T. Suntola and P. Lievonen

Mathematical Visualizations

by Petri Lievonen

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.

Contact: [email protected], [email protected]

On this page

• Cosmological Timescales
• Local Timescales and Gravitational Effects
• Energy-Momentum Relations
• System Structures and Inertial Work

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,

$$m{c}_0^2-\frac{GMm}{R}=0,$$

which we refine to a differential equation by defining

$$c_0(t)=\frac{dR}{dt},$$

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

Figure 1. Nonlinear development of the scale of the universe, using effective M = 1.758×10⁵³ kg (corresponding to a current mass density of around 4 or 5 ×10^–27 kg/m³). 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.

Figure 2. The theoretical velocity of expansion (hypervelocity) has been enormous in the past, suggesting that also processes have evolved with a great energy and pace. In this model, the hyperspeed of light is linked to the expansion velocity of space, and is currently the defined 299 792 458 m/s (in both standard and hypertime units). See also commentaries about the JWST findings, where, for example, “The objects are spinning super fast” in the early universe.

Note that as $R$ is even in $t$ (see also the matching exponent of $2/3$ in the scale factor in matter-dominated era), it naturally suggests a past symmetric development (bouncing cosmology, see also 1,2,3,4,5,6), but instead of a Big Bounce, more resembling a collapse due to its own gravity “through the eye of the needle” (perhaps related to Planck length scale; see also 1,2,3,4) to “the other side” (where some properties could get inverted, such as giving rise to the matter-antimatter asymmetry). These relations and their potential consequences could also be studied as purely algebraic definitions without physical extrapolation, perhaps represented as a scalar field.

It is interesting that by assuming (or deriving from Maxwell's equations, as Suntola has done) that the Planck constant $h$ has an inner structure where the hypervelocity $c_0$ is a factor,

$$\hslash ={\overline{h}}_0{c}_0=m_P{l}_P{c}_0,$$

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 $R$ and its hypertime derivative $c_0$ in just the right fashion for the speed of light to be experienced as constant throughout the evolution of the universe.

Figure 3. In this model, the SI second gets longer (in hypothetical hypertime units) during the course of the evolution of the universe. This is dictated by the zero-energy principle. It means that the hyperfrequencies of atomic oscillators, for example, are linked to the development of the cosmos via the fundamental energy conservation laws. Note that possible inhomogeneous developments have not been taken into account here; this is the aggregated model at the largest scales.

Figure 4. For a local observer evolving along with the rest of the universe, the speed of light (with other related processes) is experienced as a constant. This kind of attempt at modeling the observer and the observed together from a hypothetical hypertime perspective is quite unique to the Suntola framework, and offers possibilities for clarifying different timescales across the sciences.

Figure 5. In SI units, we arrive at a 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$$m${}^2$/s${}^2$] and watts [kg$$m${}^2$/s${}^3$], and in physical constants such as $c$ [m/s] and $G$ [m${}^3$/$$kg$$s${}^2$], which can be viewed as varying or constant, depending on one's conception of the second. Thus Suntola prefers to do the calculations starting from the largest scales (under this model), that is the entire cosmological history (displayed above, see also for geology). On largest scales there is a hypothetical constant timescale, which is called hypertime $t$ here. Note that the gravitational constant $G$ and total mass M [kg] is conserved under this model, and the model prefers to treat scale factor $R$ as a dimension measured in meters [m], as the definition of the meter stays constant and consistent across space and time – cosmologically retarted light travels the same physical distance (for example, 1 m) when measured with cosmologically retarted oscillators, as the SI second is defined as a count of 9 192 631 770 of specific clock cycles, and the SI meter as the distance light travels during the count of those cycles. However, there are important caveats in more local settings, as discussed later.

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 below), to beautiful logarithmic spirals that the radiative tangential momenta of light may trace with us in a four-dimensional hyperspace, see Fig. 7 next. 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.

