Are laws of nature really the same in all reference frames?

[zonde]

The equation a=GM/r² is the fundamental equation of Newtonian gravity, so any observation which falsifies Newtonian gravity falsifies that equation. Those observations include:

The anomalous precession of Mercury
The orbital decay of PSR1913+16
Geodetic precession (Gravity Probe B)
Gravitational lensing/deflection
Shapiro delay experiments

All of these experiments falsify Newtonian gravity. It is no use repeating the claim that a=GM/r² is a law of nature when nature disagrees.

When it is clear that there is significant domain where assumptions of the theory hold good you do not say that theory is falsified but instead you say that experiments have established it's domain of applicability.

So no, these experiments do not falsify Newtonian gravity. And because Newtonian gravity talk only about massive objects optical observations (Gravitational lensing, Shapiro delay) have nothing to do with it's predictions.

 

[Bjarne]

I never said it was bad. In fact, I have repeatedly said that it is a close approximation in many circumstances.
Nevertheless, it is not a law of nature.

Maybe, maybe not, - I am not convinced, but I have also not taken any decision.
Have we understood the full range of relativity?
Never say never.

Bjarne
1 second at the ISS is still 1 second, but comparable to 1 second at the Earth it is "stretching" at the ISS.
Hence 1 second at the ISS is comparable longer.
Dalespam
How exactly do you plan on making the comparison?

Sorry, - I mean the opposite
1 second at the ISS is still 1 second, (the definition is correct) but comparable to 1 second at the Earth, time (1 second) is "stretching" at the Earth.

Hence 1 second at the ISS is comparable shorter at the ISS.
When time (1 second) compared to time at a different space-time is "stretching" and speed can be said to doi the same (since speed of light always is the "same") - what prevent us from assuming that also distances are doing exactly and proportional the same,- so that also the 1 meter ruler still is one meter, but comparable stretching, exactly the same rate as speed and time. ?

So the question is, - is the "big kitchen" and "big lake" etc... (together and proportional with strech of time and speed) also "stretching" on the Earth, and comparable "shrinking" in lower gravity space time, as for example on board the ISS or on a low gravity planet etc…?

At least this would make sense, when I would imagine how it would be to jump to an (extreme) different time space reality and walk around. I would expect everything to be the same, - but not comparable the same. (except the magnitude of gravity).

 

[Dale]

When it is clear that there is significant domain where assumptions of the theory hold good you do not say that theory is falsified but instead you say that experiments have established it's domain of applicability.

I am fine with this phrasing. Nevertheless, Bjarne's scenario dealing with gravitational time dilation is outside the domain of applicability of Newtonian gravity.

And because Newtonian gravity talk only about massive objects optical observations (Gravitational lensing, Shapiro delay) have nothing to do with it's predictions.

I disagree here. While the gravitational force between a massive object and a massless object is 0 according to Newton's laws it doesn't take force to accelerate a massless object. So the acceleration of massless objects in Newtonian gravity is still given by a=GM/r².

 

[Dale]

I am not convinced

That is not an indication of a failure in physics or relativity.

1 second at the ISS is still 1 second, (the definition is correct) but comparable to 1 second at the Earth, time (1 second) is "stretching" at the Earth.

Again, how exactly do you plan on making the comparison?

 

[Bjarne]

Again, how exactly do you plan on making the comparison?

I don't understand the question..
Time, and hence 1 second is not comparable the same in different space time, - do you not agree to that?

 

[Dale]

I don't understand the question.

I am asking for a detailed description of what experiment you are planning on doing in order to compare a 1 s on the ISS to 1 s on the earth, and how are you going to interpret different possible results of that experiment in terms of comparing the seconds?

 

[Bjarne]

I am asking for a detailed description of what experiment you are planning on doing in order to compare a 1 s on the ISS to 1 s on the earth, and how are you going to interpret different possible results of that experiment in terms of comparing the seconds?

I cannot see the problem.
Clocks ticks different in different space time.
Let us say you were living in a relative very strong gravitionel field, compared to me.
Let’s say it will take 2 second on my atomic clock, - to observer one second ticking on yours atomic clock.

