139 views

Skip to first unread message

May 7, 2022, 6:38:42 AMMay 7

to

A free-falling brick is an inertial frame?

Are a pair of free-falling half-bricks an inertial frame or are they=20

two distinct inertial frame?

Are a pair of free-falling half-bricks an inertial frame or are they=20

two distinct inertial frame?

May 7, 2022, 9:00:56 AMMay 7

to

On Saturday, 7 May 2022 at 12:38:42 UTC+2, Luigi Fortunati wrote:

> > A free-falling brick is an inertial frame?

A free-falling brick is *in* "an" inertial frame. Which more precisely
> > A free-falling brick is an inertial frame?

means that we can find an inertial frame in which the brick is in

uniform motion, then just a special case is an inertial frame in

which the brick is at rest, and and even more special case is the

frame in which the brick is at rest at the origin of space. And we

might call that last one "the brick's own frame", because there is

indeed something "privileged" about it as far as that brick is

concerned: OTOH, though, notice that the fact that the brick is in

*free-fall* requires no frame to state or verify at all, it's altogether

a *true* physical property that can be verified *locally*.

> > Are a pair of free-falling half-bricks an inertial frame or are they

It should now be clear that that is simply upside down: if two

bricks are in free fall, of course one can find an inertial frame in

which both are in uniform motion, in fact infinitely many of them.

Julio

May 8, 2022, 3:41:41 AMMay 8

to

Julio Di Egidio sabato 07/05/2022 alle ore 15:00:52 ha scritto:

>>> A free-falling brick is an inertial frame?=20

As small as a brick? A half brick? A tenth of a brick?

>>> A free-falling brick is an inertial frame?=20

>

> A free-falling brick is *in* "an" inertial frame. Which more precisely

> means that we can find an inertial frame in which the brick is in

> uniform motion, then just a special case is an inertial frame in=20
> A free-falling brick is *in* "an" inertial frame. Which more precisely

> means that we can find an inertial frame in which the brick is in

> which the brick is at rest, and and even more special case is the

> frame in which the brick is at rest at the origin of space. And we

> might call that last one "the brick's own frame", because there is

> indeed something "privileged" about it as far as that brick is

> concerned: OTOH, though, notice that the fact that the brick is in

> *free-fall* requires no frame to state or verify at all, it's altogethe=
> frame in which the brick is at rest at the origin of space. And we

> might call that last one "the brick's own frame", because there is

> indeed something "privileged" about it as far as that brick is

> concerned: OTOH, though, notice that the fact that the brick is in

r

> a *true* physical property that can be verified *locally*.

How small must this be "locally"?
> a *true* physical property that can be verified *locally*.

As small as a brick? A half brick? A tenth of a brick?

May 8, 2022, 10:21:25 AMMay 8

to

On Sunday, 8 May 2022 at 09:41:41 UTC+2, Luigi Fortunati wrote:

> Julio Di Egidio sabato 07/05/2022 alle ore 15:00:52 ha scritto:

> >>> A free-falling brick is an inertial frame?

> >

> Julio Di Egidio sabato 07/05/2022 alle ore 15:00:52 ha scritto:

> >>> A free-falling brick is an inertial frame?

> >

> > A free-falling brick is *in* "an" inertial frame. Which more precisely

> > means that we can find an inertial frame in which the brick is in

> > uniform motion, then just a special case is an inertial frame in
> > means that we can find an inertial frame in which the brick is in

> > which the brick is at rest, and and even more special case is the

> > frame in which the brick is at rest at the origin of space. And we

> > might call that last one "the brick's own frame", because there is

> > indeed something "privileged" about it as far as that brick is

> > concerned: OTOH, though, notice that the fact that the brick is in

> > *free-fall* requires no frame to state or verify at all, it's altogether
> > frame in which the brick is at rest at the origin of space. And we

> > might call that last one "the brick's own frame", because there is

> > indeed something "privileged" about it as far as that brick is

> > concerned: OTOH, though, notice that the fact that the brick is in

> > a *true* physical property that can be verified *locally*.

>

> How small must this be "locally"?

> As small as a brick? A half brick? A tenth of a brick?

[Apologies for my first reply, not very constructive: please disregard,
>

> How small must this be "locally"?

> As small as a brick? A half brick? A tenth of a brick?

I am trying an actual answer.]

It's literally the *point*, which is an ideal condition of course, but one

of those that can be approximated arbitrarily well: keeping in mind

that this is classical physics, not quantum, so we don't need further

cautions.

In practice, it's just a little bit more articulated than that, it goes this

way: *you* start moving until your motion is in sync with that of the

brick, which you can check by having a rigid rod guarantee that your

distance and orientation relative to the brick does not change (and

then would you ask why and how that works?), then, by *transitivity*

of the condition of being inertial, you just check that *you* are

inertial: and that is as local as it needs be, modulo approximations

that are simply structural to doing experiments.

And I am sure most here could rephrase that in better terms, but I'd

rather invite you to take a step back and this point and reconsider

how the very progression goes: what it even means for a property

to be a *true physical property* vs e.g. an artefact of the coordinate

system.

Julio

May 8, 2022, 8:38:50 PMMay 8

to

On Sunday, 8 May 2022 at 09:41:41 UTC+2, Luigi Fortunati wrote:

> Julio Di Egidio sabato 07/05/2022 alle ore 15:00:52 ha scritto:

> Julio Di Egidio sabato 07/05/2022 alle ore 15:00:52 ha scritto:

> >>> A free-falling brick is an inertial frame?

> >

> > A free-falling brick is *in* "an" inertial frame. Which more precisely

> > means that we can find an inertial frame in which the brick is in

> > uniform motion, then just a special case is an inertial frame in

> > which the brick is at rest, and and even more special case is the

> > frame in which the brick is at rest at the origin of space. And we

> > might call that last one "the brick's own frame", because there is

> > indeed something "privileged" about it as far as that brick is

> > concerned: OTOH, though, notice that the fact that the brick is in

> > *free-fall* requires no frame to state or verify at all, it's altogether

> > a *true* physical property that can be verified *locally*.

>

> >

> > A free-falling brick is *in* "an" inertial frame. Which more precisely

> > means that we can find an inertial frame in which the brick is in

> > uniform motion, then just a special case is an inertial frame in

> > which the brick is at rest, and and even more special case is the

> > frame in which the brick is at rest at the origin of space. And we

> > might call that last one "the brick's own frame", because there is

> > indeed something "privileged" about it as far as that brick is

> > concerned: OTOH, though, notice that the fact that the brick is in

> > *free-fall* requires no frame to state or verify at all, it's altogether

> > a *true* physical property that can be verified *locally*.

>

> How small must this be "locally"?

Can I assume the first part is clear now?
> As small as a brick? A half brick? A tenth of a brick?

you?), just like everything else here, so either you have some

serious question starting from that definition, or you should

simply get a good introductory course to the matter. Otherwise

I too don't see why you are even allowed to keep posting here.

Julio

May 15, 2022, 6:10:45 AMMay 15

to

On 5/8/22 9:21 AM, Julio Di Egidio wrote:

> On Sunday, 8 May 2022 at 09:41:41 UTC+2, Luigi Fortunati wrote:

>> How small must this be "locally"? As small as a brick? A half

>> brick? A tenth of a brick?

> [... verbiage that does not answer the question]
> On Sunday, 8 May 2022 at 09:41:41 UTC+2, Luigi Fortunati wrote:

>> How small must this be "locally"? As small as a brick? A half

>> brick? A tenth of a brick?

You both miss the key concept about locally inertial frames: they are

only APPROXIMATELY inertial, and the maximum size they can be is

determined by your measurement resolution; better resolution puts a

smaller limit on the size of the region in which they can be considered

inertial.

Here's a simple example.

Suppose you start with a 4-meter-wide elevator in freefall near the

surface of the earth, and you release two small ball bearings 3 meters

apart horizontally, at rest relative to the inside of the elevator. They

will APPROXIMATELY remain at rest relative to the elevator, but we know

that they are each falling toward the center of the earth, so they will

slowly approach each other as the elevator continues to fall.

