...and since 3-dimensional space is curved, who knows about the number or orientation of dimensions... YAR!
If you'd like to check this WMAP site on the shape of the universe, Is the Universe Infinite? you'll see an interesting diagram regarding the geometry of the "bulk" of 3D space, and the WMAP mission has measured it to a very high degree of accuracy and determined that the geometry of the universe is flat (within a 2% margin of error). This space will be "curved" near a concentration of mass, and the mass gives you an orientation with which to work. I think the question of additional dimensions relates mostly to the nature of quantum mechanics...
...and since 3-dimensional space is curved, who knows about the number or orientation of dimensions... YAR!
Curved space (or, more correctly spacetime) does not change the number of dimensions. This is to say, at any point in spacetime, you can always find exactly 4 perpendicular directions, of which three will look like space and one will look like time (although, no such choice of directions is unique). What curvature means is that if you do this systematically at every point, the directions you've picked at different points will not coincide.
Let me try to give a concrete example.
I think we will all accept that the Earth's surface is curved. Imagine that you stand at the North pole and face South, looking straight ahead. Your friendly neighborhood alien watching you from her spaceship would see you looking in a direction perpendicular to the Earth's axis.
Now, imagine that you walk forward (south) until you reach the equator. In the context of our curved space, Earth's surface, you are still facing the same direction - South. But, the alien now sees you looking in a direction parallel to the Earth's axis.
Now, you may not have noticed this, but I just cheated a little in that example. The reason I say this is that I made reference to a space in which the curved one is embedded. I did that by citing the Earth's axis. and comparing orientations with it. In the context of General Relativity, that is a luxury we don't have, because there is no bigger space to compare to. So, we need ways to measure the effects of curvature without such references. So, let me demonstrate one of those using the Earth's surface as a model.
This time, we'll start at the equator. You are standing at the equator, facing East, and carrying an arrow that, for the moment, points straight ahead. Now, you walk straight ahead for about 10000 km (one quarter of the way around the world). At this point, you turn 90 degrees to your left without turning the arrow, so the arrow still points East. Again, you walk 10000 km straight forward (North), taking you to the North pole. Once again, you turn 90 degrees to the left without turning the arrow, so it now points straight behind you. One last time, you walk 10000 km straight ahead, taking you back to the very point at which you started. But, while you are at the same location you started, not everything is the same. Your arrow, which started out pointing East now points North; but, you never once turned it.
This effect is a hallmark of curved spaces. What we've done here is called "parallel transporting a vector" - we've moved your arrow around in such a way that every move we make, the arrow is pointing in a direction parallel (in the context of the space) to the direction it pointed immediately before. Yet, we were able to find a path the we could travel where coming back to the point we started at, we found that the direction of the arrow had changed. Imagine trying a similar procedure on a tabletop. No matter what path you picked, the arrow would always be in the same direction when you got back to where you started.
So, we've seen a way to look at curvature. Now, the last thing is to note that, even though the Earth's surface is curved, at every point on it, there are exactly two perpendicular directions you can choose to move and stay on the surface.
... So, we've seen a way to look at curvature. Now, the last thing is to note that, even though the Earth's surface is curved, at every point on it, there are exactly two perpendicular directions you can choose to move and stay on the surface.
A related effect is seen on a 'rhumb line' of constant bearing on the globe, which is generally not even a geodesic, let alone straight in 3-space. But on a Mercator projection it maps to a straight line, at the cost of distorting the scale and shape of the continents.
Curved space .......... can choose to move and stay on the surface.
Beautifully put!
That reminds me of another ( thought experiment ) way of demonstrating curvature. Recall that Special Relativity predicts the shortening of whichever dimension is aligned with the direction of ( relative ) motion. So if I drive a car in the usual fashion facing forwards, but really fast with respect to another observer ( say one stationary with respect to the ground ) then shortening of the front-to-back length of the car may be observed from that viewpoint. There are some subtleties as to how I would measure and demonstrate that effect but let's take it as read. The car's measurable dimensions in other axes are unchanged like the driver-side-doorhandle-to-passenger-side-doorhandle distance, and height from bottom-of-tyre-tread-to-top-of-radio-aerial. Unless the car slides sideways or somesuch!