Opposite it will take you ½ second to observe 1 second ticking on my atomic clock.

The definition of 1 second is for both of us 9,192,631,770 cesium frequency cycles per second.

But it will take a comparable long second (relative double period “of time”) for your clock to reach this number, and only a relative comparable short (½) period, and hence a short second for me to count the same numbers of frequencies.

So the comparable second is not the same. I cannot understand how you seem not to agree to that.

 

[Dale]

Clocks ticks different in different space time.

You are going in circles. I ask you how you are making the comparison and you simply assert that they are "different". Since "different" is a comparison where you find that they are not equal that still leaves the question of exactly how you are making the comparison.

Please answer the question. How are you comparing clock rates at different points in spacetime?

Let us say you were living in a relative very strong gravitionel field, compared to me.
Let’s say it will take 2 second on my atomic clock, - to observer one second ticking on yours atomic clock.

What is the experiment you are proposing here? How do you "observer" the time ticking on your clock and mine?

 

[Bjarne]

You are going in circles. I ask you how you are making the comparison and you simply assert that they are "different". Since "different" is a comparison where you find that they are not equal that still leaves the question of exactly how you are making the comparison.

Please answer the question. How are you comparing clock rates at different points in spacetime?

Satellites proves this every day

What is the experiment you are proposing here? How do you "observer" the time ticking on your clock and mine?

The same answer as above.
I cannot see any other option, - can you ?

 

[Mentz114]

Bjarne, your answers are getting more and more feeble. I think you misunderstand what is meant by "the laws of physics are the same in all inertial frames" so I'll spell it out.

If an experiment is done in lab A to test the law f=ma using billiard ball type experiments and of course measuring with local clocks and rulers, then the result will be the same as in any other lab frame, where of course, they use their local clocks and rulers.

It matters not a jot if the local clocks and rulers are 'different', both frames will verify that f=ma is true, up to a small experimental error.

Going on about seconds or centimeters being different in different frames is irrelevant.

 

[Dale]

I cannot see any other option, - can you ?

Sure, I can see at least 4 ways to compare two distant clocks:

A) Broadcast a reference signal, measure the frequency of the signal locally at each clock
B) Take a reference clock, physically transport it from one clock to the other and measure the rate of the reference locally at each clock
C) Agree on a standard physics experiment as a reference, perform it locally at each clock and measure the time for the experiment
D) Agree on an astronomical reference, and measure the time for the astronomical reference locally

Your suggested measurement of one year with two clocks is an example of D. Your mention of satellites probably refers to A. Mentz114's recent post refers to C, which encapsulates the principle of relativity. I came up with B on my own.

With D and A you will get that the Earth clock and the ISS clock run at different rates. With B and C you will get that the Earth clock and the ISS clock run at the same rates.

The beauty of GR is that it is a single law of physics which explains A, B, C, and D all together.

 

[Bjarne]

Double .

 

[Bjarne]

Sure, I can see at least 4 ways to compare two distant clocks:

A) Broadcast a reference signal, measure the frequency of the signal locally at each clock
B) Take a reference clock, physically transport it from one clock to the other and measure the rate of the reference locally at each clock
C) Agree on a standard physics experiment as a reference, perform it locally at each clock and measure the time for the experiment
D) Agree on an astronomical reference, and measure the time for the astronomical reference locally

Your suggested measurement of one year with two clocks is an example of D. Your mention of satellites probably refers to A. Mentz114's recent post refers to C, which encapsulates the principle of relativity. I came up with B on my own.

With D and A you will get that the Earth clock and the ISS clock run at different rates. With B and C you will get that the Earth clock and the ISS clock run at the same rates.

The beauty of GR is that it is a single law of physics which explains A, B, C, and D all together.

Off course I cannot disagree to that.
I was confused because I thought you did not agree to that, - when times ticks different, in different space-time, then each second also does, - whereby the comparable length off 1 second must be either stretching or shrinking. – So I expected a different kind of answer to your question, not easy to know what that possible could be.

 

[Dale]

whereby the comparable length off 1 second must be either stretching or shrinking.