Figure 6. Energies of matter (motion in the hyperspace) and energies of gravitation (integrated throughout the whole of cosmos, where gravity is assumed to act along the tangential surface of the hyperspherical space causing the gradient of the hyperscalar potential to point along the “virtual” hyperradius due to positive curvature of the space) have been quite formidable in the past, but always sum to zero. Note that these are hyperenergies, as in standard units the energies are constant due to the changing SI second.

Figure 7. Assuming that the velocity of light in ordinary tangential space is linked to the radial velocity of space (expanding zero-energy volumetric surface), the resulting logarithmic spirals of light on a circle crosscut view of the hyperspace display manifestly scale-free behavior. With these definitions, the form of the light geodesics are independent of the velocity of light, so these fundamental relations hold in both standard and hypertime units. Predicted redshift 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, where Solar velocity with respect to CMB is notable), 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 $L$ (see also apparent magnitude), using the geometrically motivated optical distance $D$ (see the same Fig. 7 above) with inverse distance squared dilution, together with a single $1/(1+z)$ apparent power dilution due to expansion of the space (the energy is conserved in an expanded wavelength of light):

$$\text{Using}D=R\frac{z}{1+z},$$$$L=L_z\frac{D_0^2}{D^2}\frac{1}{(1+z{)}^3}=L_0\frac{D_0^2}{D^2}\frac{1}{1+z}=L_0\frac{D_0^2}{R^2}\frac{1+z}{z^2},$$

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 $L_z$ and its dilution during the expansion, so actually when calculating in linear hypertime units there is an additional dilution of $1/(1+z{)}^2$, that is exactly canceled by $(1+z{)}^2$ higher luminance in the past, so the formula using only a single $1/(1+z)$ dilution factor in relative units holds when calculating with standard absolute magnitudes.

This very constrained and almost parameterless form fits very accurately to Type Ia supernova observations (Lievonen & Suntola, in preparation, see Fig. 9a), or at least as accurately as luminosity distance in standard cosmology, but there has to be extra $1/(1+z{)}^2$ dilution added to the model to fit with observations. Suntola maintains (Suntola 2018, pp. 264–275) that due to historical reasons, this extra dilution is actually in the processing of supernova observations (specific to conventions in multi-bandpass filter comparisons, see Figs. 9c and 9d later), but not in observed reality, so the following modified formula that fits the observations should be regarded as phenomenological model, but not physical:

$$L=L_z\frac{D_0^2}{D^2}\frac{1}{(1+z{)}^5}=L_0\frac{D_0^2}{D^2}\frac{1}{(1+z{)}^3}=L_0\frac{D_0^2}{R^2}\frac{1}{z^2(1+z)}.$$

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

Figure 8 (a,b,c,d). Modeling Type Ia supernova observations (latest released DES 5-year data) by integrating a redshifted supernova spectral flux distribution template (Hsiao et al. 2007) over each filter (g,r,i,z) bandwidth at each redshift 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 in the DES-SN5YR release) 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 CIDs. 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 curve corresponds to the phenomenological brightness model above, whereas the blue curve 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/cm²/Å physical units to fluxcal units in AB magnitudes): the current hyperradius R of 13.8 billion light years.

Figure 9a. SNIa distance modulus 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/H₀ = 13.8 bln ly (in parsecs to be compatible with absolute magnitude distance D₀ = 10 pc). Interestingly, the corresponding H₀ = 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.

Figure 9b. According to Suntola analysis, there may be unintentional extra (1 + 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. The button on the top right corner resets the plot.

Figure 9c. Suntola maintains (2018, pp. 264–275) that due to historical reasons, there is an unintentional extra (1 + 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.

