1. Introduction

Theory in Brief

The theory, as proposed in our first paper, is that time is a spatial dimension that obeys the same laws of physics as other spatial dimensions but is largely invisible to us. Time does not progress in a single direction but rather in all directions at once. Each particle is caught up in the expansion wave and is consequently moving in time—a spatial dimension over which we can appreciate only a small, but finite sized, sliver of a sphere at any one moment. If we consider a cross section through the centre of the expansion sphere showing one normal space dimension and one of the time space dimensions, then time will progress outwards, as shown in Figure 1, away from time t₀. Overall, the expansion wave front will be isotropic, but where large masses have amalgamated, dimples will form in the wave front, as shown in the figure. A galaxy would form a large dimple, and the stars would form smaller dimples as you zoom in further.

We ourselves will predominantly follow a single time path governed by the mass on which we “ride”, but we will interact with objects on a different time path as they expand at different velocities/arrive from different parts of the wave front.

The time dimension can, in a sense, be thought of as a dimension over the top of all the other dimensions that is equally stretching in all directions when viewed as an isotropic universe expansion. In a sense the theory is an extension to general relativity. Einstein identified that space and time were interchangeable or part of a single space-time entity. By taking this a step further and saying that time is the expansion effect we see and that stretches all the other 3 dimensions, then this simplifies the mathematics and provides explanations that are otherwise absent as we shall explain later in this paper.

Another artefact of the theory ties in the subatomic realm. As the slice of time in which we exist must have a finite size then particles/photons on the atomic-to-nothing size level have the freedom to rotate within this space and consequently can move forwards and backwards in time relative to ourselves. We will expand upon this part of the theory here.

Figure 1. Visualisation of Time expansion wave cross section cutting through a single time dimension (as shown by the radius) and a single space dimension (as shown by the circumference). This expansion wave started at time t₀ and has reached time t₁ signified by the dark black line. Two gravitationally distinct objects would expand away from each other in space as the time expansion wave increases. Where objects have clumped together then dimples in the wavefront would show up as you zoom in. The thin black line shows the observable slice of time with the grey areas showing the possibility of matter existing outside of our observable time window.

Figure 1. Visualisation of Time expansion wave cross section cutting through a single time dimension (as shown by the radius) and a single space dimension (as shown by the circumference). This expansion wave started at time t₀ and has reached time t₁ signified by the dark black line. Two gravitationally distinct objects would expand away from each other in space as the time expansion wave increases. Where objects have clumped together then dimples in the wavefront would show up as you zoom in. The thin black line shows the observable slice of time with the grey areas showing the possibility of matter existing outside of our observable time window.

Luminosity Model and Fit

Stellar luminosity data were modelled in our earlier paper to fit the observed data. As all historical light reaching us from distant luminous objects must have travelled at the speed of light and in a straight line from its point of origin to reach us, we need to consider only one spatial dimension and one temporal expansion dimension to model it. It is assumed that the light that reaches us must have travelled within one of the space dimensions perpendicular to the origin point at t₀ as visualised in Figure 1. It is also assumed that bending of any of these light ray paths as caused by gravity effects of objects that the light passes will be random and equalise out when considering the universe as a whole. This would mean that to reach us, each light beam has taken a path along the surface of the expanding sphere. Although other light beams will have taken completely different paths, each light pinpoint can be dealt with similarly. Therefore, for each photon, if we take a slice through the temporal dimension and the straight-line spatial path that each photon takes, then you will end up with a spiral path (in time) consistent with the expansion of the universe.

A different photon would require a different cross section/alignment, but the spiral path would simplify to be the same assuming that the expansion of the universe has been isotropically similar and dependent only on the distance from the Earth to the object being observed.

The magnitude of the observed light (expressed as the distance modulus m-M) of any object is related to the luminosity distance by:

$$m-M=5log{D}_L+25$$

(1)

where $D_L$ is the luminosity distance in Mpc and can be defined as

$$D_L={\left(\frac{L}{4\pi F}\right)}^{\frac{1}{2}}$$

(2)

Figure 2. Visualisation of the spiral effect of a time wave expanding universe caused by the speed of light having a fixed value. Scale for both x and y is in Mpc. Distant, and therefore historic, objects in the universe, if observed from point A will appear as if on a spiral path reaching back to time zero. The above figure was generated using the complex decay model mentioned later in this paper against luminosity data of Betoule et al. (2016) and Reis et al. (1998). Points highlighted, emanating from the centre of the spiral are, respectively as follows: Time zero, most distant observed object at z of 11.09, Most distant fitted data point, Earth.

