lördag 10 augusti 2019

The Seduction and Spell of the Lorentz Transformation

Modern physics is based on the idea of Lorentz invariance as the basic postulate of Einstein's special theory of relativity:
• Laws of physics are invariant under the Lorentz transformation between different inertial systems, that is, laws of physics are Lorentz invariant.
Recall that the Lorentz transformation connecting two inertial space-time coordinate systems $(x,t)$ and $(x^\prime ,t^\prime )$ for two observers moving with velocity $v$ with respect to each other, reads:
• $x^\prime =\gamma (x - vt)$, $t^\prime =\gamma (t - vx)$,
• $x =\gamma (x^\prime + vt^\prime )$, $t =\gamma (t^\prime + vx^\prime )$,
where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $\vert v\vert \lt 1$. We see that the space coordinate $x$ and time coordinate $t$ appear in symmetric form with a an apparent similarity between space and time, which Lorentz viewed to be a formality without physics, but Einstein took as a basis of modern physics with space mixed into time.

Which laws of physics are then formally Lorentz invariant? By the chain law, we have
• $\frac{\partial}{\partial x}=\gamma (\frac{\partial}{\partial x^\prime}-v\frac{\partial}{\partial t^\prime})$,
• $\frac{\partial}{\partial t}=\gamma (\frac{\partial}{\partial t^\prime}-v\frac{\partial}{\partial x^\prime})$,
and so
• $\frac{\partial}{\partial t}-\frac{\partial}{\partial x}=\gamma (1+v)(\frac{\partial}{\partial t^\prime}-\frac{\partial}{\partial x^\prime})$,
• $\frac{\partial}{\partial t}+\frac{\partial}{\partial x}=\gamma (1-v)(\frac{\partial}{\partial t^\prime}+\frac{\partial}{\partial x^\prime})$,
• $\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}=(\frac{\partial}{\partial t}-\frac{\partial}{\partial x})(\frac{\partial}{\partial t}+\frac{\partial}{\partial x})$
• $=\gamma^2(1-v^2)(\frac{\partial}{\partial t^\prime}-\frac{\partial}{\partial x^\prime})(\frac{\partial}{\partial t^\prime}+\frac{\partial}{\partial x^\prime})=\frac{\partial^2}{\partial t^{\prime 2}} - \frac{\partial^2}{\partial x^{\prime 2}}$.
We see that the second order wave equation
• $\frac{\partial^2u}{\partial t^2} - \frac{\partial^2u}{\partial x^2}=0$,
is Lorentz formally invariant, in the sense of reading exactly the same in the $(x^\prime ,t^\prime )$ system.  On the other hand, for the first order wave equation:
• $\frac{\partial u}{\partial t} - \frac{\partial u}{\partial x}=0$,
the multiplicative factor $\gamma (1+v)$ appears, and so only a form of restricted formal Lorentz invariance is in place.

The second order wave equation describes waves in an elastic string with clear material spatial presence or coexistence, as well as plane electromagnetic waves in a vacuum without material spatial presence.

The idea of Einstein (picked up from Lorentz) was that since the wave equation takes the same form in all inertial systems connected by the Lorentz transformation (more or less), all inertial systems are equally valid (with in particular the same speed of wave propagation/light), which Einstein declared to be the essence of the new physics of the special theory of relativity. This was the seduction of Lorentz invariance which promised to solve the mystery on an "aether" medium for the propagation of electromagnetic waves in vacuum without material presence.

For both wave equations we see the space coordinate $x$ and time coordinate $t$ appearing in symmetric form, which opens up to some invariance with respect to the Lorentz transformation with similar symmetry.

But the formal symmetry in space and time in the wave equations does not say that space and time have the same nature and can be mixed into each other. In the wave equations there is a clear distinction between space and time which is expressed in the initial condition complementing the wave equation in a mathematical description of a wave $u(x,t)$, which takes the form
• $u(x,0)=u_0(x)$ for all $x$,
for the first order equation (with also an initial condition for $\frac{\partial u}{\partial t}$ in the second order case), where $t=0$ is an initial time and $u_0(x)$ an initial wave form with extension in space. With the initial condition a clear distinction between space and time is made. This is physics which is very obvious for the elastic string but also relevant for electromagnetic waves. The initial wave form shows spatial coexistence at different points $x$ for some common initial time $t=0$.

And now comes the catch showing that the Lorentz transformation is not compatible with physics, even if the wave equation is formally (more or less) Lorentz invariant: The initial condition is not invariant under Lorentz transformation, because $(x,0)$ translates into
• $(x^\prime ,t^\prime ) =\gamma (x, -vx)$,
which does not have the form of an initial condition for $t^\prime =0$. The physics of coexistence expressed in the $(x,t)$-coordinates through the initial condition for $t=0$ does not carry over to physics of coexistence of an initial condition for $t^\prime =0$. This means that the physics expressed in the different inertial systems is different. The whole idea of relativity of expressing the same physics in different inertial systems thus collapses.

The formal symmetry of the space and time coordinates in the two forms of the wave equation misled a confused Einstein to believe that space and time could be mixed, because Einstein did not properly understand the physical meaning of the mathematics of the wave equations. The sad fact is that generations of physicists have followed in the footsteps of Einstein with a mantra of Lorentz invariance as a necessary requirement of a law of physics.

A traveling wave is a solution $u(x,t)$ of either of the above wave equations of the form
• $u(x,t)=f(x+t)$,
where $f(\cdot )$ is a function of one variable. For example $f(y)=\sin(y)$ with
• $u(x,t)=\sin(x+t)$.
The initial condition for $t=0$ would then have the form $u_0(x)=\sin (x)$ as a wave in space, while an observer sitting at $x=0$ would experience a wave in time of the form $\sin(t)$, but the observer would have no reason to mix the wave in space with the wave in time just because the mathematics looks the same. To do that as Einstein did, shows that the mathematics is misunderstood.

The second order (but not the first order) equation also has standing wave solution of the form
• $u(x,t)=sin(t)sin(x)=\sin(\gamma (t^\prime +vx^\prime))\sin(\gamma (x^\prime +vt^\prime ))$,
with seemingly stationary spatial character in $(x,t)$-coordinates, but visibly not so in $(x^\prime ,t^\prime )$ coordinates. A standing wave solution is not Lorentz invariant. A standing wave for the observer using $(x,t)$-coordinates is not a standing wave for the observer using $(x^\prime ,t^\prime )$, of course not since the observers are moving with respect to each other.

In general the equations of mathematical physics do not show the symmetry of space and time of wave equations and thus do not show any Lorentz invariance at all. The physics is the same for all observers but its mathematical description varies between moving observers, as soon as the physics has some spatial presence, which is the nature of physics.  Only Maxwell's equations for vacuum can show formal Lorentz invariance, but not in the presence of charges and not with respect to initial conditions. The equations of physics are not Lorentz invariant. Not Maxwell with charges, not Schrödinger, not MHD, not Navier, not Navier-Stokes, not anything.

The net result is that the notion of Lorentz invariance has only a purely formal mathematical meaning and carries no real physics. If the spell of Lorentz invariance can be broken, then many possibilities  to progress seem to open up. But this is not something physicist like to hear. They will cling to Lorentz invariance no matter the cost and lack of reason. It is a spell.

Getting out of the spell means understanding that Leibniz distinction between space and time is valid also for modern physics:
• space = order of coexistence.
• time = order of succession.
But the spell has such strong grip on the minds of modern physicist that not even a basic discussion is possible.