Space, Time and Coordinates in a Rotating World

The peculiarities of rotating frames of reference played an important role in the genesis of general relativity. Considering them, Einstein became convinced that coordinates have a different status in the general theory of relativity than in the special theory. This line of thinking was confused, however. To clarify the situation we investigate the relation between coordinates and the results of space-time measurements in rotating reference frames. We argue that the difference between rotating systems (or accelerating systems in general) and inertial systems does not lie in a different status of the coordinates (which are conventional in all cases), but rather in different global chronogeometric properties of the various reference frames. In the course of our discussion we comment on a number of related issues, such as the question of whether a consideration of the behavior of rods and clocks is indispensable for the foundation of kinematics, the influence of acceleration on the behavior of measuring devices, the conventionality of simultaneity, and the Ehrenfest paradox.

💡 Research Summary

The paper revisits the long‑standing puzzle of rotating reference frames and asks what truly distinguishes them from inertial frames. The authors argue that the difference does not lie in a special status of the coordinates themselves—coordinates remain conventional labels in any frame—but in the global chronogeometric structure that a rotating (or more generally accelerating) frame endows on space‑time.

First, the authors show that in a rotating frame the standard Einstein synchronization procedure fails globally. If a clock is carried around a closed loop on a rotating platform, the accumulated desynchronization demonstrates that a single, globally consistent time function cannot be defined. This is not a sign that the time coordinate has lost its conventional meaning; rather, the metric of the rotating frame possesses off‑diagonal terms (the “g0i” components) that make the time coordinate non‑integrable over closed paths. Consequently, the global temporal ordering is path‑dependent, a property absent in inertial frames.

Second, the paper critiques the traditional claim that acceleration directly deforms measuring devices such as rods and clocks, thereby altering kinematics. The authors maintain that acceleration changes the space‑time metric in which the devices reside; the devices themselves retain their intrinsic properties, but the values they read are altered because the underlying geometry has changed. Hence, the behavior of rods and clocks is not a prerequisite for defining kinematics—kinematics follows from the metric, not from the dynamical response of specific instruments.

Third, the issue of simultaneity is examined. While local simultaneity surfaces can always be chosen arbitrarily (reflecting the conventionality of simultaneity), a rotating frame lacks a globally consistent simultaneity surface. This demonstrates that the conventionality of simultaneity survives locally but is constrained globally by the geometry of the frame.

The authors then turn to the Ehrenfest paradox, which highlights the apparent contradiction between Lorentz contraction of the rim of a rotating disc and the lack of contraction at its centre. By recognizing that a rotating disc does not admit a globally defined spatial metric compatible with Euclidean geometry, the paradox dissolves: the rim and the centre belong to different congruence classes of world‑lines, and no single global distance measure can be applied across the whole disc.

In the concluding discussion the authors reaffirm that the “different status of coordinates” that Einstein inferred for general relativity was a misinterpretation. In both special and general relativity coordinates are merely a bookkeeping device; the physical content resides in the metric tensor and its global properties. Rotating and accelerating frames illustrate how the same conventional coordinates can describe space‑time with dramatically different global chronogeometric features, without requiring any revision of the conventional role of coordinates. The paper thus clarifies the conceptual foundations of kinematics in non‑inertial frames and reinforces the metric‑centric view of general relativity.