Accounting for the expansion of the universe using an energy/momentum model to construct the space-time metric

Hugh James

doi:10.12688/f1000research.108648.2

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Research Article

Revised

Accounting for the expansion of the universe using an energy/momentum model to construct the space-time metric

[version 2; peer review: 1 not approved]

Hugh James

PUBLISHED 30 Jan 2023

Author details Author details

Retired, AWE, Aldermaston, Reading, Berkshire, RG7 4PR, UK

Hugh James
Roles: Conceptualization, Writing – Original Draft Preparation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background: The success of the theories of special and general relativity in describing localised phenomena, such as objects undergoing high speed motion or located in gravitational fields, needs no further elaboration. However, when applied to the evolution of the universe several problems arise which can require an additional model, e.g., inflation during the early expansion, and adjustments to parameters to account for phenomena such as the late-time acceleration of the universe.
Methods: Focusing on the difference between the ways in which space and time are measured, this paper shows that there are two paths which allow the equations of special relativity to be produced from the same basic postulates.
Results: While the standard theory utilises the Minkowski metric, an alternate path is possible which uses an energy/momentum, or dynamic model which transforms the Minkowski metric into an Euclidean form by multiplying the coordinates by functions of γ (=(1-v²/c² )^-1/2) to derive a new space-time metric. When utilising this dynamic metric, the relativistic equations are unchanged for local phenomena such as the Lorentz coordinate transformation and the energy/momentum equation for high-velocity objects.
Conclusions: However, the derived metric alters the perceived overall structure of the universe in a manner that, for the simplest model under this system, allows the reproduction of observed cosmological features, such as the intrinsic flatness of the universe and the apparent late-time acceleration of its expansion, without the need of additional models or changes in parameter values.

Keywords

Cosmology, large-scale structure of universe, relativistic processes.

Corresponding author: Hugh James

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2023 James H. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: James H. Accounting for the expansion of the universe using an energy/momentum model to construct the space-time metric [version 2; peer review: 1 not approved]. F1000Research 2023, 11:344 (https://doi.org/10.12688/f1000research.108648.2) First published: 21 Mar 2022, 11:344 (https://doi.org/10.12688/f1000research.108648.1) Latest published: 27 Jan 2025, 11:344 (https://doi.org/10.12688/f1000research.108648.4)

Revised Amendments from Version 1

This version attempts to clarify the relationship between the standard Minkowski metric and the dynamic metric developed in the paper. This is described in a revamped Fig.1 and Eqns 7-11, plus supporting text. The addition of Tables 1 and 2 show the dynamic Lorentz transform and how it can be converted back to standard coordinates using Eqn 7. Eqns 21-23 have been also been corrected using Eqn.7. An additional exercise has been carried out in Fig.7, plus supporting text, which describes that the current tension between standard theory values of the Hubble constant for different ranges of redshift in cosmology is also reflected in the fit of the standard theory to the dynamic curve. The dynamic curve only requires one value of Hubble constant.

See the author's detailed response to the review by Boudewijn F. Roukema

Introduction

The success of the current theories of special (SR) and general relativity (GR) in describing localised phenomena, such as objects undergoing high speed motion, located in local gravitational fields or generating gravitational waves, needs no further elaboration. However, when applied to the evolution of the universe several problems arise which require either an additional model, e.g., inflation to account for the flatness or the horizon problems, or adjustments to coefficients and/or parameters to account for phenomena such as the universe’s late-time acceleration and the lack of observable mass to match the universe’s expansion¹.

In exploring an approach which attempts to answer at least some of these problems, while still satisfying the basic postulates of relativity, it will be shown that it is possible to construct a space-time metric that satisfies the Lorentz coordinate transformation while producing an alternative overall structure of the universe. Any modification of the current theory must still conform to both the observation that the speed of light (c) is a constant in all inertial frames, and to the principle that the laws of nature are the same in all inertial frames. It also must conform to the principle of consistency, i.e., any new theory has to account for the successful predictions of the Newtonian and relativistic theories that it attempts to modify.

The alternative approach taken by this paper is based on the argument that in space-time it is only the spatial dimensions that are directly observable, while a time dimension must be constructed from indirect observations such as the reliance on memory or artificial aids such as film. Current theories rely on such a construction to complete a four-dimensional space-time, which has been very successful but has some problems on a universal scale such as those outlined above. An alternative approach is discussed which assumes, for reasons explained below, that it is the energy and momentum of objects that are fundamental to their behaviour, and from which the time element of a four-dimensional space-time coordinate system can be inferred, rather than starting from a four-dimensional coordinate system against which the energy and momentum of objects can be measured. Such an approach allows the Minkowski metric to be transformed into an Euclidean form (see Equation (7)–Equation (11) below). The possible reason for this change in emphasis will be discussed, and the consequences of such a change are explored.

The simplest model of the universe that can be constructed using this dynamic metric consists of a surface, which is locally observed to consist of three spatial dimensions and a changing present, but has a non-local view in which this surface is expanding along four spatial dimensions. All observers are located on this surface and consequently have no local view of this fourth spatial dimension into which this expansion occurs, but instead interpret it as time (i.e. the changing present, see subsequent sections on the basis of the dynamic theory and also its relevance to cosmology). All gravitational effects that we can see and measure take place along this surface. Consequently when this model also approximates the mass distribution along this surface, and at any given time during the surface’s expansion, as being uniform, the gravitational potential along the surface (which is normal to its expansion) is zero. This approximation allows the surface expansion to be calculated using only special relativity and without recourse to general relativity - as will be shown in the cosmology sections. Obviously more detailed analyses using realistic mass distributions would require GR to take a role, but the current approach shows that even in its most basic form this model provides very good predictions of the universe’s expansion (Cosmology Section).

In the theory it is postulated all objects in SR have a vector equal to the universal expansion, shown as equivalent to the speed of light. This is possible as it is the surface expanding rather than the object in motion relative to the surface. The components of this vector are seen from other inertial frames as spatial motion and changes in time, as discussed more fully in this paper.

The basis of the dynamic theory in special relativity (SR)

Possible grounds for allowing an alternative to the coordinate theory

As already stated, a possible area that could lead to more than one approach arises with the inclusion of time as the equivalent of a spatial dimension. Leaving aside any philosophical questions about what space and time are, and just concentrating on how they are measured when considering relativistic effects illustrates the problem of mixing time and space to form the space-time of coordinate relativity.

Basically, relativistic effects are measured with a clock and a ruler. To state the obvious a ruler is composed of a distribution of matter which is taken to define a spatial dimension. For the purposes of relativistic measurements, the ruler is unchanging for an observer in the same inertial frame, i.e. the coordinates it provides along a spatial dimension are static within an inertial frame and any position on this ruler can be revisited at will (albeit at different times). In contrast a clock gives a continuous (and regular) conversion of potential into kinetic energy which is able to do work on its surroundings, e.g. the release in energy from a clock spring to continuously move the clock’s hands, or the decay of radioactive material to activate a counter. This is intended to model what an observer experiences as time, i.e. he is always trapped in the current moment in the sense that he can only influence or be influenced by events which happen in the present (memories of past events may be said to influence him, but can only do so in his present), but this moment is always changing^*. Since the observer is always in the present, past events can only be accessed by memory or aids such as film. This access is of a different order to the interactions granted to events in the present. An observer can visit a spatial location many times and interact with what is happening at that location. The ability to physically interact with an event in time is restricted by the fact that a temporal location can only be visited once. Since the observer only exists in the present, temporal coordinates can only be constructed by integrating experiences in the present using memories or other recording devices. The ever-changing present has no real coordinate that can be directly experienced (in the sense of spatial coordinates that can be both seen to be separated and individually revisited at will) since from within an inertial frame no dimension can be seen in what can be assumed as the time direction. The experience of time is analogous to a piece of cardboard with a vertical slit cut in it being placed over a train window. There is an ever-changing view of a segment of the landscape seen through the slit, but you cannot see what is coming and you need either a good memory or some recording apparatus to build a complete picture of where you have been. The whole directly observable view of the world is the ever-changing view through the slit.

Time in this view appears to facilitate the change of the energy state - objects appear, grow, change or vanish as they pass the slit - while space provides a framework which can facilitate a distribution of objects. Consequently, there are two routes that can be taken when attempting to mix space and time. The coordinate route of the standard theory keeps the spatial dimension unchanged while integrating the changing temporal view to give a fourth dimension which is analogous to space. The dynamic view keeps time as an ever-changing entity but requires this to be linked to objects (including other observers) which have an ever-changing spatial position. Hence the dynamic view is based on energy and momentum while the coordinate view is based on trajectories.

Since the integral of the dynamic view will give coordinates (albeit artificially constructed), and since it has already been stated that, at least locally, the two theories must match for the dynamic view to have any validity, the next two sections show where the two theories coincide, where they diverge and the derivation of the dynamic metric.

The intersection of the standard and dynamic theories

To clarify which theory is being referred to in the following, the derivatives in the standard theory result from infinitesimal changes in the dimensional coordinates and relate to trajectories. In this paper such changes are denoted by d(). In the dynamic theory D() are infinitesimal changes in coordinates which are derived from the magnitude of the change in energy experienced in the present (located in what can be constructed as the time direction - i.e., orthogonal to the spatial directions), and the momentum of objects in the spatial dimensions. Subscripts N and M denote any inertial frame, in contrast to A and B which usually have specified conditions such as which frame contains the observer and which the observed.

In the standard theory a single subscript is used to denote the frame in which the coordinates are located, e.g. dt^A, dx^B, see Equation (1) below. In the dynamic theory two subscripts are used for time, e.g. Dt_AB where the first subscript is the frame containing the observer and the second the frame containing the object being observed. For space there are three subscripts, e.g. Dx_ACB where the first and third subscripts are as in the description for time, while the middle subscript is the frame in which the ruler is located against which the spatial motion is measured. In this example A is the observer, B is the object being observed and C is the frame containing the ruler against which A measures B’s motion.

Both theories must satisfy the condition that the speed of light is the same in all inertial frames.

For the standard dimensional coordinate theory, this leads to the Minkowski metric which provides the following relationship between space and time

c^{2} d τ^{2} = c^{2} d t_{A}^{2} - d x_{A}^{2} = c^{2} d t_{B}^{2} - d x_{B}^{2} (1)

where A and B are inertial frames of reference, velocities are along the x axis and so changes in y and z are zero, and dτ is defined as the proper time interval, i.e. it is time recorded by a clock which moves with the object of interest and is an invariant quantity. In this paper, for brevity - but without any loss of generality - only spatial motions along the x axis are considered. For photons dτ = 0.

In the dynamic theory several points must be considered.

The view of all inertial frames is always done from the present of an observer, and since both theories agree that there is no such thing as a universal present, the continuous changes are seen from a single viewpoint.
The dynamic theory deals with energy and momentum, i.e. the constant change in time (where coordinates must be inferred) and space (real coordinates). For brevity, these phenomena will sometimes be referred to as “motion”. As already outlined, this motion will be denoted by Dt_NM for time – where M is the frame in which the clock resides, and N is the frame containing the observer – while Dx_NNM refers to spatial motion where M is the frame in motion relative to both a ruler (middle subscript) and observer (first subscript) located in N.

The key point in dynamic SR is that the theory assumes that the fundamental parameters are the energy/momentum (motions) of objects (which includes observers) which are used to construct a 4-vector coordinate system, and not a 4-vector coordinate system that the motion is measured against. Consequently, for the dynamic theory the above equation has to be written as

c^{2} D t_{A B}^{2} = c^{2} D t_{A A}^{2} - D x_{A A B}^{2} and c^{2} D t_{B A}^{2} = c^{2} D t_{B B}^{2} - D x_{B B A}^{2} (2)

to preserve both the existence and equivalence of two unique viewpoints - A sees B as B sees A - as well as the constant value of the speed of light from both viewpoints, i.e. if B is a photon then Dt_AB = 0, and vice versa. However, as will now be shown, the grouping of the parameters and the general invariance of the time term will be different.

Concentrating on A as the observer with the single viewpoint, then in an inertial frame the only motion that he can directly experience is that of his clock (Dt_AA). He does not experience any spatial motion (Dx_AAA = 0), and any such motion that could be seen from outside the frame will, from A’s view, be added to the spatial motion of B. This motion A sees as being relative to A’s own ruler (Dx_AAB). However, A is able to see (for this illustration and accounting for relativistic Doppler effects) B’s clock (Dt_AB). Consequently, grouping the motion that A can see of himself, and the motion that A sees of B on separate sides of the above equation gives the dynamic metric (which should be compared to the Minkowski metric in Equation (1)) as

c^{2} D t_{A A}^{2} = c^{2} D t_{A B}^{2} + D x_{A A B}^{2} . (3)

The same view can be seen from B with the appropriate change in subscripts, i.e.

c^{2} D t_{B B}^{2} = c^{2} D t_{B A}^{2} + D x_{B B A}^{2} . (4)

The first thing to note is that the invariant is now Dt_NN as this is the only parameter that N can experience of itself, and one of the main principles of SR is that nature’s laws must be the same when seen from inside an inertial frame. Since a lot of these laws are time-dependent, then if Dt_AA ≠ Dt_BB there should be an effect seen from outside the frame which cannot be ascribed to either relative velocity or gravity. No such effect has ever been reliably reported so it is a safe postulate that Dt_AA ≡ Dt_BB.

The second thing is that the only coordinates that A can directly observe are spatial. Hence the three subscripts for Dx – where the middle one relates to a ruler - compared to the two subscripts for Dt. However, as already mentioned, a time coordinate can be inferred by simply integrating Dt. If there is to be any overlap between the dynamic and coordinate theories, we need to find the relationship between the inferred and standard coordinates and, consequently, the relationship between their differentials.

In the standard coordinate theory, it is dτ that is the invariant between inertial frames, i.e. when we look for where the two theories overlap, we are looking for specific conditions where dτ = Dt_NN. This will be shown to be along the time-like hyperbola in Figure 1 where dτ is an invariant for all possible inertial frames that have their origin at O, i.e. OA=OB=OG=cdτ.

Figure 1. Inertial frames intersecting the time-like hyperbola from A’s viewpoint.

The value of gamma is constant for a given frame, but depends on the velocity of that frame relative to A - see Equation (7) and Equation (14)–Equation (16).

In Figure 1, A is the observer, and the condition dτ = Dt_NN can only occur where the observer and the event or object being observed lie entirely on their respective time axes, e.g. at A, B and G in Figure 1. This condition corresponds to dx_N = 0 in Equation (1), and Dx_NNM = Dx_NNN = 0 in Equation (3) and Equation (4). The three sets of inertial axes (A, B and G) in this figure are shown from A’s viewpoint. All frames have a common initial location at O. These positions are consistent with the dynamic hypothesis that from within any inertial frame, the only features of the frame that changes are those entirely connected with changes in time (i.e. Dt_NN).

The divisions shown on the axes in the figure are arbitrary.

