This gave me deeper understanding. Actually RIM will experience the length of circumference in accordance with PI=3.14 ( he will issue 314 signals ) but it will be Hub who will calculate/estimate Rim's "There-Now" reality projection to have longer circumference (corresponding to PI=3.16) and will be wondering why Rim is issueing his light signals more seldom than he should. Then Hub will reajust Rim's experience by relativity effects saying that even though real (according to Hub's calculations) "There-Now" projection is correponding to PI=3.16 the effects of relativity make Rim to experience shrinked reality corresponding to PI=3.14 and as well makes Rim's time-flow slower so Rim issues 314 signals instead of 316 :). So Rim sends 314 signals and Hub registers 314 signals but Hub was expecting to register 316 signals and can only blame relativity length and time dilation effecting Rim's behavior.

What is interesting here is that Hub's estimation of what is there-now (what is the circumference contemporal with himself ) is just a loosy attempt to view universe old style way as if speed of light was infinite.

Thanx Mike for explanation. btw i have to point to you one misleading statement of yours:

> "RIM will discover that he has a longer road to hoe when going around the circumference."
Actually Rim will not discover longer than normal (with PI=3.14) length , it is rather Hub who will estimate/think of Rim's reality in the way as if PI=3.16 for Rim.

Sorry my man, but it's quite correct.... :-)
I think we're stuck in some 'conceptual incongruity' between us here. It will actually take more end-to-end lengths of RIM's IFR to get around the circumference. They will both count 316. The central idea of the scenario is that a radially directed length measurement is unchanged from the non-rotating situation, while a tangentially directed measurement is contracted with respect to the non-rotating frame. ( ASIDE: We don't really need to send signals, just get both HUB and RIM to count and compare notes when RIM returns )
I'll analogise by going down some dimensions, but please beware of any expectations. Think of it this way:
(A). Draw a circle on a 'flat' plane. This is done by selecting a point ( call it the centre ) and by some measuring device mark all points on that plane that are at a given fixed distance ( the radius ) from that centre. Call it 'DISC' and measure it all up, and you'll get the standard Euclidean PI.
(B). Draw a circle on a 'sphere' shape. You select a point ( say the 'North pole' for a definite image ) and while staying on the sphere mark out the set of points which are at a constant distance from it. Call it 'CAP' and measure it. If you've used the same radius for case A, the circumference will be less and the ratio of the two will be less than PI. To check this you could get your handy scissors out, and thus remove CAP from the sphere by slicing around the circumference. Then make a number of cuts around from the edge radially into the centre ( but not quite ) and then flatten it out. Now overlie it, centre to centre, onto DISC on the plane from case A. CAP will 'splay' and leave gaps in it's rim compared to DISC.
(C). Draw a circle on a 'saddle' shape. Again select a point ( say where you'd park your groin if you sat on it ) as the centre, and yet again while staying on the saddle trace out the set of equidistant points on it. Call this 'CRISP' and measure it up. By using the same radius as in case A, this will yield a greater circumference and hence a bigger ratio than PI. Now as with case B, you could check this by removing it with scissors etc and overlying it on DISC in the plane. CRISP will be 'ruffled' like pleats on a dress, and you'll be blessed with extra rim in comparison with DISC.
Our HUB and RIM story analogises to case C, and not to case B as per some pre-conceptions. The distortion in spacetime here can't be viewed by us or HUB or RIM from without. I've cheated with cases A, B and C by embedding in a space of more dimensions. As HUB and RIM can't really do that, they make a series of local measurements and conclude their situation. For HUB and RIM geometry is what they can measure. They begin with a non-rotating ( inertial ) frame, and then spin up the wheel and introduce apparent forces. The bloody geometry changes with them in it !!
( ASIDE: I did another simplification too, in my original explanation, so I'd better 'fess up. To measure a distance in a geometry, one actually integrates a metric. A metric defines the distance between one point and a neighbouring one, 'infinitesimally' close. To measure some discrete gap you add up ( integrate ) a bunch of infinitesimal jumps along a path between the endpoints of interest. RIM was roughly approximating that. Hopefully we could imagine a limiting process whereby some distance was measured repeatedly ( all else constant ) with progressively shorter rulers. This will yield a numerical series of approximate numbers for said distance. For the radial path we would get 50, 50, 50, 50.... regardless of the ruler size, because it is a 'straight' path. For the tangential/circumferential path we would get 314, 314.1, 314.15, 314.159....... because it is a 'bent' path, and we can get as close to the real deal PI as we like by choosing a sufficiently small ruler. )
Cheers, Mike.

( edit ) I hope I wasn't too ambitious with my original example. I was trying to demonstrate a leap from Special Relativity's time and distance alterations to General Relativity's full warping. :-(

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

I think we're stuck in some 'conceptual incongruity' between us here. It will actually take more end-to-end lengths of RIM's IFR to get around the circumference. They will both count 316

You mean if there is a gravity source in the center ?

but if there is no gravity source in the center and Rim is using for example a rope tied to Hud (Hud is pulling Rim towards center) or a magnet then both Rim and Hud count 314.

You mean if there is a gravity source in the center ?

No, not at all. RIM is rotating, this means he is accelerating. The further out he goes the greater that will be. The rope pulls against that and keeps RIM with us. Einstein's Equivalence Principle ( EEP ) will allow us to locally ( wherever we choose ) consider an acceleration as gravity. RIM could EITHER say "I am being spun around a centre, what a good thing this rope stops me flying off" OR "I am dangling at the end of this rope, what a good thing it stops me from falling down under gravity". He will feel 'weightful'. Both are legitimate views, and you can pass from one to the other via EEP.
HUB's situation at the centre is not the same as RIM's. HUB is not accelerating if he is at the exact centre of rotation, which is an untrue approximation really - another 'small lie' - but it doesn't break the scenario. EEP will allow us to consider 'no acceleration' as 'no gravity'. HUB could EITHER say "I am in free fall" OR "I am far from any massive body". He will feel 'weightless'. Again go from one to the other via EEP.
I should repeat that EEP is a local rule.
Perhaps I should not have chosen this counter-intuitive example, but I thought it would be instructive. You see, the farther RIM ventures from HUB the greater is his local gravity. Most of us are used to the idea of gravity lessening from some centre!

Quote:

Can we finally agree on that ? :)

No, alas, we can't. :-)
It's hard to find a good 'gut' example, but spacetime actually does warp!! The agreement between experiment and General Relativity is so phenomenally good ( thus far at least ), it's such a damn shame it hasn't been melded with Quantum Mechanics - another counter-intuitive, but phenomenally good predictive theory ...... :-(
Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

You mean if there is a gravity source in the center ?

No, not at all. RIM is rotating, this means he is accelerating.

Oh man, if there is a gravity source at the center than ok i can agree the circumference length will probably correspond to PI=3.16 (i still need a prove of it)
but if Rim is circling around empty spot (no gravity source there) then i can not agree, sorry. If there is no gravity source Rim will experience normal flat geometry that is he will measure PI=3.14
Rim can choose the speed of circling around by choosing acceleration he wants. Does it mean he can choose how much he curves the space-time ? And what if other person is circling the same circle at different speed ? everyone has his own curved space, do they happen to live in different universes then ?

if there is a gravity center then it makes everyone experience the same curved space-time.

for me Einstein's Equivalence Principle simply means that locally one can change the forces that cause acceleration of test object with curvature of space-time (it is purely the choice of mathematical representation) but it does not mean the space time actually gets curved by those (e.g. electromagnetic or other) forces. take Coriolis force for example - it is not a real force but rather a convinient mathematical representation

take Coriolis force for example - it is not a real force but rather a convinient mathematical representation

GR takes this idea to the next level. There are no forces involved. The covariant acceleration of each world line is identically zero. Force is mass times acceleration so the covariant force is also zero.
This is the only context known in which EEP can be consistently applied. By itself the EEP says nothing about space being curved. Mikeâ€™s statement about EEP is historically accurate, debugasâ€™s is revisionist.

Oh man, if there is a gravity source at the center than ok i can agree the circumference length will probably correspond to PI=3.16 (i still need a prove of it)

A gravity source AND a rotating wheel?

Quote:

Rim can choose the speed of circling around by choosing acceleration he wants.

I think it is better to say he chooses his position on the wheel ( if he can stay on it ) and the rest follows as described.

Quote:

Does it mean he can choose how much he curves the space-time ?

No, it is the motion that determines the distortion, whether he's there or not. RIM's choice is where in that curvature he lies, and EEP locally swaps his descriptions at that site - as MarkF says EEP alone doesn't make the curvature. RIM has been travelling around the various spots measuring it all up.

Quote:

And what if other person is circling the same circle at different speed ?

Different place in the warp ( and weft ) of the fabric of spacetime.

Quote:

everyone has his own curved space, do they happen to live in different universes then ?

No, we each have our own local ( or personal ) reference frame. At least I'd rather think that I am conversing now with someone in the same universe. Is your sky purple and/or is it raining donuts? It is in mine. :-)

Quote:

if there is a gravity center then it makes everyone experience the same curved space-time.

