## Euclidean reality, or what the Theory of Relativity would have looked like if Einstein had waited another 20 years

©Witold Nawrot,   ORCID: 0000-0002-8687-2066, info@euclideanreality.com, witek@hanakom.pl

Einstein created the foundations of the Theory of Relativity in 1905-1907. At that time, a model was in place that treated particles as discrete objects (for example, balls, material points) that moved relative to each other in empty space.

Meanwhile, in 1924, Louis de Broglie postulated the existence of a wave structure of matter. In order to reconcile fire with water, i.e. the discrete nature of particles with their wave properties, Quantum Mechanics was created, which attributed wave properties to discrete objects by making a simple non-relativistic equation: Total energy = Potential energy + Kinetic energy – the Schroedinger equation and from the relativistic equation  $Latex formula$– the Klein-Gordon equation. Thanks to the wave function, it has been possible to create shudder-inducing equations for science students from these simple relationships. And all this only to preserve 19th century ideas about the discrete nature of matter.

And things could have been different and simpler.

Here we go

## Chapter 1

### “What came first – the egg or the hen?” i.e. about interactions and space

We know that interactions propagate along space dimensions. But it has not occurred to anyone for over a hundred years to consider – what is the primary concept here, Is it the rigid spatial dimensions that determine the directions in which interactions propagate or is it perhaps the directions in which interactions propagate that determine the directions we interpret as space dimensions.

Ultimately, we do not see space as such. All we see are the particles, and we see this through the interactions that these particles emit.

And here’s a problem to put some people into insomnia: If in a 10-dimensional space there are particles emitting interactions in three mutually orthogonal directions, then by observing the particles we register their motions in only three directions. We are informed of the existence of space by the motions of the observed bodies and these are always maximally three-dimensional. How many dimensions of space will we imagine by assembling a picture of reality from observations of many particles?

In the following, I will present two approaches – a classical one, in accordance with the Theory of Relativity, in which space dimensions determine the directions of propagation of interactions – on the left side of the screen, and an alternative model of space, in which the directions of propagation of interactions determine the space dimensions – in the column on the right.

In both the Minkowski and Euclidean reference frames, I will present the mutual observation of the same three bodies. Since most cases can be reduced to two-dimensional problems so, for the purposes of this presentation, we will present the problem as a two-dimensional one.

### How it is in Minkowski spacetime

In space-time, bodies can move, but the motion of bodies is a relative motion and requires the definition of a coordinate system relative to which we define this motion.

The dimension of time determines the flow of time in the observer’s system. The world lines (trajectories in space-time) of the observed bodies determine the proper times of the bodies in motion. These times depend on the velocity of the bodies relative to the observer according to the rule  of conservation of the space-time interval. The dependence of the body’s proper time on the choice of observer’s system makes it impossible to interpret time directly as motion along the fourth dimension.

The observer always measures the distance from all other bodies along spatial directions that are perpendicular to its time axis. Thus, interactions between bodies in Minkowski spacetime always propagate in a direction perpendicular to the observer’s time axis.

### And how can this be described differently?

If no external forces act on bodies, then in Euclidean space all bodies move along arbitrary chosen rectilinear trajectories with an absolute velocity equal to unity.

Velocity equal to unity means that the path a body takes in Euclidean space is a measure of the flow of time in that body’s system. In other words, the flow of time directly describes the absolute motion of a body in Euclidean space. The trajectory of a body is therefore at the same time the time axis of the reference system of that body. Changing the direction of the trajectory has no effect on the speed of the flow  of time in the body system.

In Euclidean space, interactions propagate perpendicular to the trajectory of the observed body and it is the direction of these interactions that is interpreted by the observer as the space dimension of the observer’s coordinate system. Thus, in Euclidean space, the spatial direction of the observer is perpendicular to the time axis of the observed body and not the observer as in the left column.

Observer and observed body in Minkowski spacetime. The time axis of the observed body is stretched, i.e. from the observer’s point of view, time in the observed body system flows more slowly. The slowing down of time is described by the formula obtained from the rule of conservation of spacetime interval

$Latex formula$.

A world line inclined at 450 corresponds to the speed of light

In Euclidean space (E4), both black and blue bodies move with an absolute velocity of 1. The lengths of the trajectories are measures of the time flowing in both body systems. In the left figure, the observer is marked as the black body and it interprets the direction perpendicular to the blue body as its time axis. In the figure on the right, the observer is the blue body. Observing each other, the observers see that time in the neighbouring system flows slower, but this is not the result of a deformation of the time axis but of a choice of direction interpreted as the spatial axis. From the right triangles formed by the sides Δt, Δt’, Δx, the formulae for the relative velocity of bodies are derived: $Latex formula$ and for the observed shortening of time: $Latex formula$.

Due to the identical scale of all axes in E4, the speed of light here is equal to 1

### A case of three bodies watching each other

In the figure below, we have an example of observing bodies ct2 and ct3 in the observer system ct1.

The axes of the reference system of body one, which is now the observer, are perpendicular to each other.

