Special Relativity: Physics without Physical Laws

Unphysics of sawing a woman into two pieces.

The focus of both the special and general theory of relativity is coordinate systems with the idea that  coordinate systems hide truths about the world. Special relativity concerns Euclidean space coordinate systems moving with constant velocity with respect to each other, so-called inertial systems, while general relativity is expressed in general curvi-linear space-time coordinate systems with Einstein's equations expressing a connection between space-time curvature and mass-energy distribution.

Einstein's contribution to physics with the special theory is the postulate that physical laws have the same formal expression in all inertial systems connected by the Lorentz transformation, in other words are Lorentz invariant. Einstein thus postulates a formal requirement on what is allowed to be called a physical law: It must be Lorentz invariant.

Recall that the Lorentz transformation connecting two inertial space-time coordinate systems $(x,t)$ and $(x^\prime ,t^\prime )$ moving with velocity $v$ with respect to each other, read:

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

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

Einstein's contribution to physics with the general theory is Einstein's equations which express a physical law satisfying the invariance requirement by being covariant in the sense of having the same formal expression in different space-time coordinates as if allowing a coordinate-free representation in terms of curvature and mass-energy.

Which physical laws are then Lorentz invariant?  Does Newton's 2nd law $\frac{d^2x}{dt^2}=F(x)$ for a body of unit mass moving under the force $F(x)$ in the $(x,t)$ system take the same form in the the $(x^\prime ,t^\prime )$ system? Let us check out: 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 conclude that 

• $\frac{\partial^2}{\partial t^2}=\gamma^2(\frac{\partial}{\partial t^\prime}-v\frac{\partial}{\partial x^\prime})^2$.    

Does this show that Newton's 2nd law takes the same form in the two systems? Does not seem so, but to be sure let's take an even simpler case, that of a body moving with constant velocity $V$ in the $(x,t)$ system, with motion satisfying the physical law $x=Vt$ (with the initial condition $x=0$ for $t=0$), which in the $(x^\prime ,t^\prime )$ takes the form

• $x^\prime =\frac{1-Vv}{V -v}t^\prime$.  

We conclude that only for $V=1$ does the physical law $\frac{dx}{dt}=V$ take the same form in the $(x^\prime ,t^\prime )$ system. In other words, the physical law of propagation with constant velocity $x=Vt$ is Lorentz invariant only if $V=1$, that is only if the physical law of propagation is the law of propagation of light. Of course you can save the situation by simply defining $V^\prime =\frac{1-Vv}{V-v}$ and then claim the $x=Vt$ and $x^\prime =V^\prime t^\prime$ have the same formal appearance (with and without prime), but opening this possibility would loose the meaning of Lorentz invariance in the sense that any law could be made Lorentz invariant by suitable manipulation of symbols. 

Einstein thus says that the physical law of propagation with constant speed less than 1 is not a physical law. The only physical law compatible with Lorentz invariance is the law of propagation of light at speed 1.  This means that special relativity is empty of almost all physics as a physics with the only physical law being that of propagation light.  That this is so, is clear from the only hypothesis of special relativity, which is constant speed of propagation of light. With (close to) zero real input the real output can only be (close to) zero.

Special theory thus does not contain even the most basic physics as Lorentz invariant physics, but instead a lot of unphysics such as time dilation and space contraction as a consequence of postulated Lorentz invariance.

General relativity is supposed to be a generalisation of the special theory and if the special theory is zero so is the general theory. More precisely general relativity is not Lorentz invariant, so there is a possibility that the general theory contains some physics such as Newton's 2nd law and gravitation, but without special relativity the rationale of replacing Galilean invariant Newtonian mechanics with new Einstein mechanics is missing. See Many-Minds Relativity for a generalisation of Newtonian mechanics different from Einstein's reaching into speeds comparable to the speed of light.

PS1 What about Maxwell's equations? Yes, they are Lorentz invariant insofar they express propagation of light with the same constant speed in all inertial systems, but not concerning initial conditions and the connection between electric and magnetic fields, as made clear in Chapters 5 and 17 of Many-Minds Relativity.

PS2 The starting point for Einstein in 1905 was that it is impossible to determine the speed of a train traveling in rectilinear motion with constant velocity (inertial motion) from an experiment made inside the train, if there is no possibility to look out into the environment. This is the same in Newtonian mechanics under Galilean invariance. Similarly it showed to be impossible (the null result in the Michelson-Morley experiment) to determine motion of the Earth vs a (stationary) aether medium carrying electromagnetic waves. Without an environment or aether as (stationery) reference,  inertial motion is impossible to detect; only relative inertial motion is possible to detect. But that does not hold for non-inertial motion, like rotation; an ice skating princess with closed eyes can certainly feel if she is spinning or not.

In any case the Michelson-Morley null experiment made Einstein claim that there is no aether at all, and postulated that therefore all observers independent of inertial motion must record the same (unit) speed of light, independent of any physics of propagation of light. This was not an assumption about physics,  but simply a human standard or recipe about how to measure time and space so that the speed of light comes out to be 1, independent of any physics. This made special relativity to a theory without physical content and as such without scientific meaning.

PS3 Many-Minds Relativity proposes a different way to explain the MM null result based on the following assumption with clear physics content: All observers share a common time (have identical clocks), travel with respect to each other with constant velocity and make observations in a Euclidean space coordinate system in which they are stationary. Different observers thus use different inertial systems or aethers, and there are thus as many aethers as inertial coordinate systems (in the spirit of Ebenezer Cunningham). Each observer assumes the validity of a wave equation in the observer's coordinate system which says that light propagates with unit speed in the observer's coordinate system and which effectively determines how to measure length (in light seconds). This is an assumption about physics which is consistent with the MM null result. The key question of focus in Many-Minds Relativity is then to what degree different observers will agree on lengths and motion in space.

Many-Minds Relativity is different from special relativity in that all observers use the same type of clock (with operation independent of inertial motion such as a pendulum) and thus can share a common time without time dilation by some suitable synchronisation, while all observers are tied to their own inertial system. The conundrum of special relativity of one observer making observations in two different inertial systems, then is not an issue at all and the paradoxes of special relativity all collapse to null.

To modern physicists special relativity and Lorentz invariance is viewed as such a holy cow that it cannot be subject to a critical analysis and only be swallowed without any questioning, despite all its paradoxes. My experience is that it is very difficult to find a physicist willing to enter into a discussion about special relativity and its role as pillar of modern physics.