Non-Physical Lorentz Transformation

This is a follow up of the previous post with another aspect of the non-physicality of the Lorentz transformation. 

Lorentz invariance is a holy feature of modern physics coming to expression for a wave equation:

• $\frac{\partial u}{\partial t}-\frac{\partial u}{\partial x}=0$                 (W)

with solution $u(x,t)$ depending on a 1d Euclidean coordinate $x$ and time coordinate $t$. As shown in this post, the function $u^\prime (x^\prime ,t^\prime )=u(x,t)$ expressed in the Lorentz transformed coordinates $(x^\prime ,t^\prime)$ stated in the previous post, satisfies the wave equation

$\frac{\partial u^\prime}{\partial t^\prime}-\frac{\partial u^\prime}{\partial x^\prime}=0$  (W'),

which is viewed to express Lorentz invariance: The function $u(x,t)$ satisfying the wave equation (W), in transformed coordinates $u^\prime (x^\prime ,t^\prime )$ satisfies the wave equation (W') which reads exactly the same way as (W). 

The current wisdom is thus to say that the wave equation being invariant under Lorentz transformation, expresses a physical law which takes the same form in different coordinate systems connected by the Lorentz transformation. Invariance!

After having noted that the wave equation is Lorentz invariant, Einstein bravely proclaimed that all (true) laws of physics are Lorentz invariant, which immediately forced him to throw out Newtonian mechanics since Newton's 2n Law is not Lorentz invariant, and proceed to form a new relativistic mechanics unfortunately creating lots of trouble for modern physicists.

To someone with a bit of schooling in mathematics the wave equation (W) needs a qualification into an initial value problem with $t>0$, and $u(x,0)$ as an initial value as a function with spatial extension for all $x$, thus as co-existence in space for $t=0$. 

Now comes the catch: The initial value is not Lorentz invariant, since $t=0$ and $t^\prime =0$ do not say the same. In fact, $u(x,0)=u^\prime (\gamma x, \gamma vx)$, which is not an initial value for $u^\prime (x^\prime ,0)$. 

Co-existence as extended existence in space at a specific time is fundamental. Without co-existence the world collapses into point-like isolated events in space without cohesion and meaning.  

In other words, not even the wave equation (W) is Lorentz invariant, and so the whole idea of modern physics to consider laws of physics which are Lorentz invariant, collapses. This should be a relief for all students of physics for which exams in relativistic mechanics appear as road-block.