In the Figure it can be seen that all inertial time axes with a common origin (O) must intersect the hyperbola, and so this hyperbola can be considered as the locus of intersections of all the time axes of objects undergoing different inertial spatial motions as seen from A.

Consequently, for these conditions to be met, the dynamic equivalence of the coordinate viewpoint requires that every object (and hence every observer) must be located on its own time axis, even if such an axis can only be inferred by such an observer. This corresponds to the initial definitions that were given about the dynamic condition. Every point on the above hyperbola coincides with a time axis of an object which has a given relative velocity to A, and whose velocity is different to every other point on the curve. The location of an object on its own time axis means that Dt_AA= Dt_NN. However, since this occurs at every point along this curve, in general terms, Dt_NN and dτ are both invariant along this hyperbola.

Hence, from the above along the time-like hyperbola

D t_{N N} = d τ at D x_{N N N} = 0 (5)

And since

c^{2} D t_{N N}^{2} = c^{2} d τ^{2} then c^{2} D t_{N M}^{2} + D x_{N N M}^{2} = c^{2} d t_{N}^{2} - d x_{N}^{2} (6)

Also since dx_N⁄cdt_N= Dx_NNM⁄cDt_NN=ν⁄c,

then from this and Equation (6) we can, after some manipulation, derive the following relationships

D t_{N N} = d τ

γ D t_{N N} = d t_{N} (7)

γ D x_{N N M} = d x_{N}

γ D t_{N M} = d τ = γ^{-1} d t_{N} .

Where γ = (1 – v²/c²)^–1/2. And where v/c=Dx_NNM/cDt_NN, see Equation (14)–Equation (16) and Table 1 for examples of changes in v.

Table 1. Dynamic Lorentz transforms from the viewpoints of A and B derived from Table 2 and Equation (7).

Column 1	Column 2
Dx_BBG=γ(Dx_AAG–νDt_AA):	Dx_AAG=γ(Dx_BBG+νDt_BB)
Dt_BB=γ(Dt_AA–vDx_AAG⁄c²):	Dt_AA=γ(Dt_BB+νDx_BBG⁄c²)
γ=γ₁(γ₂/γ₃)	γ=γ₁(γ₃/γ₂)

Table 2. Standard Lorentz transforms.

Table 1 derived from these and the use of Equation (7).

Column 1	Column 2
dx_B=γ(dx_A–νdt_A):	dx_A=γ(dx_B+νdt_B)
dt_B=γ(dt_A-νdx_A⁄c²):	dt_A=γ(dt_B+νdx_B⁄c²).

Hence $γ^{- 1} (d x_{N}, d t_{N}) = (D x_{N N M}, D t_{N N}) . (8)$

Consequently we can rewrite the dynamic portion of Equation (6) in terms of standard coordinates to give

c^{2} d τ^{2} = γ^{- 4} c^{2} d t_{N}^{2} + γ^{- 2} d x_{N}^{2} = c^{2} d t_{N}^{2} - d x_{N}^{2} . (9)

To express the Minkowski and dynamic metrics in more familiar nomenclature, let

μ,v = 0 to 3. Then cdt_N = dx⁰; dx_N = dx¹; dy_N = dx²; dz_N = dx³.

The Minkowski metric is

{(c d τ)}^{2} = η_{μ v} d x^{μ} d x^{v} (10)

where

\begin{array}{l} η_{μ ν} = 1 0 0 0 \\ 0 -1 0 0 \\ 0 0 -1 0 \\ 0 0 0 -1 \end{array}

While the dynamic metric is

{(c d τ)}^{2} = γ^{- 2} η_{μ v}^{'} d x^{μ} d x^{v} (11)

Where

\begin{array}{l} η_{μ ν}^{'} = γ^{- 2} 0 0 0 \\ 0 1 0 0 \\ 0 0 1 0 \\ 0 0 0 1 \end{array}

The equivalent dynamic diagram to Figure 1 is, from A’s viewpoint, given in Figure 2. However, from this Figure when B moves relative to A in the dynamic theory, it appears to A that B’s time is partly at a reduced time rate and partly as a motion against A’s spatial dimensions. In the limit a photon would appear to A as only having a spatial motion and with no time element present. Although to an imaginary observer within the photon’s frame of reference, according to the requirement of nature’s laws having to be constant, the photon’s time would appear to proceed at Dt_NN while A would only have a motion along the photon’s spatial dimension during which A’s clock would appear frozen. Hence in this limit of possible frame rotations, what appears to be time from within the frame is seen from the outside as a motion through space against the background of the outside observer’s spatial dimensions. To an observer within a frame there is no observation (as distinct from inference) of any dimension of any sort - temporal or spatial - in the time direction and so it appears that the space against which this motion takes place does not exist, i.e. it has collapsed to a point on the time axis where it corresponds to the ever-changing, dimensionless present. However, the summation of all the views from every possible frame indicates the overall framework within which any individual observation is made is of four orthogonal spatial dimensions. It is just that the individual observer can only see three of them plus a changing present (see Cosmology Section).

Figure 2. The dynamic view of inertial frames.

Similarities and differences between the theories

Time dilation. In Equation (3) it should be noted that all the observations can be made by A, so the dynamics are A’s clock as seen by A (cDt_AA), equated to B’s spatial motion (Dx_AAB) and clock (cDt_AB). Dividing through by $c^{2} D t_{A A}^{2}$ and noting v = Dx_AAB/Dt_AA, gives Dt_AB = γ^–1Dt_AA, where γ = (1 − v²/c²)^−1/2.

Hence for the standard theory where dt_A ≠ dτ

γ = d t_{A} / d τ = D t_{A A} / D t_{A B} (12)

i.e. the dynamic is the same as the standard coordinate time dilation relationship. Substituting the equivalent standard coordinates for the dynamic ones in Figure 2 (see Equation (7)) would give the same Euclidean diagram but with cdt_AA being kept constant and cdτ changing as the velocity of B is increased.

Length contraction. The key element when dealing with spatial dimensions in the dynamic theory is that no spatial dimension can be seen to exist by an observer in his own time direction. However, there will be items that have unchanging spatial coordinates within an inertial frame, and which can also be seen outside of that frame. Take frames A and B as being stationary relative to each other at some point in time. Both are equipped with a rods of equal length along the x axis. The frames are then given a relative velocity of v, and A then compares B’s rod with his own. The dynamic equations given above can be represented by a simple axis rotation equal to ζ = sin⁻¹(v/c), and by the acknowledgement that the spatial x coordinates lie in the same direction as Dx.

For the dynamic theory let any coordinate terms be represented (e.g. the rod length) by δx, δy etc. Being a coordinate within an inertial frame, this is unchanging in time, in contrast to Dx which we have defined as constantly changing to match the ever-changing present. The subscript convention is the same as for Dx, e.g. δx_ABC where A is the observer, B the frame containing the ruler and C is the frame containing an object that A wishes to measure and which lies in A’s x direction. For Euclidean-type coordinate transforms, A would see δx_BBB as having a component δx_BBB cos ζ in A’s spatial direction and δx_BBB sin ζ along A’s time axis. However, since there is no spatial dimension that can be seen by A along his own time axis, the length of B’s rod which is present in A’s inertial frame is seen by A to only be

δ x_{A A B} = δ x_{B B B} \cos ζ = γ^{- 1} δ x_{B B B} (13)

This is the same length contraction result as for the standard theory, and this length contraction is independent of time since we are using δx. The apparent contraction in coordinate space for B (seen from A), is also seen (by A) to occur in the spatial motion of frames relative to B, since the motion takes place in a space that appears to have contracted by a factor of γ⁻¹. Since only space is involved when considering just δ, and A’s clock is unaffected, then in turn the motion arising from this contraction can be assumed to occur for all times between inertial frames. This includes observations by both A and B due to the postulation of the equivalence of viewpoints in inertial frames in SR.

Both the time and space effects in special relativity are apparent effects. The word "apparent" is used as the changes in the parameters are brought about by changes in perspective rather than any fundamental change of the value of the parameter, e.g. in the length contraction discussed above, the same contraction is seen by B in A’s rod. Fundamental changes may be assumed to only occur once gravitation (or acceleration according to Einstein’s equivalence principle) is considered, but this is beyond the scope of the present paper.

It should also be noted that in the dynamic theory while Dx_NNN and Dx_NMM are always zero (the frame cannot move relative to its own ruler), δx_NNN and δx_NMM can have non-zero values.

Lorentz transformation. An additional spatial motion must be used by the dynamic theory to obtain the Lorentz transformation, as Dx_NNN is zero. For all spatial motions taking place along A’s x direction, the transformation can be derived as follows.

Take frame G moving relative to both A and B, where the spatial motion of G relative to the ruler in B is observed by B to be Dx_BBG, and the spatial motion relative to a ruler in A (observed by A) is Dx_AAG. The velocity of B (observed by A) is given by v = Dx_AAB/Dt_AA; the velocity of G relative to B (observed by B) is w = Dx_BBG/Dt_BB and the velocity of G relative to A (observed by A) is u = Dx_AAG/Dt_AA.

Let

γ_{1} = {(1 - ν^{2} / c^{2})}^{- 1 / 2} (14)

γ_{2} = {(1 - u^{2} / c^{2})}^{- 1 / 2} (15)

γ_{2} = {(1 - w^{2} / c^{2})}^{- 1 / 2} (16)

As can be seen from the subscripts, the left side of Column 1 in Table 1, and the right side of Column 2 are B’s observations of G’s motion relative to the clock and ruler in B’s frame. The reverse is true for A, with A’s observations of G being on the right of Column 1 and the left of Column 2.

Dividing the top row by the bottom in Column 1 gives

w = (u - v) / (1 - v u / c^{2}) (17)

while dividing the top row by the bottom in Column 2 gives

u = (w + u) / (1 + v w / c^{2}) (18) .

When comparing these equations with the coordinate version of the Lorentz transformation, they give both the same velocity-addition relationships and, as will be shown, the same coordinate transforms.

In Table 2, γ=γ₁ and substituting the appropriate D( ) term into d( ) using Equation (7) gives the terms in Table 1.

The 4-vector, energy and momentum. The dynamic theory gives the same energy/momentum relationship as the coordinate theory, but results in a change of interpretation of the role of rest mass energy (E₀). This result can be obtained geometrically for the dynamic theory by taking the top right-hand quadrant of Figure 2, but omitting the G axis. In the following, the terms OA, OB, OE and EB are those used in Figure 2.

Note that γ=γ₁ in the rest of the text. Let each of the sides of the triangle OBE in this figure be divided by cDt_AB and denote them by a dashed superscript, which then gives OB^/ =OB/cDt_AB; OE^/ = OE/cDt_AB. Hence triangles OBE and OB^/E^/ are similar which gives

OB^/ = OA^/ = cDt_BB/cDt_AB = cDt_AA/cDt_AB = γ;

E^/B^/ = Dx_AAB/cDt_AB = γv/c;

OE^/ = 1.

The energy/momentum vectors can then be obtained by multiplying each of the sides by B’s rest mass energy (E₀= mc²) where m is B’s mass. In turn this gives

OE^/ = E₀.

OB^/ = γE₀ = E.

E^/B^/ = γmvc²/c = pc,

where p is momentum, i.e. p = γmv.

From Figure 2, OB² = OE² + EB². Hence (OB^/)² = (OE^/)² + (E^/B^/)², and so

E^{2} = E_{0}^{2} + p^{2} c^{2} (19) .

as in the coordinate theory.

It should be noted that the rest mass energy lies in A’s time direction and so appears to be linked to the continually changing time, i.e. it is a kinetic rather than a static phenomenon.

Let κ = E/cDt_AA = E/cDt_BB where the mass term m in E = γmc²is still B’s mass. From this, κ is also equal to E₀/cDt_AB. Substituting κ into Equation (19) leads to the general conclusions that Dt_NN is proportional to E; Dt_NM is proportional to E₀and hence

D t_{N N} / D t_{N M} = E / E_{0} (20) .

This is consistent with the basic hypothesis made in the dynamic theory that the measurement of time, which underpins SR, is based on the observed expenditure of the energy of objects.

The 4-vector, acceleration and force. Strictly acceleration does not belong in SR, but since we are dealing with 4-vectors, and this is an area of significant difference in terms of interpretation between the theories, it is included here.

In the standard coordinate theory the 4-force (F) is

F^µ = dp^µ/dτ

where µ = 0, 1, 2, 3 denoting the dimension the force is acting in, i.e. F^µ is equivalent to force components acting in time(t), x, y, z.

By removing the mass we get the 4-acceleration (A^µ) which is equal to dU^µ/dτ (U equals velocity). The acceleration in the time direction is

A⁰= dU⁰/dτ = dγ/dτ.

Along the x axis the acceleration is A¹ = dU¹/dτ = d(γv)/dτ.

In the dynamic theory sinζ = BE/OB (≡ BE/OA) in Figure 2,

i.e. ζ = sin⁻¹(Dx_AAB/cDt_AA) = sin⁻¹(v/c).

Now γ = (1 − v²/c²)^−1/2or γ = cos⁻¹ζ. From this, and after some manipulation (remembering d_τ= Dt_AA from Equation (7))

A^{0} = d U^{0} / d τ = D (c o s^{- 1} ζ) / D t_{A A} = γ^{2} (v / c) D ζ / D t_{A A} (21)

and

A^{1} = d U^{1} / d τ = d (γ v) / d τ = γ^{2} c (D ζ / D t_{A A}) (22) .

As v → 0, sinζ → ζ, A⁰ → 0, but A¹ → cDζ/Dt_AA, i.e. while spatial acceleration in the Newtonian limit (γ→1) for the coordinate theory is a = dv/dt_A, in dynamic terms this is

d v / d t_{A} = cD (v / c) / D t_{A A} = c D ζ / D t_{A A} (23) .

Hence for the dynamic theory in the Newtonian regime, it is the change in angle per change in time which produces the acceleration rather than the change in velocity per unit time. The total (albeit largely unseen) velocity of the object (c) remains the same before and after the acceleration, only the direction has changed to produce a spatial velocity relative to the observer’s axes, and an apparent time dilation shown by the object’s clock.

Cosmology

The standard model and its problems

In describing the universe, the standard theory has evidence of an expanding space-time which started at a singularity and was impelled outwards by the Big Bang some 13.7 billion years ago. In the simplest model the mass of the universe is assumed to be uniformly distributed and the expanding 4D space-time is taken as analogous to the surface of an expanding balloon in which all views and expansions are along the surface. Nothing is assumed to exist outside of this surface so what can be taken in an ordinary balloon as an expansion which is normal to its surface, is only seen in the standard model as a stretching surface. There are a number of problems with this model¹ which can be summarised as

The need for an inflation model. This postulates that at very early times the universe underwent a very rapid expansion which was abruptly switched on and then off, and has never been seen since. There currently appears no solid physical explanation for this model^1,3 and the reason for adopting it is that it provides an explanation of some of the following observational conundrums.
The flatness problem. From observation the universe is very close to its critical mass density which means the curvature of space-time is near zero⁴. Since the universe is thought to start from a point, the inflation model is needed to suddenly expand to a surface whose curvature is locally insignificant.
The isotropic or horizon problem. The microwave background radiation is very uniform in all directions. Either we are somehow at a unique place in the cosmos, or all parts of the universe were in contact at the earliest times after the Big Bang. However, the areas that needed to be in contact exist along a line of sight that precludes even the speed of light to be fast enough to provide a connection. Again, the inflation theory can be used to provide an initial expansion that was far faster than the speed of light and so allowed such connections to exist at early times.
The existence of quantum fluctuations needed to provide the seeds of the current galaxies. The origin of such fluctuations requires the formation of virtual particles that are separated by space-time expanding too fast for them to recombine. An inflation-type model is needed for this expansion.