Imagine you can measure the distance between them with a resolution of 1

millimeter. Since the earth radius is 6.371E6 meters, the elevator can

fall 1,460 meters until they are 2.999 meters apart, which will take

17.2 seconds.

Suppose, instead, that you can measure the distance between them with a

resolution of 1 micron. The elevator can fall 1.46 meters until they are

2.999999 meters apart, which will take 0.54 seconds.

Clearly the limited region of spacetime over which this locally inertial

frame is approximately inertial depends on how well you can measure.

Other types of measurements will put different constraints on the size

of the region. For instance, recent measurements of optical clocks at

NIST will put a limit of a few centimeters tall before such clocks at

rest in the elevator at its ceiling and floor will cease to remain in

sync.

Julio Di Egidio continued with an unrelated quest:

> [I] invite you to [...] reconsider [...] what it even means for a

> property to be a *true physical property* vs e.g. an artefact of the

> coordinate system.

Modern physics has a simple and very general requirement: "true physical
> coordinate system.

properties" must be invariant under changes of coordinates. This is why

modern physical theories are all expressed in terms of tensors, which

are completely independent of coordinates, and therefore invariant under

changes of coordinates.

Tom Roberts

May 17, 2022, 2:45:36 AMMay 17

to

[[Mod. note -- I have rewrapped overly-long lines. -- jt]]

On Saturday, May 7, 2022 at 5:38:42 AM UTC-5, Luigi Fortunati wrote:

> A free-falling brick is an inertial frame?

If it's not rotating. Otherwise, the free-falling object is in a

rotating frame, if the frame is attached to and associated with the

object, itself. If it's not rotating, then it is locally inertial.

The distinction between the two rests on Newton's bucket thought

experiment.

> Are a pair of free-falling half-bricks an inertial frame or are they

> two distinct inertial frame?

The curvature of space-time is precisely the warping of a field of

locally inertial free-fall frames whereby those that are initially

at rest with respect to one another start to accelerate with respect

to each other. If you display them as worldlines in a 4-dimensional

graph (or in a 3-dimensional graph, where one of the spatial

dimensions is suppressed, for the benefit of the unlucky few who

are visually impaired to see in 4 dimensions) then you'll see the

worldlines for locally inertial free-fall frames - initially parallel

in a time-like direction - starting to curve into one another -

hence the "falling" action associated with gravity. In this sense,

the gravity one feels and experiences is actually a warping in time,

first and foremost, rather than a warping in space. The actual

contraction is quantified and accounted for in the Raychaudhuri

equation, which is closely related to the "geodesic deviation

equation".

In a flat space-time spatially separated locally inertial frames,

initially at rest with respect to one another, remain at rest; and

so can be said to comprise the different locations of a global

inertial frame.

All of the foregoing applies independent of paradigm - to *both*

relativistic *and* non-relativistic theory; so it is neither a

construct nor innovation of "general relativity", but rather one

which first fully emerged at the onset of general relativity and

so has been (falsely) associated with it as a characteristic feature

of it. It is a general feature of any theory of gravity that respects

the Equivalence Principle.

In fact, both Newtonian gravity and Einsteinian gravity (specifically:

the Schwarzschild solution) can be unified as a one-parameter family

of geometries, that are warped versions of the 5-dimensional Bargmann

Geometry, via the line element + constraint:

dx^2 + dy^2 + dz^2 + 2 dt du + a du^2 - 2 V dt^2 + 2aV/(1 + 2aV) dr^2 = 0

where r = root(x^2 + y^2 + z^2), dr = (x dx + y dy + z dz)/r and

V = -GM/r is the potential of a gravitating body of mass M located

at r = 0.

The extra, u, coordinate is the non-relativistic limit of (s - t)

c^2, as c goes to infinity, where s is proper time. This has meaning

... which also (by the way) shows that such things as "time dilation"

and "twin paradox" are *also* rooted in non-relativistic theory in

disguised form as u, and are not features specific to Relativity!

The u coordinate shows up, physically, as negative the action per

unit mass for an inertial particle.

