**Why theories are divided i****n an undivided world**

Scientific theories tell us,

On every single page,

The universe is divided

Into time and space,

With seconds and with metres,

Which every theory says

Are what one needs to measure with

When measuring time and spaceScientists also tell us

In everything they do,

That making science is done by making theories,

Theories of time, but also of space,

Cos that’s the only kind of theory, there is.

But the world is not divided,

Into time and space,

And does not need two instruments to measure it,

You can measure it with metres,

Or measure it with seconds,

And it makes no difference which of them you choose,

But the thing you mustn’t do,

Must never, never do

Is to you use them both at once, together‘Cos that’s what makes you think,

Makes you wrongly, wrongly think,

The universe is divided into two.It is theories that divide

Into space and time,

And they do it with their seconds and metres,

To see the world as it is,

As it really, really is,

You must only use one or the other,

Then the world that you see

The real world that you see,

Has no time, space, speeds or theories

**Thinking about divided and undivided worlds**

To find out whether the world is divided, or un-divided, I have two helpers, ST and U. I don’t know what they believe but ST has been asked to behave as though he believes the world is divided into space and time and U to behave as though she believes the world is un-divided. ST will use traditional arguments that have been deployed over the last 115 years in expounding Einstein’s 1905 theory of relativity and U will transform all data into a single kind of unit as advised in verse 5 above. ST will defend theories and in particular, Einstein’s theory, while U will aim to describe measurements rather than theories. Theories use two kinds of unit like miles per hour while measurement data are ratios with only one kind of unit which can be miles or hours.

My first question to both of them is this:

Are metres and seconds (or miles and hours) the same kind of unit or different?

ST thinks they are different and U thinks they are the same. Both answers are logical because if the world is split into space and time, metres and seconds are different due to the split, whereas if the world is not split they are the same.

For my second question we have come to a ‘speed’ camera beside a busy road. The camera’s routine is to take two photographs in quick succession of a vehicle travelling along a scale painted on the road. The camera has just measured the interval between two photographs to be one second and the photographs show two images of a vehicle 30 metres apart. The question is:

What has the camera measured?

ST thinks it has measured a relative speed of 30 metres per second. U thinks it has measured a ratio of one to ten million. U has mentally re-calibrated the camera’s clock from seconds to metres by multiplying the number of its ticks by 300 million, so finding a ratio of 30 metres to 300 million metres and a ratio of one to 10 million which she puts equal to the sine of an angle, a. She could also have recalibrated the scale painted on the road from metres to seconds by dividing the number of marks on the scale by 300 million, so finding a ratio of 30/(300 million) seconds to one second. Either way, she has used only one kind of unit and the measured ratio she has obtained is one to 10 million.

ST’s approach is right if the world is divided into space and time and U’s approach is right if it isn’t. One of them, of course, is wrong.

Fig 1 shows how ST organises the information from the camera and fig 2 shows how U does it.

The triangles in figs 1 and 2 are the same shape but AB is equal to tan(a) in fig 1 and sin(a) in fig 2, so the triangle in fig 1 is larger than the triangle in fig 2 by a factor 1/C. Fig 2 only has a triangle whereas fig 1 has a triangle plus a reference frame. Either U has missed something or ST has put something into his diagram that isn’t in the data. They have already diverged.

In drawing figs 1 and 2, ST and U are taking logical steps on the basis of different starting assumptions. If the world is split, ST and fig 1 are right though the split leaves no trace in U’s data. If the world isn’t split, U and fig 2 are right.

That is why I need both of them.

**Red shifts, blue shifts and D**

Speed cameras also measure the red-shifts of vehicles travelling away from them and the blue-shifts of vehicles travelling towards them. From those measurements they obtain a number, D, the Doppler parameter. They scan all traffic for their D values and use the results to decide which vehicles to photograph. Einstein assumed that D and relative speed, v, are related.

I ask ST and U whether they think v and D are related.

ST says yes and U says no. ST points out that Einstein obtained an equation relating D and v in his 1905 theory of relativity. U thinks Einstein was wrong. She thinks D is related to S=sin(α) in fig 2, not v in fig 1. D and S, she says, belong to the un-divided world while v is an idea associated with the idea of a divided world. To support her views, U shows how the red and blue shift data, from which D is obtained, fit snugly into the S data of fig 2.

