Physics Illusion 11: Lorentz Transformation as Holy Doctrine of Physics

A Galilean transformation gives the connection between the coordinates in one system at rest and another system translating with constant velocity with respect that at rest. Fundamental with a clear physical meaning and relevance.    

Lorentz transformation with $x^\prime$-axis in red pointing into the $(x,t)$ first quadrant making it unphysical since different parts of an object extended in space get different time coordinates and thus does not exist as an object. For this reason, Einstein only speaks about points in space-time as events which excludes interactions of objects extended in space. 


The coordinates in two space-time coordinate system $S$ and $S^\prime$ with coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ moving with constant velocity $v$ with respect to each other is in classical (non-relativistic) mechanics connected by a Galilean transformation:

• $x^\prime  =x - vt$,
• $t^\prime  =t$,  

In relativistic mechanics according to Einstein's special theory, the connection is instead postulated to be that of a Lorentz transformation: 

• $x^\prime  =\gamma (x - vt)$,
• $t^\prime  =\gamma (t - vx)$,  

where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $0 < v < 1$.

In the Galilean case the time rate in $S$ and $S^\prime$ are the same, but in the Lorentz case it differs with a factor $\gamma$, so that with $S$ motionless the time in $S^\prime$ would seem to run slower, and the other way around.

The Lorentz transformation is by physicists regarded to reveal deep truths about physics, as a result  of the fact that it leaves a wave equation in $(x,t)$ coordinates:

• $\frac{\partial^2u}{\partial t^2}- \frac{\partial^2u}{\partial x^2} = 0$

invariant under a Lorentz transformation, that is the wave equation looks precisely the same in $(x^\prime ,t^\prime )$ coordinates. 

On the other hand, a Galilean transformation brings in a second order correction term in the velocity $v$ of the form:

• $v^2 \frac{\partial^2u}{\partial x^2}$,

and therefore a Galilean transformation is not viewed to have the same dignity as the Lorentz transformation without such a correction. For human observations $v^2 < 10^{-10}$ (with the speed of light normalized to 1) the correction is miniscule and of no practical significance.

However, as strongly emphasized by Lorentz himself, the Lorentz transformation is not a transformation between physical coordinates. In particular the transformed time $t^\prime$ should not be viewed as real physical time.

This was not understood correctly by Einstein, who gave $t^\prime$, against the strong advice by Lorentz, a physical meaning and so the special theory of relativity was born based on the Lorentz transformation, which then (surprisingly against all odds) became a fundamental pillar of modern physics. Anything which is not Lorentz invariant is not modern physics!

And there we are today with the Lorentz transformation as a Holy Doctrine of Modern Physics of the same stature, and mysticism, as the Doctrine of the Holy Trinity of the Catholic Church. But a Lorentz transformation is just a very simple linear coordinate transformation and as such cannot reveal deep truths of physics, while the Holy Trinity may well represent a deep truth of religion.

Newtonian mechanics is Galilean invariant, but not Lorentz invariant, and so in Einstein's hands Newtonian mechanics had to be sacked, according to the Doctrine of Lorentz Transformation. Instead a wonderful magical world of space contraction and time dilation was born from the Doctrine, but of course inheriting the unphysical nature of the Lorentz transformation and thus only a world of illusions without the reality of Newtonian mechanics.