Special Relativity: A Simple Matter of Perspective?

Popular belief has it that Eintein's Theory of Special Relativity is something difficult. It is not. Special Relativity is nothing more than what we usually call "Perspective", but applied to space and time, and not only to space.

On this page, you will understand why the "perspective" explanation of the paradoxs is equivalent to the traditional formulation of relativity.

Trajectory and local coordinates

Consider the trajectory of an object in a plane on which there is no coordinate systems already drawn. It will just look like a plain, rather boring curve:

What coordinate systems can we use to mark points on this trajectory? If I'm somewhere on the curve, it is tempting to use a tangent coordinates system, defined as having one axis along the curve, and the other perpendicular.

For any point O on the curve, we will call proper coordinates system in O a coordinate system like this.

Two coordinates systems

If we now consider two trajectories, we need to also consider two coordinates systems. We can arbitrarily tag one system with a prime, thus calling its coordinates x’ and y’.

An interesting exercise then is to find the relation between the coordinates of a same point P in both coordinate systems. This process is called coordinate conversion. It can be shown that any such conversion involves only two particular kind of transformations: translations and rotations.

Translations compensate for the fact that the origin (zero value) of the two coordinates system do not match.
Rotations correspond to the fact that the axis are not parallel.

More precisely, it can be shown that any coordinates conversion can be expressed as the combination of at most one translation and one rotation. On our figure, to convert from the non-prime to the prime system, the translation will bring O "onto" O’, and the rotation will then rotate the Oxy axis counter-clockwise so that they superpose O’x’y’. If the axis are parallel, then only the translation is needed. If the origins are identical (O = O’), then only the rotation is needed.

Mathematical expression of coordinate conversions

A point P has coordinates that depend on the coordinates system. These coordinates are found by projecting perpendicularly to the axes, and measuring the distance along each axis.

The facts that point P has coordinates x and y is commonly written as: P=(x,y). For three dimensions, it will be written P=(x,y,z). However, the same point P has different coordinates in the two systems. That is, you can also write the same point in the prime coordinates as P=(x',y'). The values of the prime coordinates will not be the same in general as the values of the non prime coordinates. Coordinate transformations are the mathematical operations required to convert from one system to the other, for instance to compute the prime coordinates from the non prime ones.

A translation corresponds to adding a constant to each coordinate, and is the only transformation needed if axis are parallel:

x'=x-x_O
y'=y-y_O

Also note that vectors do not require this coordinate change. In the rest of this document, we will deal only with vectors, and we will therefore ignore translations, except where noted.

A rotation involves the angle a between axis in the two systems, and is the only transformation needed if the two origins are identical (O=O’) or if the transformed objects are vectors:

x'=x.cos a - y.sin a
y'=y.cos a + y.sin a

This rotation will be noted in the rest of this document R_xy(a)

Rotation for a proper axis system

For a trajectory, consider an arbitrary coordinates system Oxy, where O is on the trajectory, but possibly different from the proper axis. We will note the proper axis at O as Ox’y’. The rotation angle a between Ox’y’ and Oxy can be computed as a function of the trajectory.

For instance, if the trajectory is defined as a function of some argument s by P(s)=(X(s),Y(s)), we can compute the sine and cosine of the angle as follows:

This can be simplified to be written as a function of the slope dX/dY only:

Distance between two points

To compute the distance between two points in a given coordinates system, we use the Pythagor theorem. If we have two points P₁=(x₁,y₁) and P₂=(x₂,y₂), then the distance d(P₁, P₂) between the two points can be computed using:

However, mathematicians have been studying spaces for which the Pythagor theorem is written differently. For instance, you can consider spaces where:

These can alternatively be considered as spaces where the y coordinates is an imaginary number. That is, considering an imaginary unit i such that i²=-1, you can consider that a space where distance obeys the above equation is a space where all y coordinates are actually written iy.

Space-Time

The complete coordinates required to position a physical events are three coordinates of space, plus one coordinate of time. So the position of a physical event P is identified by P=(x,y,z,t).

However, experimental evidence shows that the distance in our 4-dimensional space requires an imaginary time. More precisely, the distance is written as:

Such a space is called Minkowski space.

Another experimental evidence is that, although we can control the displacement along the three space axis, we do not control the displacement along the time axis. In other words, the time axis above plays the role of the y axis in our 2D drawings. The time axis is the axis "along our displacement".

Lorentz transformations

Now, we can easily re-compute Lorentz equations. If we replace y by ict in the last rotation equations, we get:

using the classical notations beta and gamma of special relativity. Notice however that the angle is imaginary, which means that the above cosine is always larger than 1. When the angle is a real number, the cosine is always between -1 and +1.

The coordinate transforms become then:

... which are nothing else than the Lorentz equations at the heart of special relativity.

Conclusion

The Lorentz equations express simply a rotation in a 4-dimensional space corresponding to the imaginary "angle" between the trajectory of a moving object and the reference coordinates system corresponding to the trajectory in space-time of the "observer at rest".

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