On the Fundamental Theorem of the Theory of Relativity

Abstract

A new formulation of what may be called the “fundamental theorem of the theory of relativity” is presented and proved in (3 + 1)-space-time, based on the full classification of special transformations and the corresponding velocity addition laws. A system of axioms is introduced and discussed leading to the result, and a study is made of several variants of that system. In particular the status of the group axiom is investigated with respect to the condition of the two-way isotropy of light. Several issues which are ignored or misunderstood in the literature are emphasized.

Appendix: Proof of Theorem 3.3

It is easy to verify that if $\phi , \phi '$ are affinely equivalent cs’s, then they are also inertially equivalent.

Let us consider the converse. Suppose that every uniform worldline for $\phi$ is also uniform for $\phi '$. Since points at rest for $\phi$ are physical worldlines (Axiom 3), we have for every $\mathbf{r}_0 \in \mathbb R^3$:

$$\begin{aligned} \frac{\partial \mathbf{r}'}{\partial t}(\mathbf{r}_0 , t)=\frac{\partial t'}{\partial t} (\mathbf{r}_0, t)\mathbf{W}\end{aligned}$$

(61)

with $\mathbf{W}$ depending at most on $\mathbf{r}_0$. It follows that the transition function from $\phi$ to $\phi '$ can be re-written as

$$\begin{aligned} \left\{ \begin{array}{rcl} \mathbf{r}' &{}=&{} t' (\mathbf{r}, t)\mathbf{W}(\mathbf{r}) + \mathbf{U}(\mathbf{r}) \\ t' &{}=&{} t' (\mathbf{r}, t). \end{array}\right. \end{aligned}$$

where $\mathbf{U}: \mathbb R^3 \rightarrow \mathbb R^3$ is a suitable differentiable function.

Claim 1: $\mathbf{W}$ is constant.

In fact let $\mathbf{r}(t) = \mathbf{r}_0 + (t-t_0)\mathbf{v}$ be any uniform motion for $\phi$ with $\mathbf{v}$ physical and $\mathbb R$ as its domain. If we put $f(t):= t' (\mathbf{r}_0 + (t-t_0 )\mathbf{v}, t)$ it is easy to see that f is a diffeomorphism of $\mathbb R$ onto itself. (Let $\mathbf{v}'$ the constant velocity of the same worldline according to $\phi '$, and suppose that the interval $f(\mathbb R)$ had, say, a finite supremum b; then, if $\phi '\circ \phi ^{-1} (\mathbf{r}_0, t_0) = (\mathbf{r}'_0, t'_0)$ the subset

$$\begin{aligned} \phi \circ \phi '^{-1}\left( \{(\mathbf{r}'_0 + (t'- t'_0)\mathbf{v}')\; : \, t'\in [t'_0, b]\}\right) \end{aligned}$$

would have to be noncompact, which is absurd.) Now

$$\begin{aligned} \mathbf{v}' = \mathbf{W}+ \frac{1}{\dot{f}}(f(\mathbf{v}\cdot \nabla )\mathbf{W}+ (\mathbf{v}\cdot \nabla )\mathbf{U}) \end{aligned}$$

(62)

must not depend on t. Putting $t=t_0$ we get that there is a function $\mathbf{v}' = \mathbf{K}(\mathbf{r}_0, \mathbf{v})$ such that $\mathbf{K}(\mathbf{r}(t), \mathbf{v}) \equiv \mathbf{K}(\mathbf{r}_0, \mathbf{v})$, and

$$\begin{aligned} \dot{f}(t_0)\ (\mathbf{K}(\mathbf{r}_0, \mathbf{v}) -\mathbf{W}(\mathbf{r}_0)) - f(t_0) ((\mathbf{v}\cdot \nabla )\mathbf{W})_{\mathbf{r}_0} = ((\mathbf{v}\cdot \nabla )\mathbf{U})_{\mathbf{r}_0}. \end{aligned}$$

For every fixed $\mathbf{r}_0$ and physical velocity $\mathbf{v}$ this is an ordinary differential equation in f(t) (after notation change from $t_0$ to t) with vector coefficients:

$$\begin{aligned} \dot{f}\mathbf{A}+f \mathbf{B}= \mathbf{C}, \; \text{ where }\; \mathbf{A}:= \mathbf{K}- \mathbf{W}, \; \mathbf{B}:= -(\mathbf{v}\cdot \nabla )\mathbf{W}, \; \mathbf{C}:= (\mathbf{v}\cdot \nabla )\mathbf{U}. \end{aligned}$$

Now, if for all $\mathbf{r}_0$ we have $\mathbf{B}= {\mathbf 0}$ when we choose $\mathbf{v}$ in three linearly independent directions, then $\nabla \mathbf{W}=0$ and therefore $\mathbf{W}$ is constant. But for $\mathbf{B}$ there is no other possible value. Suppose by contradiction, that $\mathbf{B}\ne {\mathbf 0}$ for some $\mathbf{r}_0$ and $\mathbf{v}$. Then f must be a solution of a (scalar) differential equation with constant coefficients of the form $a\dot{f}+ bf = c$ with $b\ne 0$. Now $a\ne 0$, since otherwise f would be constant; it follows that f is of the type $f(t) = ke^{-bt/a} + c/b$, which clearly is not onto $\mathbb R$, no matter what the values of a, b, c are. So claim 1 is proven.