Figure 9d. Analytical curves (dashed lines) displaying the effect of K correction on a blackbody source observed through idealized filters, resulting in a (1 + 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 df = –(c/λ²) dλ, 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 $(1+z)$), would have a simple geometrical explanation: propagating electromagnetic evidence in the form of light from hyperradius interval between the beginning and end times of a supernova gets exaggerated by exactly $(1+z)$ during the expansion, as is evident from the logarithmic spirals above (Fig. 7). For a local observer, the supernova would have occurred in a standard time interval, but in linear hypertime units (where the observer is modeled along with the observed) the process would have gone faster in the past, and slower in recent history, and all this would be seen as lengthening of the light curve by $(1+z)$ compared to the observations nearer to us, exactly as is observed.

Figure 10a. SNIa light curves (vertical lines in Fig. 8c, DECam i filter) plotted from low redshift (blue in the back) to high redshift (red in the front). The vertical axis is the same (log fluxcal), but now the horizontal axis is time (days), aligned on approximative peakmjd reported in the DES 5Y data release. On higher redshifts (red, 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.

Figure 10b. Shrinking the duration of light curves by 1/(1 + 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.

It seems that it is standard practice to use both the stretch factor (displayed above in Fig. 10a), and the so-called color (related to average redshift) in regression analysis when producing the distance modulus magnitude data displayed in Fig. 9a (see Scolnic et al. 2022, p. 4, from Pantheon+ papers):

Each light-curve fit determines the parameters color ($c$), stretch ($x_1$), and overall amplitude ($x_0$), with $m_B\equiv -2.5{\log }_{10}(x_0)$, as well as the time of peak brightness ($t_0$) in the rest-frame B-band wavelength range [emphasis added]. To convert the light-curve fit parameters into a distance modulus, we follow the modified Tripp (1998) relation as given by Brout et al. (2019a):

$$\mu =m_B+\alpha x_1-\beta c-M-{\delta }_{\mu \text{-bias}}$$

where $\alpha$ and $\beta$ are correlation coefficients, $M$ is the fiducial absolute magnitude of an SN Ia for our specific standardization algorithm, and ${\delta }_{\mu \text{-bias}}$ is the bias correction derived from simulations needed to account for selection effects and other issues in distance recovery. For the nominal analysis of B22a, the canonical “mass-step correction” ${\delta }_{\mu \text{-host}}$ is included in the bias correction ${\delta }_{\mu \text{-bias}}$ following Brout & Scolnic (2021) and Popovic et al. (2021). The $\alpha$ and $\beta$ used for the nominal fit are 0.148 and 3.112, respectively, and the full set of distance modulus values and uncertainties are presented by B22a.

It could prove out to be problematic that the distance and redshift is used implicitly (via stretch and color) already in the data releases, where the cosmological models are then fitted, instead of predicting the observed spectral energy flux densities directly (as would be preferable in physics). It seems that when comparing different cosmological models (using predicted brightnesses and their inferred distances) using SNIa data, some theoretical assumptions may have been baked in already, as the language mentioning rest-frames also suggests in the above quote. It would be illuminating if the hypothetical $(1+z{)}^2$ dimming effect of K correction (in Fig. 9d) could be shown or invalidated with the analysis pipelines and data parameters in Pantheon+ (for example, using this table and its sources). This analysis is in progress, but is complicated by the fact that different physics communities intermix discussing photons (in frequency space) and spectral energy densities (in wavelength space), and have different established conventions in analysing and reporting their results.

Contrary to standard cosmology, employing these novel brightness-redshift-scale-age relations under study on this page here, many observations of the James Webb Space Telescope might make more sense under this model: instead of “too early” galaxies at redshift $z\approx 15$ (about 270 million years cosmic time in standard cosmology, see also), that redshift would correspond to about 860 million years (in standard units, and only 140 million hypothetical hyperyears, where processes would have gone much faster during that time), which is plenty more time for galaxy formation. One can study these by setting the scale at the time of emission in the logarithmic spiral plot above (Fig. 7) to observed redshift $z$, and then inspecting the other synchronized plots, such as the first one, that displays both the hypertime and the hyperradius (in billions of light years), from where the standard time can be inferred from.