When a = 0, this is Newtonian gravity, and it can be generalized

by having V be the total potential for all gravitating bodies,

rather than just for one. The geodesics for this geometry are the

orbits of Newtonian gravity. Since V is a function of the coordinates

and velocities of individual bodies, rather than a bona fide field

quantity, it is very tempting to try and quantize this geometry

directly in quantum *mechanics*. But for the fact that you still

have the self-energy and self-force problems to deal with (in starker

form, in fact) you'd almost have a full-fledged *geometric* quantum

theory for Newtonian gravity - one in which space-time itself is

quantized. But the whole "quantizing field theory as mechanics"

strategy has these same issues roadblocks, here, as did Feynmann

and Wheeler's attempt to do the same with electromagnetic theory

in the 1940's.

The case where a = 0 and V = 0 is the Bargmann geometry, which is

the natural geometric arena for non-relativistic theory.

When a > 0, this is the Schwarzschild solution in which the proper

time is given as s = t + a u, and in which c = root(1/a) is an

invariant speed (i.e. "light speed").

The term "dx^2 + dy^2 + dz^2" is the legacy of Euclidean geometry;

while "2 dt du" is the legacy of *Galileo*'s principle of Relativity,

which is where space and time *actually* became unified into the

chrono-geometry of spacetime. The secret eloping of the two, however,

went largely unnoticed until it was fully consummated by the addition

of the Poincare' term "a du^2", which turns this into a geometry

for Minkowski space. The warping of time associated with Newtonian

gravity is in the "-2V dt^2" term, while the warping of space,

itself, associated with General Relativity is limited to the

substantially smaller "2aV/(1 + 2aV) dr^2" term.

The only effects of paradigm, here, are those limited to the "Special

Relativity" term a du^2 and the "General Relativity" term

2aV/(1 + 2aV) dr^2 (so that whatever "relativistic corrections" there

are, which are made to trajectories have to come from these terms).

The main thrust of gravity - and the essential background behind your

query - resides with the "Newtonian Gravity" term -2 V dt^2.

On Saturday, May 7, 2022 at 5:38:42 AM UTC-5, Luigi Fortunati wrote:

> A free-falling brick is an inertial frame?

rotating frame, if the frame is attached to and associated with the

object, itself. If it's not rotating, then it is locally inertial.

The distinction between the two rests on Newton's bucket thought

experiment.

> Are a pair of free-falling half-bricks an inertial frame or are they

> two distinct inertial frame?

locally inertial free-fall frames whereby those that are initially

at rest with respect to one another start to accelerate with respect

to each other. If you display them as worldlines in a 4-dimensional

graph (or in a 3-dimensional graph, where one of the spatial

dimensions is suppressed, for the benefit of the unlucky few who

are visually impaired to see in 4 dimensions) then you'll see the

worldlines for locally inertial free-fall frames - initially parallel

in a time-like direction - starting to curve into one another -

hence the "falling" action associated with gravity. In this sense,

the gravity one feels and experiences is actually a warping in time,

first and foremost, rather than a warping in space. The actual

contraction is quantified and accounted for in the Raychaudhuri

equation, which is closely related to the "geodesic deviation

equation".

In a flat space-time spatially separated locally inertial frames,

initially at rest with respect to one another, remain at rest; and

so can be said to comprise the different locations of a global

inertial frame.

All of the foregoing applies independent of paradigm - to *both*

relativistic *and* non-relativistic theory; so it is neither a

construct nor innovation of "general relativity", but rather one

which first fully emerged at the onset of general relativity and

so has been (falsely) associated with it as a characteristic feature

of it. It is a general feature of any theory of gravity that respects

the Equivalence Principle.

In fact, both Newtonian gravity and Einsteinian gravity (specifically:

the Schwarzschild solution) can be unified as a one-parameter family

of geometries, that are warped versions of the 5-dimensional Bargmann

Geometry, via the line element + constraint:

dx^2 + dy^2 + dz^2 + 2 dt du + a du^2 - 2 V dt^2 + 2aV/(1 + 2aV) dr^2 = 0

where r = root(x^2 + y^2 + z^2), dr = (x dx + y dy + z dz)/r and

V = -GM/r is the potential of a gravitating body of mass M located

at r = 0.