Assuming you, or some test mass, are not large enough to distort it. :-)

Quote:

for me Einstein's Equivalence Principle simply means that locally one can change the forces that cause acceleration of test object with curvature of space-time

EEP swaps descriptions. The curvature comes from mass or motion.

Quote:

(it is purely the choice of mathematical representation) but it does not mean the space time actually gets curved by those (e.g. electromagnetic or other) forces.

Ahhhh ...... I guess that's a good philosophical point. Nonetheless if you assume the truth of the curvature of spacetime as a proposition and run through the machinery of GR to yield some prediction - you'll arrive at numbers which stunningly agree with measurements ( well, in those cases when you can calculate it all out and do the experiment ). Science says: what's real is what you measure. The LIGO's are a case in point. Have you seen a single electron? I haven't either, but we have an 'electron model' which produces great results in many, many circumstances by agreeing with experience. Whatever image/model/concept you have of the electron is only useful for science if it 'works' come testing time.

Quote:

take Coriolis force for example - it is not a real force but rather a convinient mathematical representation

We have some sugar cane and banana growers DownUnder, here and now, who would not agree. In their personal frame it's been real alright. :-(
Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: This gave me deeper

)

Sorry my man, but it's quite correct.... :-)

I think we're stuck in some 'conceptual incongruity' between us here. It will actually take more end-to-end lengths of RIM's IFR to get around the circumference. They will both count 316. The central idea of the scenario is that a radially directed length measurement is unchanged from the non-rotating situation, while a tangentially directed measurement is contracted with respect to the non-rotating frame. ( ASIDE: We don't really need to send signals, just get both HUB and RIM to count and compare notes when RIM returns )

I'll analogise by going down some dimensions, but please beware of any expectations. Think of it this way:

(A). Draw a circle on a 'flat' plane. This is done by selecting a point ( call it the centre ) and by some measuring device mark all points on that plane that are at a given fixed distance ( the radius ) from that centre. Call it 'DISC' and measure it all up, and you'll get the standard Euclidean PI.

(B). Draw a circle on a 'sphere' shape. You select a point ( say the 'North pole' for a definite image ) and while staying on the sphere mark out the set of points which are at a constant distance from it. Call it 'CAP' and measure it. If you've used the same radius for case A, the circumference will be less and the ratio of the two will be less than PI. To check this you could get your handy scissors out, and thus remove CAP from the sphere by slicing around the circumference. Then make a number of cuts around from the edge radially into the centre ( but not quite ) and then flatten it out. Now overlie it, centre to centre, onto DISC on the plane from case A. CAP will 'splay' and leave gaps in it's rim compared to DISC.

(C). Draw a circle on a 'saddle' shape. Again select a point ( say where you'd park your groin if you sat on it ) as the centre, and yet again while staying on the saddle trace out the set of equidistant points on it. Call this 'CRISP' and measure it up. By using the same radius as in case A, this will yield a greater circumference and hence a bigger ratio than PI. Now as with case B, you could check this by removing it with scissors etc and overlying it on DISC in the plane. CRISP will be 'ruffled' like pleats on a dress, and you'll be blessed with extra rim in comparison with DISC.

Our HUB and RIM story analogises to case C, and not to case B as per some pre-conceptions. The distortion in spacetime here can't be viewed by us or HUB or RIM from without. I've cheated with cases A, B and C by embedding in a space of more dimensions. As HUB and RIM can't really do that, they make a series of local measurements and conclude their situation. For HUB and RIM geometry is what they can measure. They begin with a non-rotating ( inertial ) frame, and then spin up the wheel and introduce apparent forces. The bloody geometry changes with them in it !!

( ASIDE: I did another simplification too, in my original explanation, so I'd better 'fess up. To measure a distance in a geometry, one actually integrates a metric. A metric defines the distance between one point and a neighbouring one, 'infinitesimally' close. To measure some discrete gap you add up ( integrate ) a bunch of infinitesimal jumps along a path between the endpoints of interest. RIM was roughly approximating that. Hopefully we could imagine a limiting process whereby some distance was measured repeatedly ( all else constant ) with progressively shorter rulers. This will yield a numerical series of approximate numbers for said distance. For the radial path we would get 50, 50, 50, 50.... regardless of the ruler size, because it is a 'straight' path. For the tangential/circumferential path we would get 314, 314.1, 314.15, 314.159....... because it is a 'bent' path, and we can get as close to the real deal PI as we like by choosing a sufficiently small ruler. )

Cheers, Mike.

