Re: The Elevator in Free Fall

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/