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From: Hendrik van Hees <hees@itp.uni-frankfurt.de>
Newsgroups: sci.physics.research
Subject: Re: The Elevator in Free Fall
Date: Sun, 22 Dec 2024 10:35:29 +0100
Organization: Goethe University Frankfurt (ITP)
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On 22/12/2024 09:57, Luigi Fortunati wrote:
> Jonathan Thornburg [remove -color to reply] il 21/12/2024 09:27:44 ha
> scritto:
>> ...
>> Now let's look at the same system from a GR perspective, i.e., from a
>> perspective that gravity isn't a force, but rather a manifestation of
>> spacetime curvature.  In this perspective it's most natural to measure
>> accelerations relative to *free-fall*, or more precisely with respect
>> to a *freely-falling local inertial reference frame* (FFLIRF).  An
>> FFLIRF is just a Newtonian IRF in which a fixed coordinate position
>> (e.g., x=y=z=0) is freely falling.
> 
> Can we define the interior space of the elevator as "local" or is it
> too big?
> 
> If it is too big, how big must it be to be considered "local"?
> 
> If it is shown that there are real forces inside the free-falling
> elevator, can we still consider this reference system inertial?
> 
> Are tidal forces real?
> 
> Do we mean by "freely falling bodies" only those that fall in the very
> weak gravitational field of the Earth or also those that fall in any
> other gravitational field, such as that of Jupiter or a black hole?
> 
> Luigi Fortunati.

This problems in understanding GR is, in my opinion, due to too much 
emphasis on the geometrical point of view. Of course, geometry is the 
theoretical foundation of all of modern physics, i.e., a full 
theoretical understanding of physics is most elegantly achieved by 
taking the geometric point of view of the underlying mathematical 
models. However, there's also a need for a more physical, i.e., 
instrumental formulation of its contents.

Now indeed, from an instrumental point of view, the gravitational 
interaction is distinguished from the other interactions by the validity 
of the equivalence principle, i.e., "locally" you cannot distinguish 
between a gravitational force on a test body due to the presence of a 
gravitational field due to some body. In our example we can take as a 
test body a "point mass" inside the elevator, with the elevator walls 
defining a local spatial reference frame. The corresponding time is 
defined by a clock at rest relative to this frame at the origin of the 
frame (say, one of the edges of the elevator). Now, the equivalence 
principle says that it is impossible for you to distinguish by any 
physics experiment or measurement inside the elevator, whether you are 
in a gavitational field (in our case due to the Earth), which can be 
considered homogeneous (!!!), for all relevant (small!) distances and 
times around the origin of our elevator reference frame or whether the 
elevator is accelerating in empty space. A consequence is also that if 
you let the elevator freely fall in the gravitational field of the 
Earth, you don't find any homogeneous gravitational field, i.e., free 
bodies move like free particles locally, and thus the free-falling 
elevator defines a local inertial frame of reference.

Translated to the "geometrical point of view" that means that you 
describe space and time in general relativity as a differentiable 
spacetime manifold. The equivalence principle means that at any 
space-time point you can define a local inertial frame, where the 
pseudometric of Minkowski space (special relativity) defines a 
Lorentzian spacetime geometry.

If you now look at larger-scale physics around the origin of the 
freely-falling-elevator restframe, where the inhomogeneity of the 
Earth's gravitational field become important, there are "true forces" 
due to gravity. In the local inertial frame these are pure tidal forces, 
named because they are responsible for the tides on the Earth-moon 
system freely falling in the gravitational field of the Sun.

So it's important to keep in mind that the equivalence between 
gravitational fields and accelerated reference frames in Minkowski space 
holds only locally, i.e., in small space-time regions around the origin 
of your coordinate system, in which external gravitational fields can be 
considered as homogeneous (and static). T

he physically interpretible geometrical quantities are tensor (fields), 
and the general-relativistic spacetime at the presence of relevant true 
gravitational fields due to the presence of bodies (e.g., the Sun in the 
solar system) is distinguished from Minkowski space by the non-vanishing 
curvature tensor, and this is a property independent of the choice of 
reference frames and (local) coordinates, i.e., you can distinguish from 
being in an accelerated reference frame in Minkowski space (no 
gravitational field present) and being under the influence of a true 
gravitational field due to some "heavy bodies" around you, by measuring 
whether there are tidal forces, i.e., whether the curvature tensor of 
the spacetime vanishes (no gravitational interaction at work, i.e., 
spacetime is described as a Minkowski spacetime) or not (gravitational 
interaction with other bodies present, and you have to describe the 
spacetime by some other pseudo-Riemannian spacetime manifold, which you 
can figure out by solving Einstein's field equations, given the 
energy-momentum-stress tensor of the matter causing this gravitational 
field, e.g., the Schwarzschild solution for a spherically symmetric mass 
distribution).

-- 
Hendrik van Hees
Goethe University (Institute for Theoretical Physics)
D-60438 Frankfurt am Main
http://itp.uni-frankfurt.de/~hees/