[timey (OP)]

Where does length contraction fit into this scenario?

The Earth's depth to centre of gravity is greater at sea level equator than it is at sea level poles.
This means that a clock 'would' run faster at sea level equator than a clock at sea level polar apart from the fact that:
The Earth's radius at the equator is greater than it is at the poles, and therefore centripetal motion at the equator is greater than it is at the poles, where the time dilation caused by the increased motion of radius exactly cancels out the time dilation caused by the increased altitude from centre of Earth at that radius, and clocks at sea level equator will run at the same rate as clocks at sea level polar.

SR calculations for time dilation are usually made via the Lorentz transformations which also incorporate the contraction of length.

Where does length contraction apply for the SR calculations of centripetal motion of radius?

[timey (OP)]

Any chance of someone answering the question?

[Bored chemist]

Where does length contraction fit into this scenario?

The Earth's depth to centre of gravity is greater at sea level equator than it is at sea level poles.
This means that a clock 'would' run faster at sea level equator than a clock at sea level polar apart from the fact that:
The Earth's radius at the equator is greater than it is at the poles, and therefore centripetal motion at the equator is greater than it is at the poles, where the time dilation caused by the increased motion of radius exactly cancels out the time dilation caused by the increased altitude from centre of Earth at that radius, and clocks at sea level equator will run at the same rate as clocks at sea level polar.

SR calculations for time dilation are usually made via the Lorentz transformations which also incorporate the contraction of length.

Where does length contraction apply for the SR calculations of centripetal motion of radius?

Not my field, but I think the answer may be "tangentially" - in more ways tan one.

[timey (OP)]

Ah... I have just realised that I have made one of my terminology gaffs again.
Where I have said radius, when what I mean is circumference.

But 'tangentially' suggests that if we measure the distance of circumference of the equator from West to East, and then measures the distance of circumference of the equator from East to West, that these measurements of the equatorial circumference will differ.

But by remit of logic, the distance of the equatorial circumference remains as constant.

Apart from the fact that the equatorial bulge is created by the centripetal motion, which suggests dilation, not contraction, the inline to motion measurement of the equatorial circumference doesn't actually result in a distance that is physically shorter, does it?

[Bored chemist]

I'm sat on the Earth's surface. The Earth is rotating.
My velocity, on account of that, is tangential to the Earth's surface (rather than radial)
As far as I understand it, that's the direction in which I'd  have my length contracted.

[timey (OP)]

But what about the length of the equatorial cicumferance?
Measuring the equatorial circumference, i.e. tangential, from West to East will result in differing measurement than when measuring from East to West, and I'm sorry but it just defeats the realms of physical possibility that the equatorial circumference of the Earth can be physically possessed of two differring lengths.

[Janus]

But what about the length of the equatorial cicumferance?
Measuring the equatorial circumference, i.e. tangential, from West to East will result in differing measurement than when measuring from East to West, and I'm sorry but it just defeats the realms of physical possibility that the equatorial circumference of the Earth can be physically possessed of two differring lengths.

It depends on what you mean by measuring the circumference, if you mean that someone traveling West relative to the Earth's rotation would measure a different distance than someone traveling East, then yes, but that would be because they were measuring the distance from two different rotating reference frames.    And the fact that someone at the equator and rotating with the Earth would measure a different radius to circumference ratio than someone not rotating with the Earth is due to the non-Euclidean nature of space-time in a non-rotating frame. (Even someone not rotating with the Earth would, if they measured close enough, not get exactly pi for this ratio, due to the fact that Earth's gravitational field causes space-time in its vicinity to be non-Euclidean.)

[timey (OP)]

But if 2 people were to set off from the same equatorial location, at the same time, moving at the same rate, reeling out an impossibly long architects measuring tape until they both reached the location they started from, not only would the clock at that location agree that both had completed the journey in the same amount of time, but the amount of tape reeled out would also be the same...

So what physical meaning in terms of the distance of the circumfrence of the equator does the length contraction measurement have?

Edit:
Ok, as an after thought, the equatorial circumference is 24,901 miles...
Let's say we made the impossibly long architects measuring tapes that the 2 people are reeling out exactly 24,901 miles long.
According to the length contraction mathematics the inline motion measuring tape will contract on its way round.

So the question is - will the circumference of the equator also contract inline motion, or will the inline motion person run out of measuring tape before he gets back to the starting point?

I daresay that there may be an argument where one considers that the person measuring in the other direction might end up with just the right amount of extra tape, where the point at which both the measuring reels running out of tape has a meeting point slightly to the East* of the original starting point?

*edit: I said East, that should be West of the starting point,

[Colin2B]

According to the length contraction mathematics the inline motion measuring tape will contract on its way round.

So the question is - will the circumference of the equator also contract inline motion, or will the inline motion person run out of measuring tape before he gets back to the starting point?

You have really answered your own question. Everything you try to measure will appear contracted, circumference, tape, etc.
Now apply that to your eg.

[jeffreyH]

Ok. So set two people on the equator and have them each measure out a mile. One walking in the direction of rotation and the other against. Now both aim a laser at a set angle to meet at a predetermined point say north of the equator. This should tell you what difference they have in their measurements without impossibly long measuring tapes.

[timey (OP)]

According to the length contraction mathematics the inline motion measuring tape will contract on its way round.

So the question is - will the circumference of the equator also contract inline motion, or will the inline motion person run out of measuring tape before he gets back to the starting point?

You have really answered your own question. Everything you try to measure will appear contracted, circumference, tape, etc.
Now apply that to your eg.

Actually I don't think I have answered my own question at-all.  What I have done is raised a whole bunch of questions...

Both persons have a measuring tape that is measured as 24,901 miles long, this being the circumference distance of the equator.

Q1: How was the distance of the circumference of the equator measured to be a distance of 24,901 miles?

Q2: How were the measuring tapes ascertained as being 24,901 miles long?

Person 1 sets out in the Easterly direction of inline motion and we are expecting a length contracted measurement.

Q3: Does the 24,901 mile measuring tape run out of length before person 1 gets back to the starting point?

Q4: Or does the end of the 24,901 mile measuring tape run out at the starting point because the circumference of the Earth is also contracted?

Person 2 sets out in the Westerly direction where we are expecting that no length contraction will occur.

Q5: Does the 24,901 mile measuring tape meet up with the starting point?

Q6: Or does the 24,901 mile measuring tape run out of length to the West of the starting point because the circumference distance is contracted inline motion?

Question 6 presents us with a paradox because if the answer to Question 4 is yes this is because the circumference has contracted inline.
Person 2's measuring tape will not be contracted and therefore should also extend back to the starting point, so the answer to Question 5 is also yes, but this would mean that the circumference of the Earth is possessed of 2 differing distances simultaneously.

If the answer to Question 5 is yes and the answer to Question 3 is yes, we end up with the same paradox.

If we consider that it is the measuring of the circumference that is contracted rather than the circumference itself where the answer to Question 3 is yes:

If the inline motion person 1's tape runs out West of the starting point, and person 2's tape extends West of the starting point, this would be because the answer to Question 6 is yes, where both measuring tape ends are meeting at this point West of the starting point.

If the answer to Question 6 is yes, and the answer to Question 3 is yes, this presents the same paradox in that the Earth's circumference is both contracted and not contracted.

If we considered that the length contraction was due to person 1's additional motion relative to centripetal motion, and that person 2's length dilation was due to a reduced motion relative to centripetal motion, then this would give explanation of the 2 persons running out of measuring tape West of the starting point.

Now we have eleviated the paradox but...

Q7: Considering that both measuring tapes are 24,901 miles long, where has the length contraction or length dilation of the equatorial circumference, i.e that that was being measured, occurred?

Both Question 6 and Question 7 lead us back to Question 1, where Question 1 leads to Question 2.

But your reply has raised another question:

Now apply that to your eg.

I'm sorry, but you have completely lost me... Could you explain what an eg is please?

[Janus]

But if 2 people were to set off from the same equatorial location, at the same time, moving at the same rate, reeling out an impossibly long architects measuring tape until they both reached the location they started from, not only would the clock at that location agree that both had completed the journey in the same amount of time, but the amount of tape reeled out would also be the same...

If they are reeling out the tapes as they walk, the tape will at rest relative to the Earth frame and thus give an circumference reading as measured from the Earth frame.   To measure it by their own frame they would have use a measuring device that is stationary relative to themself. (for example, assuming each knew exactly how long his stride was, and could maintain that stride perfectly for the whole trip around the Earth, by counting their steps and multiplying by the number of steps they will come up with as  having walked a different distance in after completing their circumnavigation of the Earth.

[timey (OP)]

The initial question stated the persons as circumnavigating the Earth in opposing directions at the same speed.

Therefore the inline motion person 1 is moving at 1,040mph+speed x, and the opposing direction person 2 will be moving at 1,040mph-speed x.

Within the mph consideration the rate of an hour is being held relative to the clock at the starting point which is travelling at 1,040mph.
Held relative to the clock at the starting point, +speed x will be decreasing the rate of time for person 1, and -speed x will be increasing the rate of time for person 2.

We must realise now that we are holding speed x relative to the clock at the starting point.

Let's say that both 24,901 mile long tapes are unmarked, and that we give each person a clock and a pen.
Now we will state speed x as 1 metre per second...
We ask person 1 and person 2 to make a mark on their tape for every second that passes on their own clock.

On the basis that the speed x of 1 metre per second is held relative to the clock at the starting point and remains constant:

As person 1 circumnavigates the Earth he will be marking out longer metres, and as person 2 circumnavigates the Earth he will be marking out shorter metres, so we do have a length contraction/length dilation measurement of sorts but the person 1 who is moving with the inline motion will be measuring a dilation of a metre, and the person 2 moving against the inline motion will be measuring a contracted metre.

But both persons will run out of tape at the starting point.

If we now consider that the speed x of 1 metre per second be held relative to each person's time dilated clock, and ask person's 1 and 2 while they circumnavigate the Earth to make marks on their tape for every second that passes on their clocks, person 1's speed x will be slower than person 2's speed x.
Person 1 will be marking out contracted metres, and person 2 will be marking out dilated metres.
Here we can see that the expected inline motion contraction of length is occurring despite the fact that again, both persons will run out of tape at the starting point.

But the question...

Q7: Considering that both measuring tapes are 24,901 miles long, where has the length contraction or length dilation of the equatorial circumference, i.e that that was being measured, occurred?

...still remains to be answered, where it would seem (to me at least) that a non Euclidian geometry is a function of time rather than of distance.

[jeffreyH]

The initial question stated the persons as circumnavigating the Earth in opposing directions at the same speed.

Therefore the inline motion person 1 is moving at 1,040mph+speed x, and the opposing direction person 2 will be moving at 1,040mph-speed x.

Within the mph consideration the rate of an hour is being held relative to the clock at the starting point which is travelling at 1,040mph.
Held relative to the clock at the starting point, +speed x will be decreasing the rate of time for person 1, and -speed x will be increasing the rate of time for person 2.

We must realise now that we are holding speed x relative to the clock at the starting point.

Let's say that both 24,901 mile long tapes are unmarked, and that we give each person a clock and a pen.
Now we will state speed x as 1 metre per second...
We ask person 1 and person 2 to make a mark on their tape for every second that passes on their own clock.

On the basis that the speed x of 1 metre per second is held relative to the clock at the starting point and remains constant:

As person 1 circumnavigates the Earth he will be marking out longer metres, and as person 2 circumnavigates the Earth he will be marking out shorter metres, so we do have a length contraction/length dilation measurement of sorts but the person 1 who is moving with the inline motion will be measuring a dilation of a metre, and the person 2 moving against the inline motion will be measuring a contracted metre.

But both persons will run out of tape at the starting point.

If we now consider that the speed x of 1 metre per second be held relative to each person's time dilated clock, and ask person's 1 and 2 while they circumnavigate the Earth to make marks on their tape for every second that passes on their clocks, person 1's speed x will be slower than person 2's speed x.
Person 1 will be marking out contracted metres, and person 2 will be marking out dilated metres.
Here we can see that the expected inline motion contraction of length is occurring despite the fact that again, both persons will run out of tape at the starting point.

But the question...

Q7: Considering that both measuring tapes are 24,901 miles long, where has the length contraction or length dilation of the equatorial circumference, i.e that that was being measured, occurred?

...still remains to be answered, where it would seem (to me at least) that a non Euclidian geometry is a function of time rather than of distance.

BINGO!

[guest4091]

But what about the length of the equatorial cicumferance?
Measuring the equatorial circumference, i.e. tangential, from West to East will result in differing measurement than when measuring from East to West, and I'm sorry but it just defeats the realms of physical possibility that the equatorial circumference of the Earth can be physically possessed of two differring lengths.

The moving observer experiences time dilation as a function of his relative speed. His interpretation of his time dilation, i.e. the target destination arriving early, results in his conclusion, based on his assuming a pseudo rest frame, that the world is contracted while moving past him in the opposite direction. The contraction is his perception, not the world. His frame is contracted and that makes for a seamless transition from within his frame to beyond his frame. The universe can't move since there is no reference separate from it. Since time dilation depends on speed, moving east or west at a constant ground speed will yield different time dilations which yield different perceptions of distance. Distance is a relation between objects and not a physical entity, nor a property of physical entities.
For a 1st approximation, assume a rotation speed of 1000 mph and a plane/clock speed of 400 mph. The westbound clock is moving east at 600 (relative to earth center) and the eastbound clock is moving east at 1400 mph (rtec). SR time dilation and perception will differ.

[timey (OP)]

Phyti - In light of your reply above, can you address the questions raised on this thread and in particular the question of post 6:

https://www.thenakedscientists.com/forum/index.php?topic=70240.msg512969#msg512969

[guest4091]

While searching for data on the Hafele-Keating experiment, I found this.
arXiv:1405.5174v1 Hafele-Keating experiment, by J.H.Field, associated with CERN.
He states, with a earth centered non-rotating frame, there is no length contraction.
This exception, if true, may be due to the absolute nature of rotation around a point vs linear motion. Will have to study this for a while.

You said:

it just defeats the realms of physical possibility that the equatorial circumference of the Earth can be physically possessed of two differring lengths.

Positions and relations using positions are not object properties, but perceptions. If 10 observers pass an object at 10 different velocities, you will get 10 different perceptions, but the object has not changed. Another rainbow effect!
Go back to the train scenario. The passenger drops an object, it falls vertically in a straight line to the floor. The bystander watching the train pass by sees the object (xray vision) fall in a curved path to the floor. Neither the straight line nor the curve are real physical independent things, but they are real perceptions, reality in the mind.
A trajectory is the minds history of a sequence of positions. Since no two observers can be in the same location simultaneously, each history is personal.

[jeffreyH]

This is definitely David Cooper territory. Consider a measuring tape completely around the circumference of a uniform sphere at its equator. Now unroll the strip so that it sits in flat spacetime. Now send various rockets parallel to the strip at various speeds and ask them to use the markings on the strip to calculate their speed. This then links local time, which is dilated, to the time in the frame of the measuring strip. The strip cannot contract differently for each spaceship. Especially if they are all travelling parallel to each other and overtake each other.

If we start rockets at each end and set them to pass each other then while contracting at each end the measuring tape would have to appear to stretch in the middle.

[guest4091]

This is definitely David Cooper territory. Consider a measuring tape completely around the circumference of a uniform sphere at its equator. Now unroll the strip so that it sits in flat spacetime. Now send various rockets parallel to the strip at various speeds and ask them to use the markings on the strip to calculate their speed. This then links local time, which is dilated, to the time in the frame of the measuring strip. The strip cannot contract differently for each spaceship. Especially if they are all travelling parallel to each other and overtake each other.

If we start rockets at each end and set them to pass each other then while contracting at each end the measuring tape would have to appear to stretch in the middle.

Let the distance =24k miles, time in sec, E denotes earth, A and B anauts.
A transit time at .6c =.22 s E-time. A-time = .22 *.8=.18 s. A-distance = 19.2k.
B transit time at .8c =.16 s E-time. B-time = .16 *.6=.10 s. B-distance = 14.4k.
Each is affected by time dilation and concludes the strip is contracted to reconcile the time difference between E-time and anaut time.
The marks cannot match speed since length contraction is not a linear function of speed. (see the gamma factor)