Notes

¹This relation must hold for an arbitrary choice of the origin and of the direction (ratios $\Delta x_1:\Delta x_2:\Delta x_3$) of the interval.

²In reality there are equations.

³There are thus two kinds of Cartesian systems which are designated as “right-handed” and “left-handed” systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them.

⁴The equation ${a^{\prime }}_{\sigma \tau }{{\xi }^{\prime }}_{\sigma }{{\xi }^{\prime }}_{\tau }=1$ may, by (8), be replaced by ${a^{\prime }}_{\sigma \tau }{b}_{\mu \sigma }{b}_{\nu \tau }{\xi }_{\mu }{\xi }_{\nu }=1$, from which the result stated immediately follows.

⁵The laws of physics could be expressed, even in case there were a unique direction in space, in such a way as to be co-variant with respect to the transformation (4); but such an expression would in this case be unsuitable. If there were a unique direction in space it would simplify the description of natural phenomena to orient the system of co-ordinates in a definite way in this direction. But if, on the other hand, there is no unique direction in space it is not logical to formulate the laws of nature in such a way as to conceal the equivalence of systems of co-ordinates that are oriented differently. We shall meet with this point of view again in the theories of special and general relativity.

⁶These considerations will make the reader familiar with tensor operations without the special difficulties of the four-dimensional treatment; corresponding considerations in the theory of special relativity (Minkowski’s interpretation of the field) will then offer fewer difficulties.

⁷Strictly speaking, it would be more correct to define simultaneity first, somewhat as follows: two events taking place at the points $A$ and $B$ of the system $K$ are simultaneous if they appear at the same instant when observed from the middle point, $M$, of the interval $AB$. Time is then defined as the ensemble of the indications of similar clocks, at rest relatively to $K$, which register the same simultaneously.

⁸That this specialization lies in the nature of the case will be evident later.

⁹That material velocities exceeding that of light are not possible, follows from the appearance of the radical $\sqrt{1-v^2}$ in the special Lorentz transformation (40).

¹⁰In order to avoid confusion from now on we shall use the three-dimensional space indices, $x$, $y$, $z$ instead of $1$, $2$, $3$, and we shall reserve the numeral indices $1$, $2$, $3$, $4$ for the four-dimensional space-time continuum.

¹¹The emission of energy in radioactive processes is evidently connected with the fact that the atomic weights are not integers. Attempts have been made to draw conclusions from this concerning the structure and stability of the atomic nuclei.

¹²To be summed for the indices $\alpha$ and $\beta$.

¹³It has been attempted to remedy this lack of knowledge by considering the charged particles as proper singularities. But in my opinion this means giving up a real understanding of the structure of matter. It seems to me much better to give in to our present inability rather than to be satisfied by a solution that is only apparent.

¹⁴These considerations assume that the behaviour of rods and clocks depends only upon velocities, and not upon accelerations, or, at least, that the influence of acceleration does not counteract that of velocity.

¹⁵If we multiply (80) by $\frac{{\partial x^{\prime }}_{\alpha }}{\partial x_{\beta }}$, sum over the $\beta$, and replace the $d{\xi }^{\mu }$ by a transformation to the accented system, we obtain

The statement made above follows from this, since, by (80), we must also have ${d{x}^{\prime }}_{\alpha }=g^{\sigma {\alpha }^{\prime }}{d{\xi }^{\prime }}_{\alpha }$, and both equations must hold for every choice of the ${d{\xi }^{\prime }}_{\sigma }$.

¹⁶This expression is justified, in that $A^{\mu }\sqrt{g}dx={𝔄}^{\mu }dx$ has a tensor character. Every tensor, when multiplied by $\sqrt{g}$, changes into a tensor density. We employ capital Gothic letters for tensor densities.

¹⁷The direction vector at a neighbouring point of the curve results, by a parallel displacement along the line element , from the direction vector of each point considered.

¹⁸That the centrifugal action must be inseparably connected with the existence of the Coriolis field may be recognized, even without calculation, in the special case of a co-ordinate system rotating uniformly relatively to an inertial system; our general co-variant equations naturally must apply to such a case.

¹⁹The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.

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