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 $D=Rz/(1+z)$, derived and used above, would be much more simpler to manipulate while also more accurate (under this model). It is quite well known that the integral in the light-travel distance has a special point at ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.737125$ (when ${\mathrm{\Omega }}_k={\mathrm{\Omega }}_r=0$), where the integral from zero to infinite redshift is equal to 1, and thus the light-travel distance to the edge of the observable universe equals the Hubble distance $R$. This can be calculated by taking the limit $z\to \mathrm{\infty }$ in the integral, arriving at $d_T(\mathrm{\infty })=d_{H}2\mathrm{atanh}({\sqrt{\mathrm{\Omega }}}_{\mathrm{\Lambda }})/3{\sqrt{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$ (using hyperbolic angle addition formula and taking the argument to the limit). Of course, one does not necessarily need to take the limit to infinite redhift to see that the light travel distance approaches Hubble distance $d_H$ (or $R$) with that ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ parameter value, as redshift z = 1 000, for example, is already quite far to the early universe. So it is interesting that it seems that some of the parameter values in standard cosmology (such as cosmological constant $\mathrm{\Lambda }$ at ${\mathrm{\Omega }}_{\mathrm{\Lambda }}\approx 0.7$, see also), that are claimed to be empirical findings, are quite close to values that can be derived formally just from the chosen mathematical structure of standard cosmology without referring to observations.

Furthermore, a more precise cosmological angular measure $A$ for a rigid object with diameter $a$ would then turn out to be simply

$$\text{Constant object}A=\frac{a}{D}M=\frac{a}{R}\frac{1+z}{z}M,$$

where $M$ is a possible magnification/lensing factor of hyperspherical space (see Michal Křížek, forthcoming; see also their Cosmological Section (in Czech), Cosmology on Small Scales conferences, and Mathematical Aspects of Paradoxes in Cosmology).

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

$$\text{Expanding object}A=\frac{a_D}{D}M=\frac{a}{D(1+z)}M=\frac{a}{Rz}M,$$

due to the (hypothetical) current diameter $a$ having been smaller in the past, by $1+z=R/R_S=a/a_D$ (inspect relations in Fig. 7 and compare the above angular measure to observational data in the figure below).

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

Figure 0x. Predicting apparent angular sizes of distant objects (at each redshift 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.

$$\text{Assuming}m{c}_0^2=\frac{GMm}{R}\text{and}m{c}_r(x,t)\le m{c}_0(t).$$

Local gravitation, in an idealized setting where there is a mass $M_r$ at a distance $r$, is modeled as bending of space to angle ${\varphi }_r$ (in the hyperplane spanned by hyperradius $R$ and a vector along $r$) to maintain the zero-energy balance:

$$c_{0}m{c}_r=\frac{GMm}{R}-\frac{G{M}_{r}m}{r}=\frac{GMm}{R}(1-\frac{M_r/M}{r/R})=\frac{GMm}{R}(1-\frac{G{M}_r}{r{c}_0^2})=\frac{GMm}{R_r}=c_0(m{c}_0\cos {\varphi }_r).$$

So the binding (decreasing) effect of local gravitational energy $-G{M}_{r}m/r$ on the rest energies $m{c}_0^2$ is equivalent to multiplying the rest energies with a gravitational factor $(1-\frac{G{M}_r}{r{c}_0^2})$, that is only slightly less than unity in weak gravity (when $r\gg G{M}_r/c_0^2$) and approaches zero in strong gravitational fields, observed as gravitational time dilation. The effect of the gravitational factor can then be conceptualized as lengthening of the “virtual” hyperradius $R$ to $R_r$ (where the conceptual mass equivalence of the rest of space is following along), specific to that point in space, or equivalently as rotating the hypersurface locally to angle ${\varphi }_r$ with respect to the flat space, the projection of which then contains the necessary effect on rest energy. The rotation is in the hyperplane, so the radial distance differentials along the hypersurface lengthen, whereas transverse components (such as the circumference $2\pi r$ at a radius $r$) stay unchanged. In the process, the venerated $E=m{c}^2$ has been split into hypervelocity $c_0$ (in the unchanged hyperradial direction) and a novel concept of “rest momentum” $m{c}_r$, that represents the locally available rest energy (when multiplied by $c_0$ to convert from momenta to energies, $E=c_{0}m{c}_r$). The reduction cannot be observed locally, due to all the reference energies (and thus SI units) varying in the same proportion in the vicinity.

