A possible area that could lead to more than one approach to producing an SR coordinate system arises with the need to include time as the equivalent of a spatial dimension in either of the theories (standard or dynamic) discussed in this paper. As stated in the Introduction, for the dynamic theory 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 physically 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 physically 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 definitions of observable and inferred events given in the Introduction... Hence from within an inertial frame no dimension can be directly observed in what can be assumed as the time direction. It can only be inferred.

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 an inferred 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 also 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 coordinates generated by the two theories coincide; where they diverge; the derivation of the dynamic metric, and the transformation between dynamic and standard coordinates within a single inertial frame.

To clarify which theory is being referred to in the following, the derivatives in the standard theory result from infinitesimal changes in intervals which are used to obtain 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 in which the object being observed is located. For space there are three subscripts, e.g. Dt _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 standard descriptions within a given frame by Equation (7), while both dynamic and standard coordinates are transformed between different frames by Lorentz transforms. The dynamic transform is given in Table 1 and the standard in Table 2 in the sections 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

c 2 d τ 2 = c 2 d t A 2 − d x A 2 − d y A 2 − d z A 2 = c 2 d t B 2 − d x B 2 − d y B 2 − d z B 2 ( 1 ) .

where A and B are inertial frames of reference, and d _τ 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 d _τ = 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, when the spatial component of motion lies entirely along the x axis, 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. 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 inferred from movement of objects outside the frame can, 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

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 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 hyperbolas in Figure 1a where dτ is an invariant for all possible inertial frames that have their origin at O, i.e. frames containing OA, OB and OG all have the same value of cdτ for any given hyperbola.

Since the dynamic theory has an ever-changing present, this is equivalent in Cartesian coordinates to mapping out time-like trajectories 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 ).

In Figure 1b it can be seen that all inertial time axes with a common origin ( O) must intersect the hyperbolas defining different values of cdτ, and so these hyperbolas can be considered as the loci 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 a given 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 hyperbolas

D t N N = d τ a t 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 + D y N N M 2 + D z N N M 2 = c 2 d t N 2 − d x N 2 − d y N 2 − d z N 2 ( 6 ) .

Let

D X N 2 = D x N N M 2 + D y N N M 2 + D Z N N M 2 and d X N 2 = d x N 2 + d y N 2 + d z N 2 , so d X N / c d t N = D X N N M / c D t N N = υ N / c ,

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

D t N N = d τ γ N D t N N = d t N γ N D X N N M = d X N γ N 2 D t N M = γ N d τ = d t N . ( 7 ) .

Where γ N = ( 1 − υ N 2 / c 2 ) − 1 / 2 .

Hence γ N − 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 N − 4 c 2 d t N 2 + γ N − 2 d X N 2 = c 2 d t N 2 − d X N 2 ( 9 ) .

To express the standard and dynamic coordinates in more familiar nomenclature, let

μ , v = 0 t o 3. Then c d t N = d x 0 ; d x N = d x 1 ; d y N = d x 2 ; d z N = d x 3 .

The Minkowski metric is

( c d τ ) 2 = η μ v d x μ d x v ( 10 ) .

Where in standard coordinates

η μ v = 1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1

While the dynamic coordinates transformed into standard coordinates are

η μ v = γ N − 4 0 0 0 0 γ N − 2 0 0 0 0 γ N − 2 0 0 0 0 γ N − 2

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

U 0 = d x 0 / d τ = γ N c D t N N / d τ = c γ N ; and U 1 = d x 1 / d τ = γ N D x N N M / D t N N = v 1 γ N e t c .

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

P μ = m γ N ( c , V ) ( 11 )

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).

Time dilation. It is straightforward to show that Equation (3) can be manipulated to give to give a dynamic time dilation that is the same as for the standard theory. Indeed we note that from the transform in Equation 7 we can immediately write

d t A / d τ = γ A D t A A / γ A D t A B = γ A . ( 11 ) .

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 observed 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 B B = δ x B B B c o s ζ = γ A − 1 δ x B B B ( 12 ) .

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.

