Is the free-falling elevator an inertial reference?

[Luigi Fortunati]

The first principle states that the inertial frame is the one in which
bodies maintain their state of rest.

In my animation
https://www.geogebra.org/m/qdg3kgc8
there is the rigid rod AB which remains at rest and there are the
bodies C and D which move away from their initial position due to the
(real) tidal forces.

Thus, the inertiality of the free-falling elevator is determined by the
presence or absence of tidal forces: if tidal forces are there, the
free-falling elevator is an accelerated reference, if they are not
there, it is a inertial reference.

What is the size limit that separates lifts of one type from those of
the other type?

[Luigi Fortunati]

I answer: there is no limit.

Just look at my animation to see it: body D is closer to the center of
gravity and, therefore, accelerates more.

How much is this acceleration difference? It is very small near the
Earth and extremely large near a black hole.

In any case, the difference is always there and never disappears: the
uniform gravitational field does not exist.

And I also want to point out a case of real acceleration during free
fall.

Watch the video
https://www.youtube.com/watch?v=cyPAEMQKBuo

Astronaut Samantha Cristofoletti is on the spaceship in free fall where
the gyroscope left free to turn does not remain stationary in its
initial position but tilts with respect to the spaceship, ie rotates.

Samantha Cristofoletti explains that it is the spaceship in free fall
that rotates and not the gyroscope that maintains its initial position.

Well, if the spaceship rotates, it's not an inertial reference frame.

[Jonathan Thornburg [remove -color to reply]]

Your statement

> Thus, the inertiality of the free-falling elevator is determined by the
> presence or absence of tidal forces: if tidal forces are there, the
> free-falling elevator is an accelerated reference, if they are not
> there, it is a inertial reference.

is correct.

However, in practice there are other important facts, notably:
(1) there are almost always tidal forces present, i.e., we almost never
have something that's *exactly* an inertial reference frame (IRF),
and
(2) physics experiments are always of finite accuracy, so we almost
never care about having something that's *exactly* an IRF.

This makes it useful to introduce the concept of "approximate inertial
reference frame" (AIRF), where we only require "inertial" to hold up to
some specified error tolerance. And having introduced that concept,
it's then useful to consider Luigi's question for an AIRF.

I.e., it's useful to consider the question "how large can an AIRF be"
(where "size" is measured in both space and time, see below). As we'll
see below, the answer depends on how accurately we want the property
"inertial" to hold, i.e., how approximate do we want our AIRF to be.

To explain this, let's focus on measuring the vertical positions of
bodies in the freely-falling elevator (which, we'll see, is an AIRF)
as functions of time. Let's specify an accuracy tolerance h for our
measurements (e.g., "we'll measure positions to +/- h = 1 millimeter").
Then (for a given (gravitational) tidal field) we can calculate how
long it would take for B and C to drift by that tolerance h with respect
to the stick AB. Let's call this time interval T(BC,h). Then we can
say that for observations with an accuraty tolerance of +/- h, and
durations less than T(BC,h), the elevator is an AIRF.

If, on the other hand, we instead consider a larger AIRF, say one
containing bodies E and F which start out farther apart (vertically)
than B and C, then we'll find the corresponding time T(EF,h) to be
shorter than T(BC,h). That is, if we make our AIRF larger, than
the AIRF has a shorter time-of-validity.

In general, we can write the (vertical) position of any body X in
the freely-falling elevator (relative to the AB midpoint) as a Taylor
series in time,
X(t) = x_0 + x_1 t + (1/2!) (k_2 x_0) t^2 + higher order terms
where the coefficients x_0 and x_1 depend on the body X (they are just
the initial position and velocity of the body at time t=0), but the
coefficient k_2 does NOT depend on the body X. (k_2 is a property of
the (gravitational) tidal field in and near the elevator, more precisely
it's a component of the Riemann tensor).

Neglecting the higher order terms, we can approximately write
X(t) = x_0 + x_1 t + (1/2) k_2 x_0 t^2
T(BC,h) is then the smallest t such that
(1/2) d_CD k_2 t^2 = h.
where d_CD = the initial distance from the AB midpoint to C or D.
Solving this last equation gives
T(BC,h) = sqrt((2 h)/(k_2 d_CD))

This makes it clear how the time-of-validity of the AIRF (the time T)
varies with the tolerance h, the spatial size of the AIRF (the distance
d_CD), and the (gravitational) tidal field as parameterized by k_2.

In particular, notice that in the limit d_CD --> 0, T(BC,h) --> infinity
for any fixed tolerance h, i.e., if we consider an infinitesimally small
AIRF, then for any given tolerance the AIRF effectively lasts forever.
In other words, an infinitesimally small AIRF is effectively a true IRF.

When you see the phrase "inertial reference frame" in a context where
tidal forces are present, it almost always really means an infintesimally
small AIRF. So, returing to Luigi's initial question, if you want a true
IRF, then the size limit is "infinitesimally small".

[Tom Roberts]

On 5/2/23 7:57 PM, Luigi Fortunati wrote:
> [...]

You have completely missed the essential thing about locally inertial
frames in GR: they are APPROXIMATIONS. There exists no perfectly
inertial frame anywhere in the universe we inhabit, including here on
earth. But this is physics, and measurements are never perfect, they
always have a resolution/errorbar. A region of spacetime that is small
enough so the deviations from a truly inertial frame are smaller than
measurement resolutions can be treated as if it is inertial -- in
particular, gravity can be ignored and SR can be applied (which is
enormously simpler to use than GR).

That usually means the locally inertial frame must be in freefall, and
small enough so any tidal forces present are smaller than measurement
resolutions. But not always:

For instance, at the LHC the experimental caverns are less than 100
meters in any direction. The particles they measure travel with speed
indistinguishable from c relative to the lab. So each event has a
duration less than 100m/c = 3E-7 seconds. During such an event, a truly
inertial frame initially at rest relative to the lab would fall
0.5 g t^2 = 0.5 9.8 (3E-7)^2 = 5E-13 meters
Their best detectors have resolution greater than 1E-6 meters, so for
each event they can consider the apparatus to be at rest in a locally
inertial frame, and use SR in their analysis. They analyze each event
separately, because for longer durations (> ~ milliseconds) the
difference between a locally inertial frame and their lab cannot be
neglected.

[That estimate uses the first-order contribution from
earth's gravity; higher order contributions, such as
tidal forces, are considerably smaller and can also
be neglected. Ditto for the non-inertial effects of
earth's rotation.]

This is physics, and approximations abound. It is ESSENTIAL to be able
to estimate when a given approximation is good enough. In all your
discussions of elevators you have never mentioned how accurately
measurements are made -- that is essential information for one to
determine whether the elevator can be considered to be locally inertial.
(The moderator and I have mentioned this, but you have ignored that.)

Tom Roberts

[Luigi Fortunati]

To fully understand everything you wrote, I need to adjust my animation
https://www.geogebra.org/m/qdg3kgc8
to introduce the concept of "approximate" inertial frame of reference (AIRF).

My idea is to add one or more sliders so that the user can vary any
dimension at will, enlarging or reducing it to see how the distances
vary (or don't vary) during the approximation of the AIRF towards the
infinitely small .

Do I make the dimensions of the lift variable? Or the size of bodies C and D? Or their distance? Or the time?

If you give me these indications, I will modify my animation accordingly.

[Luigi Fortunati]

Luigi Fortunati il 02/05/2023 12:57:39 ha scritto:
> Watch the video
> https://www.youtube.com/watch?v=cyPAEMQKBuo
>
> Astronaut Samantha Cristofoletti is on the spaceship in free fall where the gyroscope left free to turn does not remain stationary in its initial position but tilts with respect to the spaceship, ie rotates.
>
> Samantha Cristofoletti explains that it is the spaceship in free fall that rotates and not the gyroscope that maintains its initial position.
>
> Well, if the spaceship rotates, it's not an inertial reference frame.

I ask you for confirmation.

It seems to me that Samantha's gyroscope works *exactly* like Foucault's pendulum: does it?

[Luigi Fortunati]

Perhaps I have found the right way to represent the concept of an
"approximate" inertial frame of reference (AIRF) tending to zero.

In my animation
https://www.geogebra.org/m/k5bunuh3
there are bodies A and B initially in contact with each other.

We can choose their initial size between 0.1 and 1 and then, during free
fall, their separation is maximum when the two bodies are large and
decreases more and more when the bodies are smaller.

When the separation of bodies A and B tends to zero, the tidal forces
also tend to zero and below a certain level (for example 0.1) they can
be considered null.

[Luigi Fortunati]

The movements of Foucault's pendulum prove that the Earth rotates, the
movements of Samantha's gyroscope demonstrate that her spaceship
rotates.