# Mathematical Visualizations

*This page is a work in progress.*

We are building a research consortium and collaboration around these topics. Physics Foundations Society is a registered association (3066327-8) in Finland. Current research partners include a project in the philosophy of physics at the University of Helsinki and the Cosmological Section of the Czech Astronomical Society, where Suntola is a foreign member.

On this page

## Interactive Mathematical Illustrations of Proposed Cosmological Timescales

Historically, zero-energy universe was studied by Arthur Haas, Richard Tolman, Dennis Sciama, Edward Tryon, Pascual Jordan (see Kragh 2015, p. 8, for some context and mathematical formulation), among others (perhaps Dicke, Dirac; see Kragh 2015b). The principle and its specific mathematical form was also mentioned by Richard Feynman in his Lectures on Gravitation, p. 10 – which is based on notes prepared during a course on gravitational physics that he taught at Caltech during the 1962–63 academic year – closing that section with the comment that

All of these speculations [described here] on possible connections between the size of the universe, the number of particles, and gravitation, are not original [by me] but have been made in the past by many other people. These speculators are generally of one of two types, either very serious mathematical players who construct mathematical cosmological models, or rather joking types who point out amusing numerical curiosities with a wishful hope that it might all make sense some day.

The following interactive illustrations are based on a novel interpretation of the zero-energy principle – leading naturally to bouncing cosmologies (see also 1,2,3) and to serious rethinking of the common usage of SI units, especially the second, the metre and the kilogram, along with the various derived units – and its application to physical theory development by Tuomo Suntola as documented in his work The Dynamic Universe (2018).

Assuming that the energies of matter and gravitation are in balance in a finite universe, reminescent of an action principle and analytical mechanics,

which we refine to a differential equation by defining

the following relations hold (see 1,2,3,4,5,6,7,8).

*M*= 1.758×10

^{53}kg (corresponding to a current mass density of around 4 or 5 ×10

^{–27}kg/m

^{3}). Under this model, the current 13.8-billion-year-old universe is 13.8 billion light years in radius (hyperradius of a multidimensional space, specifically the 3-sphere). In the hypothetical linear hypertime units (scaled to current years), it corresponds to approximately 9.2 billion hyperyears of positive age.

Note that as

It is interesting that by assuming (or deriving from Maxwell's equations, as Suntola has done) that the Planck constant

one attains an exotic cosmology where the frequencies of physical oscillators, such as clocks (which also define the SI second) follow the development of the scale factor

*R = ct*universe due to the constancy of the speed of light. One should be vary of quick comparisons to standard interpretations and cosmology models derived from Friedmann equations (in the context of general relativity), as here the age vs. scale vs. redshift vs. brightness -relations are quite different and derived from conservation of energy and geometrical first principles. The behavior displayed here is linear and scale-free (see also renormalization group), which is conceptually prudent as we should be able to infer the same developmental history even in the far future without living at a preferred moment of time on some S-curve, and the expansion depicted here affects then potentially even the smallest gravitationally bound spatial scales “here-and-now”. In this model, there is no dark energy nor acceleration, and in hypertime units, the expansion is actually decelerating in a very regular fashion, as was evident in Fig. 2 . The model seems to even have some empirical support in supernova observations, provided one uses the redshift-brightness relations derived from this model (see Figs. 8,9,10 later). Note that the redshift-inferred age of far JWST observations would be quite different and perhaps favorable to galaxy development under this model (mentioned briefly also later on this page).

So under this model, it seems that one should be quite careful in distinguishing between different timescales (see also cosmic time and age of the universe), as otherwise one may mix units in a confusing fashion. The second is involved in SI units such as joules [kg

Devising a simplistic geometrical explanation for the apparent uniformity of the cosmos (see, e.g. horizon problem) leads one to consider finite, positive curvature geometries and hyperspheres, especially the expanding 3-sphere, for contemplation as the global zero-energy structure of the universe. It has many desirable mathematical properties, such as it supporting *exactly three* linearly independent smooth nowhere-zero vector fields, allowing consistent global rotations and making crosscuts along the great hypercircle arc between *any* two points in the ordinary three-dimensional space (the volumetric surface of the 3-sphere), still having that extra zeroth hyperdimension along the hyperradius as a degree of theoretical freedom at *every* point of interest. For more information about these spaces, see my presentation on Clifford algebras and other topics at the Physics & Reality 2024 conference (and 1,2,3,4). Note also how from the viewpoint of general relativity, Sean M. Carroll mentions that in the Robertson-Walker metric on spacetime, for the closed, positive curvature case “the only possible global structure is actually the three-sphere”.

