50 views

Skip to first unread message

Apr 30, 2023, 3:50:09 AMApr 30

to

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?

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?

May 2, 2023, 8:57:45 PMMay 2

to

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.

May 3, 2023, 2:56:28 AMMay 3

to

> 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.
> 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.

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".

May 3, 2023, 2:57:49 AMMay 3

to

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

> [...]

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

May 4, 2023, 3:10:46 AMMay 4

to

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.

May 4, 2023, 3:10:46 AMMay 4

to

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.
> 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.

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

May 5, 2023, 2:59:18 AMMay 5

to

"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.

May 5, 2023, 3:51:36 AMMay 5

to

movements of Samantha's gyroscope demonstrate that her spaceship

rotates.

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu