photo_guy
This was generally raised in another op's thread, but I think it deserves a subject of it's own...

"
I understand we're just 'a few short years' away from tethering satellites to earth by nano strings. So we'll be able to run material, supplies and people up and down these strings in elevators. The height of course would have to be the same 24k miles at the equator.., so we're not talking 'kite'.

Assuming the satellite loses time compared to earth at 24k miles up.., 'time dilation', a clock on the up elevator will go slower and slower as it reaches the top, and faster and faster again on it's return... When it returns it will record the total difference, relative to a clock on earth, that was accumulated from start to finish, including the max slow time spent at the top. That will be a history of the perceived time dilation it incurred. And it will keep that difference for the rest of it's life... As long as it's properly wound every day...

So assuming the two develop a substantial time dilation between them... Even though there is no physical change between the Earth's surface position, the 'vertical' string, and the satellite..., it being in sync with the earth and held by the string. Along with both pov's looking at each other. I understand the dilation is said to occur because of the difference in the two velocities...

But where and how will they come to see the Earth gaining time on them in minutes, days, etc. over an accumulating number of earth rotations.

I can visualize a good line drawing illustrating of this 'time dilation' as it stands.. Sort of an expanding funnel following the string upward to represent more and more dilation... Then comes the rounded time dilation around the satellite at the top.., the whole thing looking like an ice cream cone.

Looks good.., except for an apparent paradox...

A person in the satellite would say 'Hey, the earth is moving faster... Look at all those cars down there racing around like crazy... And.., uh ohhh..., the earth is going round faster too. Heaven forbid... How could this be...? We have a string attached to it and it will stretch too much as the earth tries to wind us up on itself... Maybe it will break after all.., and we'll be sent into space... Or more likely the earth will just wind us in tight and we'll crash on the surface...

But he says., I'm going to ignore it... It's probably just a paradox... I just won't look down any more.
"

How can this be explained..?

Robittybob1
That is so easy I'm surprised you needed to ask.

At that height/orbital time will be slowed therefore the day is longer and there is more time to to get around the Earth and time to untangle the string before it pulls them off course.
photo_guy
QUOTE (Robittybob1+Feb 19 2012, 11:21 PM)
That is so easy I'm surprised you needed to ask.

At that height/orbital time will be slowed therefore the day is longer and there is more time to to get around the Earth and time to untangle the string before it pulls them off course.

I don't think it's quite that easy...

First, the string doesn't pile up to need 'untangling'... Remember it stays taut because of cf pulling on it. Holding the satellite straight above its mooring at the equator... And for the satellite the day is not longer... Things are normal to them. Twenty four hours in a day according to their clocks and bodies... They may have a higher velocity than earth's surface but they don't care... It's just enough so they can complete one rotation in the same period the earth does one rotation... And their clocks would be recording 24 hours to that rotation *without* time slowing...

But something IS slowing the clocks physically and the passengers mentally.., to some slower amount. Whatever that amount, the satellite is in the same time dilation field.., so it too is being slowed physically. Everything is being slowed the very same amount, including their velocity... Because they can't cover the same distance they would if they don't have the required time. So all of this happens while the passengers record the ship as still doing the higher velocity. They ARE running slower but they don't know it. If they WERE to try to go higher for more velocity, time would slow even more for them. They never quite get to where they're gonig.... Because that seems to be what relativity is all about., right.?

Their atomic clocks are running just fine. So time is right on time for them... It matches their body calls for lunch and sleep just fine. But they can actually SEE the earth moving faster than they can move to catch up... That's what time dilation is all about, right..? Some physical things move slower (or faster) depending on it... Else one could look straight down and see the same spot on earth all the time... Would they not..?

They always considering themselves to be right on time... And indeed they are. As we do here on earth... Who's right..? Both are right... What happens to the string..?

Btw.., since they are truly slowed in a real way, in 'time' people say, it would give them less time to get around the earth. So that's in fact why the earth would keep creeping faster ahead of them. I'm not sure where you were getting their 'extra' time from.

They are looking at an earth rotating faster than they can keep up to it... They are *seeing* the earth move faster with their eyes too... Because it IS moving faster than they are. According to the current theory... It is covering more ground than they are. Time dilation does that, they say. Sooooo, is the string now supposed to prevent that..?

Actually.., and I am totally convinced..., because I know it happens... There ARE geo sync satellites... And while they do run physically slower than they would without 'time dilation' or whatever all around them.., they really don't separate distance wise. In fact.., and in my theory too...

That is because it is NOT time that's going slower in one area than another... So they don't physically move at different speeds in relation to each other.

Velocity causes their aging.., but it doesn't tie time into itself.., so it can't affect itself... Velocity does not change, whether time exists or not..

But the objects do AGE at different physical speeds while staying in place. They could stay tied together forever, and the passenger in the satellite would stay young and able to look down on that one spot on earth forever... But over many thousands of orbits of the sun he would stay physically connected with no change of position. He would perhaps see the jungle give way to dessert.., and perhaps the earth splitting apart to form a new sea..... Or see earth covered with a soot cloud from an asteroid hit.., for 50-100 years.., killing off most of what lives under it..

The passenger would be running in low gear, but he wouldn't realize it.. His day would be full. He would just be watching something that was going through a very speedy evolution below.

All based on a much higher than likely 'amount' of relativity for sure.., but only for emphasis...

IMHO it's the answer that has all the answers... They don't move apart because it is gravity that controls aging.., but it doesn't do a thing to time itself... Time isn't time being warped and bent as the current thought implies...

So you'll probably need more than a couple of lines to explain how time dilation does it..?

Can you explain it using time dilation..? Keep everything relative now. And I hope you can.
Robittybob1
With the GPS satellites which were traveling faster at a lower orbit they had to account for the influence of two factors:
1. The speeding up of time in a lower gravitational field.
2. The slowing of time because of the velocity to maintain that orbit.
And the second factor was greater than the first.

Now it is possible that at the speed and height of the geosynchronous orbit these two factors actually cancel each other out, for the orbit is higher and slower.
I have a feeling that this is the case, since the effects of SR are not mentioned in the case of these satellites.
photo_guy
QUOTE (Robittybob1+Feb 20 2012, 03:05 AM)
With the GPS satellites which were traveling faster at a lower orbit they had to account for the influence of two factors:
1. The speeding up of time in a lower gravitational field.
2.  The slowing of time because of the velocity to maintain that orbit.
And the second factor was greater than the first.

Now it is possible that at the speed and height of the geosynchronous orbit these two factors actually cancel each other out, for the orbit is higher and slower.
I have a feeling that this is the case, since the effects of SR are not mentioned in the case of these satellites.

Yes.., I agree the GPS's are seen to have both influences... Not sure which is winning with them...

But I'm only able to relate the paradox using geo sync orbits. Polar and other orbits would be too complicated to follow easily. I do understand the two influences... One is that mass gravity rules when the satellite is close enough to be affected by it more than it's inertia gravity can overcome it to any significant degree..

I mean like being on the mountain top.., where gravity has a little less hold.., time and aging speeds up some... But as you keep going up.., plane, balloon or satellite.., there is a slow shifting from earth mass effect to whatever velocity inertia there is. A shifting away from real earth time to a velocity rules situation. Where the 'slowing' can become more and more pronounced with increased speed.

With all satellites, once they're in free fall.., there should be no earth gravity effect on them at all. It being cancelled out by an identical quantity of centrifugal gravity. Leaving only the forward inertial gravity to affect them, and their 'time'.

So I'm assuming for the sake of the example that the inertia will be sufficient to slow the satellite time more than surface time. Actually it would make no difference which way it went... The end result would just be the opposite... Earth would be seen lagging behind in its rotation when seen from the satellite.

We could almost guarantee a large time shift with a space ship at 100,000 miles out that was watching and staying in sync with a spot on the equator.. But that's a lot of string... So a geo-sync on a leash works nicely. Easy to understand.

Can you see what I'm saying about 'time dilation' causing a paradox..?

We know it doesn't really happen as the time dilation says it should... And that's not ALL that's involved and unworkable in that situation either.

But my posit that gravity influences 'aging', while not affecting time, covers all the bases nicely...? It's predicts what *does* happen. Mine actually 'works', in all the areas I've put it too... Even explaining c being constant in all the various velocity situations out there.

But I'm open to being disappointed if you can show there's no 'time dilation' paradox.
Robittybob1
The other thing that is obvious is that from the satellite's perspective there is no relative motion between the Earth surface and the itself. Since there is no relative motion they are in the same inertial frame, and therefore have no time dilation affects between them.

Now that may not be the case in the long run for I seem to remember Einstein did say relativity would affect a rotating disk.
AlexG
QUOTE (Robittybob1+Feb 20 2012, 02:10 AM)
Since there is no relative motion they are in the same inertial frame, and therefore have no time dilation affects between them.

Both the surface of the earth and a sat in orbit are undergoing accelerated motion. Neither one is an inertial framework.

QUOTE
Now that may not be the case in the long run for I seem to remember Einstein did say relativity would affect a rotating disk.

It's not the case in the short run either.

XXIII. Behaviour of Clocks and Measuring Rods on a Rotating Body of Reference

photo_guy
QUOTE (Robittybob1+Feb 20 2012, 07:10 AM)
The other thing that is obvious is that from the satellite's perspective there is no relative motion between the Earth surface and the itself.  Since there is no relative motion they are in the same inertial frame, and therefore have no time dilation affects between them.

Now that may not be the case in the long run for I seem to remember Einstein did say relativity would affect a rotating disk.

The satellite will be rotating on it's axis perpendicular to it's view of earth as it orbits the earth. It will therefore be at rest to itself and see the earth revolving around IT, at a speed of 24k x 2 x 3.1416 per rotation. Roughly..? So there would be relative motion and 'time dilation'...

If time existed.

As for the disk.., I see no puzzle in it...

Relating to AlexG's post and the link to a K and K' example.

QUOTE (AlexG+)

It's not the case in the short run either.

XXIII.  Behaviour of Clocks and Measuring Rods on a Rotating Body of Reference

The first part of that link's 'explanation' is basically that the author can't explain it either. That he's guilty of a certain 'slovenliness of treatment' of the details. At bottom he says he does not want to go further into it with a similar example.

I suggest that he's using the wrong concept.

Because..., he resolves his case with the two clocks on the revolving disk, one placed at centre and one at edge, thusly... "To make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest). For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with respect to the body of reference."

I guess he can't figure out how a *solid* rotating body can run in two different time zones, which would indicate one part is moving physically slower than another part. Eventually to be ripped apart... Actually there would be an infinite number of *time* zones from centre/rest to the 'slowest' at the edge.

But I see no dilemma.... Simply remove the imagined concept of time and all you have left is physical 'aging'. The clock on the outside edge is simply 'slowed down' by gravity, with the amount of slowing according to the strength of it. As though its spring was nearly unwound. But as I've said before, its slowing will not relate to it's own area of 'correct time' at all.., because there is no such thing. Time if it exists at all is actually flat across the universe. IMHO of course.

But this simple switch of concepts clears up all kinds of questions. So I'm hoping it will be *strongly* considered.

As for the disk... With the rods placed around it..? Those that shrink in length as they travel in the direction they are pointing...

First of all, they shrink only during 'acceleration'. The amount of shrinkage will depend on the incremental increase of acceleration. And they will return to their normal length once that acceleration is slowed and stopped... When they reach what is to be their constant velocity. Of course they won't know the shrinkage happened.

This follows with the rods around the outside of the spinning disk... They will only shorten if the speed of the disk spin is increasing. They will return to normal when it reaches constant velocity.

So if one sees those rods as welded to the outside edge of the disk.., one can see that as a large disk that is starting to spin, and as it increases its rpm or rpusec, that the large disk will become a smaller disk... It will shrink in size.

That guy with the measuring rods around the outer edge will see then shrinking.., but only because the outer edge is also shrinking, as is every incremental point inward by an appropriate amount. And at any time the rods are laid to measure diameter.., they WILL come up with pi... Because the physical size of the disk has become smaller during acceleration. This will be seen by K... Because that is a totally separate phenomenon than 'time dilation' or my 'rate of aging'. It is strictly a matter of compression, whether linear or circular twisting. As the physical properties of the disk fight back against the *changing* of inertia.

The disk will return to normal size as it lessens it 'acceleration' and settles into it's constant velocity.

It's all easy to see when one doesn't use math, and/or trusts other's blaze marks.
boit
photo_guy
QUOTE (boit+Feb 20 2012, 04:49 PM)

It's basically the same 'situation'.., except there are more errors of thinking in the Ehrenfest explanation...

The first big reason is that it seems to be passing the buck.., by saying the physical disk will break before achieving relativistic speeds of rotation... lol

Secondly, a Born rigid cylinder CAN shrink. It will remain rigid while in one time reference point.., as it will in all other time dilation points it shrinks to... As I believe was later alluded to, according to the second paragraph in the sub link 'Ehrenfest's argument'. Not to mention my mention of shrinkage from acceleration, or 'spinning up' as it's called here...

Thirdly, if one only consider the outside edge, they are missing the point... Consider the disk as being an 'infinite' number of concentric circles.., from the outside inward. The number of circles only limited by the width of an atom and it's 'space'... But each one of them affected by contraction to the extent needed to meet the requirement of Lorentz contraction. That of course would NOT lead to a flat disk 'dishing out' either... So using a cylinder to avoid it was pointless...

Fourth.... Ehrenfest paradox - The circumference of a rotating disk should contract but not the radius, as radius is in perpendicular to the direction of motion. This is just NOT true...

Any size disk, in any time frame, will have the same relative forces applied to it... They will not be held in place by the last position they held in time... That's all gone now, you know..?

freenergy
I truly think everyone is overlooking the simplicity in this.

Unless there is a time loss calculated initially it will forever remain lost. Unless of course the rotating object, in this case earth, ever stops rotating. The longer the tether the more time lapse.

Take any spherical object and start it on a steady rotation, lets say for instance you have a balloon in front of you that is maintaining a consistant velocity and direction. Now take a piece of string and attach it to any part of the balloon. No matter what, that string and whatever object at the end will not duplicate the rotation of the balloon until the string has become taught and in a fixed location. the longer the string the more the lag before tethered object begins its time. In the end the time difference will never change or vary. That is until there is ever a decrease in speed by the balloon. Giving the tethered object the ability to close that gap but even then only until its loss in momentum forces the string taught again and it to once again rely on the balloons rotation to give it back a stable and constant motion.
photo_guy
QUOTE (freenergy+Mar 1 2012, 11:09 AM)
I truly think everyone is overlooking the simplicity in this.

Unless there is a time loss calculated initially it will forever remain lost. Unless of course the rotating object, in this case earth, ever stops rotating. The longer the tether the more time lapse.

Take any spherical object and start it on a steady rotation, lets say for instance you have a balloon in front of you that is maintaining a consistant velocity and direction. Now take a piece of string and attach it to any part of the balloon. No matter what, that string and whatever object at the end will not duplicate the rotation of the balloon until the string has become taught and in a fixed location. the longer the string the more the lag before tethered object begins its time. In the end the time difference will never change or vary. That is until there is ever a decrease in speed by the balloon. Giving the tethered object the ability to close that gap but even then only until its loss in momentum forces the string taught again and it to once again rely on the balloons rotation to give it back a stable and constant motion.

If you're talking about a sat going around the earth, the centripetal force is already gravity so we need no string to keep the sat in place. It's boosted up to orbit and once there, stays there. Its exact speed at 24 k determined only by a spot on the ground below... If a string is attached between them it will only be taught enough to keep the elevator in a straight line, for the shortest string possible, and to have the least amount of barf inside when it arrives.. So it won't be pulling on the satellite.., 'much'. It will probably require a slightly higher orbit to wash any possible string weight.., and the elevator pull-downs that occur every so often. Remembering that every action gets a reaction... But I'm sure they'll figure that out at the time..., if they ever get to build the string, the rest will be easy.

So we know the sat will keep pace with the earth. My only use of a leash was to point out that the people up there would KNOW they and their clocks are simply running slower because they can see the REAL time below. As least the real time according to those who built the clocks... To match the rotation of the earth... They will know their clock is simply 'running' slower now, by an hour and a half per earth's rotation meridian to meridian. They're not way out in space away from earth thinking the earth has actually sped up it's orbits while their clocks are staying true... The leash gives physical proof that inertial gravity is causing a slowing of physical aging of the atoms that comprise the sat and everything in it, including the clocks.

Therefore, imagined time is moot... Because it really doesn't change at all... Whatever there may be of it.. it should be fairly obvious that it is flat across the universe and that physical objects simply have different rates of operation... They 'age' differently..., compared to the time frames we have on-going here on earth.

Robittybob1
QUOTE (Robittybob1+Feb 20 2012, 07:10 AM)
The other thing that is obvious is that from the satellite's perspective there is no relative motion between the Earth surface and the itself. Since there is no relative motion they are in the same inertial frame, and therefore have no time dilation affects between them.

Now that may not be the case in the long run for I seem to remember Einstein did say relativity would affect a rotating disk.

I think I'm right about it being in the same frame as the Earth so no loss of time. The satellite being able to be tethered shows there is no relative motion between them, they can't be moving wrt each other and able to be tethered at the same time. That is a paradox in itself otherwise!
photo_guy
QUOTE (Robittybob1+Mar 1 2012, 05:11 PM)
I think I'm right about it being in the same frame as the Earth so no loss of time.  The satellite being able to be tethered shows there is no relative motion between them, they can't be moving wrt each other and able to be tethered at the same time.  That is a paradox in itself otherwise!

If the edge of a disk can be affected by speed relative to its center.., then the satellite is likewise affected compared to earth being its center...

The sat is moving faster than the earth... Everything moving through the cosmos is relative to the entire cosmos..., not really some other single object.., not even it's own center. That center has it's own relativity to the cosmos.

As for the paradox.., there is no paradox... Because there is no 'time dilation'. The satellite is simply not *aging* as fast as the earth is aging... Time, if it is supposed to exist.., remains the same for both. The sat's atoms are simply slowed by the gravitational flux of the cosmos that it is traveling through, at a higher rate of speed than the earth. Similar to the counter-emf produced in a motor... So it's clocks run slower too. A result of their atoms running slower than they did where they were made.., in earth's lesser speed and gravity.

It is the thee dimensional gravity web of the cosmos that is everywhere and constantly fluctuating. It is not *time* doing that... But more like the internet... Every computer being a part of it and contributing to the overall web. It's a network that keeps on ticking... But not using time in this case... lol
Robittybob1
QUOTE (photo_guy+Mar 2 2012, 02:46 PM)

If the edge of a disk can be affected by speed relative to its center.., then the satellite is likewise affected compared to earth being its center...

The difference I see is the satellite is affected by the two opposing effects: gravity and velocity. Whereas the disk is only affected by velocity. So one is time dilated but the other is at the point in space where the two cancel each other out.
photo_guy
QUOTE (Robittybob1+Mar 2 2012, 07:36 PM)
The difference I see is the satellite is affected by the two opposing effects: gravity and velocity.  Whereas the disk is only affected by velocity.  So one is time dilated but the other is at the point in space where the two cancel each other out.

The satellite is affected by all three... Mass, centrifugal and inertia... Mass and centrifugal cancel each other out so the sat can stay in freefall orbit.

It's the inertial gravity that's slowing their clock down. Although I don't rule out the possibility that both mass and centrifugal both play a part too... I strongly suspect they do. They balance each other out as far as keeping the sat in orbit but that's not necessarily saying they don'e affect the sat clocks's time keeping ability to some extent too...

They may be like two cars chained to something in the middle and pulling in opposite directions. The thing in the middle isn't moving either way but it would certainly be under some stress....
Robittybob1
QUOTE (photo_guy+Mar 3 2012, 01:27 AM)

The satellite is affected by all three...  Mass, centrifugal and inertia...  Mass and centrifugal cancel each other out so the sat can stay in free fall orbit.

It's the inertial gravity that's slowing their clock down.  Although I don't rule out the possibility that both mass and centrifugal both play a part too...  I strongly suspect they do.  They balance each other out as far as keeping the sat in orbit but that's not necessarily saying they done affect the sat clock's  time keeping ability to some extent too...

They may be like two cars chained to something in the middle and pulling in opposite directions.  The thing in the middle isn't moving either way but it would certainly be under some stress....

Satellites are held in their orbits at many different heights and the closer to the Earth they are the faster they are orbiting. So if relativity is correct the Space Station might be a place to confirm that there is time dilation.

But I feel you still haven't understood that the higher the orbit the slower the satellite moves, so the time dilation due to velocity declines the higher you go. But the other time-speeding-up effect of lower gravity is increased the further from the Earth you go so there is a point where the two opposing effects balance.
What height does that happen?
photo_guy
QUOTE (Robittybob1+Mar 3 2012, 01:53 AM)
Satellites are held in their orbits at many different heights and the closer to the Earth they are the faster they are orbiting.  So if relativity is correct the Space Station might be a place to confirm that there is time dilation.

But I feel you still haven't understood that the higher the orbit the slower the satellite moves, so the time dilation due to velocity declines the higher you go.  But the other time-speeding-up effect of lower gravity is increased the further from the Earth you go so there is a point where the two opposing effects balance.
What height does that happen?

I understand satellites fairly well.. I worked at Western Union's Glenwood Earth Station for a while in the late 70's. At one time there were 7 Westars up at once.

So I've always been thinking geo-sync orbits... 24 k miles up where mass and centrifugal nutralize each other and the sat has enough inertia to keep it in free fall orbit.

So inerta is what is slowing the clocks. Although I suspect mass and centrifugal still play a part... They may balance as far as keeping the sat in orbit but I'm sure they're affecting the clock's atoms to do that. Like two cars tied to something in the middle and pulling in opposite directions. The something wouldn't be moving but it would certainly be under stress.

In any case, I never got into the physics end of it at the time so I don't know if there was a relative time slowing. I don't remember any being mentioned... The only delay I knew about was up and down transmission times.

So as for your question at what height 'speed up' from height and 'slow down' from velocity washes.., it's still a question. 24 k miles up may very well be where it happens. When all three are in sync to keep the sat in one place above the earth. ?

But right now, to me, it's only just a possible. And only possible because it IS a balance point that's locked to the earth's rotation. And you don't need a ladder on a mountain top to be there.

Cause with sats at least.., their velocity is going to be what's needed to keep the mass and cf balanced. If you went real close to earth and didn't have that mountain top to put a ladder on.., a sats velocity would be too intense to get a meaningful measurement.

It certainly wouldn't be in geo sync, which I believe is what your question ultimately requires...

Robittybob1
That is right - the GPS satellites do have clocks which are slowed and the problem must only get worse as you come in closer. For your velocity rises and the G force increases as the radius is shorter.
Guiri

Time Dilation is pure fantasy.

Here's an easy question for the logically challenged.
Read the second section of Einsteins paper (1905) Electrodynamics of moving bodies.

tB-tA and t'A-tb = Time taken for the rod travelling between A and B
c-v and c+v = Velocities of the rod travelling between A and B
rAB = Length of the Rod (not distance travelled)

What is the length of the rod at rest and at velocity c?

Velocity of rod at rest is 0, therefore rAB = 0
Velocity of rod at c is c, therefore rAB = 2c

This means the rod at rest is zero length and if it travels at c then it is twice the length it has actually travelled,

Does something look wrong?
[Moderator: Your abuse of math and physical assumptions looks wrong. A rod cannot travel between locations at well-defined times because one end arrives before the other. There are other problems and a lack of clear explanation. For a better introduction to length contraction, you are required to go read and understand this post before making any assertions in physics. The penalty for not fully understanding this post before continuing will be a complete loss of faith in your good intentions by the moderator.]
Guiri
QUOTE (Guiri+May 3 2012, 10:39 PM)

Time Dilation is pure fantasy.

Here's an easy question for the logically challenged.
Read the second section of Einsteins paper (1905) Electrodynamics of moving bodies.

tB-tA and t'A-tb = Time taken for the rod travelling between A and B
c-v and c+v = Velocities of the rod travelling between A and B
rAB = Length of the Rod (not distance travelled)

What is the length of the rod at rest and at velocity c?

Velocity of rod at rest is 0, therefore rAB = 0
Velocity of rod at c is c, therefore rAB = 2c

This means the rod at rest is zero length and if it travels at c then it is twice the length it has actually travelled,

Does something look wrong?
[Moderator: Your abuse of math and physical assumptions looks wrong. A rod cannot travel between locations at well-defined times because one end arrives before the other. There are other problems and a lack of clear explanation. For a better introduction to length contraction, you are required to go read and understand this post before making any assertions in physics. The penalty for not fully understanding this post before continuing will be a complete loss of faith in your good intentions by the moderator.]

Thank you for your comment moderator,

I make no assumptions nor abuse maths, they are Einstein's straight out of his 1905 paper.

"where rAB denotes the length of the moving rod—measured in the stationary system" (not the length).

First equation
Stationary rod at origin of x-axis v = 0 and t = 0
rAB = (tB-tA).(c-v)

tB-tA = 0 rod has not moved
and c-0 = c velocity is zero

rAB = (0).(c-0) = 0

Length of rod is zero

Second equation
rAB = (t'A - tB).(c+v)

v = c
t'A - tB = assume 1 second

rAB = 1.c+c = 2c

Alternatively

tB = d/c and t'A = 2(d/c)
d/c(2c) = 2d

rAB = 2d.2c

the math is simple
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