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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."
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."
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.
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).
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?
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?
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).
Is the problem some people have with what I am saying simply the fact that
we do not know of any homogeneous gravitational fields?
If so, is it valid
to claim (2) is not true.
Or is it valid to say that F=mg and F=ma are
always distinguishable?
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?
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)?
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)?
Those interested should check the earlier discussion of this
topic. The notion of a "homogeneous gravitational field" causes much
trouble, and is best avoided.
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.