söndag 12 juli 2020

Can a Coordinate Transformation Contain Physics?

The Special Theory of Relativity SR is identical to the Lorentz coordinate transformation. The General Theory of Relativity involves as an essential step transformations of space-time coordinates. 

Many-Minds Relativity shows, in a careful mathematical analysis, that SR is empty as a theory about real physics. The argument is in brief that since SR is identical to the Lorentz transformation, and a coordinate transformation in itself does not contain any physics, SR is empty of physics. 

Why does a coordinate transformation not contain any physics? Consider for example the coordinate transformation of a pure translation from an $x$-axis to an $x^\prime$-axis, as 1d Euclidean coordinate systems, of the form $x^\prime = x - v$, with $v$ a constant.  One property of such a transformation is that distances are conserved, or are invariant, under transformation: If $x^\prime = x - v$ and $y^\prime = y - v$, then $x^\prime - y^\prime = x - y$. So far no physics has been introduced and so the coordinate transformation cannot have any physical meaning. A natural way to give physical meaning is to say the coordinate $x$ represents the position of a physical body $X$ and $y$ the position of a body $Y$. The positions of $X$ and $Y$ can then alternatively be expressed in terms of $x^\prime$ and $y^\prime$. 

Now, can we say that the transformation contains any physics of the position of the bodies? Can we say that since the transformation expresses that distances are conserved under the transformation, we can conclude a law physics expressing that translation of a rigid body does not change the shape of the body? Is this a law of physics, which may be false, or simply a law of logic or language and not physics which cannot be false? I guess, you we would agree with me that it is not a true law of physics. That a rigid body does not change shape is built in into the meaning of a rigid body, and has nothing to do with physics with the shape of a body determined by physical forces. It cannot be used to say that in a rigid body there are forces which maintain its shape. It would be like pulling a rabbit out of an empty hat.  

Choice of coordinate systems in physics directly connects to the choice of units in physics. If you believe that the choice of units contains physics, that using a meter stick instead of yard stick changes the size of physical objects, then you are well prepared to accept the wonders of SR. If you understand that this is not the case, then you will (like me and many others) have a hard time with SR. 

We can add a common time coordinate $t$ to the $x$ and $x^\prime$ systems into a Galilean space-time transformation of the form 
  • $x^\prime = x-vt$, $t^\prime =t$,
between a $(x,t)$-system and a $(x^\prime ,t^\prime)$, with the $x^\prime$-axis translating with respect to the $x$-axis with speed $v$, but still with the same lack of real physics.

The Lorentz transformation of SR is a similar linear coordinate transformation from a a space-time $(x,t)$-system to a space-time system $ (x^\prime ,t^\prime )$-system of the form 
  • $x^\prime =\gamma (x-vt)$, $t^\prime = \gamma (t-vx)$, $\gamma =\frac{1}{\sqrt(1-v^2)}$,
  • $x =\gamma (x^\prime+vt^\prime )$, $t = \gamma (t^\prime+vx^\prime )$.
where $\vert v\vert <1$, with now also the time variable beeing subject to transformation, and in particular the property that a $x = t$ is transformed into $x^\prime  = t^\prime$ and back.

In order for the Lorentz coordinate transformation to contain physics, the coordinates must be given some physical meaning. Einstein did this by associating space-time coordinates $(x,t)$ to events (of some unspecified physical nature) or rather to observations of events (of some unspecified physical nature), like a body $X$ observed to be at position $x$ at time $t$. Einstein associated the system $(x^\prime ,t^\prime )$ to represent coordinates with the $x^\prime$ axis translating with speed $v$, like in the Galilean transformation, but now with also the time variable being subject to transformation.

Einstein then created SR by drawing conclusions about physics from the Lorentz transformation, like the fact that $x=t$ is the same as $x^\prime =t^\prime$ interpreted by Einstein to mean that a light signal always propagates with speed 1 in any (inertial) coordinate system. Einstein then proceeded to derive properties of space contraction and time dilation from the Lorentz transformation and boldly claimed that he thereby had discovered new truely astonishing real physics (apparently from mysterious forces twisting space and time). 

Many early critics said that this was empty physics, for the reasons exposed above, but somehow Einstein managed to twist the brains of the physical community into believing that he had revealed deep new truths about real physics. This came to serve as footstep into modern physics, as weird empty physics based on transformations of coordinates brought to full bloom in the General Theory of Relativity in curved space-time.  

Galileo of course understood that his coordinate transformation does not contain any real physics. Why do modern physicists insist that the Lorentz transformation contains real physics?  Lorentz did not believe so, but he was quickly side-stepped by not being modern enough. 

Modern physics is in deep crisis. Why? Is it time now for post-modern physics?

The crisis is expressed in the obsession of modern physicists to focus efforts on a Theory of Everything a ToE, with the aim of explaining the basic four forces (electromagnetic, weak and strong nuclear and gravitation) as different manifestations of one basic force/interaction. No progress towards an answer has been made and the mystery of the origin of the four forces is as mysterious today as ever.   

1 kommentar:

  1. Applying the Galilean or the Lorentz coordinate transformation doesn't give any new physics, although it can be useful in calculations and for understanding situations. But finding out that space and time are connected and obey the Lorentz coordinate transformation instead of the Galilean coordinate transformation gave an insight into how the universe works and lay the foundation to new physics.