homogeneous gravitational field
on a large scale, but a non-homogeneous gravitational field on a local scale
(where local means we are experimenting with objects smaller or similar in
size to the things making up the mass array, and we are close the mass
array)?

I haven't tried it, but it sounds harder to do than I'm inclined to
attempt.

Since we are living among the stars, is the gravity we experience
non-homogenous on a local scale, but there could be homogenous gravitational
fields on a non-local scale (if the universe is flat - whatever that means)?

Well, for a couple of years now it seems that the best evidence is the
Hubble expansion of the universe is cosmically accelerating, and it still
seems to be *spatially* flat on large length scales. If this is true it
means that the Cosmological Constant is nonzero (& seems to have a value
that is about 65% of what it would have to be to both remain spatially
flat and to accelerate at the same current rate but without any matter
present in it). In *a sense* this might be construed as a kind of
homogeneous gravitational field since is a true scalar constant
independent of space and time. *But* its effect is to modify Netwon's
law of gravitational attraction at very large distances. Masses attract
each other at close by distances with a 1/r^2 force law, but at very
large distances they tend to repel and accelerate away from each other
because of the cosmic acceleration of the expansion of space. This
effect is homogeneous in the one sense that it works the same way as
viewed from anywhere in the universe (after dealing with the local
gravity of the various nearby galaxies). But it is not uniform in
another sense because the direction of this gravitational field is
radially outward from any given/chosen spatial origin and it gets
stronger with increasing distance from that origin.

Regarding Jack's sage comments:
Those interested should check the earlier discussion of this
topic. The notion of a "homogeneous gravitational field" causes much
trouble, and is best avoided.

Well said. The "notion of a 'homogeneous gravitational field'" is a
dangerous thing. Subtleties and paradoxs abound as do pitfalls for the
definitionally unwary. Proceed only with the utmost caution. If one is
not familiar with the mathematics of GR, one probably ought not try it at
home. This is completely besides the fact that such a notion is not even
useful in practice.

A "local" measurement of a gravitational field is, by definition,
a measurement at a point. It is always possible to choose a metric
that is diagonal (-1,1,1,1) at some point. That mathematical statement
defines the meaning of the statement that a <local> measurement of
a gravitational field is indistinguishable from an acceleration.

Almost, but not quite. Just making the metric diagonal (-1,1,1,1) only
says that the coordinate system has coordinates that are mutually
orthogonal at that point and that they have been scaled so that at
that spacetime point the metric coefficients become -/+ 1. Besides this
we need to make a statement about the value of the Levi-Civita
connection at that point if we want to assert the local
indistinguishability of coordinate system acceleration and gravitation.
But we *can* also choose a coordinate system that not only has makes the
metric have the above form, but *also* makes the Levi-Civita connection
vanish at that point. Such a coordinate system is (locally near that
point) a version of Cartesian coordinates for a freely falling frame at
that point, and near that point, in that coordinate system, the physics
is that of a gravity-free inertial coordinate system.

David Bowman
David_Bowman@georgetowncollege.edu