The extra, u, coordinate is the non-relativistic limit of (s - t)

c^2, as c goes to infinity, where s is proper time. This has meaning

... which also (by the way) shows that such things as "time dilation"

and "twin paradox" are *also* rooted in non-relativistic theory in

disguised form as u, and are not features specific to Relativity!

The u coordinate shows up, physically, as negative the action per

unit mass for an inertial particle.

When a = 0, this is Newtonian gravity, and it can be generalized

by having V be the total potential for all gravitating bodies,

rather than just for one. The geodesics for this geometry are the

orbits of Newtonian gravity. Since V is a function of the coordinates

and velocities of individual bodies, rather than a bona fide field

quantity, it is very tempting to try and quantize this geometry

directly in quantum *mechanics*. But for the fact that you still

have the self-energy and self-force problems to deal with (in starker

form, in fact) you'd almost have a full-fledged *geometric* quantum

theory for Newtonian gravity - one in which space-time itself is

quantized. But the whole "quantizing field theory as mechanics"

strategy has these same issues roadblocks, here, as did Feynmann

and Wheeler's attempt to do the same with electromagnetic theory

in the 1940's.

The case where a = 0 and V = 0 is the Bargmann geometry, which is

the natural geometric arena for non-relativistic theory.

When a > 0, this is the Schwarzschild solution in which the proper

time is given as s = t + a u, and in which c = root(1/a) is an

invariant speed (i.e. "light speed").

The term "dx^2 + dy^2 + dz^2" is the legacy of Euclidean geometry;

while "2 dt du" is the legacy of *Galileo*'s principle of Relativity,

which is where space and time *actually* became unified into the

chrono-geometry of spacetime. The secret eloping of the two, however,

went largely unnoticed until it was fully consummated by the addition

of the Poincare' term "a du^2", which turns this into a geometry

for Minkowski space. The warping of time associated with Newtonian

gravity is in the "-2V dt^2" term, while the warping of space,

itself, associated with General Relativity is limited to the

substantially smaller "2aV/(1 + 2aV) dr^2" term.

The only effects of paradigm, here, are those limited to the "Special

Relativity" term a du^2 and the "General Relativity" term

2aV/(1 + 2aV) dr^2 (so that whatever "relativistic corrections" there

are, which are made to trajectories have to come from these terms).

The main thrust of gravity - and the essential background behind your

query - resides with the "Newtonian Gravity" term -2 V dt^2.

May 20, 2022, 8:44:14 AMMay 20

to

thing as an inertial frame because no matter how small you make two

points in space and how close together they are they will always be

seperate “inertial” frames. Being pulled towards the Center of gravity

from slightly different angles. So an inertial frame cannot actually

exist in 3D space.

May 20, 2022, 11:15:36 AMMay 20

to

On 5/20/22 8:44 AM, Lou wrote:

> On Sunday, 15 May 2022 at 11:10:45 UTC+1, Tom Roberts wrote:

>> the key concept [is that] locally inertial frames [...] are
> On Sunday, 15 May 2022 at 11:10:45 UTC+1, Tom Roberts wrote:

>> only APPROXIMATELY inertial [...]

> An interesting point you make.

Einstein began thinking about gravity.

> It follows that there is no such

> thing as an inertial frame

manifold of GR with any nonzero energy density, there are no EXACTLY

inertial frames. But in physics, where measurements are always inexact,

approximately inertial frames can be very useful, because in a locally

inertial frame one can use SR rather than GR, which GREATLY simplifies

the analysis. The point is to make sure the approximation involved

affects answers less than measurement resolutions.

[For example, at the LHC no individual event lasts more

than 1 microsecond. A truly inertial frame that is at

rest in the lab at the start of the event will fall at

most a few picometers during the event. As their

detectors have resolutions at least a million times

larger than that, the difference between the lab frame

and that inertial frame is completely negligible.]

> So an inertial frame cannot actually

> exist in 3D space.

space.

Tom Roberts

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu