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

Possible grounds for pursuing alternative coordinate systems

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. This paper leaves aside any philosophical questions about what space and time “really” are and just concentrates on the phenomena that humans consider as providing realistic analogues of such phenomena, i.e., as discussed below, changes in energy are considered to be analogues of changes in time (clocks) while a spatial dimension can be considered as corresponding to a distribution in mass (rulers). It is these analogues that are used to provide quantifiable tests of relativistic theories. They also illustrate the problems of mixing time and space to form the space-time of standard, or indeed any, theory of 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 correspond to 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 his 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 his 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 his changing present, temporal coordinates can only be inferred by integrating experiences of the changing present using memories or other recording devices. The ever-changing present has no real coordinate that can be directly observed in the sense of spatial coordinates, there are only directly observed changes in time - see above definitions. Hence from within an inertial frame no dimension can be directly observed 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.

Changes in time in this view appear to facilitate changes in 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 dimensions.

Since the integral of the dynamic view will give space-time coordinates (albeit artificially constructed), and since it has already been stated that, at least locally, the standard and dynamic 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 can be used to construct 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. These are changes along trajectories. 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 more than one subscript is used to describe the direct observation within a single frame (A in what follows) of how other frames appear relative to A’s measuring apparatus, e.g. two subscripts are used for time, Dt_AB, where the first subscript is the frame containing the observer and the second the frame from which the object being observed originates. 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 relative to A’s space and C is the frame containing the ruler against which A measures B’s motion and again is relative to A’s space. Coordinates are transformed between dynamic and Cartesian descriptions within a given frame by Equation (7), while both dynamic and Cartesian coordinates are transformed between different frames by Lorentz transforms. The dynamic transform is given in Table 1 and the Cartesian in Table 2 below.

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 Cartesian relationship between space and time

where A and B are inertial frames of reference, 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 – the total component of spatial motion is usually assumed to be lying along the x axis. For photons dτ = 0.

In the dynamic theory several points must be considered.

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

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 infinitesimal changes in dynamic coordinates (which should be compared to the changes in standard coordinates in Equation (1)) as

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

The first thing to note is that the invariant is now Dt_NN as this is the only changing 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 between two recorded events. 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 coordinate systems overlap, we are looking for specific conditions where dτ = Dt_NN. This will be shown to be along the time-like hyperbola in Figure 1a where dτ is an invariant for all possible inertial frames that have their origin at O, i.e. OA= OB= OG= cdτ.

Since the dynamic theory has an ever-changing present, this is equivalent in Cartesian coordinates to mapping out a trajectory such as OB in Figure 1a.

In Figure 1a, 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. along OA, OB, OF and OG in Figure 1a. This condition corresponds to dx_N = 0 in Equation (1), and Dx_NNM = Dx_NNN = 0 in Equation (3) and Equation (4). The four sets of inertial axes (A, B, F 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 Cartesian coordinate viewpoint requires that every object (and hence every observer) must be located on its own time axis, even though 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

And since

c²$D{t}_{NN}^2$ = c²dτ² then from Equation (1) & Equation (3),

Also, since dx_N/cdt_N = Dx_NNM/cDt_NN = v/c,

then from this and Equation (6) we can, after some manipulation, derive the following relationships between dynamic and Cartesian coordinates within a given frame (N) as

Where γ_N = (1–v²/c²)^–1/2, and γ_N is constant for a given inertial frame.

Hence ${\gamma }_N^{-1}(d{x}_N,d{t}_N)=(D{x}_{NNM},D{t}_{NN}).(8).$

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

To express the Cartesian and dynamic coordinates 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

Where in Cartesian coordinates

While the dynamic coordinates are

From the above it is straightforward to show the dynamic theory gives the same 4-momentum vector as the standard theory, i.e.

let U⁰ = dx⁰/dτ = γ_N cDt_NN/dτ = cγ_N; and U¹ = dx¹/dτ = γ_N Dx_NNM/Dt_NN = v_xγ_N, etc.

Then U^μ = γ_N(c,V) where V is the complete spatial vector, and by multiplying by the mass (m) of the object under investigation in frame M, we get

which is the 4-momentum of the standard theory. The main difference between the theories is that one corresponds to an Eulerian form while the other does not. This gives rise to the following relationship between spatial motion and a changing present.

The equivalent dynamic diagram to Figure 1a 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).

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 $c^{2}D{t}_{AA}^2$ and noting v = Dx_AAB/Dt_AA, gives Dt_AB = ${\gamma }_A^{-1}$Dt_AA, where γ_A = (1–v²/c²)^–1/2.

Hence for the Cartesian coordinates where dt_A ≠ dτ

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_A 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 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 a complete four-dimensional Euclidean-type coordinate transform, 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

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 ${\gamma }_A^{-1}$. 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. This is the transformation of the same type of coordinate (dynamic or Cartesian) between different frames, in contrast to Equation (7) where the transformation is between coordinate systems in the same frame.

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

γ₁ = (1–v²/c²)^–1/2; γ₂ = (1–u²/c²)^–1/2 and γ₃ = (1–w²/c²)^–1/2

Where in Table 1, γ= (γ₁ γ₂)⁄γ₃ for Column 1, and γ= (γ₁ γ₃)⁄γ₂ for Column 2.

Dividing the top row by the bottom in Column 1 gives

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

When comparing these equations with the Cartesian 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 1 multiply both sides of each equation by γ and substituting the appropriate d() term from Equation (7). This gives substitute Table 2 in standard Cartesian coordinates which corresponds to the standard transformation.

The 4-vector, energy and momentum. As already shown the dynamic theory gives the same 4-momentum relationship as the coordinate theory, but this 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

as in the Cartesian 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 = γmc² is still B’s mass. From this, κ is also equal to E₀/cDt_AB. Substituting κ into Equation (15) leads to the general conclusions that Dt_NN is proportional to E; Dt_NM is proportional to E₀ and hence

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 standard model and its problems

In describing the universe, the standard theory has evidence of an expanding space-time which started from a singularity and was impelled outwards by the Big Bang some 13.8 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

And some items which cannot be explained by inflation.

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 Cartesian 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 FLRW fits to observations. The reasons for this approach have been outlined at the start of this paper.

The simple dynamic model

In the simplest model the dynamic theory assumes the universe started from 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 velocity 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.

Quite when this expansion transforms to a Minkowski space-time is not clear other than it is assumed by the dynamic theory to occur soon after the Big Bang. Also, like the standard theory, quantum effects should be important in these very early stages but are not considered in this paper. Equally speculation on what exists in the space outside the expanding surface is again outside the scope of the current paper.

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 are 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).

The trajectories of all objects are assumed in this simple model to be constant and are seen from outside their frames of reference to obey R = cT, R being the dynamic spatial coordinate of the surface at time cT, and T is the dynamic time coordinate relative to the Big Bang, i.e. $cT=c{\int }_0^{T}D{t}_{NN}$ which is the integral of Dt_NN along N’s trajectory as inferred by an observer located within N’s frame of reference. $R={\int }_0^{R}D{r}_{NNP}$ and is the position of a photon in frame P, emitted at the Big Bang, in N’s spatial axes while $D{r}_{NNP}^2=D{x}_{NNP}^2+D{y}_{NNP}^2+D{z}_{NNP}^2$ (see above for nomenclature). All inferred surfaces (see Figure 4) are the loci of all such trajectories originating from the Big Bang.

Figure 4. Simple dynamic model of the evolution 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):-

Although the time dimension does not exist when observed from within a given reference frame, 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 an inferred dimension and, as indicated above, is carried out by integrating the radial motion to provide a dimensional construct in which events lie relative to A.

There are three types of time shown in the following diagram (Figure 4). Solid blue lines are inferred 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.

The second type (solid red line) is the directly 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 (vertical lines such as CF in the figure), only allows A to see B in B’s past. This is the only directly observable surface due to observations of the Doppler effect, and as such its position is linked to the change in scale factor with time (see below for a more detailed analysis of the scale factors relating to the universe). Such observations always occur in A’s continually changing present (which is equivalent in Cartesian terms of laying down a time-like trajectory) since this is the only location at which A can interact with events or phenomena such as photons. However, that the observation is of B’s past allows A to construct B as a point on A’s own time axis, e.g. OF in Figure 4.

The third type (broken red line) is also inferred and results from the correction needed for A to obtain the location of the observed surface in terms of B’s “proper” time relative to the Big Bang, i.e. the removal of the time dilation term and the assumption that the change in time along trajectories such as OB will proceed at the same rate as occurs along OA while keeping the observed scale factor the same. The removal of the time dilation term removes the link with recessional velocity, and thus the curve’s position is linked to the scale factor rather than its time derivative. For this to happen OF would be located at OD on B’s trajectory (surface FDJ is at constant s) and in turn OD would be located at OG on A’s trajectory. This broken red line will be referred to as the corrected surface and is always relative to the observer. This observer is A in Figure 4, but the same diagram can be applied to an observer located on any other trajectory.

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 velocity 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 (T₀ is the integral of Dt_NN over the period from the Big Bang to the present - see above). Note that the value of T₀ is frozen for this analysis, and changes along the trajectories are indicated by d( ) rather than D( ) under these conditions despite still referring to dynamic coordinates. 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.

The details of Figure 4 are summarised in the following bullet points.

A’s construction of the surfaces required by the simple model.

Comparison of the predictions of the dynamic and coordinate theories

The evolution of the universe is calculated in terms of the scale factor s of the universe at time T after the Big Bang, which can be expressed as

Since a dynamic coordinate is being used (e.g. T or R) it should be emphasized that this is along a given trajectory (e.g. OA or OB), but inferred surfaces (e.g. FDJ in Figure 4) link constant values of s. This parameter can measure the amount of stretch in the surface between then (T) and now (T₀), and is often used when comparing experiment and theory The connection between the observed and inferred surfaces, and the scale factor is given by the three phenomena discussed below :-

Let H(T) be the Hubble parameter at time T after the Big Bang and H₀ is the present value of the parameter. Then H₀ = 1/T₀ and along a trajectory that is frozen at R₀, T₀ ≡ R_L, T_A we have H(T) = s^–1 ds/dT. From A’s viewpoint H(T) varies along the time axis (OA) and at a point along this axis R_J/T_F = dR_J/dT_F = c, the velocity of light. The observed surface on which A sees B is at R_J = cT_F. Since s = R_J/R₀ then ds/dT = $R_0^{-1}$ dR_J/dT and so

H(T_F) = s^–1 ds/dT_F = $R_J^{-1}$ dR_J/dT_F = c/R_J = 1/T_F. Hence H(T_F) = 1/T_F. So

Let H(T) be designated as H. In the standard coordinate theory

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.

Figure 5 shows a comparison of the two theories (Equation (26) and Equation (27)).

Figure 5. The comparison of the standard coordinate and dynamic coordinate 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.

Figure 6. The comparison of the standard coordinate and dynamic coordinate models of the universe’s deceleration and late-time acceleration (Equation 28 and Equation 29).

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. This would allow the dynamic theory to 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.

Figure 8. Comparison of standard theory with the dynamic corrected curve.

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 differential ds/dT of A’s observation of B in both the standard and dynamic theories is given by H₀/H = $T_0^{-1}$s(ds/dT)^–1. Hence for the dynamic theory, substituting this into Equation (26) gives

In the standard theory a similar substitution in Equation (27) gives

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

The match between the theories means that it is possible that dark energy (the accelerating universe represented by Ω_Λ) could be 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 Standard theory from the effects of gravity - and then in the relatively recent past beginning to accelerate. This is shown above (Figure 5 & Figure 6), along with the position, in terms of redshift, where the dynamic theory predicts the acceleration to start. Comparisons of this acceleration start (z_a) with results from various models based on both the Standard theory and observational astronomy are given in Table 3 below.

For A to adjust B’s times to a non-time dilated system to get comparative absolute times for s for both A and B relative to the Big Bang, it should be noted that s occurs on the same inferred surface (FDJ) when considered from within the appropriate trajectory, i.e. it is located at F on trajectory OA, and at D for trajectory OB. Consequently, in Figure 4, A’s adjusted estimate, i.e. from the inferred corrected surface containing B’s absolute location, would be at G.

By a similar reasoning that led to Equation (26), OG/OA = OD/OB = s cosζ, or

The standard curves can be obtained from any text on cosmology and can be summarised as follows; comparisons with dynamic theory are given in Figure 8.

Radiation-only model; s = (T/T₀)^1/2.

Einstein-de Sitter (matter-only); s = (T/T₀)^2/3.

Dark energy only; s = e^{H₀(T–T₀)}.

Combined Friedmann equations; s = (Ω_M/Ω_Λ)^1/3 sinh^2/3(T/T_Λ).

where T_Λ = (4/3Λc²)^1/2 and Λ ≈ 10^–35 secs^–2.

The key elements in the dynamic theory are: -

The following current cosmological problems have possible dynamic solutions

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 Sections 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 directly 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 must 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 a directly observed surface (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 (28) 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.

In the SR dynamic theory

The consequences of these assumptions are

In General Relativity (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 which 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 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 us 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 results can be summarised as follows:-

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.