The tangent of the hypersurface crosscut is then

$$\tan {\varphi }_r=\tan {\cos }^{-1}(1-\frac{G{M}_r}{r{c}_0^2})=\frac{\sin {\varphi }_r}{\cos {\varphi }_r}=\frac{\sqrt{1-{(1-\frac{G{M}_r}{r{c}_0^2})}^2}}{1-\frac{G{M}_r}{r{c}_0^2}},$$

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

$$\int \tan {\cos }^{-1}(1-\frac{G{M}_r}{r{c}_0^2})dr=\frac{2G{M}_r}{c_0^2}(\sqrt{\frac{2r{c}_0^2}{G{M}_r}-1}-{\tanh }^{-1}\sqrt{\frac{2r{c}_0^2}{G{M}_r}-1})+C,$$

or integrating from a reference distance $r_0$,

$$\mathrm{\Delta }R={\int }_{r_0}^r\tan {\cos }^{-1}(1-\frac{G{M}_r}{r{c}_0^2})dr=\frac{2G{M}_r}{c_0^2}(\sqrt{\frac{2r{c}_0^2}{G{M}_r}-1}-\sqrt{\frac{2r_0{c}_0^2}{G{M}_r}-1}-{\tanh }^{-1}\frac{\sqrt{\frac{2r{c}_0^2}{G{M}_r}-1}-\sqrt{\frac{2r_0{c}_0^2}{G{M}_r}-1}}{1-\sqrt{\frac{2r{c}_0^2}{G{M}_r}-1}\sqrt{\frac{2r_0{c}_0^2}{G{M}_r}-1}}).$$

Let's also note that in general relativity, the Schwarzschild solution differs from the above by a square root and a factor of two in the critical radius, so integrating GR solution to model the “bending of spacetime in extra hyperdimension” so that the local projection $c_0\cos {\phi }_r$ would correspond to gravitational time dilation $c_r=c_0\sqrt{1-2GM/c_0^2}$ could be achieved instead by surface integral

$$\int \tan {\cos }^{-1}\sqrt{1-\frac{2G{M}_r}{r{c}_0^2}}dr=2\frac{2G{M}_r}{c_0^2}\sqrt{r-\frac{2G{M}_r}{c_0^2}}+C.$$

The following illustrates the resulting hypothesized “dent” $\mathrm{\Delta }R$ in zero-energy surface volume in this idealized setting around a black hole (or any mass center when radially far in weak gravity), measured in meters and scaled to the unit critical radius $G{M}_r/c_0^2$. In an Euclidean four-dimensional space, the direction of virtual hyperradius is always present as orthogonal to any regular space direction, so the diagram should be read as a hyperplane crosscut spanned by the direction of the hyperradius $R$ and a vector along $r$ towards the mass center (in flat space). It is a symmetric picture from which ever direction in ordinary three-dimensional space we approach the mass center. Going forward, this is quite crucial to get consistently operated on, and it has not been formulated in a more expressive language, such as differential geometry or geometric algebra yet. For comparison, the GR solution (where $\mathrm{\Delta }R=c_0\mathrm{\Delta }t$ and $c_r=c_0\cos {\phi }_r$, interpreted crudely under this model) is plotted with a gray dashed line (and dotted line is its artificial extension, as the solution would otherwise end abruptly at the Schwarzschild critical radius, where the projection vanishes, so transforming to alternative coordinates would be needed for comparison there).

Figure 11. The hypothetical dent in hyperspace of the volumetric zero-energy surface (in hyperplane crosscut view) around a mass center M_r according to the ΔR(r) relation above. For a test mass 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, as if the test mass would be “lifted” slightly from the hypersurface due to speed (see the last equation on this page). In a circular orbit, the kinematic effect would not be as strong, as the orbital velocities are quite moderate. The diagram also depicts with dual vectors on the “underside” of the surface how the virtual mass equivalence of the rest of space could be conceptualized as moving further away as the test mass approaches the mass center; that virtual hyperradial distance R_r has been scaled down from approximative Hubble distance (currently 13.8 billion light years) for visualization purposes. 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 at rest 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 (middle 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 middle yellow line), which changes with gravitational state (distance $r$) and is not affected by the velocities of relatively small masses moving in space. The bottom one (Fig. 12c) relates the velocities to a local free-falling (non-inertial) observer (darkest yellow line), where the additional observer-specific dilation due to motion causes the velocities to be observed as even greater. One may be able to salvage the locally (in “local spacetime patch”) observed invariant speed of light by assuming (or deriving under this model, as will be done later), that actually objects 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.

Figure 12a. Various velocities at a distance r from a mass center M_r with respect to the flat space. Solid lines display the predictions from the model studied here, whereas dashed lines are their Schwarzschild equivalents (but partly interpreted under this model). The darkest yellow lines are not actually velocities, but plotted here for convenience (it is the combined effect of gravitation and motion in free-fall, resembling the equivalence principle). Yellow lines are related to gravitational time dilation and thus speed of light (in coordinate time). Blue lines are velocities of free fall from a state of rest at a distance, and their radial and hyperradial components are plotted with gray. Red lines are orbital velocities (with no hyperradial component). See text for details.

Figure 12b. The same velocities with respect to the local gravitational state “hovering” at each distance. Many velocities seem to reach the local speed of light, which is extraordinary behavior (and test masses cannot maintain their integrity way before that), so it seems that orbital velocities (in solid red) could be the most characteristic and stable features here. There could be a way to take the orbital velocity as fundamental (so that total diffused mass M_r is really there, in the slow orbits), analyzing some kind of spiraling capture dynamics in the evolving universe. That kind of investigation has not been attempted yet.

Figure 12c. Again the same velocities, but now plotted as observed locally by a free-falling (starting from far away or at orbit) observer at each distance 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 – v²/c²)^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. The interpretations of the Schwarzschild coordinates (dashed lines, and accompanied velocities) should be treated with caution here, as only some of them are exactly known from GR studies, and in different coordinates (such as using null geodesics), the meaning of critical radius would change considerably. Some Schwarzschild velocities, such as free-fall coordinate velocity (dr/dt) and orbital velocities (red dashed lines, in their “distant observer” coordinate form, “hovering” observer proper form, and an orbiting observer proper form), should be exact according to our knowledge, but even they may contain errors, because we have conflicting information from text books (see also 1,2,3,4). It is of interest that in Schwarzschild coordinates, the Newtonian orbital velocity depicts also the orbital velocity in coordinate time (for a far-away observer), but cannot support stable circular orbits nearer the critical radius (as depicted by the abrupt ending of red dashed line at the photon sphere).

Some special points of Schwarzschild geodesics have been highlighted in Fig. 11, such as between 1.5 (so-called photon sphere) and 3 times the Schwarzschild radius $2G{M}_r/c_0^2$, which are already challenging for sustaining stable orbits in Schwarzschild metric (discussed also briefly later).

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 $r$ (middle plot, Fig. 12b), the maximum orbital velocity is attained at two times the critical radius, which also happens to be the radius for the minimum period for circular orbits, as the calculated period (from the perspective of a far-away observer) is

$$P_r=\frac{2\pi }{c_0}\frac{G{M}_r}{c_0^2}{\left(\frac{G{M}_r}{r{c}_0^2}(1-\frac{G{M}_r}{r{c}_0^2})\right)}^{-3/2},$$

which can be compared to Kepler's third law of planetary motion. In the limit, as $r\to \mathrm{\infty }$, the prediction for the relations between orbital period and orbital radius are equal, as they should. In fact, the familiar relation $P_r^2\propto r^{\prime 3}$ holds with $r^{\prime }=r/(1-G{M}_r/r{c}_0^2)=r/\cos {\varphi }_r$ which is the tangential distance in Fig. 11.

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 perhaps 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 $(2+\sqrt{2})G{M}_r/c^2$ is geometrically special in Suntola framework, as there the space has been bended and rotated to exactly 45 degrees with respect to the expanding hyperradius, and it happens to correspond to approximative radius of neutron stars (about 5–11 km, using current estimated variability of neutron star mass).

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 $-\pi$ inside the critical radius. But note that the spatial and energetic relations as defined have broken down, so care should be taken to make progress in interpreting this.

Figure 13. Plot of the hypersurface integral in the complex domain. One could also compare the figure to the logarithmic integral function and Ramanujan–Soldner constant, but here the association is quite distant. (An even more distant association is in relation to the fine-structure constant, see 1,2,3,4,5, where in the last one the minimum kinematic factor in circular orbit has been experimented with.)

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 ($r_0\gg G{M}_r/c_0^2$ and $r\gg G{M}_r/c_0^2$), the above asymptotically approaches

$$\mathrm{\Delta }R={\int }_{r_0}^r\tan {\cos }^{-1}(1-\frac{G{M}_r}{r{c}_0^2})dr\approx \frac{2G{M}_r}{c_0^2}(\sqrt{\frac{2r{c}_0^2}{G{M}_r}}-\sqrt{\frac{2r_0{c}_0^2}{G{M}_r}})=2\sqrt{\frac{2G{M}_r}{c_0^2}}(\sqrt{r}-\sqrt{r_0}),$$

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 $r_0=a_0(1-e_0^2)$, where $a_0$ and $e_0$ are (projections of) orbital elements in flat space, to calculate characteristic hyperradial deviations, such as between apsides, where $r_{min}=a_0(1-e_0)$ and $r_{max}=a_0(1+e_0)$ (see semi-major and semi-minor axes), to arrive at the approximative hyperradial distance range of

$$\mathrm{\Delta }R=2\sqrt{\frac{2G{M}_r}{c_0^2}{a}_0}(\sqrt{1+e_0}-\sqrt{1-e_0})$$

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)

$$-⟨\frac{d{P}_b}{dt}⟩=\frac{54\sqrt{2}\pi G^{5/3}{m}_1{m}_2(m_1+m_2{)}^{-1/3}}{c^5{(1-e_0^2)}^2}(\sqrt{1+e_0}-\sqrt{1-e_0}){\left(\frac{P_b}{2\pi }\right)}^{-5/3},$$

whereas GR presents

$$-⟨\frac{d{P}_b}{dt}⟩=\frac{192\pi G^{5/3}{m}_1{m}_2(m_1+m_2{)}^{-1/3}}{5c^5{(1-e^2)}^{7/2}}(1+\frac{73}{24}{e}^2+\frac{37}{96}{e}^4){\left(\frac{P_b}{2\pi }\right)}^{-5/3},$$

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 $e\to 0$ (circular orbit), as DU predicts invariant periods for circular orbits, whereas GR predicts (at least for the above approximation) that also circular orbits decay and emit gravitational waves. The actual modeling machinery behind these claims (what kind of physical binary situations do the models apply to, and what is the effect of approximations made) and their observational evidence are under investigation, to prevent too early conclusions.

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

With this construction, the “rest momentum” $m{c}_0$ is decomposed in free fall into two orthogonal components: $m{c}_r$ is the local reduced rest momentum in the bended space at a distance $r$, and $m{v}_{esc}$ is the momentum of abstract free fall, or alternatively the escape momentum, that would be needed to escape back to flat space (see the components in Fig. 11):