4D Spacetime is Worth Only 2D!

Spacetime was created for co-transformation of space and time, while maintaining their separate identities. Since its inception, however, transformation of space has been restricted to only one direction i.e. the direction of motion of the observer, even when it was aligned in a different direction, with the same time paired. The restriction thus permits only one dimension for space, just like its co-parameter time which too is endowed, though rigidly, with only one dimension. This leads to only two dimensions for spacetime and showcasing it in more than 2D is rendered infructuous to that extent. The article discussed it in detail.

Introduction:

Spacetime is constituted of two entities i.e. the physical space and the virtual time, and both carry equal weightage, going by its structure. The structure does not resolve time into components at all, though we do so for space. The time, remaining unified, treats space also in the same manner i.e. unified. Therefore, by resolving space into three components, we add two infructuous dimensions, which unnecessarily compound complexity of the transformation expressions.

The above mentioned restriction, of considering only the space (distance) component which is parallel to the direction of motion of the observer, provides all the more justification to treat space as one dimension. It may be recalled that it (the restriction) pervades explicitly in the Special Relativity, and consequently, also in the General Relativity.

We have the choice of selecting a system of space and time coordinates in the benchmark (stationary) frame, wherein the quantities are ascertainable by, in Einstein’s own words, “measurements with meter-rods and clocks”. However, the freedom is not available to a (relatively) moving observer, as orientations of his/her space axis, as well as the time axis, are already decided by the velocity he/she possesses.

The most preferred system of space coordinates in the stationary frame is obviously the mutually perpendicular 𝑥-𝑦-𝑧 axes, and the resultant/unified space quantity is √(𝑥2 + 𝑦2 + 𝑧2). When the space is treated as one dimension in this direction, and the time as the other, with its scale of 1⁄𝑐 in any direction normal to it, the square of the two coordinates combine to become (𝑐2𝑡2 − 𝑥2 −𝑦2 − 𝑧2). This represents a spacetime point. The term formed by intervals of the constituents i.e. (𝑐2𝑑𝑡2 − 𝑑𝑥2 − 𝑑𝑦2 − 𝑑𝑧2), is the square of the spacetime length with two spacetime points at its ends.

Thus the structure stipulates only two dimensions for spacetime. The following discussion clarifies it more.

Orientation of Space Element — Physical vs. Graphical:

Before we proceed further, let us recall the two.

The time element, being virtual, has only graphical, and no physical, orientation. However, the space element has both. It is quite intuitive to represent the 3D space graphically with the corresponding 3 coordinates. However, when it comes to spacetime, the two lose their one-to-one (component-wise) correlation, due to the strict rules the time dimension, being virtual, imposes on the space element as a whole.

The spacetime graph, in the stationary frame, may well be chosen in such a way that the graphical representation of the space element coincides with it physically. However, in the moving frame, since the time axis has necessarily to be the world line of the moving frame, the space element has also to be graphically in a definite direction, which would always be different from the physical orientation.

For example, in respect of a uniformly moving observer along the physical space element, though the transformed space element is collinear with its parent, its graphical representation orients in a different direction, for the reasons mentioned above.

Special Relativity:

The (above mentioned) restriction, as stipulated by Einstein in his simplified derivation, manifests itself in the following four relations, worked out by him, with 𝑥-axis being the direction of velocity of the observer 𝑣, and 𝑥, 𝑦 and 𝑧 being the coordinates of the event (of light signal) under transformation, at the instant 𝑡. The primed parameters are, obviously, of the moving frame/observer.

The above bifurcation, by cutting off the infructuous part, makes the transformation relations more objective, revealing the true structure of (transformable) spacetime, which is 2D.

The restriction, by flattening all spaces into one direction i.e. along the observer’s motion, ensures conformity to the Lorentz Transformation condition.

It is pointed out here that even if, contrary to the above restriction, the entire distance 𝑟, at an inclination to the direction of the observer’s motion, was taken for transformation, the structure of spacetime would still be 2D, with both the directions forming a plane. The expressions for transformation would, however, be much lengthier, and it would also not be possible to conform to the Lorentz Transformation Condition in its present form, with the given postulate. [This has been addressed in detail, in my book.]

General Relativity:

The General theory has been built up on Special Relativity, and therefore, it automatically inherits the restriction/requirement of space flattening, as stated above. This is obvious from the principle adopted in separating gravitation from uniform motion, out of the transformation relations of General Relativity. The details are briefly discussed below.

In his paper “THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY”, Einstein started formulation of the theory with an infinitesimally small linear spacetime element 𝑑𝑠, with two infinitely proximate spacetime points at its ends, as follows.

As explained in introduction, the above relation meant — plotting the space element on an axis directed along it, and time 𝑑𝑥4 plotted along any direction/axis normal to the space axis.

If the orientation of the above 3D line was different from that of motion of the observer, the requirement could be met by, as per the existing rules, considering the line’s component parallel to the latter, and ignoring the rest.

Thus flattening of space, along the direction of observer’s motion, is implicitly included in the above mentioned values of the transformation matrix.

As for the non-uniform motion (causing gravitation), it could be broken into small stretches of uniform velocities. When, in a stretch of uniform velocity, the given space element was found aligned in a different direction, its component along the direction of velocity was to be taken, ignoring all other components.

As pointed out in the introduction, the velocity of observer enforces a particular orientation of the space and time axes in the moving frame, to maintain invariance of the spacetime element defined in the benchmark (stationary) frame. So, a non-uniform motion of observer carries with it changing orientations of space and time axes, in accordance with every change in the magnitude of velocity.

It may be noted that the change in orientation of the axes is only on account of the magnitude i.e. speed, and never due to change in direction of motion, as per the existing rules.

With every change in the direction of motion, the component of space element to be taken for transformation would change, as the angle between the two would change. However, the transformed space element would display transformation always along the direction of motion of the observer, and the graphical orientation of the axes, dependent on magnitude, would remain unchanged.

The discussion above brings out that even in General Relativity, spacetime could be defined by two dimensions. Breaking the space element into three components does not serve any purpose, as its co-parameter, time, considers space as a whole in transformation.

Therefore, the General Relativity would get simpler and more objective, if its mother relation, shown as eq.(1) above, was recast as follows, and developed therefrom.

The above relation is in the form spacetime transforms, and it is in 2D.

Conclusion:

As practitioners of science, it should be our endeavour to define the structures of physical phenomena with minimum number of variables, so as to make their representation more objective and easier to comprehend.

Expressing the space element of spacetime in three dimensions is of little use when its co-parameter, time, does not go into components while churning out its own, as well the former’s, transformation. This instead adds to complexity of the expressions.

With the above discussion, it is established that spacetime works with only two dimensions in Special Relativity, as well as in the General Relativity. The non-uniform motion of the latter may be broken into small segments of uniform velocities to make the structure more objective and easier to comprehend/solve.

The existing General Relativity exercise, though attempting to generalize the phenomena of spacetime transformation, lacks in presenting a scheme/algorithm of even how the “new” system of coordinates would be oriented for a given velocity, leave aside the relation between the two systems of coordinates. Statement of a general relation in mathematical terms, with a little or no utility in physical terms, is of little use.