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 coordinate systems (standard and dynamic) 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

γ 1 = ( 1 − v 2 / c 2 ) − 1 / 2 ; γ 2 = ( 1 − u 2 / c 2 ) − 1 / 2 and γ 3 = ( 1 − w 2 / c 2 ) − 1 / 2

Where in Table 1, γ = ( γ ₁ γ ₂)/ γ ₃ for Column 1, and γ = ( γ ₁ γ ₃)/ γ ₂ for Column 2.

Dividing the top row by the bottom in Column 1 gives

w = ( u − v ) / ( 1 − v u / c 2 ) ( 13 )

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

u = ( w + u ) / ( 1 + v w / c 2 ) ( 14 ) .

When comparing these equations with the standard 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 ( E ₀). 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 = γ ₁ 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 ( E ₀ = mc ²) where m is B’s mass. In turn this gives

OE′ = E ₀.

OB′ = γE ₀ = E.

E′ B′ = γmvc ²/ c = pc,

where p is momentum, i.e. p = γmv.

From Figure 2, OB ² = OE ² + EB ². Hence ( OB′) ² = ( OB′) ² + ( E′ B′) ², and so

E 2 = E 0 2 + p 2 c 2 ( 15 ) .

as in the standard 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, which is seen from other frames of reference as having components of both spatial velocity and time. As discussed above, these components have a vector addition which always adds up to the velocity of light and so within a frame, which cannot observe any spatial motion of itself, this velocity is entirely expressed as a changing time i.e. E ₀ is a kinetic rather than a static phenomenon.

In the next section it will be hypothesised that the time directions of all frames of reference lie in the local directions of the radial expansion of the dynamic model of a 4D universe. In this simple model this is the only motion that is allowed. The 3D space is everywhere orthogonal to this direction and, as discussed below, an observer can only see other frames moving in his space due to the curvature of the universe in this model and the radial expansion of this surface (see Figure 3). Consequently, despite having a total motion equivalent to the velocity of light, this is only seen from within a frame as temporal changes and is driven entirely by the expansion of the 3D surface of this universe along a radial direction in the fourth dimension. There are no objects moving along the 3D surface independently of this expansion, i.e. as previously discussed, all objects are time-like and none are space-like.

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 model ¹ which can be summarised as

And some items which may not 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 standard 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 and basis for this approach have been outlined at the start of this paper.

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 and in effect the model is the opposite of the standard theory. There nothing exists apart from a 4D surface undergoing expansion, while gravitation, spatial motion etc. takes place along this surface. In the dynamic theory, assuming uniform mass, the expansion only occurs radially into a 4D space. For the model discussed in more detail below, our 3D space exists as a surface which is orthogonal to the radial expansion. This radial expansion is experienced as time from within an observer’s reference frame, but with all other frames seen as having their spatial components of motion (and gravitational effects) along this 3D surface in the same fashion as already discussed in the previous sections on SR (see Figure 3). The observer has no observation of a fourth spatial dimension; it can only be inferred from clocks as previously discussed.

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 and the 3D spatial surface is analogous to a sphere, 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 increasing angles to A’s time direction. This 3D spatial surface is assumed to be flat, i.e., there is zero intrinsic curvature along this surface while the curvature exists in the fourth dimension, seen locally as time, and non-locally as providing an increasing velocity of the 3D surface in relation to A, so providing a stretching surface (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 (and inferred) 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. c T = c ∫ 0 T D t N N which is the integral of Dt _NN along N’s trajectory as inferred by an observer located within N’s frame of reference. R = ∫ 0 R D r N N P and is the position of a photon in frame P, emitted at the Big Bang, in N’s spatial axes while D r N N P 2 = D x N N P 2 + D y N N P 2 + D z N N P 2 All inferred surfaces are coloured blue (see Figure 4) and are the loci of all such trajectories originating from the Big Bang.

It is worth re-emphasising that the above trajectories relate to the expansion of these surfaces and not to what we would consider to be velocities along the three spatial dimensions of the surface. The surface of the universe can be postulated as everywhere (in the dynamic non-gravitational simple model) expanding radially at the speed of light. But this is in the fourth dimension which can only be inferred in terms of trajectory and observed as changes in time. No particle with a rest mass could move along the surface with this velocity. As explained in the SR section, this leads to all particles having a time-like trajectory, and this is the only trajectory linked to the radial expansion of the universe that can be inferred from within a reference frame that also contains the observer (see Figure 3 and Figure 4).

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 an inferred “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. It is constructed by integrating the radial motion to provide a dimension in which events lie relative to A.

There are two types of surface 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.

It should be re-emphasised that from A’s viewpoint his time axis is an inferred construct where the observable surface (solid red line) is seen in A’s present and consists of a single view of all other trajectories where the time dilated surface (solid red line), the object’s trajectory (e.g. OB) and the photon trajectory (e.g. CF) intersect. From the photon trajectory A calculates B as apparently appearing at a location OF in time relative to the Big Bang. Although the result of every intersection is seen at A, he has (in the current theory) artificially spread out these locations along an inferred dimension (OA). The method by which the events are calculated is discussed below.

Also note that temporal simultaneity occurs along the inferred surfaces such as ABL and ECK, and not along the vertical lines which connect different times such as OF and OC between the observed surface and the observer.

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

The evolution of the universe is calculated in terms of the scale factor a of the universe at time T after the Big Bang, which can be expressed in Figure 4 as

a = c T F / c T 0 = R J / R 0 ( 18 ) .

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 a. This parameter can measure the amount of stretch in the surface between then ( T) and now ( T ₀), 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 location where three phenomena, discussed below, meet ( C) in the example shown in Figure 4 :-

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.

A’s observations of B are that, because v/c is constant along OB (according to A), a location on OB is related to a location on OA by OA = OB cosζ, e.g.

cos ⁡ ζ = O G / O D = O F / O C = O E / O B , ( 21 ) ,

where in Figure 4, OD; OC and OB are located on trajectory OB. Equally OG; OF and OE are located on OA.

However, the inferred geometric relationships in Figure 4 obtained from the concentric surfaces can be summarised as

OA = OB; OC = OE; OD = OF and by substituting such values into Equation (21), we can write

OF/ OE = OE/ OA which lie entirely on the OA trajectory.

Multiplying through by OE/OA gives OF/OA = ( OE/OA) ²

and since a = OF/ OA, then OE/ OA = a ^1/2. These values of a also apply to a = OD/ OB and a ^1/2 = OC/ OB.

Hence A’s view of location OC, where the Doppler effect, time dilation surface and photon trajectory intersect, can be obtained from ( Equation 21) as

OF/ OC.( OC/ OB) = OE/OB.( OC/OB) or

O F / O B = a 1 / 2 cos ⁡ ζ ( 22 ) .

Consequently from ( Equation 20 and Equation 22) the key equation for the dynamic theory can now be expressed as

T F / T 0 = 2 a 3 / 2 / ( 1 + a 2 ) ( 23 ) .

Let H( T) be the Hubble parameter at time T after the Big Bang and H ₀ is the present value of the parameter. Then H ₀ = 1/ T ₀ and along a trajectory that is frozen at R ₀, T ₀ ≡ R _L , T _A , we have H( T) = a ^–1 da/dT.

From A’s viewpoint H( T) varies along the time axis ( OA) and at a point along this axis, e.g. DR _J/DT _F = dR _J/dT _F = R _J/T _F = c, ( c being the velocity of light).

The observed surface on which A sees B is at R _J = cT _F . Since a = R _J/R ₀ then d a / d T = R 0 − 1 d R J / d T , and so

H ( T F ) = a − 1 d a / d T F = R J − 1 d R J / d T F = c / R J = 1 / T F . Hence H( T _F ) = 1/ T _F . So

H 0 / H ( T F ) = 2 a 3 / 2 / ( 1 + a 2 ) ( 24 ) .

Let H( T) be designated as H. In the standard (GR-based) coordinate theory for a flat universe, we have the first Friedmann equation which is

[ 1 R d R d T ] 2 = 8 π G 3 [ Ω M a − 3 + Ω R a − 4 + Ω Λ ] − k c 2 R 2

where G is the universal gravitational constant and k is the curvature parameter which is zero for a flat universe.

H 0 = ( 8 π G ρ 0 / 3 ) 1 / 2 where ρ ₀ is the current average universal density. So for the standard theory

H 0 / H = ( Ω M a − 3 + Ω R a − 4 + Ω Λ ) − 1 / 2 ( 25 ) ,

where Ω _M is the current normalised density of matter. The Standard Cosmology Model (SCM) value quoted in 9 is 0.32 for this parameter; Ω _R is the normalised density of radiation equal to 5×10 ^–5; 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 (24) and Equation (25)).

Considering the simplicity of the model used in the dynamic theory (red line), it is remarkably close overall to the SCM coordinate theory’s values for the coefficients (black line, in the Figure) considering SCM is based on GR for a flat universe and incorporating the lambda coefficient while the dynamic theory is based on SR and the simplest assumptions about the universe’s geometry.

At the early time low expansion of the universe (low a or high z), a good fit is obtained to the dynamic theory by the SCM’s coefficients when this curve is reduced by 10%. This is shown in Figure 6 (blue dotted curve). Consequently, to fit to the dynamic curve over the complete range of a two SCM curves are needed, the low a (high z) curve needing a value of H ₀ which is 10% lower than its value obtained from nearby (high a) cosmological data. This might allow the dynamic theory to explain the current “tension” between values of H ₀ obtained from the cosmic microwave background (high z) and from more recent (low z) cosmological feature (e.g. 10).

It should be noted from Figure 6 that when the dynamic curve is divided by T ₀ - 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 da/dT of A’s observation of B in both the standard and dynamic theories is given by H( T) = a ^–1 da/dT. Hence for the dynamic theory, substituting this into Equation (24) gives

T 0 d a / d T = 0.5 ( a − 1 / 2 + a 3 / 2 ) ( 26 ) .

In the standard theory a similar substitution in Equation (25) gives the SCM scale velocity as

T 0 d a / d T = ( Ω M a − 1 + Ω R a − 2 + Ω Λ a 2 ) 1 / 2 ( 27 ) .

These equations are shown in terms of the scale factor ( a) in Figure 6 and z _a , the redshift at the start of the recent universe’s acceleration, 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 scale velocity ( da/dT) 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. In the dynamic theory this is due to the model’s geometry which is independent of gravity (see previous discussion) and yet which still provides a good fit to the trend of the SCM, as shown above in Figure 5 & Figure 6). Also the position, in terms of redshift, where the dynamic theory predicts the acceleration to start to accelerate ( z _a ) agrees well with results from various models based on both the standard theory and observational astronomy, and are given in Table 3.

The position versus time of the universe’s radius is given by the integral of Equation 26 for the dynamic theory, and the integral of the combination of Friedmann’s first equation (above) with his acceleration equation

1 R d 2 R d T 2 = − 4 π G 3 ( ρ + 3 p c 2 )

where ρ and p are the density and pressure in the universe at time T. Assuming pressure- free matter, negligible radiation density and spatial flatness gives an integral for the standard theory of

a = A 1 / 3 s i n h ⁡ 2 / 3 ( T / T Λ ) ( 28 )

where A = Ω M / Ω Λ a n d T Λ = ( 4 / [ 3 Λ c 2 ] ) 1 / 2 a n d Λ ≈ 10 − 35 s − 2 . To convert T _Λ to units of time it has to be multiplied by c.

Then T/ cT _Λ = T/ T ₀.( T ₀/ cT _Λ) = KT/T ₀. Here K is a dimensionless constant equal to 13.8Gyr divided by (4/3Λ) ^1/2. Using cT _Λ ≈ 3.72 x10 ¹⁷ s allows Equation 28 to have a scale factor of unity at T = T ₀ in Figure 7. Hence Equation 28 can be rewritten as

T / T 0 = 0.855 s i n h − 1 ( 1.4577 a 3 / 2 ) ( 29 ) .

The equivalent curve from the dynamic theory is obtained by the rearrangement and integral of Equation 26, i.e.,

2 ∫ 0 a a 1 / 2 d a / ( 1 + a 2 ) = ∫ 0 T d T / T 0 .

This gives

T T 0 = 1 2 l n [ a + 1 − 2 a a + 1 + 2 a ] + 2 [ t a n − 1 ( a − 1 2 a ) + π 2 ] ( 30 ) .

The key elements in the dynamic theory are: -