The observer observes and measures the distances to bodies ct2 and ct3 along a direction perpendicular to the time axis of his coordinate system. At time ct1=3, the observer measures a distance from body ct2 equal to A and a distance from body ct3 equal to B.

At the same time, the requirement of perpendicularity of the time and spatial axes of the observer’s reference system enforces a deformation of the time axis of bodies in motion – here bodies ct2 and ct3 – resulting from the necessity of preserving the space-time interval in the transition from one observer’s system to another. The deformation of the times is equal to respectively:

$Latex formula$   and   $Latex formula$

In Euclidean space, a body observes the motions of other bodies along the directions in which these bodies send interactions. Thus, these are directions perpendicular to the trajectory (not the world line since there are no distinguished space and time dimensions) of the body under observation and not of the observer.

Thus, in the figure below, the observer is the body t1 (we do not denote ct1 because all dimensions of Euclidean space are of equal scale, so c=1.The observer t1 observes bodies t2 and t3.

In the figure below – left, observer t1 observes the motions of body t2. The observer measures the distance from body t2 along the direction in which body t2 sends interactions i.e. along the direction perpendicular to the trajectory of body t2. The observer t1 notes that at time 3, body t2 was at a distance A from him.

In the figure below – on the right, observer t1 observes the motions of body t3 but in this case observer t1 measures the distance from body t3 along the direction in which body t3 sends out interactions i.e. along the direction perpendicular to the trajectory of body t3. Observer t1 notes that at time 3 body t3 was at a distance B from him.

It turns out that, regardless of the way of observing problem is presented, both the Minkowski model and the Euclidean space model give identical results. An observer t1 observing bodies t2, t3 or ct2,ct3 in Minkowski spacetime, in both models observes that at time 3 body t2 was at distance A from the observer and body t3 at distance B. To have a complete view of the problem let us see what happens if we now change the observer system from body t1 to body t2

Changing the observer in Minkowski spacetime involves defining a new time axis for the observer system and defining a new spatial axis.

The time axis is still vertical and the spatial axis is horizontal but now the time axis shows the time of body ct2 and the spatial axis now shows the distances of bodies ct1 and ct3 from body ct2. Currently the observer ct2 sees that at time 3 the body ct1 is at distance C and the body ct3 is at distance D.

Now, in turn, the time axes of the systems in motion are deformed, i.e. now ct1 and ct3 (previously it was the axes of ct2 and ct3).

While in the case of Minkowski spacetime, changing the observer required a new choice of coordinate system axis, in the proposed Euclidean space the trajectories of all bodies remain the same.

Simply, for the observer t2, we conventionally take its trajectory as the time axis of the observer, while the distances from bodies t1 and t3 are now measured along the directions in which bodies t1 and t3 send interactions, i.e., in directions perpendicular to the trajectories of t1 and t3, respectively.

It can be seen that at time 3, observer t2 measures a distance from body t1 equal to C and a distance from body t3 equal to D.

In Euclidean space, a change of observer causes neither a change in the trajectory system – i.e. the axis of the bodies’ times nor a deformation of the coordinates

So with two methods we arrive at two identical results. Was it worth creating a Euclidean space in order to describe the same thing more than 100 years after the development of the Theory of Relativity?

In Minkowski spacetime, a change in the observer’s system causes a change in the deformation of the coordinates of bodies in motion. The dependence of this deformation on the choice of reference system precludes the creation, on the basis of Minkowski spacetime, of a description of reality independent of the choice of observer. But, after all, the Universe exists whether we observe it or not. In addition, in Minkowski spacetime we have three dimensions the same and one slightly different. Covariant notation admittedly masks this minor inconvenience in the same way that perfume masked the lack of soap use in the recent past. Apparently it is ok, but something is not quite right here.

And now the Euclidean reality model – from the right column. Apparently it describes the same thing, but no longer from the point of view of the observer, but regardless of the observer’s choice, and in addition all dimensions are identical and whether a given distance in Euclidean space defines a temporal or spatial distance is no longer a question of the properties of space, but of observation.

If one compares the Theory of Relativity to a building, the Euclidean space model describes its foundations, but does not negate the structure itself.

If a body observes other bodies along the directions in which the surrounding bodies send interactions, then in our non-relativistic world these directions are perpendicular to our trajectory in Euclidean space (which is the time axis of our coordinate system) i.e. exactly as described by Minkowski spacetime. And we live peacefully until a relativistic particle appears. However, we measure the distance to the relativistic particle along a different direction in Euclidean space than the direction along which we measure the distances of all other non-relativistic particles. But we do not have any reference points to provide us with this information, So we assume that we measure the distance from the relativistic particle along the same direction as the other particles. And so we arrive at the left column i.e. Minkowski spacetime.

And in case anyone hasn’t yet noticed the essence of the idea of combining an absolute frame of reference such as Euclidean space with the relativity of motion – simply put, in Euclidean space the concept of relative velocity does not mean velocity relative to space, but as the figures in the right-hand column show V=x/t=sinφ so it is a measure of the angle of inclination of a trajectory and angle is relative and has nothing to do with the motion of bodies along their trajectories with absolute velocity V=1

(22-5-2024), To be continued . For the impatient, I invite you to visit my channel on