So combining, the assumptions lead from having the motion and gravitation in a fascinating eternal balance (Fig. 6), to beautiful logarithmic spirals that the radiative tangential momenta of light may trace with us in a four-dimensional hyperspace, see Fig. 7 below. The horizon problem is thus inverted, and while the observable universe can still be conceptualized around us, from a hypothetical hyperspace perspective everything (including us) is really at the “edge” of the multidimensional space, which is developing at tremendous velocity towards future possibilities, while also grounded by everpresent gravity and various consequences of past actions (but note that causality is a difficult subject; classical physics is clearly being preferred here, and contingencies are being investigated). Under this model, when we look out into the distant space (tangentially), we are actually *looking in* to the center of the hyperspace – and conceptually also past it towards the eternities.

*z*of light from a hypothetical emitter along with other important geometric relations are printed on the interactive diagram. The great distances could even appear discretized here, as conceptually we could see very, very dim traces of revolutions of light around the universe on top of each other, each fulfilling the whole sky due to spherical lensing effects (at redshifts 1 +

*z*=

*e*

^{nπ}, where even and odd

*n*mark antipodal points).

Zooming in to the present in Fig. 7, and making a variation to the hyperradius (and thus to hypertime), could give us a glimpse how light cones could perhaps be mapped to this presentation, provided one keeps in mind the distinction between position and momentum spaces (see also various astronomical kinematic velocities and their current estimates), as in the gravity model studied here, electromagnetic radiation has momentum only in the tangential direction of the 3-sphere (that is, “ordinary three-dimensional space”) and gets a “free ride” along the expansion, having no rest mass in the vacuum. This crucial idea is studied a bit further in relation to the energy-momentum relations at the bottom of this page.

Also discussions on the nature of causality could get interesting, as the picture above presents naturally the energy and information being attainable only as conveyed by the logarithmic spirals at the speed of light (at a maximum), but there are also other geometric relations present – due to the expanding spherical space, spirals have met in the past and will meet also in the very distant future, even if at the present most light cones seem separate. Also the model suggests that there is a theoretical possibility for some scalar potentials being instant across the universe “right now” in some sense, similar as in standard gravity the static field potential is “instant” (the force between inertial charges pointing towards the instant location, not to the retarted location, due to how the Lorentz-transforms operate, also in general relativity, but see [1],[2],[3][4]). As a force is the gradient of energy, and changing the energy necessitates conveying (abstract) mass (under this model), that can only propagate at the speed of light, these questions about the character of physical potentials, principle of locality, and possible “action at a distance” are complicated and under study. Already that short note about action at a distance contains suggestive ideas, such as John Wheeler and Richard Feynman “interpret Abraham–Lorentz force, the apparent force resisting electron acceleration, as a real force returning from all the other existing charges in the universe”, reminescent of Mach's principle, but for electrodynamics. In the model under study here, it is assumed that as one cannot suddenly move any mass instantly, there is also necessarily always some slowness and inertia with regard to moving (or changing) potentials, irrespective of their range. These important but difficult structural questions are also discussed a bit more nearer to the bottom of this page.

Notice that in that same circle crosscut view of the hyperspherical space, there is the prediction for apparent brightness

the actual form of which is still contested. Note that there is the added complication of greater energetic state of the earlier universe, which affects both absolute luminance

This very constrained and almost parameterless form fits very accurately to Type Ia supernova observations (Lievonen & Suntola, in preparation), or at least as accurately as luminosity distance in standard cosmology, but there has to be extra

The following is a research preview (Lievonen and Suntola, in preparation):

*z*(emitted far in the past), and applying the derived apparent brightness

*L*(

*z*) relation to arrive at physical flux predictions (max envelope curves) through each filter at the current hyperradius

*R*. Calibrated fluxes (in log

*fluxcal*units) through DECam g,r,i,z filters are plotted against redshift

*z*(

*redshift_final*or

*zHD*) up to

*z*= 1.2, using color alpha channel to indicate the approximative

*peakmjd*date.

*Photflag*detect, no quality cuts, SNIa candidates filtered to a selection of

*CID*s.

*Fluxcal*has been inverse transformed in the data release to correspond to top-of-atmosphere fluxes normalizing for galactic extinction, host galaxy surface brightness differences, among other effects. We do not know yet whether some wavelength-dependent processing has been done to the observations in the release – the

*fluxcals*correspond to AB magnitudes with a zero-point of 27.5. Predictions using several different 1/(1 +

*z*)

^{n}dilution factors have been plotted as envelope curves, where the red one corresponds to the phenomenological brightness model above, whereas the blue one would be the more correct (geometrically motivated above) brightness model. The model predicts quite nicely the maximum envelope curve of the supernova observations (the peak of each vertical light curve), with essentially just a single parameter

*R*(apart from the values in the empirical spectral template at 10 parsecs, SNIa absolute magnitude of –19.253 from Pantheon+, and a standard conversion factor of 48.60 applied in transforming the predictions in erg/s/cm

^{2}/Å physical units to

*fluxcal*units in AB magnitudes): the current hyperradius

*R*of 13.8 billion light years.

*mu*(

*MU_SH0ES*vs.

*zHD*from Pantheon+ data release) supports the model studied here. Hubble–Lemaître diagrams are among the most important tools in modern empirical cosmology. In the model under study, there is only a single parameter, the current hyperradius

*R*=

*c*/

*H*

_{0}= 13.8 bln ly (in parsecs to be compatible with absolute magnitude distance

*D*

_{0}= 10 pc). Interestingly, the corresponding

*H*

_{0}= 70.8 km/s/Mpc would be in the middle of the current Hubble Tension. The standard “–2.5 log” comes from the definition of magnitudes. The lone observation with the blue error bar at

*z*= 2.903 is arXiv:2406.05089. Note that there may be some horizontal and vertical bias due to interactive plotting; this is a research preview.

*z*)

^{2}dimming in

*all the reported SNIa data across all the surveys globally*due to the way how multi-filter observations are combined to a single magnitude value, instead of predicting the observed physical fluxes through each filter separately. The plot above displays instead the (1 +

*z*)

^{2}brightened data (for magnitudes, less is brighter), which fits then nicely to the hypergeometrically motivated brightness prediction, and could thus depict the true bolometric magnitude. This line of inquiry is under investigation.

*z*)

^{2}dilution in the processing of supernova observations (specific to K correction conventions in multi-bandpass filter fusion), but not in observed (bolometric) reality. This would obviously affect the very basis of dependable observations in all the Hubble–Lemaître diagrams on SNIa worldwide, and would have serious consequences for the empirical basis in astrophysics. See, for example, how the diagram is the very first figure (in log

*z*scale) in the Big-Bang Cosmology chapter in the Review of Particle Physics. SNIa absolute magnitude of –19.253 here is from Pantheon+. The observed peak magnitudes through different filters (dashed curves) are from Tonry et al. (2003, Table 7). Filter designations BVRIZJ (here) and griz (in Figs. 8 a–d) correspond to different photometric systems, where the older ones are usually described by energy-based transmission curves, whereas modern systems are count (photon) based. Note that there may be some horizontal and vertical bias due to interactive plotting; this is a research preview.

*z*)

^{2}dimming envelope curve. Inspect Hogg (2002, p. 4), together with (1,2) for analysis by Suntola. As an example, the black dots are reported K correction values from Riess et al. (2004, Tables 2 and 3). It does not sound impossible to get to the roots of this already quite well defined and studied issue, as the physics of photometric filters is a well-known subject. This would require the expertise of specialists in magnitude systems and their calibration (reported zero points, etc.), and other related subjects (such as chromatic corrections), however, to disentangle the factors. Note also that the differentials of frequencies and wavelengths are related by d

*f*= –(

*c*/

*λ*

^{2}) d

*λ*, so there are several quadratically varying factors present in redshifted wavelengths (and AB magnitudes are defined on constant flux density per unit frequency). The data analysis pipelines are usually well reported, but there may be important details in the complex procedures applied even to the reported raw data, and may be crucial in the eventual understanding of dilemmas such as Figs. 8 (a–d).

All in all, if this total geometric model turns out to be valid and useful in many contexts, it could have interesting consequences for our discussions about time, space, and motion in general. For example, the so-called *cosmological time dilation*, where supernova light curves are empirically observed as taking a longer duration in the past (dilated by

*fluxcal*), but now the horizontal axis is time (days), aligned on approximative

*peakmjd*reported in the DES 5Y data release. On higher redshifts (emitted in the distant past) the light curves are observed as broadened, the supernova explosion seemingly taking a longer duration. We acknowledge this may not yet be the best representation, as usually one fits a computational light curve model to this raw, noisy data, so the relation is then more clear. Also one needs to ascertain that this apparent cosmological time dilation is not an artefact of dimmer observations at higher redshifts; the logarithmic scale should keep the shapes of the curves comparable.

*z*) seems to homogenize them. This would be in line with the model studied here, where the expansion of space would cause the hyperradius interval between the emitted light at the start and end events of the supernova to get exaggerated by exactly this amount by the time of observation. See the text for some details, and study Fig. 7 closely to familiarize with the proposed geometric relations affecting these phenomena. Note also that with the definitions above (related to hypervelocity of light, and thus the hyperradial velocity affecting the SI second and rapidity of processes in general), the hyperradius interval traveled during the SNIa explosion is constant, irrespective of the time when the explosion happened in the cosmic history.

Employing these novel brightness-redshift-scale-age relations, many observations of the James Webb Space Telescope might make more sense under this model: instead of “too early” galaxies at redshift

Various distance measures utilized in reasoning about the dimensions of the observable universe would get updated; the comoving distance would be related to the length of the hyperradius-normalized circular arc in the logarithmic spiral plots above (Fig. 7), and the so-called proper distances would be then related to the non-normalized, expanding arcs between the points of interest.

Instead of integrating out the light-travel distance, the optical distance

Furthermore, a more precise cosmological angular measure

where

What is highly intriguing (and needs a thorough investigation) is that when interpreting observational data using these geometric angular measures, it seems that many astronomically interesting objects, instead, could turn out to be expanding with the space, as the apparent angular size is then

due to the (hypothetical) current diameter

The following figure displays work-in-progress in interpreting these angular measures.

*z*, in log-log scale). Most of the components in this figure are under development:

- Blue curves predict observed angular sizes of objects that expand with the space. So under this model, it seems as if galaxies expand with the space, which would be contrary to what is the long-time consensus in astrophysics.
- Red curves predict the angular sizes of constant objects. There are various alternative versions for research purposes. The one with a turnover point is the angular diameter distance in standard cosmology (ΛCDM), which does not seem a good predictor for this particular data.
- The possible 3-sphere lensing effect M = ln(1 +
*z*) / sin(ln(1 +*z*)), predicting peaking at antipodal points, is under study. Note also that such magnification could affect the inferred velocities of distant phenomena. - The data (open circles and black dots) is from Nilsson, Valtonen, Kotilainen, and Jaakkola (1993, p. 469, Fig. 5). (See also FINCA)
- Interpreting the angular sizes (observed sizes in calibrated pixels) from JWST are under study (the dim ellipsoid at the bottom). There are various factors which can affect the interpretations. There is material, for example, in [1] and [2].

So in this model, the galaxies and planetary systems could be expanding after all (along with the rest of the space, as so called kinematic and gravitational factors are conserved in this gravitational model), and so would the stars and planets as gravitationally bound objects to some extent. “Electromagnetically bound” systems (Suntola articulation), such as atoms, would not expand along with the space, but “unstructured matter”, which also light represents due to wavelength equivalence of energy of radiation in this framework, would expand and dilute with the space, dictated by the presented zero-energy principle in the evolving universe. See also the works by Heikki Sipilä, such as Cosmological expansion in the Solar System. Finding out the consequences and implications (using the understanding of Suntola framework) is a difficult task, involving thermodynamics and contemplations about elastic SI units, among other interesting questions.

## Interactive Mathematical Illustrations of Proposed Local Timescales and Gravitational Effects

The following displays various important proposals contained in Suntola's work. Visualizations are under development.

Local gravitation, in an idealized setting where there is a mass

The tangent of the hypersurface crosscut is then

which can be integrated to arrive at the hypersurface crosscut shape of

or integrating from a reference distance

The following illustrates the resulting hypothesized “dent” *any* regular space direction, so the diagram should be read as a hyperplane crosscut spanned by the direction of the hyperradius

*M*according to the ΔR(r) relation above. For a test mass

_{r}*m*, the always present hypermomentum (in yellow) along the direction of expansion (hyperradius) is decomposed to two orthogonal components: the momentum of free-fall (equivalent to the escape momentum, in blue), and the local rest momentum (in darker yellow, indicating the local gravitational energetic state and thus the local speed of light in this gravitational model, where the gravitational time dilation and radial stretching of space have been taken into account). Zooming in to the test mass, the even darker yellow represents the local rest momentum when the mass is in actual free-fall from a great distance (thus invariant in its non-inertial coordinate system, see also geodesics), where the motion causes even more time dilation. In a circular orbit, the effect would not be as strong, as the orbital velocities are quite moderate. See the text for more details.

The plots below display various velocities related to this gravity model. They are ordered from more global timescale to more local. Specifically, the uppermost plot (Fig. 12a) displays velocities with respect to the flat space (static observer far from the critical radius, but ignoring light propagation delays). From that perspective, which is actually the most common one when we are almost always modeling physical phenomena from far away, the speed of light (darker yellow line) slowing down near mass centers is not too exotic, as that is also the prediction in general relativity. Note, however, that various forms of the equivalence principle are not taken as axioms here, and will most probably not hold in these strong gravitational fields in this gravitational model. The middle diagram (Fig. 12b) relates then the velocities to that local time standard (proper speed of light, that darker yellow line), which changes with gravitational state (distance *expand* uniformly at velocity, thus completely nullifying the effect of extra time dilation at velocity. It does not affect any longer distances (just their appearance), but that locally one will be perhaps able to measure the speed of light as invariant and isotropic (at least for a roundtrip time-of-flight calculation and optical interference studies, which could be affected by this hypothetical expansion at velocity, and also Doppler effects could become relevant). In Suntola studies, first-principles motivations, plausible physical mechanisms, and an understandable worldview are sought after, resulting in quite straightforward mathematics.

*r*from a mass center

*M*with respect to the flat space. See text for details. The dotted line is not actually a velocity, but plotted here for convenience (it is the combined effect of gravitation and motion, resembling the equivalence principle).

_{r}*r*, taking into account extra kinematic time dilation. This picture is complicated by the fact that locally SI meter seems to expand with velocity (under this model), so actually the velocities are inferred as constant irrespective of own velocity, so the same as in upper Fig. 12b. The same invariance applies when normalizing for observers in circular orbits (calculating the kinematic factor (1 – v

^{2}/c

^{2})

^{1/2}from red orbital velocities in Fig. 12a). This seems to imply that locally matter

*expands*on the move (thus enabling invariant experiments), but also inferred further distances seem to shrink at velocity, as light travels a longer distance in the same locally observed time frame. These figures and their implications are under study.

In blue, there is the velocity of free fall (equivalent to escape velocity), which is related to the sine of the angle of rotation of the hypersurface volume. The velocity of free fall saturates at the speed of light at the critical radius, which seems very nice and regular. However, do note how the “4D-well” (black line in the earlier Fig. 11) extends arbitrarily far in the direction of the hyperradius, possibly even to the origin of the hyperspace where different black holes could be connected (at least in the early history of the cosmos). But as the hyperspherical space is expanding and the hypersurface volume is thus developing vertically in the crosscut picture at the speed of light (by definition of the hypervelocity), it seems likely that a falling object can at maximum stay at the same absolute hyperradius distance, not travel "backwards in time" to the origin. Also the horizontal and vertical components (gray lines) of the escape velocity have been plotted for convenience. Newtonian escape velocity (blue dashed line) displays unphysical velocities nearer the critical radius, as is well-known.

Some special points of Schwarzschild geodesics have been highlighted, such as between 1.5 (so-called photon sphere) and 3 times the Schwarzschild radius

In the gravity model studied here, orbital velocities (red line, in circular orbits) stay nice and regular down to the critical radius (Suntola 2018, pp. 142–163). Maximum orbital velocity is attained at four times the critical radius (Fig. 12a). Relative to gravitational state at a distance

which can be compared to Kepler's third law of planetary motion. In the limit, as

For a local free-falling (non-inertial) observer, the situation is complicated as the velocities seem to increase without limit (as reference frequencies decrease towards zero, affecting SI second), but at the same time the SI meter expands (both definitionally and physically, in this model), thus the observational velocities being inferred as constant, and further distances being measured as shrunk as the light travels a longer distance in the observationally same time frame. Also for an observer in a circular orbit (red line in the above figures), the situation is quite remarkable in lower orbits, as the reference frequencies and oscillators come to a standstill the closer one orbits the critical radius, so distances are inferred as warped and shunk in a complicated but manageable way, but note that the actual orbital velocities seem to come to a standstill, so the kinematic term is approaching unity (no kinematic time dilation). Matter seems to dissolve into some exotic form of mass-energy. Suntola claims that these kind of slow orbits (see the same red line from a point of view in the distant flat space in Fig. 12a, and compare to the spatial picture in Fig. 11) maintain the mass of the black hole – it is quite a different picture than a pointlike singularity, as here photons can climb very slowly also up from the critical radius (which is half of that of the Schwarzschild event horizon).

Some claims, such as the escape velocity having a well-behaving form down to critical radius, could simplify the common physical picture in the long term. However, the velocities plotted to local time standards imply some rather exotic physics, where the velocity of free fall can meet and exceed the local proper velocity of light (which is decreasing near mass centers in this gravitational model), as Suntola claims that in a free-fall, the relativistic mass increase is not necessary, as the energy is taken from the hyperrotation of the space itself (maintaining the zero-energy principle). It could mean that on the atomic level, new kinds of mass-energy conversions could be possible around black holes around critical radius, that are perhaps hitherto undertheorized. Distance

It is also interesting to plot the surface integral outside of its domain of applicability using complex values, where mathematically the surface seems to have imaginary values (orange line) starting from

As a mathematical curiosity, also the period formula shown previously can be plotted in complex domain, where the period is pure imaginary inside the critical radius.

When operating far from the critical radius (

which can be used to approximate the volumetric surface shape (the hyperradial dimension) in many calculations. For example, Suntola utilizes semi-latus rectum ℓ as a reference distance

during an (approximative) Keplerian orbit.

It is interesting how the resulting prediction for the rate of period decrease of two bodies orbiting one another (thus emitting gravitational radiation) seems more compact in this gravity model than in general relativity, even though they are both employing approximations. The DU solution is (Suntola 2018, pp. 162–163)

whereas GR presents

which are perhaps surprisingly similar (which is reassuring, as they are derived using quite different means). However, they have very different predictions when orbital eccentricity

I recommend taking a moment here contemplating the gravity of the above statements.

With this construction, the “rest momentum”

Solving the above for

When taking the gradient of the scalar potential – again in an idealized setting around a single large mass, as is usual when analysing orbital dynamics (sums of these potentials has not been analyzed apart from discretizing the space to nested energy frames) – it seems that local gravitational force gets augmented with a cosine term due to distance

In addition to distance

These discussions on hypothetical reduced rest momentum and local timescales (variable proper speed of light, as observed from the distance), lead us to kinematic and gravitational time dilation, that are both routinely taken into account in satellite operations. To see the components, study, for example, these two images (from popular sources [1], [2], with texts and markings kept verbatim, and the model studied here plotted on the image):

*r*

_{0}is Earth radius, see also middle yellow line in Fig. 12a). Red line is also a simple fraction (orbital velocity

*v*is calculated using an accurate formula, see red line in Fig. 12a, and

_{r}*v*

_{0}is the velocity at the equator). The total effect on hyperfrequencies of clocks (blue line) is just a simple fraction of the product of the aforementioned effects – thus Suntola proposes that gravity is actually multiplicatively separable (!), where the “rest momentum”

*mc*is modulated by both the gravitational factor and the kinematic factor. Green dotted line is a crude estimation of gravitational time dilation inside Earth with linearly increasing density and only each inner shell affecting the potential energy (as usual when integrating a scalar potential in 3D).

_{0}*test*general relativity. The other spacecraft on this chart (except for the ISS, whose range of points is marked "theory") carry atomic clocks whose proper operation

*depend on*the validity of general relativity.” (the emphasis and link in the original) The simple equations are exactly the same as in the previous figure, just the scales of the axes are different. Compare to the more complicated equations usually presented, where the factors do not have as straightforward interpretations as this model seems to have here. Note also how in the proposed factorized structure of gravity, possible common factors from “larger frames” will cancel away when taking the fractions.

Note that the experimentally confirmed slight gravity speedup (gravitational time dilation) of clocks seems to imply that the speed of light is necessarily also slightly higher up there, as the “speed of light in a locale is always equal to

Note also that the orbital speed slowdown (kinematic time dilation) is *not* calculated with respect to each observer, but with respect to the Earth-centered inertial (ECI) coordinate frame (see, for example, geocentric celestial reference system (GCRS) and Geocentric Coordinate Time TCG). All the system components are eventually referred to a common coordinate time scale. So judging from this picture, it seems quite evident that the reciprocity of time dilation is *not true* in most physical settings (outside of thought experiments in special relativity, or in experiments where the temporal and spatial scales are so small that these effects can be ignored in some otherwise symmetric setting). According to these pictures, if a clock is taken to a higher satellite orbit where it has less orbital velocity, its hyperfrequency will be sped up (interpreting the kinematic factor), and if a clock is lowered to a lower satellite orbit where it needs to have more orbital velocity to stay on orbit, its hyperfrequency will be slowed down (again interpreting just the kinematic factor). This happens from the point of view of this common frame irrespective of any observers, and there cannot be reciprocity of kinematic time dilation here, as observed from these satellites (or anywhere else, for that matter), the situation cannot somehow reverse without major paradoxes appearing down the line. With a slowed down clock, one necessarily measures the other velocities as higher, not the other way around as would be needed for reciprocity. We are happy to discuss any empirical evidence proving otherwise, but for now, we do not have reasons to believe that warping or contracting the space, rotating coordinate systems, or some other hypothetical effect would somehow salvage the reciprocity here. So for now, these are treated as common confusions due to apparently mixing event-centric kinematic and properly system-centric dynamic (i.e. energy conserving) descriptions, and will be studied later on this page.

In terms of the hypothetical gravitational model studied here, the hyperfrequencies of clocks and oscillators are predicted to follow the reduction in local “rest momentum”, due to both motion and gravitation effects in the local “energy frame” (see again Fig. 14a), which is regarded as an objective and totality-oriented, as opposed to observer-oriented, concept.

The reduction is modeled as two factors (kinematic and gravitational), formally affecting the “rest mass” and “speed of light” separately, but actually modulating the “rest momentum” as a total:

Note the striking similarity (when multiplied by

It is related to kinetic energy, potential energy, and total energy, and it makes the relativistic action functional proportional to the proper time of the path in spacetime.

Note that the critical radius

whereas in DU space, the above could be interpreted as