And some items which cannot be explained by inflation.

Dark matter. There is a lack of observable mass in the universe needed to account for its expansion using the standard general relativity theory. There also appears to be too little mass to account for the rotation of galaxies. The shortfall is sometimes postulated as being due to a so-far unobserved particle and is often referred to as dark mass. To account for the dynamics of the universe’s expansion using the standard theory, this dark mass would have to consist of about 25 percent of the matter in the universe.
Dark energy. It has recently been found^5,6 that the expansion of the universe appears to be accelerating, rather than decaying as would be expected if the expansion were only controlled by gravitational effects (or at least, effects due to the standard theory of gravitation^**). This energy would be equivalent to about 70 percent of the matter needed to model the universe using the standard theory.

The dynamic theory will attempt to answer all of the above points based on the different structure of the universe generated by the differences between coordinate and dynamic relativistic models. In the dynamic theory no further models, such as inflation, will be needed. The next segment will construct the simplest model of the universe that is consistent with the dynamic theory. The last segment will compare quantitative predictions from this theory with observations.

The simple dynamic model

In the simplest model the dynamic theory assumes the universe started as a singularity embedded in four spatial (not space-time) dimensions at, what we consider, the time of the Big Bang. This provides a uniform radial expansion in the four dimensions if we assume that mass is uniformly distributed throughout the history of the expansion. This is assumed to be analogous to the expanding shell of a 3D sphere, where the mass can be taken as lying at the centre of the sphere, despite all the material being located in the shell, and there is no gravitational potential along the shell normal to the radial expansion vector.

Time comes into being as soon as the expansion starts. This is the radial expansion vector which is related to the speed of light (see above). In effect the model is the opposite of the standard theory. There nothing exists apart from a 4D surface and expansion, gravitation etc. takes place along this surface. In the dynamic theory, assuming uniform mass, the expansion only occurs radially into a 4D space. From the model outlined in the first few Sections, in our local surroundings 3D beings such as ourselves see this radial expansion as time acting on 3D space, but with no observation of a fourth spatial dimension.

Because this is a radial expansion, an observer (A) sees a non-local galaxy (such as B) as having a spatial motion away from him as the radial expansion occurs at an angle to A’s time direction. This is very similar to the behaviour of inertial frames in special relativity section. To reiterate, the motion and the curvature is assumed to be normal to a three-dimensional spatial surface. This 3D spatial surface is assumed to be flat, i.e., there is zero intrinsic curvature along this surface while there is extrinsic curvature in the fourth dimension which is locally seen as time, and non-locally seen as providing an increasing velocity of the 3D surface in relation to A, so providing a stretching surface. However, it should be emphasised that the curvature is only in the fourth dimension (see Figure 3).

Figure 3. Simple dynamic model of the current state of the universe.

There are some points which have to be attached to the simple 4D picture of the dynamic universe (see also Figure 3):-

There is no significant motion along the surface normal to the radial expansion. Otherwise matter will tend to clump and the uniform mass model will tend to be invalid. Obviously this does happen to an extent in the real universe as galaxies etc. exist. But in this model we are taking the simplest possible mass distribution where in spatial terms it is assumed that most of the universe can be treated as not having clumped matter.
There is no overall gravitational potential along the 3D surface as it is assumed analogous to a uniform shell of material which forms the surface of a 3D sphere in Newtonian space, where the potential gradient lies along the radius but is zero along the surface normal to this radius. Locally the same comment about non-uniformity applies as in the previous bullet.
As a consequence of the last point General Relativistic effects can, to a first approximation, be ignored when considering the observed overall motion of this surface, in contrast to the standard theory where they dominate due to all forces having to lie along a 4D surface^***.
In the standard theory the redshift in B, as seen by A, could be generated by either a receding velocity or a gravitational redshift (or a combination of both). From the previous bullets, in the dynamic theory it is assumed that to fit the simple theory the redshift is overwhelmingly provided by a receding velocity, which in turn results from the angle the radial velocity at any location makes with an observer. Hence, to reiterate previous bullets, it is assumed that there is little matter (compared to the mass of the universe) that is trapped in intense local gravitational fields (in a black hole or around stars) or moving at high speeds along the 3D surface.
The expansion velocity of the universe would be observed to be constant (and equal to the speed of light) once the perspective effects due to curvature have been subtracted, despite the overall expansion probably decreasing. This apparently contradictory state of affairs results from there being no direct way for 3D observers to measure changes in the actual 4D radial expansion rate from within the universe. Any real change in radial motion will be equally seen in both the passage of time and the expansion of space (see Figure 3), so that a phenomenon such as velocity remains unchanged.

Ironically having spent most of the first part of this paper trying to emphasise that the time dimension does not exist - at least not when observed from within a given reference frame - now most of the remaining analysis requires the construction of a “time dimension” in order to situate the relative position of bodies and events, e.g. such as where and when A locates galaxy B, or where events are located relative to the Big Bang. This is carried out by integrating the radial motion to provide a dimensional construct in which events lie relative to A.

There are two types of time shown in the following diagram (Figure 4). Solid black lines are “actual” surfaces which consist of the locus of 4D trajectories from the Big Bang to the present. Each trajectory is locally constructed, e.g. OA and OB in the Figure. The present is the surface constructed from the termination of these trajectories such as at A and B. This surface is where the 4D sphere lies, but is not observable other than locally since information about B can only arrive at A in B’s past. These surfaces are defined in time by integrating cDt_NN (γ = 1 see previously for nomenclature).

Figure 4. Simple dynamic model of the evolution of the universe.

The second type (red line) is an observed surface. Observer A sees time dilation effects in B because of the galaxy’s recessional velocity, and these, plus the time delay for photons from B to reach A, only allows A to see B in B’s past (see below for a more detailed analysis). Such observations always occur in A’s present since this is the only time at which A can interact with events or phenomena such as photons. However, that the observation is of B’s past is assessed as being so by A through the methodology discussed below, and which allows A to construct B as a point in A’s own past, i.e. at the location shown on A’s time dimension. These surfaces are defined in time by integrating γcDt_AB.

The quantitative dynamic model

To quantify the simple dynamic model we can draw the Figure 4 from the assumptions previously discussed, i.e. that each part of the universe’s surface is expanding radially outwards at a function of the speed of light and that the trajectories of OA and OB since the Big Bang are straight. These straight lines result from the unchanged velocity (see above) of objects seen by an observer to be moving with this surface. Note that in this simple model the only observable movement on a universal scale is due to the expansion of this surface, high velocity objects along the surface are ignored. From the viewpoint of these observers the surface resulting from the locus of the endpoints of these trajectories is located at cT₀ (T₀is the integral of Dt_NN over the period from the Big Bang to the present - see previous Section). The principal observer is located in the present at position A, although the same figure will be obtained if we swap to B being the principal observer.

At the risk of some repetition, the details of Figure 4 are summarised in the following bullet points.

A’s view of the universe originates from the “observed” surface (red line) whose observed and actual surfaces can be calculated as described in the last bullet point below.
An imaginary superhuman, who can see in four dimensions, will see the “actual” surface moving radially out from the Big Bang. He will see all observers located on the “present” surface (although they will only have A’s view of the rest of the universe), and the past is located at previous positions of the present surface as it expanded out from the Big Bang. These are the concentric surfaces represented by C’C and DD’ on A’s time axis. Note that T = 0 is the location of the Big Bang while T = T₀is the present. Also R = 0 is the Big Bang location in what A sees as a spatial axis, while R = cT₀ is the actual (but for A not observable - see above) spatial position of the present surface.
Again it should be emphasised that from A’s viewpoint his time axis is an imaginary construct where the locations of “actual” past surfaces are obtained by integrating cDt_AA in the dynamic theory. The observation of past events - such as the most recent view of galaxy B - will occur in A’s present because this involves the integration of γcDt_AB, and as explained above, this will be observed by A in his present despite this view being from B’s past. A’s construction of a time axis helps visualise the order of these events, so that A sees B as existing at D’ (T/T₀= 0.55) in the figure (the detailed construction of this is explained below and the next section), while B sees herself at T/T₀= 1. For brevity A’s time axis will be referred to as if it were a real axis in the rest of this paper.
A will see the time of events, such as his view of B, given by clocks located at the event, i.e. in this example the clock is moving with B, hence the time is a function of γcDt_AB. The exact form of this function is derived below. Photons have Dt_AP = 0 and so always travel along vertical trajectories in this figure (e.g. CD’ is a photon path). A will see actual surfaces only along his time axis (e.g. at D’ and C’).
Note that each surface links cT to R, so that for A in the present, T₀is the time from the Big Bang while R₀(= cT₀) is the spatial radius of the present surface.

How does A observe B?

Let us assume that A and B had clocks travelling with them that were set to zero at the Big Bang and have been accumulating time ever since. Because of the fourth-dimension curvature, A sees B moving at velocity v/c along A’s spatial axis. Because B has a straight radial trajectory, this velocity has not changed over time.
We are only dealing with special relativity so that A’s observation of B is from C where OC/OB = OC′/OA = (1 − v²/c²)^1/2, i.e. to A the galaxy at B has not travelled as far along OB as it should due to the time dilation factor of the apparent velocity.
Note the actual surface CC’, the observed surface (red line) and B’s trajectory all coincide at C (shown in Figure 4 within the limits of the drawing package). The photon from C travels vertically (CD’) to reach A to show B appearing at D in A’s past. In geometric terms the actual surfaces are constructed as OA = OB; OC′ = OC; OD′ = OD.
To calculate the observed and actual surfaces let cT₀=OB; cT=OD′; cT₁=OC′; X=CD′ and X₀=BC′.
Then we can write for the observed surface
$X / R_{0} = {[T / T_{0} - {(T / T_{0})}^{2}]}^{1 / 2} (24)$
For the actual surface, AB is given by
$X_{0} / R_{0} = {[1 - (T_{1} / T_{0})}^{2}]^{1 / 2} (25)$

Comparison of the predictions of the dynamic and coordinate theories

How does the dynamic theory answer the problems laid out at the start of this Section which are faced by the coordinate theory?

The evolution of the universe. The scale factor s of the universe at time T after the Big Bang can be expressed as

s = c T / c T_{0} \equiv R / R_{0} . (26)

Hence it can measure the amount of stretch in the surface between then (T) and now (T₀), and is often used when comparing experiment and theory. Let s be the scale factor resulting from the ratio, shown in Figure 4, of B’s apparent position on A’s time axis (OD’) and A’s present (OA), i.e. s = OD′/OA. (Equally B locates A along her time axis (OB) as being on the same surface, DD’, but with s = OD/OB). However, when concentrating on A, A’s estimation of the real position of B (OB), and the ratio of the apparent to real positions (OD’/OB), are not entirely along A’s time axis (OA). Hence his view of the apparent value of s for B on surface DD’ is affected by three phenomena discussed below

The relativistic Doppler redshift.
The redshift z is related to the frequency of light emitted at B from surface DD’ (f_e) and the frequency observed at A (f₀) by 1 + z = f_e/f₀. The coordinate relativistic Doppler redshift is then related to the recessional velocity by
1+z = ((1+v/c)/(1−v/c))^1/2.
The redshift has the same function as if the object being measured is embedded in a surface which is stretching between it and the observer.
Then 1 + z = f_e/f₀ = OA/OD^′ = OB/OD^′= OB/OD = 1/s.
By combining the two versions of 1+z and rearranging we get
$v / c = (1 - s^{2}) / (1 + s^{2}) . (27)$
This only applies along B’s trajectory (OB) from A’s viewpoint as A cannot see himself in motion, only B appears to have motion - see the Special Relativity (SR) Section.
The trajectory reduction.
As already mentioned, A will see B’s trajectory foreshortened due to the space-time dilation effects of B’s relative velocity. B will have only apparently travelled to OC rather than OB where OC/OB = (1 − v²/c²)^1/2. However, we already have v/c as a function of s from the Doppler redshift, and so to make the two phenomena consistent, A’s ultimate view of OD′/OB has to incorporate OC/OB (= OC′/OB), i.e.
$O C^{'} / O B = {(1 - {[(1 - s^{2}) / (1 + s^{2})]}^{2})}^{1 / 2} . (28)$
The photon travel.
From A’s viewpoint B is on surface CC’, from which a photon has to be emitted to reach A. This is along CD’ which is a time OC′ − OD′ in the past from the surface CC’ on which (from A’s viewpoint) B should be located. We note that OC′−OD′ lies entirely along A’s time axis and so is purely a geometric term when looking at Figure 4. Let angle DOD′ = ζ.
Then in geometric terms, cosζ = OD′/OC = OD′/OC′ = OC′/OA.
Hence OD′/OA = (OC′/OA)² = s, and so along A’s time axis

O C^{'} / O A = O D^{'} / O C^{'} = s^{1 / 2} . (29)

We have already defined the scale factor s to be OD′/OA along A’s time axis. However, A sees the scale factor for B between the same two actual surfaces as

OD′/OB = (OD′/OC′)(OC′/OB) = s^1/2(1 − [(1 − s²)/(1 + s²)]²)^1/2.

Let OD′ = cT and OB = cT₀ and after some simplification A’s view of B is

T / T_{0} = 2 s^{3 / 2} / (1 + s^{2}) . (30)

Let H(T) be the Hubble parameter at time T after the Big Bang and H₀is the present value of the parameter. H₀ = 1/T₀and H(T) = s⁻¹ds/dT. From A’s viewpoint H(T) varies along the time axis (OA) and along this axis R/T = dR/dT = c, the velocity of light (each actual surface is at R = cT). Since s = R/R₀then ds/dT = $R_{0}^{- 1}$ dR/dT and so

H (T) = s^{- 1} d s / d T = R^{- 1} d R / d T = c / R = 1 / T . (31)

Let H(T) be designated as just H. Now H =1/T and so H/H₀= T₀/T. Note that in the dynamic theory H is different to the coordinate theory where H = 2/(3T). This difference occurs because in the coordinate theory the variation in H is based on the density of the universe, in contrast to the dynamic theory which is based on geometric effects, i.e. in the coordinate theory

H_{0} / H = {(Ω_{M} s^{- 3} + Ω_{R} s^{- 4} + Ω_{Λ})}^{− 1 / 2} (32)

where Ω_M is the current normalised density of matter (the Standard Cosmology Model (SCM) value quoted in 9 is 0.32); Ω_R is the normalised density of radiation (5×10⁻⁵ and so is ignored for what follows); and the SCM value for Ω_Λ(the normalised dark energy coefficient) is 0.68.

In the dynamic theory the variation in H is based on the geometry of the universe and, since this is based on special rather than general relativity, this is due to perspective effects (as discussed further below) and so is more apparent than real.

In the dynamic theory, from the above relationships A’s view of B is

H_{0} / H = 2 s^{3 / 2} / (1 + s^{2}) . (33)

Remembering that s = 1/(1 + z), Figure 5 shows a comparison of the two theories (Equation (32) and Equation (33)).

Figure 5. The comparison of the coordinate and dynamic models of the universe’s evolution.

Considering the simplicity of the model used in the dynamic theory, it is remarkably close overall to the SCM coordinate theory’s values for the coefficients (red dashed line, in the Figure). A rather better fit to the higher z values is given by the blue line (coefficients given in the Figure) as will be explored below and in Figure 6 when the relationships are differentiated. The dynamic theory (solid black line) comes just from the geometry of the universe’s structure that was obtained by changing from coordinate to dynamic representations of space and time as outlined previously.

Figure 6. The comparison of the standard coordinate and dynamic models of the universe’s deceleration and late-time acceleration.

At the high z values a good fit is obtained to the dynamic theory by the SCM’s coefficients when the curve is reduced by 10%. This is shown in Figure 7 (green curve). Consequently, to fit to the dynamic curve over the complete range of z two SCM curves are needed (see Figure 7), the high z curve needing a value of H₀ which is 10% lower than its value obtained from nearby (low z) cosmological data. If there is any merit in the dynamic theory this may explain the current “tension” between values of H₀ obtained from the cosmic microwave background (high z) and from more recent (low z) cosmological feature (e.g.10).

Figure 7. The fit of two SCM’s with different values of H₀ to a single dynamic theory curve.

The important points to take away from the dynamic theory are: -

The change in Hubble parameter shown in Figure 5 is obtained from A’s observation of B. However, these are not the actual surfaces. For that H₀/H = T/T₀ = s. Very much like in special relativity A’s view of B, i.e. T/T₀ = 2s^3/2/(1+s²), is due to perspective rather than any real relationship. B’s view of A will be identical.
The match between the dynamic and coordinate theories in describing the universe’s evolution means that if there is any merit in the dynamic solution, there is no need to consider that the overwhelming proportion of dark mass (represented by Ω_M in the coordinate theory) exists. However, some must be present (or some currently unknown physical phenomenon must apply) as the rotation of galaxies requires more mass than is observable.
The match can also mean that dark energy (the accelerating universe represented by Ω_Λ) is also just a feature of perspective. This follows from the differential of the observable surface that shows the universe slowing down at large redshifts - as would be expected in the coordinate theory from the effects of gravity - and then in the relatively recent past beginning to accelerate. This is shown below (Figure 6), along with the position, in terms of redshift, where the dynamic theory predicts the acceleration to start. Comparisons of this start (z_a) with results from various models based on both the coordinate theory and observational astronomy are given in Table 3 below.

The differential ds/dT of A’s observation of B in the dynamic theory is given by H₀/H = $T_{0}^{- 1}$ s(ds/dT)⁻¹. Hence

T_{0} d s / d T = 0.5 (s^{- 1 / 2} + s^{3 / 2}) . (34)

This should be compared to the coordinate acceleration of

T_{0} d s / d T = {(Ω_{M} s^{- 1} + Ω_{R} s^{- 2} + Ω_{Λ} s^{2})}^{1 / 2} . (35)

These equations are shown in terms of redshift (z) in Figure 6 and (z_a) in Table 3 below.

It should be noted from Figure 6 that when the dynamic curve is divided by T₀ - the current age of the universe for an observer - the shape of the curve (i.e. the late time acceleration) is independent of the epoch in which the observation is made.

The flatness problem. In the dynamic theory the curvature is always caused by the radial velocity. In turn this is always in the time direction for human observers - a direction in which locally they cannot see any sort of dimension. This is illustrated in qualitative terms by Figure 3 in which it is assumed that A can see the current (according to both A’s and B’s clocks) position of B with no account taken of time dilations or delays. See previous Section for a more quantitatively precise description of A’s view of B in the dynamic theory. In Figure 3, A is again the main observer and B a distant galaxy. The real (i.e., total) motion of B lies along B’s time axis, which is orthogonal to the 4D surface. A sees only the spatial component of B’s real motion as a velocity lying parallel to A’s spatial dimension, and hence, from A’s viewpoint, projected onto it. In the Figure, a photon would have its time direction aligned with A’s spatial axis and its space direction along A’s time axis.

The dynamic universe has an extrinsic curvature, the curvature being confined to the fourth dimension. However, to any observer on this 4D surface one of these dimensions is missing (it is in the time direction) and the three remaining spatial dimensions having no intrinsic curvature appear flat.

Consequently, it is possible for this 3D space to appear flat from the viewpoint of any observer located on the 4D surface (B and A can be interchanged in terms of what each sees of the other). The only view an observer such as A has of this curvature is the projection of the radial velocity onto his 3D space by distant objects.

The horizon problem. This is simply solved in the dynamic theory because all elements of the universe are in contact in the 4D space at the location of the Big Bang. While the coordinate theory has to deal with connections between various parts of the universe within an expanding surface when trying to explain the isotropic nature of the microwave radiation, the dynamic theory postulates that all parts were initially in contact and expanded radially outwards.

Dark mass and dark energy. The dark energy needed to explain the late-time acceleration of the universe in the coordinate theory, appears in the dynamic theory to be due to the perspective created by the presence of observed rather than actual surfaces (see above), i.e. in the dynamic theory there is no requirement for it to exist.

A lot of the dark mass also appears to be due to problems with the perspective and can be explained away as not required in the dynamic theory. However, this may not be true for all of it as there remains some unanswered questions about the rotation of galaxies for instance.

Inflation-type behaviour. Working through the various relationships obtained previously shows that Equation (34) can be written as

d R / d T = 0.5 c (R_{*}^{- 1 / 2} + R_{*}^{3 / 2}) (36)

where R_* = R/R₀, i.e. the radius of the universe at time T from the Big Bang, normalised by the current radius. As can be seen dR/dT = c at the present time, and tends to infinity for values of R close to zero at T = 0. Hence the inflation-type behaviour is implicit in the model without resorting to a separate phenomenon.

Discussion and conclusions

The basis of this paper results from a paradox. The way we measure space and time show that they are two very different entities. And yet it has been shown beyond doubt that they need to be fused into space-time in order to accurately describe our universe. One may argue that “real” space and time are far removed from the way we measure them, and this could well be the case. However, at its most basic construction these two entities have been defined by the ruler and the clock, and on this has been erected one of the most successful theories in science. However, it is a theory which on a cosmic scale has produced conundrums. It has been argued in this paper that such conundrums have arisen because there is more than one way that this fusion of space and time can be achieved.

The use of a dynamic theory to create such a fusion allows for a metric which is the Minkowski metric transformed into Eulerian form by multiplying the coordinates by a function of γ (see Equation 7–Equation 11). The two metrics give the same local dynamic descriptions of motion for special relativity (SR), but has them occurring against the background of a different universal structure. The dynamic theory arises from the following summary of the chain of arguments:-

In the SR dynamic theory

There is no directly observable dimension in time when seen from within an inertial frame, and changes in time are the only dynamic property of the frame that can be experienced by an observer from within this frame. However, both spatial and temporal changes relating to that frame can be seen from other frames.
Only spatial coordinates are present in all the observed frames. A spatial position can physically be visited many times, a temporal event can only be physically visited once.
Spatial coordinates can be observed but temporal coordinates have to be constructed from second-hand experiences, such as memory or films, coupled with the calculation of the integral of the change in time displayed by clocks.
It is argued that the changes in the present are measured by changes in energy, and so it is possible to choose energy and momentum to construct a space-time coordinate system rather than assume a coordinate system exists against which energy and momentum can be measured.

How does such a system tie up with a coordinate description to give the same predictions for SR phenomena such as high-speed objects?

Changes in coordinates can be grouped together by the Minkowski metric (Equation 10), where dτ is an invariant for inertial frames. Grouping together the dynamic properties gives the dynamic metric Equation (11),
where Dt_NN is the invariant. Both Minkowski and dynamic metrics apply along the time-like hyperbola in Figure 1 (Equation (6)). The coordinates can be transformed between the Minkowski and dynamic metrics where Dt_NN=dτ by using
γ^–1(dx_N, dt_N) = (Dx_NNM, Dt_NN).
Both theories give the same Lorentz transformation (Table 1 and Table 2).
Both theories give the same relativistic energy/momentum equation (Equation (19)).

Table 3. Values of z_a from both theories and observations.

z_a	Comments	Reference
0.732	Shape independent of T₀.	Dynamic Theory
0.619	Value at current epoch.	SCM⁹
0.784	Best coordinate fit	(high z, Figure 6)
0.752+/-0.041	Astronomical data	4
0.72+/-0.05 to	5 models plus	11
0.84+/-0.03	38 measurements of H(z)	11

Where does the dynamic theory part company with the standard coordinate theory?

It is shown that time from a viewpoint within a frame is a constant motion along a space dimension from a viewpoint outside that frame.
In the limit this spatial motion is equivalent to the speed of light, while the space along which this motion takes place is rolled up into a point for an observer inside the frame who experiences this phenomenon as time (i.e. there is no observable dimension in one’s own time).
This indicates a universe of four spatial dimensions, which are always seen as three spatial dimensions and a continuously changing present.
In this theory, relative velocity is obtained from differences in the spatial direction of the motion in the fourth dimension, and not by changing its real magnitude.
SR phenomena such as time dilation result from a change of perspective, not a change in magnitude. Consequently, while frame A sees frame B as having a slower clock than his own, frame B has exactly the same view of A.

In GeneralRelativity (GR)

Counter-intuitively GR does not play a role in the simplest dynamic cosmology model outlined in the next segment. The reason is that GR is important in local phenomena such as the space-time surrounding massive objects, but on the cosmological scale in the dynamic theory, observations such as the Hubble redshift can be treated as geometric-based phenomena. These arise from the presence of a fourth spatial dimension, as discussed above.

In dynamic cosmology

The extra spatial dimension along which objects are in motion, that is derived from the SR model in the dynamic theory, allows a simple description of the universe:-

In the simplest model that can be constructed, the dynamic theory assumes the universe started as a singularity embedded in four spatial (not space-time) dimensions at - what we consider to be- the time of the Big Bang.
This provides a uniform radial expansion in the four dimensions if it is assumed that mass is uniformly distributed throughout the expansion.
Time comes into being as soon as the expansion starts. This is the radial expansion vector which is related to the speed of light. In the simple dynamic model, assuming uniform mass and negligible motion along the universe’s surface, the expansion only occurs radially into a 4D space. See Figure 3 and the accompanying section for how the radial expansion is experienced locally as time, and from a distance as a spatial velocity.
Equally, for such a model, there is assumed to be no overall gravitational potential along the 3D surface. It is assumed to be analogous to a uniform shell of material which forms the surface of a 3D sphere in Newtonian space, where the potential gradient lies along the radius but is zero along the surface.
In the standard theory the redshift in a distant galaxy, as seen from Earth, could be generated by either a receding velocity or a gravitational redshift (or a combination of both). From the previous two bullets, in the dynamic theory it is assumed that to fit the simple theory the redshift is overwhelmingly provided by a receding velocity, which in turn results from the angle the radial velocity at any location makes with an observer (Figure 3). Hence, it is also assumed that there is little matter (compared to the mass of the universe) that is trapped in intense local gravitational fields (in a black hole or around stars) or moving at high speeds along the 3D surface.

In effect the model is the opposite of the standard theory. There nothing exists apart from a 4D surface, and expansion, gravitation etc. takes place along this surface. From the model outlined above, in our local surroundings 3D beings such as ourselves see this radial expansion as time acting on 3D space, but with no observation of a fourth spatial dimension. The universe, from some outside perspective, is a 3D surface located in a 4D space into which it is expanding.

The quantitative version of the model is derived in the cosmology sections. The detailed mathematics is left for the main text but the results can be summarised as follows:-

There is excellent agreement between the dynamic theory and the best observations of the evolving universe (Figure 5, Figure 6 and Table 3). Consequently, if there is any merit in the dynamic theory the implications can be summarised by the following bullets.
Very much like in special relativity Earth’s view of a distant galaxy’s motion (i.e. the redshift) is due to perspective. The galaxy’s view of Earth will be identical.
The match between the dynamic and coordinate theories in describing the universe’s evolution means that there is no need to consider that the overwhelming proportion of dark mass exists. However, some must be present (or some currently unknown physical phenomenon must apply) as the rotation of galaxies requires more mass than is observable.
The match means that dark energy is just a feature of perspective and not real, or at least cannot be determined by current observations. (One drawback of the dynamic theory is that there is no direct way for 3D observers to measure changes in the actual 4D radial expansion rate from within the universe. Any real change in radial motion will be equally seen in both the passage of time and the expansion of space, so that a phenomenon such as velocity remains unchanged.)
Comparisons of the theory’s start of the universe’s late time acceleration (z_a) with results from various models based on both the coordinate theory and observational astronomy show good agreement.
It should be noted from Figure 6 that when the universe’s acceleration curve is normalised to the current age of the universe, the shape of the curve (i.e., the late time acceleration) is independent of the epoch in which the observation is made.
The inflation model is not needed to account for the flatness and horizon problems, although it is implicit in the dynamic model. Near the time of the Big Bang the expansion tends to infinite velocity to an observer at any point (except in a black hole) on the current 3D surface of the universe. Consequently, it can be deduced to have the right conditions to supply the quantum fluctuations needed to provide the seeds of the current galaxies. The origin of such fluctuations requires the formation of virtual particles that are separated by space-time expanding too fast for them to recombine.
When comparing the fit of the standard and dynamic models to the history of the universe in terms of its acceleration and deceleration (Figure 7), the single dynamic curve has to be fitted by two SCM curves over the complete range of z. The two SCM curves differ by a factor of 10% in the value of H₀ between fitting the high z and low z regimes. This difference is also observed when fitting the standard theory to cosmological data and is currently described as the Hubble tension¹⁰. This is not present in the dynamic formulation which has a single value of H₀.

To sum up, this paper attempts to show that it is possible to have an alternative view of space-time which agrees with the standard theory for local phenomena but provides a different view of the overall structure of the universe. This in turn provides some possible answers to some of the current problems faced in cosmology. It is far from the complete answer to every conundrum, but it is hoped that it may provide some alternative pathways to explore in the future (or from the dynamic viewpoint, in a present which is not currently being accessed!).

Data

No data are associated with this article.

Author endorsement

Professor John Curtis confirms that the author has an appropriate level of expertise to conduct this research, and confirms that the submission is of an acceptable scientific standard. Professor John Curtis declares they have no competing interests. Affiliation: Atomic Weapons Establishment; Visiting Professor UCL Mathematics Department.

Acknowledgements

The author wished to thank those who initiated invaluable discussions and gave advice during the early stages of the production of this paper.

Footnotes

^*This view has a long history, e.g. Heraclitus of Ephesus (c.535-c.475BC) has an ever-present change being a fundamental essence of the universe²

^**There are theories which postulate non-standard gravitational behaviours and a large number of alternative theories exist to explain the apparent effect of dark energy. See a summary and references in 7

^***See, however⁸, for example where an eleven dimensional space is discussed.

Faculty Opinions recommended

References

1. Peebles PJE, Ratra B: The Cosmological Constant and Dark Energy. Rev Mod Phys. 2003; 75: 559–606. Publisher Full Text
2. Beris AN, Giacomin AJ: Everything flows. Appl Rheol. 2014; 24(5): 1–13. Publisher Full Text
3. Earman J, Mosterin J: A Critical Look at Inflationary Cosmology. Philosophy of Science. 1999; 66(1): 1–49. Publisher Full Text
4. Suzuki N, Rubin D, Lidman C, et al.: The Hubble Space Telescope Cluster Supernova Survey. V. Improving the Dark Energy Constraints Above z > 1 and Building an Early-Type-Hosted Supernova Sample. Astron J. 2012; 746(1): 85. Publisher Full Text
5. Perlmutter S, Aldering G, Goldhaber G, et al.: Measurements of Ω and Λ from 42 High-Redshift Supernovae. Astron J. 1999; 517(2): 565–586. Publisher Full Text
6. Riess AG, Filippenko AV, Challis P, et al.: Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron J. 1998; 116(3): 1009–1038. Publisher Full Text
7. Rashki M, Fathi M, Mostaghel B, et al.: Interacting dark side of universe through generalized uncertainty principle. arXiv: 1901.05766v2. 2019. Publisher Full Text
8. Rostami T, Jalalzadeh S: Why the measured cosmological constant is small. arXiv: 1510.02068v1. 2015. Publisher Full Text
9. Planck Collaboration, Ade PAR, Aghanim N, et al.: Planck 2013 results. I. Overview of products and scientific results. Astron Astrophys. 2014; 571: 48. Publisher Full Text
10. Di Valentino E, Mena O, Pan S, et al.: In the realm of the Hubble tension—a review of solutions. Class Quantum Grav. 2021; 38(15): 153001. Publisher Full Text
11. Farooq O, Madiyar FR, Crandall S, et al.: Hubble parameter measurement constraints on the redshift of the deceleration-acceleration transition, dynamical dark energy, and space curvature. arXiv: 1607.03537v2. 2016. Publisher Full Text

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Version 4

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Author details Author details

Retired, AWE, Aldermaston, Reading, Berkshire, RG7 4PR, UK

Hugh James
Roles: Conceptualization, Writing – Original Draft Preparation, Writing – Review & Editing

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No competing interests were disclosed.

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James H. Accounting for the expansion of the universe using an energy/momentum model to construct the space-time metric [version 2; peer review: 1 not approved]. F1000Research 2023, 11:344 (https://doi.org/10.12688/f1000research.108648.2)

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Reviewer Report 01 Nov 2023

Boudewijn F. Roukema, Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University of Torun, Torun, Poland

Not Approved

https://doi.org/10.5256/f1000research.142944.r161776

Revision 2 of this paper has not addressed the main concerns of lack of clarity (properly quantified and defined assumptions, method, results). It remains extremely difficult to try to guess what the author is proposing, and there are too many errors and verbose expressions without definitions to make it possible to guess the author's intended line of argument. There is no proper mathematical definition of the proposed model, and no separation between the model and its interpretations. The changes add material, but do not make it possible for the reader to work through the basic assumptions and method of the paper prior to trying to understand the new material.

As an example of a specific wrong physical claim that remains uncorrected, the sentence "The basis..." in paragraph 2 of the Introduction still contains an assertion that tachyons exist: "an observer ... can only influence or be influenced by events which happen in the present".

Some specific examples of problems in version 2 follow. Their correction would be necessary, but *not* sufficient, for the paper to constitute a research paper in this field.

The Introduction points the reader to Eqs (7)-(11); this requires the reader to first work through the equations starting with (1). The interpretation of (1) differs from standard SR, since the author says that d\tau is necessarily a proper time interval, contradicts himself by saying that it can be zero, and then neglects the case where (d\tau)^2 is negative, in which case d\tau is not a proper time interval.

Even more confusing in this same paragraph is that the author states that "only spatial motions along the x axis are considered". In the context of (1), which is the only equation or convention introduced by this point of the text, the lack of a definition of "spatial motion" forces the reader to speculate what this might mean. The apparently intended meaning is that the only worldlines that are considered are those "along the x axis", i.e. with constant t_A and t_B values, i.e. tachyons. On the contrary, if the author's intended meaning is that he wishes to present a pseudo-Riemannian manifold, and projections within that manifold, then that has to be presented and defined unambiguously.

The definition of Dt_{AB} prior to (1) remains obscure: there is no indication here of how a coordinate in two frames (A, B) relates to choices of coordinate systems A and B. The subscript C for the frame of a ruler "against which spatial motion is measured" leaves Dx_{ACB} undefined.

Trying to interpret the text after (4), we find that "the only coordinates that A can directly observe are spatial." This contradicts Eq (1), which has a time coordinate t_A, and also implies that clocks don't exist in the proposed model.

At three places in and around Fig. 1, a spacelike hyperbola is incorrectly described as timelike. This is wrong: a spacelike hyperbola is spacelike; it is not timelike.

Trying to continue through to (7)-(11), there is no clear way for the reader to guess the definition of the manifold being proposed.

Returning to the introduction:

Paragraph 4 of the introduction describes what appears to be the global model with the phrase "a non-local view in which this surface is expanding along four spatial dimensions"; but "is expanding" implies the existence of a fifth dimension that is timelike (let us call it x^5). The following sentence implies that one of the four spatial dimensions is to be reinterpreted as time (let us call it x^4), in which case the model must be five-dimensional with two time dimensions: x^4 and x^5.

The sentence in this paragraph "Consequently when this model ... " defines a gravitational potential which is apparently a vector field, since it refers to the potential being "normal to ...", but lacks a definition of the way in which a vector potential might be used to model gravity. The (undefined) vector potential is apparently directed in the \tau direction ("normal to its expansion").

Paragraph 5: "a vector equal to the universal expansion, shown as equivalent to the speed of light." Expansion in FLRW models is modelled as a scalar function of a time parameter; it is not a vector. The speed of light is not a vector either. Equating any of these three things ((i) a vector; (ii) universal expansion; (iii) the speed of light) to one of the other is mathematically wrong in the standard model, and this paper does not make it clear how any of these three things can be equated.

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Observational cosmology; galaxy formation; large-scale structure; cosmic topology; inhomogeneous cosmology.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

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Author Response 09 Nov 2023

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

09 Nov 2023

Author Response

The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this ... Continue reading The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this paper. The standard theory deals with coordinates and their derivatives while the dynamic theory in the paper deals with momentum (space) and energy (time) from which coordinate derivatives can be obtained. The derivatives can be linked to each other (eqs.7-11) and the reasons for differences are discussed primarily in the paragraph before Eqs. 3 & 4.

2. The paragraph referred to is simply trying to make the point that the way humans obtain a quantified measure of space and time (by rulers and clocks) has fundamental differences between these measures despite needing to amalgamate them to obtain space-time. The comment about the observer always being in his present simply reflects the human experience of time. A spatial coordinate can be visited many times, a temporal coordinate only once.
At no point do I assert tachyons exist. In the model (see under "Cosmology") the universe consists of a surface made up of the locus of all inertial frames moving outwards from a central point at what can be constued as the speed of light. This is the motion of the universe's surface, not motion along that surface which is what tachyons would require.

4. Eq.1 is the standard SR which for brevity assumes the spatial motion is along x, which saves writing
-dx²-dy²-dz² each time. According to Foster & Nightingale ("A short course in general relativity", p165, Longman, 1979) "the path through spacetime...is called its world line..and the proper time interval d\tau between points...relative to some frame K..is defined by.." and then follows my Eq.1, or in its complete form my Eq.10. The italics are their own. Further "..the proper time tau is the time recorded by a clock which moves along with the particle" Again their italics. And again on the same page "..for a particle at rest in K [or in my case A] the proper time tau is..the coordinate time t ...measured by stationary clocks in [A]." Substitute ds² for d\tau² in my Eq.1. It can then be positive, negative or zero. I have chosen the positive one as it joins up time-like intervals (Fig.1, OA, OB, OG). I have concentrated on this as A is the observer's frame in which d\tau=dt.

5. Better in the paper would be "only changes in spatial coordinates along.." and refers to the brevity discussed in 4. At that point I had not started discussing the dynamic theory.

6. Since the paper deals mainly with special relativity (SR), A, B and C are seperate inertial frames with different relative velocities and their own spatial coordinate frames and clocks. Assuming time on clocks in each frame can be seen from any other frame, then Dt_AB has A as the observer looking at the clock in frame B and comparing it to their own clock. Consequently Dt_AB (the observed clock) = Dt_AA (the observer's clock) multiplied by gamma^-1 (see text). Equally it is assumed the observer can see other frames moving relative to a ruler. Hence Dx_AAB has A observing B's change in spatial position relative to a ruler in A's frame. Dx_ACB is A observing B's space changes against a ruler in C's space. The links between observers are still given by the Lorentz transforms (Table 1 & Eqs. 14-18. Note the typo in Eq.18, the right side should be (w+v)/(1+vw/c²)).

7. This was meant to emphasise the differences between models. The standard model (Eq.1) assumes both spatial and temporal coordinates are present (dtA, dxA) while the dynamic model (Eq.4) says the only coordinates that can be directly observed are spatial, e.g. divisions on a ruler. Time coordinates have to be constructed since clocks only give changes in time, and other methods are needed to give (Dt_AA, Dx_AAB) but NOT Dx_AAA - see text.

8. See comments under 4.

9. Eq.11 defines the dynamic metric while Eq.9 defines the line element. Both are expressed in terms of standard coordinates. Fig.1 shows the Minkowski hyperbola from standard SR but constructed as the locus of the time axes of inertial frames with increasing velocities relative to a given observer (A). Eq.9 shows such a construction can also be expressed in an Euclidean form where lines of increasing velocity intersect the hyperbola.

10. The model only has 4 spatial dimensions, each frame having 3 of space and one which takes on the aspects of time. Within a frame, however, the time dimension is only inferred from memory and aids such as film. As far as direct observations and interactions with your surroundings go, the temporal dimension appears rolled up into a point which we call the present. From outside the frame (i.e. from a frame with a relative velocity to the first) this time appears as motion along a spatial set of coordinates (see paragraph after Eq.11). The observer sees a velocity but the observed sees dimensionless changing time. Consequently a "non-local" view has a universe with a surface described in 2, and under "Cosmology" in the paper.

11. The model is based on SR since any gravitational effect normal to the universe's expansion would equally affect the apparent motions of an observer in both space and time, leading to a constant expansion velocity (see Fig.3).

12. In the paper it should have been the velocity of light since it has a definite direction.
The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this paper. The standard theory deals with coordinates and their derivatives while the dynamic theory in the paper deals with momentum (space) and energy (time) from which coordinate derivatives can be obtained. The derivatives can be linked to each other (eqs.7-11) and the reasons for differences are discussed primarily in the paragraph before Eqs. 3 & 4.

2. The paragraph referred to is simply trying to make the point that the way humans obtain a quantified measure of space and time (by rulers and clocks) has fundamental differences between these measures despite needing to amalgamate them to obtain space-time. The comment about the observer always being in his present simply reflects the human experience of time. A spatial coordinate can be visited many times, a temporal coordinate only once.
At no point do I assert tachyons exist. In the model (see under "Cosmology") the universe consists of a surface made up of the locus of all inertial frames moving outwards from a central point at what can be constued as the speed of light. This is the motion of the universe's surface, not motion along that surface which is what tachyons would require.

4. Eq.1 is the standard SR which for brevity assumes the spatial motion is along x, which saves writing
-dx²-dy²-dz² each time. According to Foster & Nightingale ("A short course in general relativity", p165, Longman, 1979) "the path through spacetime...is called its world line..and the proper time interval d\tau between points...relative to some frame K..is defined by.." and then follows my Eq.1, or in its complete form my Eq.10. The italics are their own. Further "..the proper time tau is the time recorded by a clock which moves along with the particle" Again their italics. And again on the same page "..for a particle at rest in K [or in my case A] the proper time tau is..the coordinate time t ...measured by stationary clocks in [A]." Substitute ds² for d\tau² in my Eq.1. It can then be positive, negative or zero. I have chosen the positive one as it joins up time-like intervals (Fig.1, OA, OB, OG). I have concentrated on this as A is the observer's frame in which d\tau=dt.

5. Better in the paper would be "only changes in spatial coordinates along.." and refers to the brevity discussed in 4. At that point I had not started discussing the dynamic theory.

6. Since the paper deals mainly with special relativity (SR), A, B and C are seperate inertial frames with different relative velocities and their own spatial coordinate frames and clocks. Assuming time on clocks in each frame can be seen from any other frame, then Dt_AB has A as the observer looking at the clock in frame B and comparing it to their own clock. Consequently Dt_AB (the observed clock) = Dt_AA (the observer's clock) multiplied by gamma^-1 (see text). Equally it is assumed the observer can see other frames moving relative to a ruler. Hence Dx_AAB has A observing B's change in spatial position relative to a ruler in A's frame. Dx_ACB is A observing B's space changes against a ruler in C's space. The links between observers are still given by the Lorentz transforms (Table 1 & Eqs. 14-18. Note the typo in Eq.18, the right side should be (w+v)/(1+vw/c²)).

7. This was meant to emphasise the differences between models. The standard model (Eq.1) assumes both spatial and temporal coordinates are present (dtA, dxA) while the dynamic model (Eq.4) says the only coordinates that can be directly observed are spatial, e.g. divisions on a ruler. Time coordinates have to be constructed since clocks only give changes in time, and other methods are needed to give (Dt_AA, Dx_AAB) but NOT Dx_AAA - see text.

8. See comments under 4.

9. Eq.11 defines the dynamic metric while Eq.9 defines the line element. Both are expressed in terms of standard coordinates. Fig.1 shows the Minkowski hyperbola from standard SR but constructed as the locus of the time axes of inertial frames with increasing velocities relative to a given observer (A). Eq.9 shows such a construction can also be expressed in an Euclidean form where lines of increasing velocity intersect the hyperbola.

10. The model only has 4 spatial dimensions, each frame having 3 of space and one which takes on the aspects of time. Within a frame, however, the time dimension is only inferred from memory and aids such as film. As far as direct observations and interactions with your surroundings go, the temporal dimension appears rolled up into a point which we call the present. From outside the frame (i.e. from a frame with a relative velocity to the first) this time appears as motion along a spatial set of coordinates (see paragraph after Eq.11). The observer sees a velocity but the observed sees dimensionless changing time. Consequently a "non-local" view has a universe with a surface described in 2, and under "Cosmology" in the paper.

