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Apparent Difficulties in SR and Einstein's Attempted Resolution

In the previous article (Part 1/4) we reviewed the basic assumptions of SR, represented in assumptions we call (A1) and (A2).

  • (A1) that the laws of physics are the same in all nonaccelerated reference frames, i.e. if $K'$ is a coordinate system moving uniformly (and devoid of rotation) with respect to a coordinate system $K$, then natural phenomena run their course with respect to $K'$ according to exactly the same laws as with respect to $K$.

  • (A2) that light in vacuum propagates along straight lines with constant velocity $c = 300,000 ~km/sec$.

The conjunction $(A12):=(A1) \wedge (A2)$ leads to intellectual difficulties, as Einstein himself admits. But these difficulties are only apparent incompatibilities according to Einstein which are resolved by postulating the Lorentz transformations. The assertion that Lorentz formulae resolve all apparent difficulties in (A12) follows from two claims:

  • Claim (i): that inertial frames $K, K'$ are related by Lorentz transformations.
  • Claim (ii): that the law of propagation of light (A2) is Lorentz invariant. I.e. if (A2) is satisfied in $K$ , then (A2) is satisfied in every Lorentz translate $K′=\lambda.K$.

The key issue is verifying whether (A2) really follows from the Claims (i) and (ii). This is the subject of Einstein's spherical wave front argument which we examine below.

Critique of Einstein's Proof of the Compatibility of (A12) with Lorentz Transformations

We were led to reviewing Einstein's SR by reading the articles of S.J. Crothers and R. Bryant, which allege that Einstein's spherical wave proof contains significant errors.

Einstein's argument looks to derive Claim (ii) as a mathematical consequence of Claim (i). The starting point is the fundamental property of Lorentz transformations, that the quadratic expression $$x^2+y^2+z^2 = c^2 t^2$$ is invariant with respect to Lorentz transforms. In other words, the null cone $N$ is Lorentz covariant. Thus if $$(\xi, \eta, \zeta, \tau)=\lambda. (x,y,z,t)$$ for a Lorentz transform $\lambda$, then we have

  • [Eq1] $$\xi^2 + \eta^2+\zeta^2 = c^2 \tau^2.$$

But what does the expression [Eq1] represent?

Here is Einstein's first error: his argument misidentifies [Eq1] with "the equation of a sphere." The error is subtle. But the fact is that [Eq1] is strictly speaking a three-dimensional cone in the four independant variables $\xi, \eta, \zeta, \tau$. The cone indeed contains many spherical two-dimensional subsets, but our point is that a second independant equation is necessary to specify these spheres. And these independant equations cannot be chosen Lorent invariantly!

Penrose in his textbook with Rindler is quite careful here. The Penrose approach is to projectivize [Eq1] and obtain a projective sphere with a uniquely defined Lorentz invariant conformal structure.

But again we must emphasize that there is no canonical metric invariant with respect to the Lorentz group on the projectivization. In otherwords, the null sphere has a Lorentz invariant conformal structure, but it does not have a Lorentz invariant Riemannian structure.

For instance the standard round sphere $S$ centred at the origin simultaneously satisfies [Eq1] and additionally the equation $$\frac{1}{2}d(\xi^2+\eta^2+\zeta^2)=\xi d\xi+\eta d\eta +\zeta d\zeta=0.$$ In short, the round sphere requires that two quadratic forms $\xi^2+\eta^2+\zeta^2$ and $c^2\tau^2$ be simultaneously constant.

This leads us to Einstein's second error, which is the failure to observe that the Lorentz invariance of the quadratic form $h=x^2+y^2+z^2-c^2t^2$ in no way implies the Lorentz invariance of $h_1:=x^2+y^2+z^2$ and $h_2:=c^2 t^2$.

Indeed the quadratic forms $h_1, h_2$ are degenerate, with nontrivial radicals $rad(h_1)=\{x=y=z=0\}$ and $rad(h_2)=\{t=0\}.$ The radicals are linear subspaces of ${\bf{R}}^4$. But if $h_1, h_2$ are invariant, then $rad(h_1)$ and $rad(h_2)$ are also nontrivial invariant subspaces. This contradicts the fact that the standard linear representation of the Lorentz group acts irreducibly on ${\bf{R}}^4$.