It follows that $\mathbf{r}' (\mathbf{r},t) = t'(\mathbf{r},t) \mathbf{W}+ \mathbf{U}(\mathbf{r})$ with $\mathbf{W}$ constant. Note that $\mathbf{U}(\mathbf{r})$ cannot be constant, otherwise

$$\begin{aligned} \det \left( \frac{\partial \mathbf{r}'}{\partial \mathbf{r}}\right) = \det \left( \mathbf{W}(\nabla t')^T\right) = 0, \end{aligned}$$

since every $3\times 3$ matrix of the form $\mathbf{a}\mathbf{b}^T$ is singular, which would contradict Axiom 5.

Thus (62) becomes:

$$\begin{aligned} \mathbf{K}(\mathbf{r}_0, \mathbf{v}) = \mathbf{W}+ \frac{1}{\dot{f}}(\mathbf{v}\cdot \nabla )\mathbf{U}. \end{aligned}$$

(63)

By comparing the functional dependence of the various terms of this equation we get that $\dot{f}$ must be independent of t ; since

$$\begin{aligned} \dot{f}= (\nabla t')(\mathbf{r}_0 , t)\cdot \mathbf{v}+ \frac{\partial t'}{\partial t} (\mathbf{r}_0 , t) \end{aligned}$$

this means that neither $\displaystyle \frac{\partial t'}{\partial t}$ (put $\mathbf{v}={\mathbf 0}$), nor $\nabla t'$ may depend on t. It follows that $t' = g(\mathbf{r}) + \alpha t$ for some $\alpha >0$ and $g: \mathbb R^3 \rightarrow \mathbb R$ differentiable.

Claim 2: Both functions $\mathbf{U}$ and g are affine.

Equation (63) can be re-written as

$$\begin{aligned} \mathbf{K}(\mathbf{r}_0 , \mathbf{v}) = \mathbf{W}+ (\nabla g(\mathbf{r}_0)\cdot \mathbf{v}+ \alpha )^{-1}((\mathbf{v}\cdot \nabla )\mathbf{U}) (\mathbf{r}_0). \end{aligned}$$

By substituting $\mathbf{r}_0$ with $\mathbf{r}(t)$ and differentiating with respect to t, we obtain zero; if we neglect the irrelevant denominator and evaluate for $t=t_0$ we get:

$$\begin{aligned} {\mathbf 0} = \sum _{\alpha ,\beta =1}^3 \left( (\nabla g(\mathbf{r}_0)\cdot \mathbf{v}+\alpha ) \frac{\partial ^2\mathbf{U}}{\partial x^{\alpha }\partial x^{\beta }}(\mathbf{r}_0) -\frac{\partial ^2 g}{\partial x^{\alpha }\partial x^{\beta }}(\mathbf{r}_0)\frac{\partial \mathbf{U}}{\partial x^{\gamma }} (\mathbf{r}_0)v^\gamma \right) v^{\alpha }v^{\beta }. \end{aligned}$$

By re-arranging the terms we have:

$$\begin{aligned} \alpha \sum _{\alpha ,\beta =1}^3 \frac{\partial ^2\mathbf{U}}{\partial x^\alpha \partial x^\beta }v^\alpha v^\beta =-\sum _{\alpha ,\beta ,\gamma =1}^3 \left( \frac{\partial g}{\partial x^\gamma }\frac{\partial ^2\mathbf{U}}{\partial x^\alpha \partial x^\beta } - \frac{\partial ^2 g}{\partial x^\alpha \partial x^\beta }\frac{\partial \mathbf{U}}{\partial x^\gamma }\right) v^\alpha v^\beta v^\gamma . \end{aligned}$$

Now, for every fixed $\mathbf{r}_0$ at the lefthand side there is a quadratic vector polynomial in the $v^\alpha$, while at the righthand side there is a cubic polynomial – both homogeneous with respect to $v^\alpha$: this is possible, for every $\mathbf{v}$ in an open neighborhood of ${\mathbf 0}$ (Axiom 8), if and only if both polynomials vanish for every $\mathbf{r}_0$, that is, if all their coefficients are identically zero:

$$\begin{aligned} \frac{\partial ^2 \mathbf{U}}{\partial x^\alpha \partial x^\beta } ={\mathbf 0},\; \frac{\partial ^2 g}{\partial x^\alpha \partial x^\beta }\frac{\partial \mathbf{U}}{\partial x^\gamma } ={\mathbf 0}. \end{aligned}$$

(64)

From the first equality we have that $\mathbf{U}$ is an affine map: $\mathbf{U}(\mathbf{r}) = X\mathbf{r}+ \mathbf{a}$, so we can write

$$\begin{aligned} \left\{ \begin{array}{rcl} \mathbf{r}' &{}=&{} (g(\mathbf{r}) + \alpha t) \mathbf{W}+ X\mathbf{r}+ \mathbf{a}, \\ t' &{}=&{} g(\mathbf{r}) + \alpha t + \ell .\end{array}\right. \end{aligned}$$

But by Axiom 5 it must be $0 <\det \displaystyle \left( \frac{\partial \mathbf{r}'}{\partial \mathbf{r}}\right) = \det (\mathbf{W}(\nabla g)^T + X)$ thus $X\ne 0$. Since, as we have seen, U is nonconstant, at least one of the vectors $\displaystyle \frac{\partial \mathbf{U}}{\partial x^\gamma }$ is nonzero; thus the second equation in (64) implies

$$\begin{aligned} \frac{\partial ^2 g}{\partial x^\alpha \partial x^\beta } \equiv 0, \end{aligned}$$

which means that also g is an affine map, and Claim 2 is proven; this ends also the proof of the theorem.