( edit ) I hope I wasn't too ambitious with my original example. I was trying to demonstrate a leap from Special Relativity's time and distance alterations to General Relativity's full warping. :-(

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: I think we're stuck in

)

You mean if there is a gravity source in the center ?

but if there is no gravity source in the center and Rim is using for example a rope tied to Hud (Hud is pulling Rim towards center) or a magnet then both Rim and Hud count 314.

Can we finally agree on that ? :)

## RE: You mean if there is a

)

No, not at all. RIM is rotating, this means he is accelerating. The further out he goes the greater that will be. The rope pulls against that and keeps RIM with us. Einstein's Equivalence Principle ( EEP ) will allow us to locally ( wherever we choose ) consider an acceleration as gravity. RIM could EITHER say "I am being spun around a centre, what a good thing this rope stops me flying off" OR "I am dangling at the end of this rope, what a good thing it stops me from falling down under gravity". He will feel 'weightful'. Both are legitimate views, and you can pass from one to the other via EEP.

HUB's situation at the centre is not the same as RIM's. HUB is not accelerating if he is at the exact centre of rotation, which is an untrue approximation really - another 'small lie' - but it doesn't break the scenario. EEP will allow us to consider 'no acceleration' as 'no gravity'. HUB could EITHER say "I am in free fall" OR "I am far from any massive body". He will feel 'weightless'. Again go from one to the other via EEP.

I should repeat that EEP is a local rule.

Perhaps I should not have chosen this counter-intuitive example, but I thought it would be instructive. You see, the farther RIM ventures from HUB the greater is his local gravity. Most of us are used to the idea of gravity lessening from some centre!

No, alas, we can't. :-)

It's hard to find a good 'gut' example, but spacetime actually does warp!! The agreement between experiment and General Relativity is so phenomenally good ( thus far at least ), it's such a damn shame it hasn't been melded with Quantum Mechanics - another counter-intuitive, but phenomenally good predictive theory ...... :-(

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: RE: You mean if there

)

Oh man, if there is a gravity source at the center than ok i can agree the circumference length will probably correspond to PI=3.16 (i still need a prove of it)

but if Rim is circling around empty spot (no gravity source there) then i can not agree, sorry. If there is no gravity source Rim will experience normal flat geometry that is he will measure PI=3.14

Rim can choose the speed of circling around by choosing acceleration he wants. Does it mean he can choose how much he curves the space-time ? And what if other person is circling the same circle at different speed ? everyone has his own curved space, do they happen to live in different universes then ?

if there is a gravity center then it makes everyone experience the same curved space-time.

for me Einstein's Equivalence Principle simply means that locally one can change the forces that cause acceleration of test object with curvature of space-time (it is purely the choice of mathematical representation) but it does not mean the space time actually gets curved by those (e.g. electromagnetic or other) forces. take Coriolis force for example - it is not a real force but rather a convinient mathematical representation

## RE: take Coriolis force for

)

GR takes this idea to the next level. There are no forces involved. The covariant acceleration of each world line is identically zero. Force is mass times acceleration so the covariant force is also zero.

This is the only context known in which EEP can be consistently applied. By itself the EEP says nothing about space being curved. Mikeâ€™s statement about EEP is historically accurate, debugasâ€™s is revisionist.

## RE: Oh man, if there is a

)

A gravity source AND a rotating wheel?

I think it is better to say he chooses his position on the wheel ( if he can stay on it ) and the rest follows as described.

No, it is the motion that determines the distortion, whether he's there or not. RIM's choice is where in that curvature he lies, and EEP locally swaps his descriptions at that site - as MarkF says EEP alone doesn't make the curvature. RIM has been travelling around the various spots measuring it all up.

Different place in the warp ( and weft ) of the fabric of spacetime.

No, we each have our own local ( or personal ) reference frame. At least I'd rather think that I am conversing now with someone in the same universe. Is your sky purple and/or is it raining donuts? It is in mine. :-)

Assuming you, or some test mass, are not large enough to distort it. :-)

EEP swaps descriptions. The curvature comes from mass or motion.

Ahhhh ...... I guess that's a good philosophical point. Nonetheless if you assume the truth of the curvature of spacetime as a proposition and run through the machinery of GR to yield some prediction - you'll arrive at numbers which stunningly agree with measurements ( well, in those cases when you can calculate it all out and do the experiment ). Science says: what's real is what you measure. The LIGO's are a case in point. Have you seen a single electron? I haven't either, but we have an 'electron model' which produces great results in many, many circumstances by agreeing with experience. Whatever image/model/concept you have of the electron is only useful for science if it 'works' come testing time.

We have some sugar cane and banana growers DownUnder, here and now, who would not agree. In their personal frame it's been real alright. :-(

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal