## Wednesday, February 27, 2013

### The relativity theorem

There is a relativity principle, postulate, and theorem, and they get confused. I explain the difference.

The principle of relativity
is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.
It goes back to ancient debates about the motion of the Earth. J.C. Maxwell coined the word "relativity", and pointed out the difficulties for electrodynamics. Poincare was among the first to argue that the principle applies to electrodynamics, and defined it in 1904:
The principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion.
The Lorentz aether theory includes:
A fundamental concept of Lorentz's theory in 1895[A 1] was the "theorem of corresponding states" for terms of order v/c. This theorem states that a moving observer with respect to the aether can use the same electrodynamic equations as an observer in the stationary aether system, thus they are making the same observations.
Lorentz improved this to cover all velocities in his 1899 and 1904 papers.

The idea of using the same equations is mathematically sloppy. The better statement is not just that the equations are formally the same, but that the equations have a geometric formulation that works for all observers. The variables in the equations are really tensors where a routine change of variables gives them meaning for different observers. This modern concept is called Lorentz covariance:
In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance is a related concept, covariance being a measure of how much two variables change together. Lorentz covariance (from Hendrik Lorentz) is a key property of spacetime that follows from the special theory of relativity.
The covariance was discovered by Poincare in 1905, and geometrically explained by Minkowski in 1907:
In particular, Lorentz's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous "Æther"; it shows that this fact is connected with the covariance of the original equation, at certain simultaneous transformations of the space and time co-ordinates; these transformations have obtained from H. Poincaré the name of Lorentz-transformations. The covariance of these fundamental equations, when subjected to the Lorentz-transformation, is a purely mathematical fact; I will call this the Theorem of Relativity; this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light. ...

Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite connection may be styled the Principle of Relativity.
So the relativity theorem is the mathematical covariance of physical variables under a change of observers. The relativity principle is the physical idea that observers in different frames see the same physics.

So what was Einstein's contribution to this? His famous 1905 relativity paper says:
They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, ...
Einstein is referring to Lorentz's 1895 theorem of corresponding states, as the 1895 paper only proved it to first order in velocity. Einstein is ignoring Lorentz's 1899 and 1904 papers proving it for all velocities. This postulate is why Lorentz said in his 1906 Columbia U. lectures on relativity that "Einstein simply postulates what we have deduced". Lorentz proved his theorem of corresponding states for Maxwell's equations, and Einstein just assumed it as a postulate and confusingly called it the relativity principle.

Einstein achieved some phony simplicity with this approach, but it was useless. There was nothing new, mathematically or physically, in postulating what someone else had proved. In 1908, everyone was convinced of the superiority of the Poincare-Minkowski approach, and adopted Lorentz covariance as the cornerstone to relativity. The Einstein approach was scrapped, and Lorentz covariance continues to be one of the central physics concepts today.