11. The model is based on SR since any gravitational effect normal to the universe's expansion would equally affect the apparent motions of an observer in both space and time, leading to a constant expansion velocity (see Fig.3).

12. In the paper it should have been the velocity of light since it has a definite direction.
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Author Response 09 Nov 2023

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

09 Nov 2023

Author Response

The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this ... Continue reading The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this paper. The standard theory deals with coordinates and their derivatives while the dynamic theory in the paper deals with momentum (space) and energy (time) from which coordinate derivatives can be obtained. The derivatives can be linked to each other (eqs.7-11) and the reasons for differences are discussed primarily in the paragraph before Eqs. 3 & 4.

2. The paragraph referred to is simply trying to make the point that the way humans obtain a quantified measure of space and time (by rulers and clocks) has fundamental differences between these measures despite needing to amalgamate them to obtain space-time. The comment about the observer always being in his present simply reflects the human experience of time. A spatial coordinate can be visited many times, a temporal coordinate only once.
At no point do I assert tachyons exist. In the model (see under "Cosmology") the universe consists of a surface made up of the locus of all inertial frames moving outwards from a central point at what can be constued as the speed of light. This is the motion of the universe's surface, not motion along that surface which is what tachyons would require.

4. Eq.1 is the standard SR which for brevity assumes the spatial motion is along x, which saves writing
-dx²-dy²-dz² each time. According to Foster & Nightingale ("A short course in general relativity", p165, Longman, 1979) "the path through spacetime...is called its world line..and the proper time interval d\tau between points...relative to some frame K..is defined by.." and then follows my Eq.1, or in its complete form my Eq.10. The italics are their own. Further "..the proper time tau is the time recorded by a clock which moves along with the particle" Again their italics. And again on the same page "..for a particle at rest in K [or in my case A] the proper time tau is..the coordinate time t ...measured by stationary clocks in [A]." Substitute ds² for d\tau² in my Eq.1. It can then be positive, negative or zero. I have chosen the positive one as it joins up time-like intervals (Fig.1, OA, OB, OG). I have concentrated on this as A is the observer's frame in which d\tau=dt.

5. Better in the paper would be "only changes in spatial coordinates along.." and refers to the brevity discussed in 4. At that point I had not started discussing the dynamic theory.

6. Since the paper deals mainly with special relativity (SR), A, B and C are seperate inertial frames with different relative velocities and their own spatial coordinate frames and clocks. Assuming time on clocks in each frame can be seen from any other frame, then Dt_AB has A as the observer looking at the clock in frame B and comparing it to their own clock. Consequently Dt_AB (the observed clock) = Dt_AA (the observer's clock) multiplied by gamma^-1 (see text). Equally it is assumed the observer can see other frames moving relative to a ruler. Hence Dx_AAB has A observing B's change in spatial position relative to a ruler in A's frame. Dx_ACB is A observing B's space changes against a ruler in C's space. The links between observers are still given by the Lorentz transforms (Table 1 & Eqs. 14-18. Note the typo in Eq.18, the right side should be (w+v)/(1+vw/c²)).

7. This was meant to emphasise the differences between models. The standard model (Eq.1) assumes both spatial and temporal coordinates are present (dtA, dxA) while the dynamic model (Eq.4) says the only coordinates that can be directly observed are spatial, e.g. divisions on a ruler. Time coordinates have to be constructed since clocks only give changes in time, and other methods are needed to give (Dt_AA, Dx_AAB) but NOT Dx_AAA - see text.

8. See comments under 4.

9. Eq.11 defines the dynamic metric while Eq.9 defines the line element. Both are expressed in terms of standard coordinates. Fig.1 shows the Minkowski hyperbola from standard SR but constructed as the locus of the time axes of inertial frames with increasing velocities relative to a given observer (A). Eq.9 shows such a construction can also be expressed in an Euclidean form where lines of increasing velocity intersect the hyperbola.

10. The model only has 4 spatial dimensions, each frame having 3 of space and one which takes on the aspects of time. Within a frame, however, the time dimension is only inferred from memory and aids such as film. As far as direct observations and interactions with your surroundings go, the temporal dimension appears rolled up into a point which we call the present. From outside the frame (i.e. from a frame with a relative velocity to the first) this time appears as motion along a spatial set of coordinates (see paragraph after Eq.11). The observer sees a velocity but the observed sees dimensionless changing time. Consequently a "non-local" view has a universe with a surface described in 2, and under "Cosmology" in the paper.

11. The model is based on SR since any gravitational effect normal to the universe's expansion would equally affect the apparent motions of an observer in both space and time, leading to a constant expansion velocity (see Fig.3).

12. In the paper it should have been the velocity of light since it has a definite direction.
The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this paper. The standard theory deals with coordinates and their derivatives while the dynamic theory in the paper deals with momentum (space) and energy (time) from which coordinate derivatives can be obtained. The derivatives can be linked to each other (eqs.7-11) and the reasons for differences are discussed primarily in the paragraph before Eqs. 3 & 4.

2. The paragraph referred to is simply trying to make the point that the way humans obtain a quantified measure of space and time (by rulers and clocks) has fundamental differences between these measures despite needing to amalgamate them to obtain space-time. The comment about the observer always being in his present simply reflects the human experience of time. A spatial coordinate can be visited many times, a temporal coordinate only once.
At no point do I assert tachyons exist. In the model (see under "Cosmology") the universe consists of a surface made up of the locus of all inertial frames moving outwards from a central point at what can be constued as the speed of light. This is the motion of the universe's surface, not motion along that surface which is what tachyons would require.

4. Eq.1 is the standard SR which for brevity assumes the spatial motion is along x, which saves writing
-dx²-dy²-dz² each time. According to Foster & Nightingale ("A short course in general relativity", p165, Longman, 1979) "the path through spacetime...is called its world line..and the proper time interval d\tau between points...relative to some frame K..is defined by.." and then follows my Eq.1, or in its complete form my Eq.10. The italics are their own. Further "..the proper time tau is the time recorded by a clock which moves along with the particle" Again their italics. And again on the same page "..for a particle at rest in K [or in my case A] the proper time tau is..the coordinate time t ...measured by stationary clocks in [A]." Substitute ds² for d\tau² in my Eq.1. It can then be positive, negative or zero. I have chosen the positive one as it joins up time-like intervals (Fig.1, OA, OB, OG). I have concentrated on this as A is the observer's frame in which d\tau=dt.

5. Better in the paper would be "only changes in spatial coordinates along.." and refers to the brevity discussed in 4. At that point I had not started discussing the dynamic theory.

6. Since the paper deals mainly with special relativity (SR), A, B and C are seperate inertial frames with different relative velocities and their own spatial coordinate frames and clocks. Assuming time on clocks in each frame can be seen from any other frame, then Dt_AB has A as the observer looking at the clock in frame B and comparing it to their own clock. Consequently Dt_AB (the observed clock) = Dt_AA (the observer's clock) multiplied by gamma^-1 (see text). Equally it is assumed the observer can see other frames moving relative to a ruler. Hence Dx_AAB has A observing B's change in spatial position relative to a ruler in A's frame. Dx_ACB is A observing B's space changes against a ruler in C's space. The links between observers are still given by the Lorentz transforms (Table 1 & Eqs. 14-18. Note the typo in Eq.18, the right side should be (w+v)/(1+vw/c²)).

7. This was meant to emphasise the differences between models. The standard model (Eq.1) assumes both spatial and temporal coordinates are present (dtA, dxA) while the dynamic model (Eq.4) says the only coordinates that can be directly observed are spatial, e.g. divisions on a ruler. Time coordinates have to be constructed since clocks only give changes in time, and other methods are needed to give (Dt_AA, Dx_AAB) but NOT Dx_AAA - see text.

8. See comments under 4.

9. Eq.11 defines the dynamic metric while Eq.9 defines the line element. Both are expressed in terms of standard coordinates. Fig.1 shows the Minkowski hyperbola from standard SR but constructed as the locus of the time axes of inertial frames with increasing velocities relative to a given observer (A). Eq.9 shows such a construction can also be expressed in an Euclidean form where lines of increasing velocity intersect the hyperbola.

10. The model only has 4 spatial dimensions, each frame having 3 of space and one which takes on the aspects of time. Within a frame, however, the time dimension is only inferred from memory and aids such as film. As far as direct observations and interactions with your surroundings go, the temporal dimension appears rolled up into a point which we call the present. From outside the frame (i.e. from a frame with a relative velocity to the first) this time appears as motion along a spatial set of coordinates (see paragraph after Eq.11). The observer sees a velocity but the observed sees dimensionless changing time. Consequently a "non-local" view has a universe with a surface described in 2, and under "Cosmology" in the paper.

11. The model is based on SR since any gravitational effect normal to the universe's expansion would equally affect the apparent motions of an observer in both space and time, leading to a constant expansion velocity (see Fig.3).

12. In the paper it should have been the velocity of light since it has a definite direction.
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VERSION 1

PUBLISHED 21 Mar 2022

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Reviewer Report 18 Jul 2022

Boudewijn F. Roukema, Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University of Torun, Torun, Poland

Not Approved

https://doi.org/10.5256/f1000research.120051.r139580

The aim of this paper is, apparently, to propose a differential pseudo-Riemannian manifold (a topological manifold M' endowed with a pseudo-Riemannian metric g', i.e. (M', g') that is an alternative to Minkowski spacetime (M, g) as a model for special relativity. However, the language and equations make it very difficult to guess what (M', g') is being proposed. It is unclear if the author is trying to express the relation between different coordinate systems for a single spacetime, or describe a relation between two or three spacetimes. Physically, the introductory section includes the experimentally absurd assertion that bradyons do not exist; and the so-far experimentally unestablished claim that tachyons exist ("... an observer...can only influence or be influenced by events which happen in the present").

The lack of a clear proposal for the paper's main aim - an alternative to Minkowski spacetime - makes it impossible to judge the proposal's implications for cosmological spacetimes. The author also appears to suggest that the proposed spacetime is identical to Minkowski spacetime (abstract: "the relativistic equations are unchanged for local phenomena such as the Lorentz coordinate transformation and the energy/momentum equation for high-velocity objects"), in which case there is no new proposal.

I would suggest that the author first work through Taylor & Wheeler's "Spacetime physics"¹ - a geometrical approach to special relativity, as opposed to the historical approach, before attempting to propose an alternative to Minkowski spacetime. The alternative model should be clearly defined using standard mathematical terminology of differential geometry, or of another field of mathematics if the model is not differentiable pseudo-Riemannian 4-manifold.

Some specific examples of problems follow:

The abstract states that the "equations" of special relativity following from (M', g') are identical to those for (M, g). It incorrectly states that a coordinate framework is a fundamental requirement of the theory. Special relativity assumes that bradyonic particles exist as timelike world lines in (M, g) independent of any coordinate system.

The paragraph "Basically, relativistic effects..." is an extremely confusing way of presenting the worldline of an observer in Minkowski spacetime. It asserts the existence of tachyons and rejects the existence of bradyons: "...what an observer experiences as time, i.e. he is always trapped in the current moment in the sense that he can only influence or be influenced by events which happen in the present." Influencing or being influenced by events at a constant time in a chosen reference frame (x, t) requires the use of tachyons, where these tachyons have world lines that exist along a constant t and cover an interval in x from the observer to the event being influenced or that influences the observer.

The paragraph "Time in this view appears..." fails to explain what (M', g') is being proposed. The sentence "The dynamic view keeps time as an ever-changing entity..." fails to say what time changes with respect to. If the change is with respect to time, then the simplest interpretation is that time t changes with respect to time t, yielding dt/dt = 1. This gives no difference from Minkowski spacetime, and would imply that the phrase "time... as an ever-changing entity" is an obfuscatory way of saying in words that dt/dt = 1.

The section "The intersection of the coordinate and dynamic theories" appears intended to present (M', g'), but fails to define energy and momentum. In Minkowski spacetime, these are defined in terms of the 4-momentum, and for a bradyon as the mass times the 4-velocity. We cannot know what the author's definition is prior to the definition of the spacetime itself.

More fundamentally, while the line element of metric can be expressed in terms of small changes in the values of coordinates, the author's proposal appears to combine two coordinate systems for time, and three coordinate systems for space. This leaves the reader with no help in understanding what (M', g') is being proposed. If three different metric spaces are being proposed using three different line elements written with a single coordinate system, then that should be done. The relation between the three metrics would also have to be defined.

The uses of the verbs "see" and "exist" in the text are generally confusing for the presentation of an alternative spacetime geometry to Minkowski spacetime.

One example of confusing phrases in the section "The simple dynamical model" is "This provides a uniform radial expansion in the four dimensions..." There is *no* radial expansion in any standard (Friedmann-Lemaitre-Robertson-Walker, FLRW) model, since an FLRW universe has no centre. In some cases, the embedding of the spatial section of an FLRW universe in a higher dimensional space X is useful for intuition and for calculations, but this does not imply any physical significance for the space X. The author's apparent inclusion of a "fourth dimension", which appears to be either spacelike or timelike at different points in the text, with an unclear relation to (M', g'), does not help to clarify what "radial expansion" might be intended to mean. (The observer's past time cone is radial by construction, with the observer at the centre, but constructing a spacetime by starting with the observer still requires unambiguously defining the general characteristics of the class of spacetimes that are under consideration).

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

References

1. Taylor E, Wheeler J, Bowen J: Spacetime Physics: Introduction to Special Relativity, 2nd ed.American Journal of Physics. 1993; 61 (3). Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Observational cosmology; galaxy formation; large-scale structure; cosmic topology; inhomogeneous cosmology.

CITE

Report a concern

Author Response 04 Aug 2022

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

04 Aug 2022

Author Response
- The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.
- It does not assert the non- existence of bradyons and
... Continue reading
The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.

It does not assert the non- existence of bradyons and the existence of tachyons, this is based on a misconception of how DS is constructed (see below).

A Cartesian coordinate system is used throughout, and only inertial frames (IFs) are considered.

The alternative spacetime is based on the differences of how space and time are measured, and leaves aside any consideration of what they might “really” be.

Space coordinates are based on a ruler - a distribution of mass to which coordinates can be assigned. A clock output is an energy release which can be equated to a continuously changing time, and from which temporal coordinates can be constructed.

Minkowski converts clock outputs to temporal coordinates to obtain a non-Euclidean spacetime (Eqn.1).

DS partners momentum to the clock energy release. An observer in an IF (e.g. labelled A) can only see a clock changing and not the IFs momentum. He also cannot see a dimension in what can be hypothesised as the time direction (orthogonal to his x,y,z space). He exists in an ever-changing present - not at a constant time (and hence nothing to do with tachyons). From this changing present can be constructed an imaginary time dimension by using memories or recording devices. A temporal coordinate in this dimension can only be physically visited once, unlike the physically real spatial coordinates which can be visited many times.

As A cannot use his own momentum, he uses his observations of a different IF (B). The combination of A’s time and A’s observation of B’s energy and momentum gives A’s Euclidean DS spacetime (Eqn.3).

The observation of these phenomena indicates that both (energy release and momentum) are relative, i.e. A’s clock is relative to (A’s observations of B’s) clock and spatial motion, which means that for a single observer dt is not just a change in time.

The verbs “see” and “exist” refer to physically-based phenomena such as spatial coordinates or a changing time, and not to an imaginary time dimension.

The derivation of A’s spacetime leads to the hypothesis that in DS it is the clock co-located with the observer that provides the invariant, in contrast to Minkowski where it is the moving clock that is invariant.

DS confines any bradyon to its own time direction (OA, OB, OG in Fig.1) and actually located at A, B, G in that Figure, which is on the time-like hyperbola provided by Minkowski. Hence only along this hyperbola can the two spacetimes (Euclidean and non-Euclidean) intersect to agree on Lorentz transformations, the 4-momentum etc. Away from this they are different and Fig.2 shows how each IF is seen from within as 3 spatial dimensions and a changing time in which any dimension has rolled up to a point. From outside this IF the changing time is seen as a motion along a spatial dimension.

This leads to the hypothesis that the universe consists of 4 spatial dimensions which, in the simplest case, has bradyons travel radially outwards from a common centre. These bradyons form a surface which is orthogonal to the radial motion, and although this radial motion is at the speed of light, this is the rate of expansion of the universe. The observed bradyon motion is the component of this expansion along the surface when seen from a different IF (Figs 3 & 4), and so does not contravene the requirement that bradyons should travel slower than the speed of light. From within the IF this radial motion is experienced as the changing time.

This is not an FLRW model, but results from an FLRW model can be compared to those from DS (Figs 5 & 6, Table 1).

The use of only IFs and not general relativity in constructing this model is discussed in the Conclusions and Discussion of the paper.
The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.

It does not assert the non- existence of bradyons and the existence of tachyons, this is based on a misconception of how DS is constructed (see below).

A Cartesian coordinate system is used throughout, and only inertial frames (IFs) are considered.

The alternative spacetime is based on the differences of how space and time are measured, and leaves aside any consideration of what they might “really” be.

Space coordinates are based on a ruler - a distribution of mass to which coordinates can be assigned. A clock output is an energy release which can be equated to a continuously changing time, and from which temporal coordinates can be constructed.

Minkowski converts clock outputs to temporal coordinates to obtain a non-Euclidean spacetime (Eqn.1).

DS partners momentum to the clock energy release. An observer in an IF (e.g. labelled A) can only see a clock changing and not the IFs momentum. He also cannot see a dimension in what can be hypothesised as the time direction (orthogonal to his x,y,z space). He exists in an ever-changing present - not at a constant time (and hence nothing to do with tachyons). From this changing present can be constructed an imaginary time dimension by using memories or recording devices. A temporal coordinate in this dimension can only be physically visited once, unlike the physically real spatial coordinates which can be visited many times.

As A cannot use his own momentum, he uses his observations of a different IF (B). The combination of A’s time and A’s observation of B’s energy and momentum gives A’s Euclidean DS spacetime (Eqn.3).

The observation of these phenomena indicates that both (energy release and momentum) are relative, i.e. A’s clock is relative to (A’s observations of B’s) clock and spatial motion, which means that for a single observer dt is not just a change in time.

The verbs “see” and “exist” refer to physically-based phenomena such as spatial coordinates or a changing time, and not to an imaginary time dimension.

The derivation of A’s spacetime leads to the hypothesis that in DS it is the clock co-located with the observer that provides the invariant, in contrast to Minkowski where it is the moving clock that is invariant.

DS confines any bradyon to its own time direction (OA, OB, OG in Fig.1) and actually located at A, B, G in that Figure, which is on the time-like hyperbola provided by Minkowski. Hence only along this hyperbola can the two spacetimes (Euclidean and non-Euclidean) intersect to agree on Lorentz transformations, the 4-momentum etc. Away from this they are different and Fig.2 shows how each IF is seen from within as 3 spatial dimensions and a changing time in which any dimension has rolled up to a point. From outside this IF the changing time is seen as a motion along a spatial dimension.

This leads to the hypothesis that the universe consists of 4 spatial dimensions which, in the simplest case, has bradyons travel radially outwards from a common centre. These bradyons form a surface which is orthogonal to the radial motion, and although this radial motion is at the speed of light, this is the rate of expansion of the universe. The observed bradyon motion is the component of this expansion along the surface when seen from a different IF (Figs 3 & 4), and so does not contravene the requirement that bradyons should travel slower than the speed of light. From within the IF this radial motion is experienced as the changing time.

This is not an FLRW model, but results from an FLRW model can be compared to those from DS (Figs 5 & 6, Table 1).

The use of only IFs and not general relativity in constructing this model is discussed in the Conclusions and Discussion of the paper.
Competing Interests: There are no competing interests Close
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Author Response 04 Aug 2022

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

04 Aug 2022

Author Response
- The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.
- It does not assert the non- existence of bradyons and
... Continue reading
The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.

It does not assert the non- existence of bradyons and the existence of tachyons, this is based on a misconception of how DS is constructed (see below).

A Cartesian coordinate system is used throughout, and only inertial frames (IFs) are considered.

The alternative spacetime is based on the differences of how space and time are measured, and leaves aside any consideration of what they might “really” be.

Space coordinates are based on a ruler - a distribution of mass to which coordinates can be assigned. A clock output is an energy release which can be equated to a continuously changing time, and from which temporal coordinates can be constructed.

Minkowski converts clock outputs to temporal coordinates to obtain a non-Euclidean spacetime (Eqn.1).

DS partners momentum to the clock energy release. An observer in an IF (e.g. labelled A) can only see a clock changing and not the IFs momentum. He also cannot see a dimension in what can be hypothesised as the time direction (orthogonal to his x,y,z space). He exists in an ever-changing present - not at a constant time (and hence nothing to do with tachyons). From this changing present can be constructed an imaginary time dimension by using memories or recording devices. A temporal coordinate in this dimension can only be physically visited once, unlike the physically real spatial coordinates which can be visited many times.

As A cannot use his own momentum, he uses his observations of a different IF (B). The combination of A’s time and A’s observation of B’s energy and momentum gives A’s Euclidean DS spacetime (Eqn.3).

The observation of these phenomena indicates that both (energy release and momentum) are relative, i.e. A’s clock is relative to (A’s observations of B’s) clock and spatial motion, which means that for a single observer dt is not just a change in time.

The verbs “see” and “exist” refer to physically-based phenomena such as spatial coordinates or a changing time, and not to an imaginary time dimension.

The derivation of A’s spacetime leads to the hypothesis that in DS it is the clock co-located with the observer that provides the invariant, in contrast to Minkowski where it is the moving clock that is invariant.

DS confines any bradyon to its own time direction (OA, OB, OG in Fig.1) and actually located at A, B, G in that Figure, which is on the time-like hyperbola provided by Minkowski. Hence only along this hyperbola can the two spacetimes (Euclidean and non-Euclidean) intersect to agree on Lorentz transformations, the 4-momentum etc. Away from this they are different and Fig.2 shows how each IF is seen from within as 3 spatial dimensions and a changing time in which any dimension has rolled up to a point. From outside this IF the changing time is seen as a motion along a spatial dimension.

This leads to the hypothesis that the universe consists of 4 spatial dimensions which, in the simplest case, has bradyons travel radially outwards from a common centre. These bradyons form a surface which is orthogonal to the radial motion, and although this radial motion is at the speed of light, this is the rate of expansion of the universe. The observed bradyon motion is the component of this expansion along the surface when seen from a different IF (Figs 3 & 4), and so does not contravene the requirement that bradyons should travel slower than the speed of light. From within the IF this radial motion is experienced as the changing time.

This is not an FLRW model, but results from an FLRW model can be compared to those from DS (Figs 5 & 6, Table 1).

The use of only IFs and not general relativity in constructing this model is discussed in the Conclusions and Discussion of the paper.
The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.

It does not assert the non- existence of bradyons and the existence of tachyons, this is based on a misconception of how DS is constructed (see below).

A Cartesian coordinate system is used throughout, and only inertial frames (IFs) are considered.

The alternative spacetime is based on the differences of how space and time are measured, and leaves aside any consideration of what they might “really” be.

Space coordinates are based on a ruler - a distribution of mass to which coordinates can be assigned. A clock output is an energy release which can be equated to a continuously changing time, and from which temporal coordinates can be constructed.

Minkowski converts clock outputs to temporal coordinates to obtain a non-Euclidean spacetime (Eqn.1).

DS partners momentum to the clock energy release. An observer in an IF (e.g. labelled A) can only see a clock changing and not the IFs momentum. He also cannot see a dimension in what can be hypothesised as the time direction (orthogonal to his x,y,z space). He exists in an ever-changing present - not at a constant time (and hence nothing to do with tachyons). From this changing present can be constructed an imaginary time dimension by using memories or recording devices. A temporal coordinate in this dimension can only be physically visited once, unlike the physically real spatial coordinates which can be visited many times.

As A cannot use his own momentum, he uses his observations of a different IF (B). The combination of A’s time and A’s observation of B’s energy and momentum gives A’s Euclidean DS spacetime (Eqn.3).

The observation of these phenomena indicates that both (energy release and momentum) are relative, i.e. A’s clock is relative to (A’s observations of B’s) clock and spatial motion, which means that for a single observer dt is not just a change in time.

The verbs “see” and “exist” refer to physically-based phenomena such as spatial coordinates or a changing time, and not to an imaginary time dimension.

The derivation of A’s spacetime leads to the hypothesis that in DS it is the clock co-located with the observer that provides the invariant, in contrast to Minkowski where it is the moving clock that is invariant.

DS confines any bradyon to its own time direction (OA, OB, OG in Fig.1) and actually located at A, B, G in that Figure, which is on the time-like hyperbola provided by Minkowski. Hence only along this hyperbola can the two spacetimes (Euclidean and non-Euclidean) intersect to agree on Lorentz transformations, the 4-momentum etc. Away from this they are different and Fig.2 shows how each IF is seen from within as 3 spatial dimensions and a changing time in which any dimension has rolled up to a point. From outside this IF the changing time is seen as a motion along a spatial dimension.

This leads to the hypothesis that the universe consists of 4 spatial dimensions which, in the simplest case, has bradyons travel radially outwards from a common centre. These bradyons form a surface which is orthogonal to the radial motion, and although this radial motion is at the speed of light, this is the rate of expansion of the universe. The observed bradyon motion is the component of this expansion along the surface when seen from a different IF (Figs 3 & 4), and so does not contravene the requirement that bradyons should travel slower than the speed of light. From within the IF this radial motion is experienced as the changing time.

This is not an FLRW model, but results from an FLRW model can be compared to those from DS (Figs 5 & 6, Table 1).

The use of only IFs and not general relativity in constructing this model is discussed in the Conclusions and Discussion of the paper.
Competing Interests: There are no competing interests Close
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Version 4

VERSION 4 PUBLISHED 21 Mar 2022

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Version 4 (revision) 27 Jan 25		read
Version 3 (revision) 30 Sep 24	read
Version 2 (revision) 30 Jan 23	read
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Boudewijn F. Roukema, Nicolaus Copernicus University of Torun, Torun, Poland
Jackson Levi Said, University of Malta, Msida, Malta

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19 Aug 2025 | for Version 4

Jackson Levi Said, University of Malta, Msida, Malta

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Not Approved

The manuscript claims to resolve some open problems in relativity theory. Overall, the work is not robustly formulated and contains informal language throughout. A more formal analysis should involve more explicit model suggestions and some rudimentary form of data analysis since the principal issues identified are observational in nature. The confused language is unclear throughout. However, any conflict within special relativity should be explicitly shown as should any proposal beyond this theoretical framework. The author should revisit the foundations of their proposal in more detail, and formulate a better defined research hypothesis and proposal.

For these reasons, I do not recommend this manuscript for indexing in its present form. Below, I detail the precise issue to be reconsidered by the authors.

Section 1 - Introduction
The opening assumes special and general relativity as foundational building blocks, as they should be. However, the introduction then goes into the need to resolve some problematic features of special relativity. These are not explained explicitly, nor are they well defined in terms of observables associated with observers in this framework. This leaves the motivation for the work ambiguous while the research hypothesis is vague in nature. The author should describe in more detail what supposed problem appears in not giving a “good fit to the corresponding astronomical data” means and at what statistical level this appears at. In its present form, these statements are very vague and not well referenced.

Section 2 - The basis of the dynamic theory in Special Relativity (SR)
In this section, the supposed problematic features of SR are described. The first subsection qualitatively explains the potential problem in coordinate systems as related to the temporal dimension. The description is inexact in nature and lacks robustness. The author should consider putting in a more robust general analysis of the technical problem in observables, otherwise this remains a qualitative supposition. The second subsection mostly includes entry-level definitions of the Minkowski metric in textbooks and the line element. This background is not needed since it is well known. Similarly, length contraction and time dilation as well as the other properties in the final subsection are well known in the literature. Indeed, these are not so much related to cosmology except in a foundational sense.

Section 3 - Cosmology
This section contains more literature review and some original results. This opens with a discussion of the known problems in standard cosmology. These are only partially reviewed and in no great depth. The author may want to consider more recent observed issues with the theory including cosmic tensions. The next subsection describes the so-called model being proposed in this work. This is very qualitative and not exactly in nature. A more formal attempt is provided in the third subsection, but this is similarly informal and very circumstantial. Any model beyond special or general relativity should conform to the robust mathematical rigor or either framework. Section four goes back to Lorentz transformations while the fifth section covers other special relativistic effects. These do not go beyond special relativity in any meaningful way. The section closes with a minor discussion of some of the open challenges to standard cosmology. However, these are not tackled in any serious way. The author should consider a numerical analysis involving observational data.

Section 3 - Discussion and conclusions
The work closes with a short summary of the main results and a long list of observations both in SR an general relativity using these suggestions made earlier in the work.

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

No

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

I am a cosmologist working on the theoretical development of models to resolve open issues in observational cosmology.

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35 Views

18 Oct 2024 | for Version 3

Boudewijn F. Roukema, Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University of Torun, Torun, Poland

35 Views Cite this report Responses(1)

Not Approved

Version 3 of the submitted paper clarifies one issue: the author's intention is that his model is for a Minkowski metric. Thus, his intended model is a different spacetime to most of the FLRW (Friedmann-Lemaitre-Robertson-Walker) cosmological spacetimes (apart from at least one special case). It appears from the Version 3 abstract that the intention is to re-interpret a Minkowski spacetime and relate it to astronomical observations that have been used over the past century to constrain the FLRW models. However, despite the author excluding most of the FLRW models, later in the text he states that his re-interpretation of Minkowski spacetime includes a singularity. This is a mathematical contradiction. Moroever, he interprets astronomical observations within the context of the FLRW models that he has excluded.

The text has ambiguities throughout. It does not present a well-defined line of reasoning. Most of my specific concerns have not been dealt with by text corrections by the author (even the assertion that tachyons exist remains in Version 3 of the text). The revisions to this submitted paper show no sign of converging. This paper is not a research paper in physics/astronomy/cosmology.

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Observational cosmology; galaxy formation; large-scale structure; cosmic topology; inhomogeneous cosmology

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Author Response

25 Oct 2024

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

R= reviewer's comment; A= author's response

R: However, despite the author excluding most of the FLRW models, later in the text he states that his re-interpretation of Minkowski spacetime includes a singularity. This is a mathematical contradiction.
A: The first sentence under the "Cosmology" heading is badly phrased. It should have read "In describing the universe, the standard theory has evidence of an expanding space-time which started from a singularity". The theory does not assume the Minkowski spacetime includes a singularity. Elsewhere it's stated this spacetime occurs after the singularity begins to expand, as explained in the text “In the simplest model the dynamic theory assumes the universe started as a singularity embedded in four spatial (not space-time) dimensions at, what we consider, the time of the Big Bang. This provides a uniform radial expansion in the four dimensions if we assume that..." (My italics). You can argue about the physics at this point (i.e. when can the singularity be considered as an expanding shell of radiation and start obeying Minkowski), much as the standard theory also has problems in this area since Quantum Mechanics must play an important role here.

R: Moreover, he interprets astronomical observations within the context of the FLRW models that he has excluded.
A: The FLRW Standard Cosmology Model (Fig.5) and the combined Friedmann equations (Fig.8) are assumed to currently give the best fits to the astronomical observations. Consequently, the fact that the dynamic theory gives a good fit to such models implies it also gives a good fit to the corresponding astronomical data without the paper being sidetracked into extensive explanations and attributions of this data.

R: The text has ambiguities throughout.
A: Difficult to respond to without specific instances being listed. If these are the same ambiguities raised for Version 2, I did respond at some length. The major error in that version, as the reviewer pointed out, was my assumption that I was dealing with different metrics rather than different coordinate systems. As stated above this has been clarified. One major unresolved issue is the reviewer’s insistence that I am dealing with tachyons, which is discussed below.

R: It does not present a well-defined line of reasoning.
A: I have probably included too much description and elements that more properly belong in sidebars. To try to precis the reasoning: -

Minkowski spacetime can be described by two different sets of coordinates, Cartesian (standard) and dynamic (current paper). The translation between the coordinate systems is given by Eq.7 (see * below).
The reasoning behind the dynamic coordinates leads to a different cosmological spacetime compared to FLRW models, where in the simplest case a 4D universe can be constructed using Special Relativity (SR) and without recourse to General Relativity (GR).
Even the simple model using the dynamic theory provides a good fit to the FLRW models based on GR (see above) but without needing some of the additional models (e.g. inflation) or difficulties (e.g. late time acceleration) that the standard theory has.
The dynamic theory simply points out some of the difficulties of using what are basically clocks and rulers to provide quantifiable tests of a selection of cosmological observations, using theories which have GR as a basis, and suggests a possible alternate view.
Obviously gravity must play an important part in most of cosmology, but an SR-based theory provides a surprisingly good fit to a number of scenarios where GR-based theories face problems.

* There is a typo in the last term of the last line of this equation. The term should have dt_N divided by gamma, not multiplied by it.

R: Most of my specific concerns have not been dealt with by text corrections by the author (even the assertion that tachyons exist remains in Version 3 of the text).
A: The reviewer’s idea that I am dealing with tachyons appears to arise from two points he makes in his Version 2 review. In my Version 2 reply these come under points I have labelled 2 and 5, and I will attempt to further clarify these (for the record I believe there is no solid evidence tachyons exist).
Point 2. He quotes my text “an observer…can only influence or be influenced by events which happen in the present.” Which he then interprets as my belief that tachyons exist. This is probably due to the clumsy way I originally phrased this. In the standard theory this would mean a single unchanging time coordinate which would have to be connected to the rest of the universe by spacelike worldlines, i.e. tachyons. However, what I should have made clear is this is the dynamic theory where the observer is in an everchanging present, which when translated into standard coordinates (Eq.7) gives timelike worldlines as illustrated in Fig.1b. The observer moves along these worldlines in standard coordinates, but to the observer in the dynamic theory he sees only the changing present. Eq.7 & Fig 1b show how the different ways of describing the observer can be equated and transformations between the two obtained - but only for timelike worldlines.
Point 5. Similarly he takes my meaning as the only worldlines considered are along the x axis, i.e. time coordinates are constant and again this would imply spacelike worldlines. This misunderstood my formulation of Eq.1 (which is the Minkowski standard equation) where what I was trying to say was the spatial content of this equation assumed all spatial coordinate changes (i.e. velocity vectors) in this paper could be reduced to lying in the x dimension, rather than having to write out the root of x² + y² + z² each time I wanted to use it.

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Reviewer Report

60 Views

01 Nov 2023 | for Version 2

Boudewijn F. Roukema, Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University of Torun, Torun, Poland

60 Views Cite this report Responses(1)

Not Approved

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Reviewer Expertise

Observational cosmology; galaxy formation; large-scale structure; cosmic topology; inhomogeneous cosmology.

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Author Response

09 Nov 2023

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

The paragraph numbering corresponds to the Reviewer's paragraphs.

1. The verbosity was intended to provid sufficient context for the differences between the standard theory and that put forward in this paper. The standard theory deals with coordinates and their derivatives while the dynamic theory in the paper deals with momentum (space) and energy (time) from which coordinate derivatives can be obtained. The derivatives can be linked to each other (eqs.7-11) and the reasons for differences are discussed primarily in the paragraph before Eqs. 3 & 4.

2. The paragraph referred to is simply trying to make the point that the way humans obtain a quantified measure of space and time (by rulers and clocks) has fundamental differences between these measures despite needing to amalgamate them to obtain space-time. The comment about the observer always being in his present simply reflects the human experience of time. A spatial coordinate can be visited many times, a temporal coordinate only once.
At no point do I assert tachyons exist. In the model (see under "Cosmology") the universe consists of a surface made up of the locus of all inertial frames moving outwards from a central point at what can be constued as the speed of light. This is the motion of the universe's surface, not motion along that surface which is what tachyons would require.

4. Eq.1 is the standard SR which for brevity assumes the spatial motion is along x, which saves writing
-dx²-dy²-dz² each time. According to Foster & Nightingale ("A short course in general relativity", p165, Longman, 1979) "the path through spacetime...is called its world line..and the proper time interval d\tau between points...relative to some frame K..is defined by.." and then follows my Eq.1, or in its complete form my Eq.10. The italics are their own. Further "..the proper time tau is the time recorded by a clock which moves along with the particle" Again their italics. And again on the same page "..for a particle at rest in K [or in my case A] the proper time tau is..the coordinate time t ...measured by stationary clocks in [A]." Substitute ds² for d\tau² in my Eq.1. It can then be positive, negative or zero. I have chosen the positive one as it joins up time-like intervals (Fig.1, OA, OB, OG). I have concentrated on this as A is the observer's frame in which d\tau=dt.

5. Better in the paper would be "only changes in spatial coordinates along.." and refers to the brevity discussed in 4. At that point I had not started discussing the dynamic theory.

6. Since the paper deals mainly with special relativity (SR), A, B and C are seperate inertial frames with different relative velocities and their own spatial coordinate frames and clocks. Assuming time on clocks in each frame can be seen from any other frame, then Dt_AB has A as the observer looking at the clock in frame B and comparing it to their own clock. Consequently Dt_AB (the observed clock) = Dt_AA (the observer's clock) multiplied by gamma^-1 (see text). Equally it is assumed the observer can see other frames moving relative to a ruler. Hence Dx_AAB has A observing B's change in spatial position relative to a ruler in A's frame. Dx_ACB is A observing B's space changes against a ruler in C's space. The links between observers are still given by the Lorentz transforms (Table 1 & Eqs. 14-18. Note the typo in Eq.18, the right side should be (w+v)/(1+vw/c²)).

7. This was meant to emphasise the differences between models. The standard model (Eq.1) assumes both spatial and temporal coordinates are present (dtA, dxA) while the dynamic model (Eq.4) says the only coordinates that can be directly observed are spatial, e.g. divisions on a ruler. Time coordinates have to be constructed since clocks only give changes in time, and other methods are needed to give (Dt_AA, Dx_AAB) but NOT Dx_AAA - see text.

8. See comments under 4.

9. Eq.11 defines the dynamic metric while Eq.9 defines the line element. Both are expressed in terms of standard coordinates. Fig.1 shows the Minkowski hyperbola from standard SR but constructed as the locus of the time axes of inertial frames with increasing velocities relative to a given observer (A). Eq.9 shows such a construction can also be expressed in an Euclidean form where lines of increasing velocity intersect the hyperbola.

10. The model only has 4 spatial dimensions, each frame having 3 of space and one which takes on the aspects of time. Within a frame, however, the time dimension is only inferred from memory and aids such as film. As far as direct observations and interactions with your surroundings go, the temporal dimension appears rolled up into a point which we call the present. From outside the frame (i.e. from a frame with a relative velocity to the first) this time appears as motion along a spatial set of coordinates (see paragraph after Eq.11). The observer sees a velocity but the observed sees dimensionless changing time. Consequently a "non-local" view has a universe with a surface described in 2, and under "Cosmology" in the paper.

11. The model is based on SR since any gravitational effect normal to the universe's expansion would equally affect the apparent motions of an observer in both space and time, leading to a constant expansion velocity (see Fig.3).

12. In the paper it should have been the velocity of light since it has a definite direction.

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Reviewer Report

107 Views

18 Jul 2022 | for Version 1

Boudewijn F. Roukema, Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University of Torun, Torun, Poland

107 Views Cite this report Responses(1)

Not Approved

Is the work clearly and accurately presented and does it cite the current literature?

Partly
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

References

1. Taylor E, Wheeler J, Bowen J: Spacetime Physics: Introduction to Special Relativity, 2nd ed.American Journal of Physics. 1993; 61 (3). Publisher Full Text

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No competing interests were disclosed.

Reviewer Expertise

Observational cosmology; galaxy formation; large-scale structure; cosmic topology; inhomogeneous cosmology.

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Author Response

04 Aug 2022

Hugh James, Retired, AWE, Aldermaston, Reading, RG7 4PR, UK

The paper defines a possible alternative space time (Dynamic space time, or DS) to that of Minkowski.
It does not assert the non- existence of bradyons and the existence of tachyons, this is based on a misconception of how DS is constructed (see below).
A Cartesian coordinate system is used throughout, and only inertial frames (IFs) are considered.
The alternative spacetime is based on the differences of how space and time are measured, and leaves aside any consideration of what they might “really” be.
Space coordinates are based on a ruler - a distribution of mass to which coordinates can be assigned. A clock output is an energy release which can be equated to a continuously changing time, and from which temporal coordinates can be constructed.
Minkowski converts clock outputs to temporal coordinates to obtain a non-Euclidean spacetime (Eqn.1).
DS partners momentum to the clock energy release. An observer in an IF (e.g. labelled A) can only see a clock changing and not the IFs momentum. He also cannot see a dimension in what can be hypothesised as the time direction (orthogonal to his x,y,z space). He exists in an ever-changing present - not at a constant time (and hence nothing to do with tachyons). From this changing present can be constructed an imaginary time dimension by using memories or recording devices. A temporal coordinate in this dimension can only be physically visited once, unlike the physically real spatial coordinates which can be visited many times.
As A cannot use his own momentum, he uses his observations of a different IF (B). The combination of A’s time and A’s observation of B’s energy and momentum gives A’s Euclidean DS spacetime (Eqn.3).
The observation of these phenomena indicates that both (energy release and momentum) are relative, i.e. A’s clock is relative to (A’s observations of B’s) clock and spatial motion, which means that for a single observer dt is not just a change in time.
The verbs “see” and “exist” refer to physically-based phenomena such as spatial coordinates or a changing time, and not to an imaginary time dimension.
The derivation of A’s spacetime leads to the hypothesis that in DS it is the clock co-located with the observer that provides the invariant, in contrast to Minkowski where it is the moving clock that is invariant.
DS confines any bradyon to its own time direction (OA, OB, OG in Fig.1) and actually located at A, B, G in that Figure, which is on the time-like hyperbola provided by Minkowski. Hence only along this hyperbola can the two spacetimes (Euclidean and non-Euclidean) intersect to agree on Lorentz transformations, the 4-momentum etc. Away from this they are different and Fig.2 shows how each IF is seen from within as 3 spatial dimensions and a changing time in which any dimension has rolled up to a point. From outside this IF the changing time is seen as a motion along a spatial dimension.
This leads to the hypothesis that the universe consists of 4 spatial dimensions which, in the simplest case, has bradyons travel radially outwards from a common centre. These bradyons form a surface which is orthogonal to the radial motion, and although this radial motion is at the speed of light, this is the rate of expansion of the universe. The observed bradyon motion is the component of this expansion along the surface when seen from a different IF (Figs 3 & 4), and so does not contravene the requirement that bradyons should travel slower than the speed of light. From within the IF this radial motion is experienced as the changing time.
This is not an FLRW model, but results from an FLRW model can be compared to those from DS (Figs 5 & 6, Table 1).
The use of only IFs and not general relativity in constructing this model is discussed in the Conclusions and Discussion of the paper.

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Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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Accounting for the expansion of the universe using an energy/momentum model to construct the space-time metric

Abstract

Keywords

Revised Amendments from Version 1

Introduction

The basis of the dynamic theory in special relativity (SR)

Possible grounds for allowing an alternative to the coordinate theory

The intersection of the standard and dynamic theories

Figure 1. Inertial frames intersecting the time-like hyperbola from A’s viewpoint.

Table 1. Dynamic Lorentz transforms from the viewpoints of A and B derived from Table 2 and Equation (7).

Table 2. Standard Lorentz transforms.

Figure 2. The dynamic view of inertial frames.

Similarities and differences between the theories

Cosmology

The standard model and its problems

The simple dynamic model

Figure 3. Simple dynamic model of the current state of the universe.

Figure 4. Simple dynamic model of the evolution of the universe.

The quantitative dynamic model

Comparison of the predictions of the dynamic and coordinate theories

Figure 5. The comparison of the coordinate and dynamic models of the universe’s evolution.

Figure 6. The comparison of the standard coordinate and dynamic models of the universe’s deceleration and late-time acceleration.

Figure 7. The fit of two SCM’s with different values of H0 to a single dynamic theory curve.

Discussion and conclusions

In the SR dynamic theory

Table 3. Values of za from both theories and observations.

In GeneralRelativity (GR)

In dynamic cosmology

Data

Author endorsement

Acknowledgements

Footnotes

References

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Figure 7. The fit of two SCM’s with different values of H₀ to a single dynamic theory curve.

Table 3. Values of z_a from both theories and observations.