Translation:On the Theory of Relativity (Kaluza)

by Theodor Kaluza

On the Theory of Relativity.

Th. Kaluza (Königsberg/Pr.).

Not read by the author himself due to illness.

Regarding the peculiar position attained by the rotation of a rigid body (in the sense of Born) in the theory of relativity, it seemed to be interesting to investigate two closely related questions, on one side the question concerning the geometry to be established upon a body moving in this way, and then the question concerning the laws of time-comparison which hold at this place.

In accordance with the relativity principle, in general one has to see the geometry of the relevant orthogonal-intersection of the worldline-bundle as the "proper geometry" (at a certain proper time) of a body moving in any way; furthermore one can admit two space-time points as "simultaneous", when they are located upon the same orthogonal-intersection. In the case of rigid translation, these orthogonal-spaces are linear; the proper geometry is Euclidean throughout. It is essentially different in the simplest special case of rigid motion (see Herglotz, Noether), i.e. in the case of rigid rotation. If one remains in $R_{3}$ for simplicities sake (a circular disc rotating in its plane around the center, angular velocity = 1), then one has a bundle of coaxial helical lines of same pitch ("helical bundle"). The helical bundle is not straightly intersected [anorthotom]; consequently we are at first not speaking about a proper geometry in the previous sense, nor about a "rest shape" of the rotating disc. One can only speak about a constantly Euclidean "individual geometry" and a "individual shape", related to a certain point of the disc (see the theory of linear complexes). For example, if one has three points 1, 2, 3, and if one draws the connecting line 23 in the individual geometry of 1, then 23 in general doesn't appear as a line in the geometries of 2 and 3.

However, it is possible to formulate the general geometry, which shortly shall also be called proper geometry. For that, one has to state the following requirement: If one considers any point P of the disc, then the lines of the proper geometry which pass through P, shall also be lines of the individual geometry of P. The proper geometry is formed in the helical bundle as follows: the individual helical lines are the points, and the lines are one-dimensional manifolds of helical lines which are orthogonally intersected by one and the same line. The uniqueness postulate of the line requires special attention at this place. If one intersects (perpendicular to the axis) the helical bundle with the plane (individual geometry of the center), then the lines of the proper geometry represent themselves in this plane as spirals of type

$\varphi -\varphi _{0}=\arccos {\frac {r_{0}}{r}}\pm r_{0}{\sqrt {r^{2}-r_{0}^{2}}}$

($r$ and $\varphi$ are polar coordinates; depending on the reality relations, the upper or lower sign holds).

The task, to lay a "proper line" (i.e. one of the previous spirals) through two points of the plane, leads to the equation for $\Psi$

$\Phi =\Psi +R\sin \Psi ,$

which is known from astronomy as "Kepler's equation". The question concerning the uniqueness of the solutions is to be affirmed for $R<1$ (simple geometric interpretation). Since the radius of a rotating disc is not allowed to exceed a certain finite limit due to the confinement to subluminal velocity, one can conclude that the uniqueness postulate of the line is indeed satisfied (which were not generally the case when superluminal velocities are admitted). A closer examination then shows the proper geometry of the rotating disc as a non-Euclidean, specifically Lobachevskian geometry.

As arc-length one has to define:

$\int {\sqrt {1+{\frac {r^{2}}{1\pm r^{2}}}\left({\frac {d\varphi }{dr}}\right)^{2}dr}}$ ("proper arc"),

as content

$\int {\frac {r^{2}}{\sqrt {1\pm r^{2}}}}d\varphi$ ("proper content")

The second of the questions posed at the beginning, gives rise to the following considerations:

If a time-comparison is done between two disc points, by comparing the two points directly at one occasion, and at another occasion by inclusion of a intermediary point located outside of the connecting line of both points, then the results of both time-comparisons will mutually deviate: Upon the rotating disc, the time comparison appears to be as depending on the path. If a point is "differentially" compared with itself along a closed curve, then the point indicates a time-difference with respect to itself ("error of closure"). A more detailed investigation gives the magnitude of the error of closure (measured in proper time) as equal to the double proper-content of the time-comparison curve. From the existence of the error of closure, the theoretical possibility of a demonstration of Earth's rotation by purely optical or electromagnetic experiments is given. (No contradiction against the theory of relativity.) Yet the idea can probably not realized practically; since it is, in the best case, only about $2\cdot 10^{-7}$ sec.

(Received September 23, 1910.)