41 minutes ago, joigus said:

Just an innocent question, before I become acquainted with the finer points of the discussion: How is this thing 'fractal'?

That is a very good question, and thank you for your interest. I think it captures the essence of the model, which involves the intricate interplay of infinite time scales and the dynamic, self-similar nature of the universe. The term "fractal" conveys the idea of repeating patterns and self-similarity, while "topology" emphasises the geometric and structural aspects of space-time.

This name was chosen to effectively communicate the novel concepts underlying this model and aligns well with the visual representation it describes. It also highlights the unique approach it takes in exploring the universe's infinite cycles and dynamic scaling.

I thought it might resonate well with those familiar with cosmology and mathematics and draw attention to the novel ideas presented.

2 hours ago, Rincewind said:

That was the first time I used the quote system, and I got it wrong. I hope you can forgive me for that.

Regarding ET, in my posts, ET stands for Ephemeris Time, not Earth Time. Ephemeris Time became the standard in 1960 and was highly accurate for defining time relative to celestial mechanics. While modern time standards like Atomic Time (AT) offer superior precision for scientific applications, ET is more than sufficient for this thought experiment because it provides a stable framework for event ordinality, unlike Earth Rotation Time, which is subject to irregularities such as tidal friction and earthquakes.

Originally adopted in 1952. The definition updated in 1960. And for scientific purposes (many of the applications of precise time, especially back then, are scientific. Commerce generally didn’t require the same level of precision)

1 hour ago, Rincewind said:

which involves the intricate interplay of infinite time scales and the dynamic, self-similar nature of the universe.

There are infinitely many time scales. But that doesn't involve fractal dimension necessarily. How does your idea involve Haussdorff dimension?

I'm intrigued.

4 hours ago, Rincewind said:

That was the first time I used the quote system, and I got it wrong. I hope you can forgive me for that.

Regarding ET, in my posts, ET stands for Ephemeris Time, not Earth Time. Ephemeris Time became the standard in 1960 and was highly accurate for defining time relative to celestial mechanics. While modern time standards like Atomic Time (AT) offer superior precision for scientific applications, ET is more than sufficient for this thought experiment because it provides a stable framework for event ordinality, unlike Earth Rotation Time, which is subject to irregularities such as tidal friction and earthquakes.

The key idea of this thought experiment isn't about debating which time standard is superior; it's about exploring the relationship between time measurement, relativistic effects, and event order in a synchronised planetary system. I would love to continue engaging with this fascinating subject, and I appreciate the discussion so far!

Oh no problem :)

12 hours ago, swansont said:

The rotation rate would still vary, owing to varying mass distribution on the planet (precipitation, rotation of weather systems, tectonic activity, etc.) and you’d expect it to slow from tidal friction.

And the precision of the measurement of the rotation would still be significantly worse than atomic clocks.

We use the rotational period of the earth because we live here. It’s convenient and what we’re used to, so change isn’t likely. Kinda like how the US resists the metric system. There’s nothing special about it when it comes to physics. It’s not “universal” by any stretch.

I appreciate your points about real-world rotation irregularities, but this thought experiment is designed to remove those variables by considering an idealised, perfectly rotating Earth mass planet. One that is so spherical that if it were not for the chip-scale atomic clocks, we would not think it was rotating. The goal is not to argue for Earth's rotation as a fundamental time standard for duration, but rather to explore how relativistic time dilation affects synchronised orbital timekeeping. In this framework, local variations in time measurement exist, yet event ordinality remains intact from a broader perspective.

This thought experiment aligns with Einstein's "ether sermon" as he later described it to Lorentz, where he did not deny Lorentz’s ether:

" According to the general theory of relativity, space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it." Albert Einstein 5 May 1920 at the University of Leiden.

In this framework, local variations in time measurement exist due to relativistic effects, yet event ordinality remains intact from a broader perspective because the structure of space-time itself provides a consistent reference for event sequencing, an idea well-aligned with Einstein’s view on the ether-like framework necessary for measurement."

9 hours ago, joigus said:

There are infinitely many time scales. But that doesn't involve fractal dimension necessarily. How does your idea involve Haussdorff dimension?

I'm intrigued.

Thank you for another great question. In a thought experiment involving a Tsiolkovsky-type space elevator, descending from a geostationary satellite (GEO), we observe that atomic time (AT) slows down due to changes in gravitational potential while ET remains invariant. This suggests an inverse relationship where ET seconds speed up relative to AT seconds, slowing down. The same absolute rotation is observed more slowly on the satellite due to atomic clocks ticking faster on the equator, highlighting the conflation of faster clocks with faster time and interpreting relative measures as absolute ideals, as the slower Earth clock shows that the same absolute ET day has passed faster in AT.

A change in AT implies a change in the meter’s length, since the SI meter is defined by the speed of light in a vacuum. As Henri Bergson’s critique of SR in his 1922 book “ Durée et Simultanéité”  pointed out for an extra-terrestrial travelling Relativistically, the measurement of the speed of light would be the same as his stationary measure but only because their units of time were different “I am measuring with a ruler whose length he sees as changing”

The SI metre establishes the scale of the universe. In General Relativity, Einstein considered the coordinate speed of light to be faster in a relative way where the gravitational potential is lower, as on the GEO satellite relative to the Earth. Our light-speed rulers shrink as we descend the elevator, indicating a changing scale. This uniform shrinkage is more pronounced in gravity wells or when travelling at relativistic velocities, when considering special relativity (SR)

The rotation is absolute and synchronous for the GEO satellite and the Earth. They share the same absolute time displacement, which reflects the absolute and universal temporal nature of ET. AT is a quantitative measure of duration, the property of time that relates to speed. If the scale change is universal, then the gravitational time dilation reflects the local peculiar change in the speed of that change. We consider that the decrease in AT duration offsets the increase in rotational inertia.

The shrinking scale in SI distances can be interpreted as expanding space relative to matter while space remains static. This makes the apparent expansion of space relative to matter in the present. If we privilege ET as a close approximation for the universal concept of absolute Newtonian time, descending the elevator speeds up the apparent space expansion as light-speed rulers have less AT in which to shrink and speed up the rotation of the Earth, as AT clocks slow down. As one ascends the elevator,  the speed at which the universal light-speed rulers shrink slows down, indicating a slower apparent expansion of space and Earth rotation. Neither of these changes the total energy of the system. Note that at any now moment in time (ET), all light-speed rulers are the same size throughout our universe, shrinking in ET unison at peculiar AT rates.

The expanding universe was predicated on the redshift-to-distance relationship, first observed by Vesto Slipher theoretically described by Alexander Freidman, and later established by Edwin Hubble, By privileging ET as the time concept and AT as the peculiar and emergent duration quantity, we can unify SR’s Lorentz transformations, GR’s relative light speed anisotropy, and the redshift-to-distance relationship, as the emergent effects of the universal changing scale resulting in matter shrinkage relative to space in this model. This shrinkage of matter, like Olivera’s matter evanescence, is a fundamental property of the universe with a similar scaling effect, not only revealing the nature of dark energy but for the nature of all the apparent expansion of the universe. This is from Oliveira’s paper "A Self Similar Model of the Universe Unveils the Nature of Dark Energy": With dark energy, cosmic expansion is neither the consequence of a cosmic event, like a Big Bang (although this may contribute), nor of some exotic substance, but of a fundamental property; and as a fundamental property, dark energy has to be embedded in fundamental physical laws. While introducing a parameter to account for it may be appropriate for the mere purpose of fitting selected observations, it is not totally satisfactory from an epistemological point of view."

Relative and Absolute Measures: This thought experiment highlights the interplay between relative measures (AT) and absolute ideals (ET). While the scale of the universe appears to change uniformly in ET, it is subject to relativistic and gravitational effects in AT, speeding up or slowing down with relativistic time dilation as the light-speed rulers shrink monotonically over cosmic time ET. Consistency: The consistent shrinkage of light-speed rulers throughout the universe in ET suggests a coherent framework for understanding cosmic scales and distances, regardless of local gravitational influences. In Oliveira’s self-similar model, the H_0 is fixed, whereas in the FTS model AT can vary in differing gravitational potentials and, as such, cannot be a universal parameter. The apparent expansion of the universe is due to the scalar change in duration and distances. The apparent expansion of the universe is measured in km s⁻¹ Mpc⁻¹. A parsec (pc) is a unit of length that measures the distance to astronomical objects outside our solar system. It is defined as the distance at which one astronomical unit (AU) subtends an angle of one arcsecond, a background-dependent unit related to ET. The apparent expansion is determined by the redshift-to-distance relationship—the speed at which the redshift changes and is measured in relative atomic SI units km s⁻¹. The second is an AT measure of duration. How fast the universe expands, like the Earth’s rotational speed, depends on the gravitational potential acting on the measurement instrument. Therefore, the measurement of the H_0 will be slower for JWST, where AT clocks tick faster relative to ET, and measured faster for HST, where AT clocks tick slower in its LEO. Different instruments at different locations with AT clocks ticking simultaneously (simultaneity being the property of ET) at different relative speeds (an AT measure) will measure different values for the H_0, and a contributory factor in the Hubble Tension.

I apologise for the length of the reply, but I did not have the time to write a shorter version. In answer to your question, we can imply a Haurssdouff dimension with the concept of a universal isomorphic change of scale that makes it aligns the strength of gravity with the length of the ruler that we measure the coastline of Britain.

So how does your idea involve fractal dimensions? I'm every bit as intrigued as I was before. Fractals involve boundaries that are not measurable.

Every answer that you give me is more and more profusely worded and farther and farther off-target wrt my question.

6 hours ago, Rincewind said:

I appreciate your points about real-world rotation irregularities, but this thought experiment is designed to remove those variables by considering an idealised, perfectly rotating Earth mass planet. One that is so spherical that if it were not for the chip-scale atomic clocks, we would not think it was rotating. The goal is not to argue for Earth's rotation as a fundamental time standard for duration, but rather to explore how relativistic time dilation affects synchronised orbital timekeeping. In this framework, local variations in time measurement exist, yet event ordinality remains intact from a broader perspective.

Chip-scale atomic clocks are among the least precise atomic clocks. Their utility is their size, not their stability.

Not think it was rotating? The stars wouldn’t appear to move? Foucalt’s pendulum wouldn’t work? Surely you don’t think so.

A perfectly spherical earth would have more relativistic effects than the oblate spheroid we live on. The deformation means the kinematic time dilation from rotation, which varies with latitude, cancels with the gravitational time dilation on the geoid.

7 hours ago, Rincewind said:

In a thought experiment involving a Tsiolkovsky-type space elevator, descending from a geostationary satellite (GEO), we observe that atomic time (AT) slows down due to changes in gravitational potential while ET remains invariant. This suggests an inverse relationship where ET seconds speed up relative to AT seconds, slowing down. The same absolute rotation is observed more slowly on the satellite due to atomic clocks ticking faster on the equator, highlighting the conflation of faster clocks with faster time and interpreting relative measures as absolute ideals, as the slower Earth clock shows that the same absolute ET day has passed faster in AT.

You seem to be under the impression that relativity only affects atomic clocks, which is not the case. Relativity affects time. Atomic clocks have sufficient precision to measure the effects, but if the effects were sufficiently pronounced, other clocks could measure it.

7 hours ago, Rincewind said:

A change in AT implies a change in the meter’s length, since the SI meter is defined by the speed of light in a vacuum. As Henri Bergson’s critique of SR in his 1922 book “ Durée et Simultanéité”  pointed out for an extra-terrestrial travelling Relativistically, the measurement of the speed of light would be the same as his stationary measure but only because their units of time were different “I am measuring with a ruler whose length he sees as changing”

A change in time implies a change in length. That’s relativity. This is not dependent on the definition of the meter (i.e. it was true prior to 1983)

7 hours ago, Rincewind said:

The shrinking scale in SI distances can be interpreted as expanding space relative to matter while space remains static.

This is very much not what relativity says

45 minutes ago, swansont said:

A perfectly spherical earth would have more relativistic effects than the oblate spheroid we live on. The deformation means the kinematic time dilation from rotation, which varies with latitude, cancels with the gravitational time dilation on the geoid.

This is interesting.

4 minutes ago, joigus said:

This is interesting.

Yep.

Neil Ashby derives the effect in his living reviews of relativity paper. I recall asking a colleague if we had to compensate our clocks for latitude, and they said no but weren’t sure why, so I dug into it.

(Einstein got this wrong in his original SR paper, in assuming the earth was a rigid sphere, and GR not yet being developed.)

22 minutes ago, Rincewind said:

This toy model offers a way to examine relativistic effects from a fresh angle, one that may provide insights into scaling transformations beyond conventional interpretations

Moderator Note

You posted this in Astronomy and Cosmology. We expect mainstream physics to be discussed here. Non-mainstream science goes in the Speculations section, where you must comply with its rules

Your most recent post has been placed there.

11 hours ago, swansont said:

Moderator Note

You posted this in Astronomy and Cosmology. We expect mainstream physics to be discussed here. Non-mainstream science goes in the Speculations section, where you must comply with its rules

The cosmology I'm trying to discuss depends on the physical framework of the model under consideration.