To generate fig 3, a zig-zag line of which LMA etc, is part, is drawn inside the ABK triangle of fig 2. The lengths of the zigs and zags grow steadily, a, aD, aD^{2}, aD^{3} etc, thereby defining D. By using standard theorems of trigonometry one can then use the diagram in fig 3 to obtain all the equations on the right in fig 3. The first 8 equations relate D to S and C while equations [9] and [10] describe how the motion of two vehicles relative to each other can be calculated from the motion of each vehicle relative to the camera. Equation [9] applies when the vehicles are moving away from each other. Then each sees the other to be red-shifted. Equation [10] applies when they are approaching each other. Then each sees the other to be blue-shifted.

Fig 3 is not a theory. It consists of data from measurements. It shows in ways that are easily tested by measurements, how the measured numbers, S, (DC), (D^{-1}C) and C are related to each other and how they are all related to red and blue shifts. It is a development from fig 2. Fig 1 has nothing similar.

Fig 3 is amazing. Fig 4 illustrates fig 3 by giving three sets of 7 numbers obtained from fig 3 for the three D values, 3/2, 2 and 3.

If any of the seven numbers is known for a particular value of D, the other six can be calculated. All are ratios and there is only one independent number. A measurement of 8/13 means 8 metres to 13 metres or 8 seconds to 13 seconds. Either will do.

I ask ST and U whether they think there is a logical path back from fig 3 to fig 1. ST thinks there is a logical path to fig 1 and that Einstein found it in 1905. U is very sure there isn’t a logical way back to fig 1 because the diagram in fig 3 and equations (1-12) have no time or distance. We decide to look more carefully at Einstein’s theory but first I will make a general point.

**The crunch issue; **

**Theories introduce time as an extra (fourth) dimension. It is present as the time axis in fig 1. It is absent from figs 2 and 3 which are based on measurements. Einstein claimed to have traced the equations of fig 3 back to reference frames with time axes. That looks like a miracle so how did he do it? **

**Einstein’s 1905 theory of relativity **

In 1905 Einstein set out to make a space and time theory from some of the equations in fig 3. To adapt the equations to his ideas he made some changes. Equations [E1] and [E2] are his versions of [9] and [10] in fig 3.

v

_{R}/c=(v_{1}/c+v_{2}/c)/(1+v_{1}v_{2}/c^{2}) [E1]v

_{B}/c=(v_{1}/c-v_{2}/c)/(1-v_{1}v_{2}/c^{2}) [E2]

In [E1] and [E2] Einstein replaced the symbol, S, in fig 3 by v/c where v is to be interpreted as a relative speed and c as the speed of light. The units of v and c are metres per second.

I ask ST and U whether [E1] and [E2] are OK. ST says yes because v/c is a ratio like S. U says no because S has only one kind of unit while v and c each have two, metres and seconds.

We move on to consider Einstein’s second step. This introduces two pairs of equations known as the Lorentz transformation into the theory. To create the first of these pairs, D in equation [5] in fig 3 is replaced by t’/t (time in two reference frames) and D^{-1} in equation [6] by t/t’. At the same time 1/C is replaced by γ (gamma). Those replacements convert equations [5] and [6] in fig 3 to [E3] and [E4].

t’=γ(t+(vt/c)) [E3]

t=γ(t’-(vt’/c)) [E4]

Equation [E5] is obtained by substituting one of these into the other.

^{γ2}=1/((1-(v/c)^{2}) [E5]

Equation [E5] tells us that if you know the value of v/c you also know the value of g. Alternatively, if you know the value of γ, you also know the value of v/c. Only one of them is independent. The new symbol t might have been introduced but a new dimension has not been created so it is wrong to identify it with the time axis of a reference frame. The time axis in fig 1 is an invention.

While mulling over this point, we noticed the inconsistency error in Einstein’s theory.

S is replaced by v/c in [E1] and [E2] but in [E3] and [E4] S is replaced by γv/c. These are only consistent if γ=1 and in that case v/c=0.

At this point everything becomes clear. It is possible to define two reference frames but not two reference frames in relative motion. It is possible to define relative motion, S, but not the relative motion of two reference frames. Fig 2 is correct in defining what can be measured and fig 1 is wrong because in addition to containing a triangle like fig 2, it contains a time axis for which there is no justification in observations.

Table 2 is valid only when γ=1 and v=0.

**Correcting the inconsistency**

If t’/t replaces CD and t/t’ replaces C/D then the two steps in Einstein’s theory are consistent because S is replaced by v/c in both.

From table 3 one obtains

t’=(t+(vt/c)) [E3A]

t=(t’-(vt’/c)) [E4A]

Equation [E5] is obtained by substituting one of these into the other.

1=1/((1-(v/c)

^{2}) [E5A]

The only solution to [E5A] is v=0.

In table 2 the inconsistency is removed by putting g equal to 1 and v equal to 0. In table 3 the inconsistency is already removed and one finds v=0.

There is thus no logical route from fig 3 to fig 1 and more generally there is no logical route from observations to theories.

**Energy and Mass**

I ask U and ST about energy and mass. U says there is no difference. M is total energy, MC potential energy and MS kinetic energy, and M^{2}=M^{2}C^{2}+M^{2}S^{2} is just Pythagoras’ theorem. ST says that total energy, E, is equal to γM, where M is potential energy and kinetic energy is γMv. Apart from ST using a triangle which is γ times larger than U’s triangle – an unimportant difference – these statements are equivalent. We are now in the part of Einstein’s theory which is not about inserting the unreal concepts of space and time. We are in the part that is about an undivided world. This part of Einstein’s theory works because it is based on observations but U’s presentation of it is much simpler. The symbols γ and t are unnecessary distractions.

**Einstein’s theory of Proper and Dilated Time**

In this theory, one of each pair of observers records the proper time, t, between a pair of events and the other records the dilated time, t’=γt. The one who records the proper time is present at both events while the one who records the dilated time is not present at both events. The one who records the proper time records a contracted distance, d/γ, and the one who records the dilated time records the proper distance, d, The classic illustration is a pair of twins, one of whom goes on a journey and then returns. The journey is defined by 3 events: travelling twin leaves home, travelling twin turns round, travelling twin arrives home. Stay-at-home twin is only present at the first and third events, When the travelling twin arrives home he is – according to the theory – younger than his brother because he has experienced the proper time of the outward and inward legs of the journey and his brother has experienced the dilated times. That conclusion is surprising to say the least. It is known as the twin paradox.

Since an event has space and time coordinates, Einstein’s theory of proper and dilated time is a theory of a divided world and has no connection whatsoever with the diagram and equations of observations in fig 3. It is not relevant to anything in this article.

In the Lorentz transformation, Einstein equated t’/t with D. In this theory he equates it with γ.

In fig 3, all symbols belong to a pair of objects. None of the symbols can be identified with one object. The world consists of pairs of objects. There is no evidence that single objects exist.

To relax, here is a little fantasy:

Einstein in ElysiumS is Sylvia, shy and retiring,

D is Diana, back from the hunt,

One is Venus, vain and voluptuous,

All are here at the sacred font,

And none of them, none of them,

Has a stitch on.C is a hunter, just passing by,

Engrossed in flaunting his manhood,

When all unbeknownst he catches the eye,

Of celestial huntress, DNow here comes Einstein all dishevelled,

With the seating plan

For the May Day revels,But oh my goodness where’s my phone,

Albert lad has found a throne,

And put Diana on her own

While C has been put with S and one,

A pair who here are very well known

To be an item.The air is thick with spells and magic,

Space and Time and weirder things,

Lengths contracting, times distending

Twins getting younger and older together,

Spaces curving as light-waves bend.Where are the naiads, dryads, satyrs,

Who can set Diana free

And bring her to her longed-for lover,

The dim but handsome COnly then will calm return,

To Elysium’s groves and bowers,

And then when space and time are fled,

Simpler science will flower.

**Stan Clough**

**Emeritus Professor of Physics, School of Physics and Astronomy, University of Nottingham, UK.**