So I still don't know how you plan on doing the comparison. For the fourth or fifth time, what method are you using to do the comparison? Feel free to simply use one of my four or come up with your own, but be specific.

 

[Mentz114]

Bjarne, I find your post #154 difficult to follow. It is not logically argued and poses more questions than it gives answers.

Have you seen this ?

http://www-istp.gsfc.nasa.gov/stargaze/Smass.htm

It is about experiments carried out on a space station.

There is a universe of evidence that the laws of nature are the same everywhere. We know that atoms continue to emit and absorb the same pattern of spectral lines which have been recognised in millions of astronomical bodies. On a large scale, the GR cosmological models explain most of what we can measure about the universe.

What you are proposing is 'new' physics. If the laws of nature really are laws of nature, they must apply everywhere. If there is experimental evidence to support it, then they must be modified, like relativity altered Newtonian concepts.

Your position is impossible to argue unless you can produce this experimental evidence.

 

[Bjarne]

So I still don't know how you plan on doing the comparison. For the fourth or fifth time, what method are you using to do the comparison? Feel free to simply use one of my four or come up with your own, but be specific.

I cannot see this is neccesery
Either time is ticking different, or it is not.
According to relativity time is ticking different.
Many experiments and experiences confirm this.
So I still cannot see any point with the question.

 

[Dale]

I cannot see this is neccesery
Either time is ticking different, or it is not.
According to relativity time is ticking different.
Many experiments and experiences confirm this.
So I still cannot see any point with the question.

No, according to relativity, if you use methods B or C to do the comparison then time is the same, and only if you use methods A or D to do the comparison then is it different. There is no "hidden reality" under which you can simply say time is different or it is not. There are only experimental measurements, and relativity correctly predicts those.

 

[Bjarne]

B) Take a reference clock, physically transport it from one clock to the other and measure the rate of the reference locally at each clock

What do you mean by "measure the rate of the reference"

 

[Dale]

I mean to take the reference clock, place it locally next to the test clock (either the Earth clock or the ISS clock), and measure the duration of one "tick" of the reference clock using the local test clock.

 

[Bjarne]

So I still don't know how you plan on doing the comparison. For the fourth or fifth time, what method are you using to do the comparison? Feel free to simply use one of my four or come up with your own, but be specific.

B + C is local measurement
A + D is comparable measurement
I agree to all that, but it has not much with the question to do.
I do off course mean A+D

Do we know (for sure) whether distances are comparable different too, so as time and speed is (see A+D), - in different space-time.
I have read that mass-energy conservation will changes the Bohr radius, which also must apply to relativity?

 

[Dale]

I do off course mean A+D

Finally, a straight answer to such a basic question. It is a shame that it took so many posts to get to that.

Do we know (for sure) whether distances are comparable different too, so as time and speed is (see A+D), - in different space-time.

I see how to compare the time on two distant clocks using methods A or D. How would you propose to compare the different distances?

Please, let's not waste another dozen or more posts chasing this down. Just describe your experimental method of comparing two spatially separated distances as clearly and succinctly as possible.

 

[Bjarne]

Finally, a straight answer to such a basic question. It is a shame that it took so many posts to get to that.

I wrote comparable difference from the start.
It confused me you could misunderstand that so much.

I see how to compare the time on two distant clocks using methods A or D. How would you propose to compare the different distances?

Please, let's not waste another dozen or more posts chasing this down. Just describe your experimental method of comparing two spatially separated distances as clearly and succinctly as possible.

Good question.
I believe it is impossible.
Maybe only mathematical possible.
Already we compare space with a stretching rubber band, - or use the expression "curvature"
What does that mean?
Can we get closer to an understandable nature of that?
What happens with the ruler in such “bended” space? ( what do we know or not know)
At least seen from a mathematical perspective, is space stretching like the rubber band?

 

[Dale]

I wrote comparable difference from the start.
It confused me you could misunderstand that so much.

Comparable doesn't have a well-defined meaning in terms of experiment. I.e. there is no experiment called the "comparable" experiment. Writing "comparable" over and over and over was not helpful to the conversation and was one of the main sources of delay. I am not a mind reader to know that by the word "comparable" you meant something like A or D and not something like B or C.

Good question.
I believe it is impossible.

Then there is no use discussing about whether or not lengths change at different locations.

Already we compare space with a stretching rubber band, - or use the expression "curvature"
What does that mean?
Can we get closer to an understandable nature of that?
What happens with the ruler in such “bended” space? ( what do we know or not know)
At least seen from a mathematical perspective, is space stretching like the rubber band?

The word "curvature" does not refer to space stretching like a rubber band. Curvature refers to deviations from (pseudo) Euclidean geometry.

Consider the difference between geometry on a sphere and geometry on a plane. On a plane, two straight lines which are parallel at one point remain parallel and never intersect. On a sphere, two neighboring lattitude lines are straight lines which are parallel at the equator and intersect at the poles. On a plane the sum of the interior angles of a triangle are 180°, but on a sphere the sum of the interior angles of a triangle are greater than 180°. This is the kind of thing that is meant by "curvature".

A rubber band which is stretched into a triangle shape on a flat table still has interior angles which sum to 180°, regardless of the stretching of the band.

 

[Bjarne]

Consider the difference between geometry on a sphere and geometry on a plane. On a plane, two straight lines which are parallel at one point remain parallel and never intersect. On a sphere, two neighboring lattitude lines are straight lines which are parallel at the equator and intersect at the poles. On a plane the sum of the interior angles of a triangle are 180°, but on a sphere the sum of the interior angles of a triangle are greater than 180°. This is the kind of thing that is meant by "curvature".

A rubber band which is stretched into a triangle shape on a flat table still has interior angles which sum to 180°, regardless of the stretching of the band.

Sorry, but it is still very difficult for me to understand how, - reality differences’ - at all is possible (in this case) when distances not is affected too.

Let’s return to the example, - the ISS and the Earth orbiting the exact same orbit around the Sun. – And both exactly 1 orbit.
Both places / both observers (A & B) will locally agree that it will take 1 year, - 31536000 second.
They will also agree about their orbit local speed is average exactly 30000 m/s

• I have claimed that distance cannot be the same.
• You have claimed that these 2 observers follows different path.

1. Do you mean these object follows 2 comparable different path?
2. Do you mean these path locally seems to be exactly the same ?
3. Or how would you describe the “different path”?

Because observer A (on the ISS) he will say it took exactly 31536000 ISS-second to complete one orbit, and the ISS-speed was exact averagely 30000m/s

Observer B on the Earth, orbiting the exact same orbit round the Sun will also say that that it took 31536000 earth-second to complete one orbit and the Earth-speed was exact averagely 30000m/s.

But according to the Shapiro delay experiments we know that the comparablespeed between the ISS and the Earth not is the same, - (even though it locally is the same) .
For exsample, - if it was possible for A and B to reflect a light beam on the Sun and get it back, it would take the exact same local time, and the exact same local speed.
But the comparable time and speed would not be the same.

Let us now say that the comparable speed difference is proven to be 50% .

Option 1.
The only way we mathematical can understand what happens here is that distances also must be 50% comparable different.
This is simple logic. Then both local and comparable different reality can both be true at the same time. - (Seen from both observer A+B+E).

Option 2.
is that we from Earth would see the ISS orbiting the Sun double as fast as the Earth (according to the exaggerated exsample) .
That would contradict that an external observer (E) would see both objects moving with the same (his) speed and using the same (his) time to complete one orbit.

If you disagree, to both these options, - can you please be more detailed according to;

• What you mean with that these 2 objects follows a “different path”?
• How is this at all possible without assuming that comparable distance differences also must be a fact.?

This REALLY confuse me. – Sorry that I not is so quick to understand this.

 

[Dale]

Sorry, but it is still very difficult for me to understand how, - reality differences’ - at all is possible (in this case) when distances not is affected too.

If you cannot come up with an experiment for comparing distances then it doesn't make sense to claim that distances are not affected. You cannot know anything about the distances if you cannot experimentally compare them, so you cannot claim that they are affected and you also cannot claim that they are not affected.

Let’s return to the example, - the ISS and the Earth orbiting the exact same orbit around the Sun. – And both exactly 1 orbit.
Both places / both observers (A & B) will locally agree that it will take 1 year, - 31536000 second.

No, this is option D from my post 151. They will measure 1 astronomical year to be a different number of seconds.

They will also agree about their orbit local speed is average exactly 30000 m/s

How does each measure that?

• I have claimed that distance cannot be the same.
• You have claimed that these 2 observers follows different path.

These claims do not contradict each other.

1. Do you mean these object follows 2 comparable different path?
2. Do you mean these path locally seems to be exactly the same ?
3. Or how would you describe the “different path”?

You like to put the word "comparable" in bold as though somehow putting it in bold magically helps give it some meaning. What is a "comparable different path"? I have never heard the term and as far as I know it has no meaning.

As far as how I would describe the different paths, I would use a 4D coordinate system. I would then write the different paths in terms of parameterized worldlines in the coordinate system. Probably the ecliptic coordinate system with the GPS time coordinate would be the easiest.

But according to the Shapiro delay experiments we know that the comparablespeed between the ISS and the Earth not is the same, - (even though it locally is the same) .

How do you experimentally compare speeds?

If you write "comparable time" then I know what you mean since we already discussed it. But we have not discusses "comparable distances" nor "comparable speeds", so don't bother to use those terms when you have not defined them. When you say "comparable speed" what experiment are you thinking about performing to compare the speed?

For exsample, - if it was possible for A and B to reflect a light beam on the Sun and get it back, it would take the exact same local time

No, this is not correct. This is known as Shapiro delay.

and the exact same local speed.

This is true, the local speed of the light beam is c.

But the comparable time and speed would not be the same.

Let us now say that the comparable speed difference is proven to be 50% .

Again, please define the experiment for comparing speeds.

Option 1.
The only way we mathematical can understand what happens here is that distances also must be 50% comparable different.

Again, what experiment are you using to compare distances?

Option 2.
is that we from Earth would see the ISS orbiting the Sun double as fast as the Earth (according to the exaggerated exsample) .
That would contradict that an external observer (E) would see both objects moving with the same (his) speed and using the same (his) time to complete one orbit.

How would we measure the speed of the ISS and how would the external observer measure both the Earth speed and the ISS speed?

If you disagree, to both these options

Neither of the options is clear enough for me to either agree or disagree with them.

• What you mean with that these 2 objects follows a “different path”?
• How is this at all possible without assuming that comparable distance differences also must be a fact.?

This REALLY confuse me. – Sorry that I not is so quick to understand this.

They follow different paths because the coordinates of their path are different in any coordinate system you might choose.

Again, you put an undefined term in bold as though that helps to convey meaning. It does not, it is simply irritating. What is a "comparable distance difference" and how would you measure it?

Please do not use the word "comparable" or any related word in our further discussions unless you have clearly defined the experiment you would perform to make the comparison.

 

[Bjarne]

Dalespam
They will measure 1 astronomical year to be a different number of seconds.

Yes off course, this was a blunder..

If you cannot come up with an experiment for comparing distances then it doesn't make sense to claim that distances are not affected. You cannot know anything about the distances if you cannot experimentally compare them, so you cannot claim that they are affected and you also cannot claim that they are not affected.

Ok so this is then still an open question. This is what confused me.
The conclusion is then, - it seems more to discover, - but whether it is possible, is then a different question.

Thanks’ a lot for your patience, and detailed explanation.

 

[Dale]

Ok so this is then still an open question.

I wouldn't even call it an open question yet. I would call it an undefined term. Until you have defined all of the terms you don't even have a properly formed question.

Thanks’ a lot for your patience, and detailed explanation.

You are welcome. I will be glad to continue the discussion if needed.

 

[Bjarne]

I wouldn't even call it an open question yet. I would call it an undefined term. Until you have defined all of the terms you don't even have a properly formed question..

Since we know the comparable length of one second is relative stretching or shrinking in different spacetime, for exsample relative to one of ours seconds, we know 1 of 3 factors.

Hence the options are whether speed and /or distance does "the same", and whether that happens proportional.

Which is the most likely option.

• Is it that only speed follows the relative (proportional) stretch of time (and hence relative slowing or speeding)?
• Or is distance also involved in the same relative proportional change?

Of course there could be other combination, according to what these 2 other factors does or not does, - but I think these 2 are the most likely.

I know you twist these words, - but I also think you know what i mean.

 

[Dale]

I know you twist these words, - but I also think you know what i mean.

It is hard to claim that someone is twisting words which are undefined. I certainly do not know what you mean. In fact, I believe that you do not know what you mean either.

Note, this is not a personal criticism, this is a common problem when you are learning a new and difficult subject, the concepts are not clear and words often have subtly different meanings in the new context. It is just a simple fact that you need to be aware of and have patience with. People will be willing to help you, but you need to recognize that communication about this topic is inherently difficult. You need to ask people to define terms when you are confused and you need to be willing to define terms when asked.

GR is mathematically guaranteed to be a self-consistent framework. So, any time that you use an English description and come up with some inconsistency you immediately know that you are not correctly translating between the math and the English. That is typically due to using a poorly defined term or due to making a statement which is well defined and gramatically correct in English but is not mathematically correct.

Here, we have the first case. You use words like "comparable", "the same", "stretch", etc. but without a rigorous definition. Any inconsistencies you come up with are due to that, not due to the theory.

 

[Bjarne]

On the one hand I have heard that when people not get confused first time they hear about reality they haven’t understood it, so I am maybe legally excused (even though it is not first time).
On the other hand it must also be possible to explain it to grandmother according to Einstein.

I think I begin to understand a lot more, especially what we don't know.

Still I would be happy to hear some ideas / opinions (if any) about how the third factor, “distance” in space time, and whether these too possible can be a changing factor too (in the same way as time)
I mean how this factor most likely can be globally understood, - there must at last have been speculations about it, what is the opinion of the majority, - are there different theories, etc.. or is that what we don’t know a dead end. If possible explained in a “grandmother adaptable language” first at all, then later I believe it is easier to understand the math behind, if any.

 

[Bjarne]

Bjarne Wrote
Either speed, distance and/or the definition of 1 second can then not be the same. It is a simple mathematical necessity. I haven't got an excact answer where the "camel is buried"

I am just asking the simple question; - how would these space time realities look like (what would be the comparable difference) if I could jump between them, and see the differences?

I am not an expert in his field, but it is easy to see that the full story can’t have been told so far.

I also did not got the answer (yes or no) to whether the orbit circumference for example of the Earth really can be 2 different sizes, (the same orbit), according to the example mentioned above..

And if this is so, how can the law of gravity apply for both without contradictions.DaleSpam Wrote
The two paths are not the same, so the answer is "yes, two different paths may have different lengths".

Each path is a 1D line (a helix) in a 4D curved spacetime. You can easily come up with coordinate systems where the two paths have two coordinates which are identical and constant, and you can parameterize the paths by one of the remaining coordinates, but they will always be different in at least the final coordinate. They are simply different paths with different lengths, no contradictions nor confusions.

You certainly haven't demonstrated any conflict with the law of gravity. I don't even know what you think would be contradicted here.

Let’s say observer ‘A’ and ‘B’ live in a high-rise flat.
'A' at 1st. floor and 'B' at 10th.
Between the Sun and the Earth there is a measurement tape.
Because time is ticking different for ‘A’ and ‘B’ they cannot agree about the speed and /or the distance to the Sun (the circumstance).
Because ‘A’ and ‘B’ live at the same planet they cannot be traveling with different speed.
Because ‘A’ and ‘B’ both can see the same measurement tape (between the Sun and the Earth) the distance (circumstance) of the Sun can also not be different. Because time multiplied with speed = distance (circumstance) we do in fact have a dilemma here.
So now they must both be following the same path.
The only logical possible explanation is that ‘A’ relative to ‘B’ lives in a completely transformed reality.
This mean when time is stretching, then distances and speed is doing the same.
Or ?

 

[Dale]

Hi Bjarne, welcome back! It has been quite a while.

Let’s say observer ‘A’ and ‘B’ live in a high-rise flat.
'A' at 1st. floor and 'B' at 10th.
...
So now they must both be following the same path.

These two statements are mutually contradictory. If they are following the same path in spacetime then one cannot be on the 1st floor at the same time that the other is on the 10th floor. I.e. as described their spacetime paths are different by approximately 30 m or so, therefore they are not following the same path in spacetime.

Not only are the paths approximately 30 m different, but that 30 m is a significant difference since there is a measurable amount of time dilation between the two paths.

we do in fact have a dilemma here.

Agreed. Which of the two mutually contradictory premises do you wish to keep?

 

[Bjarne]

Hi Bjarne, welcome back! It has been quite a while.

These two statements are mutually contradictory. If they are following the same path in spacetime then one cannot be on the 1st floor at the same time that the other is on the 10th floor. I.e. as described their spacetime paths are different by approximately 30 m or so, therefore they are not following the same path in spacetime.

Not only are the paths approximately 30 m different, but that 30 m is a significant difference since there is, according to you, a measurable amount of time dilation between the two paths.

Agreed. Which of the two mutually contradictory premises do you wish to keep?

The observers are following different paths, yes, - but the observed, - the orbit of the Earth is not following 2 different paths..
Let’s say ‘A' and 'B' live at the North Pole i the same high-rise flat
Hence speed and orbit distance of the Earth, - must be the same for both observers.
Speed of the Earth must be the same, since both are at the same planet. The planet cannot be moving with 2 different speeds at the same time.
Distance is also the same.. The measurement tape between the Sun and the Earth would prove for both A and B, that distance is the same.
This means ‘A’ and ‘B’ cannot say that these factors are different.
Hence still the conclusion mentioned above, - that both speed and distance transform proportional with time, - seems to be the most and only logical.

 

[Dale]

The observers are following different paths, yes, - but the observed, - the orbit of the Earth is not following 2 different paths.

That isn't how clocks work. They only measure the length of their own path through spacetime, not some other objects path. A's clock measures the length of A's path through spacetime, not the length of B's path nor the length of Earth's path. Similarly for B.

Also, since Earth is not a point like object in this scenario, the Earth does not have a single path through spacetime unless you define one specific point as the reference point.

The planet cannot be moving with 2 different speeds at the same time.

Why not? Speed is a frame variant quantity, so it can have as many different values at the same time as you have reference frames.

For instance, suppose one police officer is on the side of the road and another is driving on the road and suppose that they each measure the speed of the same car at the same time. One may get 100 km/h and the other may get 0 km/h. Both are valid measurements of the speed of the car, but in different frames.

 

[Bjarne]

That isn't how clocks work. They only measure the length of their own path through spacetime, not some other objects path. A's clock measures the length of A's path through spacetime, not the length of B's path nor the length of Earth's path. Similarly for B.

Let's say it really was possible to tie a tape measure to the North Pole of the Sun and to the North Pole of the Earth in the other end.
A and B is in this example only observers to the Earth orbiting the Sun.
Both can see the radius / circumstance of the orbit of the Earth and both agree that the orbit of the Earth (as just defined) really is the same for both observers, - simple because both can observe this is how the tape measure proves it to be.
So both must agree that distance / circumstance of the radius/orbit of the Earth, is observed from both observers perspective to be the same.

Why not? Speed is a frame variant quantity, so it can have as many different values at the same time as you have reference frames.
For instance, suppose one police officer is on the side of the road and another is driving on the road and suppose that they each measure the speed of the same car at the same time. One may get 100 km/h and the other may get 0 km/h. Both are valid measurements of the speed of the car, but in different frames.

A and B live at the North Pole in the same building ( ‘A’ at 1st and ‘B’ at 10th floor) . They are not moving relative to each other and also not relative to the Earth.
A and B and the Earth is all exactly following the same orbit, and hence in the same frame.

We could also say that also at the North Pole of the Sun there was a similar building, and from each floor a tape measure to the building / same floors at Earth's North Pole.
A and B would agree that all tape measure was the same length.

How can you then say they are in different frames / moving relative to each other?