To summarize the two errors in Einstein's famous argument:

  • the cone $\xi^2 + \eta^2+\zeta^2 = c^2 \tau^2$ is misidentified as the equation of a sphere.

  • the Lorentz invariance of $\xi^2 + \eta^2+\zeta^2 = c^2 \tau^2$ does not imply the Lorentz invariance of the forms $\xi^2 + \eta^2+\zeta^2$ and $c^2 \tau^2$.

In otherwords, the numerical equality between $\xi^2 + \eta^2+\zeta^2$ and $c^2\tau^2$ is always maintained, but the actual values attained by the expressions is not Lorentz invariant.

We are repeating ourselves because this is somewhat subtle point which is easily overlooked.

Elementary Computations in ${\bf{R}}^{1,1}$

To illustrate the above ideas, it's convenient to examine some elementary computations in two variables. Thus we consider ${\bf{R}^2}$ with $(x,t)$ and $(\xi, \tau)$ coordinates.

For numerical convenience we set $c:=1$.

Thus $h=x^2-t^2$ is a quadratic form on ${\bf{R}}^2$ invariant with respect to the one-dimensional Lorentz group $G=SO(1,1)_0$ generated by $$a_\theta:=\begin{pmatrix} \cosh \theta & \sinh \theta \\ \sinh \theta & \cosh \theta \end{pmatrix}$$ for $\theta\in {\bf{R}}$.

In two dimensions the null cone $$N=\{x^2-t^2=0\}$$ projectivizes to a $0$-dimensional sphere consisting of two projective points represented by the affine lines $x-t=0$ and $x+t=0$.

The round $0$-dimensional sphere $\{x^2=1\}$ consists of two vectors in the null cone, namely $$\begin{pmatrix} 1 \\ 1\end{pmatrix}, ~~~\begin{pmatrix} -1 \\ 1\end{pmatrix}.$$

Left translating these vectors by $a_\theta$ we find the translates $$\begin{pmatrix} \xi \\ \tau \end{pmatrix}=\begin{pmatrix} \cosh \theta+\sinh \theta \\ \cosh \theta+\sinh \theta \end{pmatrix}~~\text{and} \begin{pmatrix} -\cosh \theta+\sinh \theta \\ \cosh \theta-\sinh \theta \end{pmatrix}.$$

But evidently $$\xi^2 \neq x^2=1^2=1$$ and $$\tau^2 \neq t^2=1$$ when $\theta\neq 0$. Thus the quadratic forms $h_1=x^2$ and $h_2=t^2$ are not $a_\theta$-invariant. Likewise we find the image of the unit sphere $x^2=1$ does not correspond to a spatial sphere in $(\xi, \tau)$ coordinates.

These trivial computations have the effect of falsifying the alleged Lorentz invariance of spherical lightwaves. However the ``slope" of $\begin{pmatrix} 1 \\ 1\end{pmatrix}$ and $$a_\theta.\begin{pmatrix} 1 \\ 1\end{pmatrix}=\begin{pmatrix} \cosh \theta+\sinh \theta \\ \cosh \theta+\sinh \theta \end{pmatrix}$$ is identically equal to $c=1$ in accordance with (A2).

To further illustrate earlier comments on the nonexistence of canonical parameters on $N$, consider that for any $C^1$ monotonic function $$ f: {\bf{R}} \to {\bf{R}} $$ we obtain a curve $$\epsilon(t)=\epsilon_f(t)=\begin{pmatrix} f(t) \\ f(t) \end{pmatrix}=f(t) \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$ for $t\in {\bf{R}}$.

Then $\epsilon_f(t)$ is supported on a straight line in $N$, and the "photon" $\epsilon_f$ can be said to propagate in a straight line with constant speed $c=1$.

It might be said that $\epsilon_f$ is not uniform in the parameter $t$, but we argue that there exists no Lorentz invariant definition of "uniform parameter" on the null cone.

This is related to our discussion in Part 1 on Levi-Civita's attempt to derive a geodesic equation for light propagation, i.e. the hypothesis that light travels with zero acceleration. We reasoned that this is an arbitrary additional hypothesis, and not a consequence of the formal assumptions (A12).

In the remaining Parts 3/4 and 4/4 of this article, we consider the homogeneous wave equations and Lorentz transformations. As we have previously indicated, we find (A12) incompatible with both the wave and particle models of light.