*Post by 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?

Your statement

*Post by Luigi Fortunati*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".