Now let's compare two observers, Hubble and Rimmer, each of which has a one Imperial Foot Ruler ( IFR ) that they have measured as identical when laid side by side together. Place both 'Hub' and 'Rim' and their IFRs at the central axis of a sufficiently large but initially non-rotating wheel. ( They can check for non-rotation by performing some dynamic experiment anywhere on or about the wheel and finding no 'apparent' forces. That is we have an inertial frame ). Let there be spokes out to the circumference. Suppose now that Rim takes his IFR and marks out the distance to the edge along a spoke and finds it to be 50 Imperial Feet. He then proceeds to do the same for the whole circumference too, and not surprisingly comes up with a little over 314 Imperial feet. So let's calculate a number 314/(50*2) and call it PI.
Bring Rim back to Hub at the centre and spin things up, really get that wheel whizzing 'round, maybe give the guys ( Rim especially ) some shackles and gear to stay on. Keep the time rate of change of angle constant. Send Rim out to do it all again. He and Hub will observe 50 IFR's worth of radial distance. From Hub's ( and Rim's ) viewpoint there's no shortening of the length of Rim's IFR as all motion is perpendicular to that ( because the IFR is aligned with the radial direction ). Now going around the outer part of the wheel is different, as Rim's IFR is shortened from Hub's position, because it is now laid out with it's length along the direction of motion ( in a tangential direction ). There will be somewhat more than the previous 'a little over 314 Imperial feet', say 316 to be definite. Rim will probably scratch his head over this, for even without re-comparing IFRs, he'll wonder why it took more of his IFR lengths this time to circumnavigate. Now calculate a new number 316/(50*2) and call it PIED.
So what's happened? Without any Hollywood special effects the value of the ratio of the circumference to the radius has changed, and all by 'mere' rotation! Curvature of space!? Tangential shrinkage of objects!? Hmmmmmm.. you will note that we have an 'apparent' force in this rotating frame, which is why Rim in particular needs some grappling equipment lest they part company abruptly.
Let us say that in addition to the outermost circumferential path, there were also constructed other circular tracks at differing distances but each centred on the axis of rotation. If Hub could get Rim to go out again and repeat the measuring process with each track, they could calculate the circumference to radius ratio for each. Yes, you guessed it, they would find a greater deviation from PI with our PIED's as Rim measures up progressively more distantly from Hub. What also increases as you progressively move outward from Hub? Quite right! It's that 'apparent' force which may actually throw dear Rim off the whole kit'n'kaboodle. ( The difference in force strength for an object which significantly spans these different tracks would be 'tidally affected' ).
Now if Rim is still taking instructions, you could put him out at some fixed radius with one of these circular tracks to measure. Get him to repeat the processes etc.. this time with different angular rates for each measurement session ( but a constant value during a given session). How do the PIED's vary with angular rate of change now? Yup, the faster you go 'round the greater the deviation from PI.
Clearly our Dynamic Duo could claim that the ratio: circumference/radius increases with apparent force, or if you like acceleration. There is a time effect as well - Rim will have aged slightly less than Hub - as could be demonstrated by comparing upon Rim's return to the centre any clocks that were synchronised prior to his departure outwards. ( This test was more or less done on a trans-Atlantic flight a few decades ago using atomic clocks ). I guess that will be Rim's reward, though it will be pitifully small, for his exertions. That ageing differential will be accentuated with increasing distance from Hub and/or faster rotation rate. Hub's and Rim's experiences are not symmetric. This is analogous to the so-called Twin's Paradox.
All you need now to completely jump to General Relativity is the Equivalence Principle. This allows you to locally swap an acceleration for gravity, and thus predict distance and time behaviour as you go 'deeper' you go into the gravity well, or further out from Hub if you like! For me, the real buzz of Einstein's brilliant insight was the converse - that gravity can be thought of as an 'apparent' force due to choice of frame!
I suppose we could be more elaborate/dynamic with our variation of this theme, like steadily increasing rotation rates during measurements, perhaps cutting a rope or two and letting Rim experience 'free fall' ( for a while ), or ( try ) letting the very outer wheel rim approach the speed of light. But I sense a staff revolt, as they're sick of vomiting due to motion sickness, poor/no pay, and no worker's insurance or site allowances. I've made a few simplifying assumptions, and some small lies, but they don't affect the qualitative result.
Whew! 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
OK, I said I'd come back to the notion of energy in curved spacetime:
Basically, it's hard to define one except in special circumstances. Very generally, you can define energy in a way that is related to how things change with time. If you live in flat spacetime - and locally our neighborhood is a pretty good approximation - it's very easy to pick one timelike direction out of the four and look for changes along it. But if you're near a couple of orbiting black holes, for instance, spacetime is wrapping around itself in a complicated way and it's hard even to pick out a good time coordinate.
We can still talk about the energy or the mass of the orbiting black holes if we're far away from the system, where spacetime has gotten pretty flat. When we're that far out, the small residual curvature of spacetime due to the black holes looks just like what we'd get if we replaced the holes by lumps of matter with a certain mass. That's what we mean when we say the mass or the energy of the black holes, although there's no matter there.
Nice explanation of curvature, Solomon. That business with the arrow changing direction even though you came back where you started is basically how LIGO works. The light bounces down the arms and comes back where it started, but it's picked up a phase shift due to the curvature from a passing gravitational wave.
"Rim will probably scratch his head over this, for even without re-comparing IFRs, he'll wonder why it took more of his IFR lengths this time to circumnavigate. Now calculate a new number 316/(50*2) and call it PIED."
Actually i must disagree. RIM will measure the same 314/(50*2) and the value of PI will still hold true because space shrinks/dilate in the direction u move (or am i missing something here?). IT IS Only Hub that will see deviations with his sending light signals back and force.
Let's imagine instead Rim is sending light signals to the center each meter he passes. How many signals will Rim send per one full circle ? 314 or 316 ?
Rim will send 314 but Hub will see 316 because he gets the signals with delay (or am i missing something here?)
Actually i must disagree. RIM will measure the same 314/(50*2) and the value of PI will still hold true because space shrinks/dilate in the direction u move (or am i missing something here?). IT IS Only Hub that will see deviations with his sending light signals back and force.
Let's imagine instead Rim is sending light signals to the center each meter he passes. How many signals will Rim send per one full circle ? 314 or 316 ?
Rim will send 314 but Hub will see 316 because he gets the signals with delay (or am i missing something here?)
It's weird stuff, isn't it? Don't worry, it's simply counter-intuitive and we have no 'natural' language for it - which often gives rise to subtle 'conceptual incongruities' with explanations. We never experience it, which is both a blessing and a curse ..... :-)
HUB will see RIM's IFR's ( when held transverse to a radial arm ) subtend a narrower angle. There will be more of them, end to end, for a full circle's worth.
RIM will discover that he has a longer road to hoe when going around the circumference. They'll both agree on the radial distance out, 50 units, and how many of RIM's IFR's fit into the circumference. They'll agree in their PIED calculation, thus the 'curvature of space'.
( ASIDE: As RIM himself is also affected in his own personal dimensions, then any comparison between himself and his own IFR will be the same as at other stages. A similiar effect occurs with time, and RIM won't know that his thinking has slowed, until he compares clocks with HUB upon return ).
It makes a mockery of the idea of a 'rigid body' too. At different radii from HUB the material composing our wheel is progressively stretched as we go out - which is fortunately a pretty intuitive result. The internal forces ( basically electromagnetic ) that hold the caboodle together will be challenged to supply sufficient cohesion in both the radial and tangential directions.
Some of my 'small lies' were that it would all hold together, and RIM would survive under the G's, for an effect of order ( 316 - 314 ) / 314 = 2/314 at 50 odd feet out. I doubt both, but I didn't actually calculate anything. :-)
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
>"HUB will see RIM's IFR's ( when held transverse to a radial arm ) subtend a narrower angle."
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.
...and since 3-dimensional
)
...and since 3-dimensional space is curved, who knows about the number or orientation of dimensions... YAR!
RE: ...and since
)
If you'd like to check this WMAP site on the shape of the universe, Is the Universe Infinite? you'll see an interesting diagram regarding the geometry of the "bulk" of 3D space, and the WMAP mission has measured it to a very high degree of accuracy and determined that the geometry of the universe is flat (within a 2% margin of error). This space will be "curved" near a concentration of mass, and the mass gives you an orientation with which to work. I think the question of additional dimensions relates mostly to the nature of quantum mechanics...
What's YAR? Yet another reboot? :)
RE: What's YAR? Yet
)
Yet Another Rightangle?
You're Absolutely Right?
Yonder Asymmetric Reality?
Yell At Riemann?
:-)
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: ...and since
)
Curved space (or, more correctly spacetime) does not change the number of dimensions. This is to say, at any point in spacetime, you can always find exactly 4 perpendicular directions, of which three will look like space and one will look like time (although, no such choice of directions is unique). What curvature means is that if you do this systematically at every point, the directions you've picked at different points will not coincide.
Let me try to give a concrete example.
I think we will all accept that the Earth's surface is curved. Imagine that you stand at the North pole and face South, looking straight ahead. Your friendly neighborhood alien watching you from her spaceship would see you looking in a direction perpendicular to the Earth's axis.
Now, imagine that you walk forward (south) until you reach the equator. In the context of our curved space, Earth's surface, you are still facing the same direction - South. But, the alien now sees you looking in a direction parallel to the Earth's axis.
Now, you may not have noticed this, but I just cheated a little in that example. The reason I say this is that I made reference to a space in which the curved one is embedded. I did that by citing the Earth's axis. and comparing orientations with it. In the context of General Relativity, that is a luxury we don't have, because there is no bigger space to compare to. So, we need ways to measure the effects of curvature without such references. So, let me demonstrate one of those using the Earth's surface as a model.
This time, we'll start at the equator. You are standing at the equator, facing East, and carrying an arrow that, for the moment, points straight ahead. Now, you walk straight ahead for about 10000 km (one quarter of the way around the world). At this point, you turn 90 degrees to your left without turning the arrow, so the arrow still points East. Again, you walk 10000 km straight forward (North), taking you to the North pole. Once again, you turn 90 degrees to the left without turning the arrow, so it now points straight behind you. One last time, you walk 10000 km straight ahead, taking you back to the very point at which you started. But, while you are at the same location you started, not everything is the same. Your arrow, which started out pointing East now points North; but, you never once turned it.
This effect is a hallmark of curved spaces. What we've done here is called "parallel transporting a vector" - we've moved your arrow around in such a way that every move we make, the arrow is pointing in a direction parallel (in the context of the space) to the direction it pointed immediately before. Yet, we were able to find a path the we could travel where coming back to the point we started at, we found that the direction of the arrow had changed. Imagine trying a similar procedure on a tabletop. No matter what path you picked, the arrow would always be in the same direction when you got back to where you started.
So, we've seen a way to look at curvature. Now, the last thing is to note that, even though the Earth's surface is curved, at every point on it, there are exactly two perpendicular directions you can choose to move and stay on the surface.
RE: ... So, we've seen a
)
A related effect is seen on a 'rhumb line' of constant bearing on the globe, which is generally not even a geodesic, let alone straight in 3-space. But on a Mercator projection it maps to a straight line, at the cost of distorting the scale and shape of the continents.
RE: Curved space ..........
)
Beautifully put!
That reminds me of another ( thought experiment ) way of demonstrating curvature. Recall that Special Relativity predicts the shortening of whichever dimension is aligned with the direction of ( relative ) motion. So if I drive a car in the usual fashion facing forwards, but really fast with respect to another observer ( say one stationary with respect to the ground ) then shortening of the front-to-back length of the car may be observed from that viewpoint. There are some subtleties as to how I would measure and demonstrate that effect but let's take it as read. The car's measurable dimensions in other axes are unchanged like the driver-side-doorhandle-to-passenger-side-doorhandle distance, and height from bottom-of-tyre-tread-to-top-of-radio-aerial. Unless the car slides sideways or somesuch!
Now let's compare two observers, Hubble and Rimmer, each of which has a one Imperial Foot Ruler ( IFR ) that they have measured as identical when laid side by side together. Place both 'Hub' and 'Rim' and their IFRs at the central axis of a sufficiently large but initially non-rotating wheel. ( They can check for non-rotation by performing some dynamic experiment anywhere on or about the wheel and finding no 'apparent' forces. That is we have an inertial frame ). Let there be spokes out to the circumference. Suppose now that Rim takes his IFR and marks out the distance to the edge along a spoke and finds it to be 50 Imperial Feet. He then proceeds to do the same for the whole circumference too, and not surprisingly comes up with a little over 314 Imperial feet. So let's calculate a number 314/(50*2) and call it PI.
Bring Rim back to Hub at the centre and spin things up, really get that wheel whizzing 'round, maybe give the guys ( Rim especially ) some shackles and gear to stay on. Keep the time rate of change of angle constant. Send Rim out to do it all again. He and Hub will observe 50 IFR's worth of radial distance. From Hub's ( and Rim's ) viewpoint there's no shortening of the length of Rim's IFR as all motion is perpendicular to that ( because the IFR is aligned with the radial direction ). Now going around the outer part of the wheel is different, as Rim's IFR is shortened from Hub's position, because it is now laid out with it's length along the direction of motion ( in a tangential direction ). There will be somewhat more than the previous 'a little over 314 Imperial feet', say 316 to be definite. Rim will probably scratch his head over this, for even without re-comparing IFRs, he'll wonder why it took more of his IFR lengths this time to circumnavigate. Now calculate a new number 316/(50*2) and call it PIED.
So what's happened? Without any Hollywood special effects the value of the ratio of the circumference to the radius has changed, and all by 'mere' rotation! Curvature of space!? Tangential shrinkage of objects!? Hmmmmmm.. you will note that we have an 'apparent' force in this rotating frame, which is why Rim in particular needs some grappling equipment lest they part company abruptly.
Let us say that in addition to the outermost circumferential path, there were also constructed other circular tracks at differing distances but each centred on the axis of rotation. If Hub could get Rim to go out again and repeat the measuring process with each track, they could calculate the circumference to radius ratio for each. Yes, you guessed it, they would find a greater deviation from PI with our PIED's as Rim measures up progressively more distantly from Hub. What also increases as you progressively move outward from Hub? Quite right! It's that 'apparent' force which may actually throw dear Rim off the whole kit'n'kaboodle. ( The difference in force strength for an object which significantly spans these different tracks would be 'tidally affected' ).
Now if Rim is still taking instructions, you could put him out at some fixed radius with one of these circular tracks to measure. Get him to repeat the processes etc.. this time with different angular rates for each measurement session ( but a constant value during a given session). How do the PIED's vary with angular rate of change now? Yup, the faster you go 'round the greater the deviation from PI.
Clearly our Dynamic Duo could claim that the ratio: circumference/radius increases with apparent force, or if you like acceleration. There is a time effect as well - Rim will have aged slightly less than Hub - as could be demonstrated by comparing upon Rim's return to the centre any clocks that were synchronised prior to his departure outwards. ( This test was more or less done on a trans-Atlantic flight a few decades ago using atomic clocks ). I guess that will be Rim's reward, though it will be pitifully small, for his exertions. That ageing differential will be accentuated with increasing distance from Hub and/or faster rotation rate. Hub's and Rim's experiences are not symmetric. This is analogous to the so-called Twin's Paradox.
All you need now to completely jump to General Relativity is the Equivalence Principle. This allows you to locally swap an acceleration for gravity, and thus predict distance and time behaviour as you go 'deeper' you go into the gravity well, or further out from Hub if you like! For me, the real buzz of Einstein's brilliant insight was the converse - that gravity can be thought of as an 'apparent' force due to choice of frame!
I suppose we could be more elaborate/dynamic with our variation of this theme, like steadily increasing rotation rates during measurements, perhaps cutting a rope or two and letting Rim experience 'free fall' ( for a while ), or ( try ) letting the very outer wheel rim approach the speed of light. But I sense a staff revolt, as they're sick of vomiting due to motion sickness, poor/no pay, and no worker's insurance or site allowances. I've made a few simplifying assumptions, and some small lies, but they don't affect the qualitative result.
Whew! 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
OK, I said I'd come back to
)
OK, I said I'd come back to the notion of energy in curved spacetime:
Basically, it's hard to define one except in special circumstances. Very generally, you can define energy in a way that is related to how things change with time. If you live in flat spacetime - and locally our neighborhood is a pretty good approximation - it's very easy to pick one timelike direction out of the four and look for changes along it. But if you're near a couple of orbiting black holes, for instance, spacetime is wrapping around itself in a complicated way and it's hard even to pick out a good time coordinate.
We can still talk about the energy or the mass of the orbiting black holes if we're far away from the system, where spacetime has gotten pretty flat. When we're that far out, the small residual curvature of spacetime due to the black holes looks just like what we'd get if we replaced the holes by lumps of matter with a certain mass. That's what we mean when we say the mass or the energy of the black holes, although there's no matter there.
Nice explanation of curvature, Solomon. That business with the arrow changing direction even though you came back where you started is basically how LIGO works. The light bounces down the arms and comes back where it started, but it's picked up a phase shift due to the curvature from a passing gravitational wave.
Hope this helps,
Ben
"Rim will probably scratch
)
"Rim will probably scratch his head over this, for even without re-comparing IFRs, he'll wonder why it took more of his IFR lengths this time to circumnavigate. Now calculate a new number 316/(50*2) and call it PIED."
Actually i must disagree. RIM will measure the same 314/(50*2) and the value of PI will still hold true because space shrinks/dilate in the direction u move (or am i missing something here?). IT IS Only Hub that will see deviations with his sending light signals back and force.
Let's imagine instead Rim is sending light signals to the center each meter he passes. How many signals will Rim send per one full circle ? 314 or 316 ?
Rim will send 314 but Hub will see 316 because he gets the signals with delay (or am i missing something here?)
RE: Actually i must
)
It's weird stuff, isn't it? Don't worry, it's simply counter-intuitive and we have no 'natural' language for it - which often gives rise to subtle 'conceptual incongruities' with explanations. We never experience it, which is both a blessing and a curse ..... :-)
HUB will see RIM's IFR's ( when held transverse to a radial arm ) subtend a narrower angle. There will be more of them, end to end, for a full circle's worth.
RIM will discover that he has a longer road to hoe when going around the circumference. They'll both agree on the radial distance out, 50 units, and how many of RIM's IFR's fit into the circumference. They'll agree in their PIED calculation, thus the 'curvature of space'.
( ASIDE: As RIM himself is also affected in his own personal dimensions, then any comparison between himself and his own IFR will be the same as at other stages. A similiar effect occurs with time, and RIM won't know that his thinking has slowed, until he compares clocks with HUB upon return ).
It makes a mockery of the idea of a 'rigid body' too. At different radii from HUB the material composing our wheel is progressively stretched as we go out - which is fortunately a pretty intuitive result. The internal forces ( basically electromagnetic ) that hold the caboodle together will be challenged to supply sufficient cohesion in both the radial and tangential directions.
Some of my 'small lies' were that it would all hold together, and RIM would survive under the G's, for an effect of order ( 316 - 314 ) / 314 = 2/314 at 50 odd feet out. I doubt both, but I didn't actually calculate anything. :-)
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
>"HUB will see RIM's IFR's (
)
>"HUB will see RIM's IFR's ( when held transverse to a radial arm ) subtend a narrower angle."
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.