Figure 2. Visualisation of the spiral effect of a time wave expanding universe caused by the speed of light having a fixed value. Scale for both x and y is in Mpc. Distant, and therefore historic, objects in the universe, if observed from point A will appear as if on a spiral path reaching back to time zero. The above figure was generated using the complex decay model mentioned later in this paper against luminosity data of Betoule et al. (2016) and Reis et al. (1998). Points highlighted, emanating from the centre of the spiral are, respectively as follows: Time zero, most distant observed object at z of 11.09, Most distant fitted data point, Earth.

L is the intrinsic luminosity of the object, and F is the observed flux. For nearby objects (those for which the time that has passed is minimal and there has been little increase in radius), this would approximate to the distance around an arc of our spiral model, $D_{Spiral}$. For objects that are further away, then distortion will occur through two effects:

As the light stretches, the energy of the light diminishes, which causes a reduction in intensity. The wavelength of light will increase relative to $r_{now}/r_{then}$ as the circumference becomes proportionally larger for the same angle, so consequently, the energy of the light will drop in intensity relative to $r_{then}$/$r_{now}$.

Second, the stretch will mean that the photon arrival rate within a beam of photons will be reduced again according to $r_{then}$/$r_{now}$.

The consequence of this is shown below:

$$m-M=5log({\left(1+z\right)}^2{D}_{Spiral})+25$$

(3)

And

$$D_L={\left(1+z\right)}^2{D}_{Spiral}$$

(4)

where z is the observed redshift. The redshift observed for any point around the spiral will be directly related to the size of the universe at that point, i.e., the circumference. As the circumference of our expansion sphere is directly proportional to the radius, the stretching effect or redshift is:

$$z+1=\frac{r_{now}}{r_{then}}$$

(5)

The spiral distance of a light beam’s origin to earth can be seen to be the integral of the light travel distance:

$$D_{spiral}=\underset{r_{now}}{\overset{r}{{\displaystyle \int }}}\frac{c}{v_H}dr$$

(6)

where $v_H$ is the observed rate of increase in r or the resultant velocity over time (we have used H in $v_H$ to denote that this is the observed expansion velocity derived from Hubble expansion) and r is the radius of the time expansion sphere.

If the rate of expansion was constant (i.e., $v_H$ is a constant and independent of r), then:

$$D_{spiral}=\frac{c}{v_H}\left(r_{now}-r\right)$$

(7)

Note, at this point in the theory, we are not reliant on the expansion wave being linked to time directly as a basic assumption – this idea is brought out rather from the conclusions drawn later.

2. Results

Figure 3. Best fit modelled time expansion wave allowing a fixed speed of expansion (orange solid line) and a varying speed of expansion (red solid line) versus observational data taken by Reiss et al. (1998) and Betoule et al. (2014) (points).

Figure 3. Best fit modelled time expansion wave allowing a fixed speed of expansion (orange solid line) and a varying speed of expansion (red solid line) versus observational data taken by Reiss et al. (1998) and Betoule et al. (2014) (points).

Figure 5. Exponential decay curves showing various different possible fits.

Figure 5. Exponential decay curves showing various different possible fits.

3. Discussion

Dark Energy and FLRW model

The FLRW model of the universe in basic terms uses a total density parameter Ω_total which can be split up into constituents of matter and dark matter (combined as Ω_M), radiation (Ω_rad) and “dark energy” (Ω_Λ), such that Ω= Ω_M+ Ω_rad+ Ω_Λ= ρ/ ρ_critical, where ρ is density and ρ_critical is the density at which the universe is spatially flat. The current consensus with this model is that the universe is flat, i.e., Ω_total=1 and that the radiation term can be excluded at least to model the later universe so that we need consider only matter and “dark energy”. The currently accepted best fit values using this model are where Ω_M=0.3, Ω_Λ=0.7. The various fits to this model can be seen in Figure 7 alongside our current models.

As can be seen the Ω_M=1, Ω_Λ=0, matter dominated model, falls below the observations – i.e., predictions are that the objects should be brighter and closer (lower modulus) than they are. Mathematically Ω_M can be deemed a deceleration term, Ω_Λ an acceleration term. So, to balance against observation, more acceleration is required and Ω_M=0.3, Ω_Λ=0.7 is the best fit. The full Freidman equation can be seen below:

$$H^2={\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho -\frac{k{c}^2}{a^2}+\frac{\Lambda }{3}$$

(15)

Where H is the time dependent Hubble constant, a is the scale factor of the universe relative to today, ρ is the density, G is the gravitational constant, and k is a constant and can be +1, 0 or -1. k defines the curvature of the cosmological model, with 0 signifying a flat universe. $\Lambda =3H_0^2{\mathsf{\Omega }}_{\Lambda }$ and is the cosmological constant. Einstein originally added the cosmological constant to allow the equation to predict a static universe, but when the universe was seen to be expanding, he realized it was no longer required. Ironically the term has been revived to do the opposite and provide a repulsive term which is believed to be associated with dark energy. We can derive the Freidman equation from our model, as we did in our first paper, but when we do this then we do not derive the cosmological constant. Also, if the universe is flat, then we are only left with the first term of equation 10. This would be consistent with a universe that expands indefinitely with the rate of expansion dictated by the distribution of energy and matter.

Figure 7. Duplication of Figure 4 with addition of Friedman fits for Cosmological models Ω_M=0.3, Ω_Λ=0.7 and Ω_M=1 for comparison.

Figure 7. Duplication of Figure 4 with addition of Friedman fits for Cosmological models Ω_M=0.3, Ω_Λ=0.7 and Ω_M=1 for comparison.

In FLRW theories, the assumption is that the initial impetus from the big bang has diminished and no longer influences the dynamics. Consequently, only the terms related to the density of matter and other contemporary energy types are considered. In a flat universe, the kinetic energy term vanishes, leaving only the density-dependent terms. The model used in time expansion theory assumes that the universe has always been expanding and works out the shape of the universe and the way light would arrive with us if the rate of expansion were constant based on the universe having a fourth expansion dimension and then from that derives how the expansion rate has varied.

From our model we can calculate $H^2$ as a function of r, the radius of the expansion sphere. Then ignoring the curvature (which FLRW theories assume is zero after inflation) and the cosmological constant from equation 10 for a moment we can use this to determine the density of the matter from the Friedman equation as a function of radius:

$$\rho =\frac{3H^2}{8\pi G}=\frac{3{\left(\frac{v_H}{r}\right)}^2}{8\pi G}=\frac{3v_H^2}{8\pi G{r}^2}$$

(16)

We can also work out the volume of the universe in 3D space - the circumference of our sphere would be the extent of 3D space in all directions. Multiplying the density and volume you can determine the mass contained within the universe as a function of radius. We also know that if every deflection from expansion in the universe leads to a time dilation and creation of mass, and if we assume for a moment this is the only retardation mechanism, then it follows that the mass of the universe,

$$M_u\propto \frac{c}{v_H}$$

(17)

We can then set the value today of $M_u$ in equation 17 from the value in equation 16 and see how the two compare over radii of the expansion sphere and use the difference to give us insight on how the two theories compare.

Results can be seen in Figure 8 over the range for which luminosity data is available. The plot shows that the mass of the universe derived by the simple $\frac{c}{v_H}$mechanism predicts a larger mass compared to simple Freidman derivation in the radius of expansion sphere since r=45 Mpc with a peak at around 65 Mpc diminishing to today. What this tells us is that if we were to analyse the data using the Freidman equations then a repulsive expansion term would be required to match the data in the recent universe. In our model we are getting at the velocity of expansion more directly by allowing the expansion dimension to behave in the same way as the other 3 dimensions and when we do this, we see the velocity of expansion continue to slow.

So, with this simple exercise linking the two theories, we now have a reason for the inconsistency between conclusions. We reach our conclusions as our model directly extracts a speed of expansion in contrast to FLRW which relies on linking to the Friedman equation from the start.

However, the results of the above comparison are perhaps valid beyond providing an explanation for the discrepancy.

Figure 8. Mass of universe comparison using simple $\frac{c}{v_H}$ divided by Freidman calculation derived from our results as described in the text.

Figure 8. Mass of universe comparison using simple $\frac{c}{v_H}$ divided by Freidman calculation derived from our results as described in the text.

The implications of the variation of mass on our theory can be interpreted in a few ways. First off, the comparison may be over simplified, and the mass of the universe may not be directly related to the speed of expansion or there may be other factors to consider. Another conclusion might be that the universe is not completely flat after inflation. Lastly, gravity itself might vary over time. A flat universe would basically mean that the kinetic energy of expansion always balances the gravitational pull back. The large-scale structure of galaxies and the uniform CMB data do indicate that the post inflationary universe points towards a flat universe. It could just be that the universe is almost flat but the small percentage has this effect on the universe. Alternatively, in our model, as gravity comes about because of the slowing down of the universe from the speed of light, then it follows that the force of gravity experienced by all objects in the universe may indeed vary over time. There is no observational evidence of this. But the fact that the balance between density, size and speed of expansion of the universe (in our simple calculation at least) leads to a variation of mass of only between 5 and 10% in the last 10 billion years might prompt alternative theories to posit it as a genuine constant when it is in fact varying by this degree over billions of years. If gravity is fluctuating though, it might indicate that after the inflation period, the universe has existed at the critical density balanced by variation in the gravity constant which comes about due the balance of matter in the universe.

Resonance in Time

The theory results in one quantified result that we can compare to observations, and that is the current average expansion speed of the time dimension. The best value for this speed, $v_H$, is 6.87 ± 0.36 × 10⁶ ms^-1, as discussed earlier. In a universe that is created from such simple building blocks then it seems likely this rate is responsible for other “constants” in the universe.

Water molecules confined within a wave exhibit a tendency to rotate at a velocity consistent with the traveling speed of the wave. Therefore, if we are all caught up on a wave of time travelling at $v_H$, the particles caught up within the wave will perhaps try and spin at this velocity. If this were the case, the time to repeat a spin cycle in time would be $2\pi r_p/v_H$, where $r_p$ is the radius of the “particle” orbit within the time wave and $v_H$ is the current velocity of time. In the 3D world from which we observe this, we cannot perceive the width of the time dimension, so all we see within 2 of the 3 dimensions of “real” space would be a repeating phase of $2r_p$. That is, the wavelength at which we observe “radiation” should be $2r_p$. The frequency of repeat would therefore be:

$$f=\frac{v_H}{2r_p}$$

(18)

Therefore, if a particle spins in time, it will have an apparent spin speed in the 3D world $\frac{2r}{2\pi r/v_H}=\frac{v_H}{\pi }=$ of 2.19 × 10⁶ ms^-1. This velocity is approximately 1/137 times the speed of light. This is the value of the fine structure constant, which is known to be responsible for quantifying the strength of electromagnetic interaction. This finding ties together with our qualitative argument that this rotation in time is what causes the electromagnetic effect.

The velocity is also the most likely velocity (or Bohr model velocity) at which the electron travels in the first orbit of the hydrogen atom. Therefore, perhaps the speed of time effectively creates a resonant speed of rotation of the electron that matches the speed we see.

The oscillation in time gives us the energy stored in the time dimension as spin, which is then available to be brought into the 3D world we know upon interaction. The presence of electrons in the real world might be the result of this resonance with respect to the time dimension. The ideal lining up of the time dimension with an idealised spin generated from the passing of time would lead to the perfect Bohr model orbital. This would then answer one of the mysteries of the world about why the fine structure constant has its value. It would also indicate that as the speed of time slows – then the fine structure constant would decrease accordingly indicating it was different in the past.

Once you have the fine structure constant, then the value of Planck’s constant, and electric charge are consequently set. If the energised particle takes on spin speed of $v_H$, the current velocity of time and we assume all the energy available is twice the kinetic energy (potential energy is negligible or mirrored in the kinetic energy originating from the wave)

(19)

Combining 18 and 19:

$$E=m_p{v}_{H}2r_{p}f$$

(20)

If we take the particle to be an electron and radius to be the Bohr radius then we find that $m_e{v}_{H}2r_e=$6.624 x 10^-34 which is the Planck Constant. Other particles such as the photon may be affected and be allowed to spin at different rates, but it seems from above that they are somehow tied to the energy above.

We now have compelling evidence that our theory can accurately account for gravity and the electromagnetic force —the oscillation in time of charged particles will create an attractive or repulsive force which will interact with other charged particles with a force related to the fine structure constant which in turn appears to be related to the velocity of time. Once combined then the overlap of this oscillation releases a favourable amount of energy back to the expansion dimension. We will now for completeness touch on the strong and weak nuclear interactions.

4. Final Conclusions

References

1. Jarvis, G. Time blast wave theory for the expansion of the universe. J Mod Appl Phys. 2022, 6, 1–17. [Google Scholar] [CrossRef]
2. Riess AG, et al. 1998 Astron.J. 1998, 116, 1009–1038. [Google Scholar] [CrossRef]
3. Betoule M, et al. Astron Astrophys. 2014, 568, 22–53. [Google Scholar] [CrossRef]
4. Alan, H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D. 23, 347 – Published 15 January 1981. [CrossRef]
5. Planck Collaboration. 2016 Planck 2015 results XIII. Cosmological parameters. A&A 594, A13 (2016). 2016. [Google Scholar] [CrossRef]
6. Riess, AG, Casertano S, Yuan W, Macri LM, Scolnic D. Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics Beyond Lambda CDM. The Astrophysical Journal 2019, 876, 85. [Google Scholar] [CrossRef]
7. Mather, J. C., et al. Measurement of the Cosmic Microwave Background Spectrum by the COBE FIRAS Instrument. The Astrophysical Journal 1994, 420, 439. [Google Scholar] [CrossRef]
8. Hawking S. Nature, Vol. 248, No. 5443, pp.30-31, 1 March 1974. 1 March; -31.
9. Corbelli E, Salucci P. The extended rotation curve and the dark matter halo of M33. Monthly Notices of the Royal Astronomical Society 2000, 3111, 441–447. [Google Scholar] [CrossRef]
10. Young T. The Bakerian lecture. On the theory of light and colours. Proc. R. Soc. Lond. 1832, 163–167. [Google Scholar] [CrossRef]
11. Einstein, A. Ann. Phys. 17, 132, 1905.
12. Parker S. A Single-Photon Double-Slit Interference Experiment. Am. J. Phys. 1971, 39, 420–424. [Google Scholar] [CrossRef]
13. Tirole R, Vezzoli S, Galiffi E, Robertson I, Maruice D, Tilmann B, Maler SA, Pendry JB, Sapienza R. Double Slit time diffraction at optical frequencies. Nature Physics 2023, 19, 999–1002. [Google Scholar] [CrossRef]
14. P G Merli, G F Missiroli and G Pozzi. Am. J. Phys. 1976, 44, 306–307. [Google Scholar] [CrossRef]
15. Stern, O. Ein Weg zur experimentellen Nachprüfung der Richtungsquantelung im Magnetfeld. Zeitschrift für Physik, 7, 249-253. Zeitschrift für Physik 1921, 7, 249–253. [Google Scholar] [CrossRef]
16. J. Adam et al. (STAR Collaboration), Measurement of e+e− Momentum and Angular Distributions from Linearly Polarised Photon Collisions. Phys. Rev. Lett. Published 27 July 2021. 2021, 127, 052302. [Google Scholar] [CrossRef] [PubMed]
17. Bell, J. S. On the Einstein‒Podolsky‒Rosen Paradox. Physics 1964, 1, 195–200. [Google Scholar] [CrossRef]
18. Aspect A., Grangier P., Roger G. Experimental Realisation of Einstein‒Podolsky‒Rosen-Bohm Gedanken experiment: A New Violation of Bell Inequalities. Physical Review Letters 1982, 49. [Google Scholar] [CrossRef]
19. B. Hensen et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 2015, 526, 682–686. [Google Scholar] [CrossRef] [PubMed]