Einsteins Uniformly Rotating Disk and the Hole Argument

Einstein’s first mention of the uniformly rotating disk in print was in 1912, in his paper dealing with the static gravitational fields. After the 1912 paper, the rotating disk problem occurred in Einstein’s writings only in a 1916 review paper, “The Foundation of the General Theory of Relativity”. Einstein did not mention the rotating disk problem in any of his papers on gravitation theory from 1912 until 1916. However, between 1912 and 1914 Einstein invoked the Hole Argument. I discuss the possible connection between the 1912 rotating disk problem and the Hole Argument and the connection between the 1916 rotating disk problem and the Point Coincident Argument. Finally, according to Mach’s ideas we see that the possibility of an empty hole is unacceptable. In 1916 Einstein replaced the Hole Argument with the Point Coincidence Argument and later in 1918 with Mach’s principle.

💡 Research Summary

The paper investigates the historical and conceptual links between Einstein’s rotating‑disk thought experiment and two of his most famous philosophical arguments – the Hole Argument and the later Point‑Coincidence Argument – while also bringing Mach’s principle into the discussion. The author begins by noting that Einstein first mentioned a uniformly rotating disk in print in 1912, within a paper on static gravitational fields. In that early work the disk served as a concrete illustration of how measurements of length and time differ in a non‑inertial (rotating) frame compared to an inertial one. By calculating the circumference of the rotating disk and showing that it is larger than (2\pi) times the radius, Einstein demonstrated that the geometry perceived by observers depends on their state of motion, thereby hinting at the relativity of coordinate choices. Although the Hole Argument had not yet been formulated, the 1912 discussion already raises the question of whether physical quantities are tied to the coordinates we use or to invariant relationships among events.

The second part of the analysis focuses on the period 1914‑1916, when Einstein explicitly introduced the Hole Argument. The argument exploits the freedom to perform arbitrary diffeomorphisms inside a matter‑free region (the “hole”) while keeping the metric fixed outside. Because the Einstein field equations are generally covariant, such a transformation yields a mathematically distinct solution that nevertheless agrees with all observable data outside the hole. This seemed to threaten determinism: the same external conditions could correspond to multiple interior geometries. The paper argues that the rotating‑disk example anticipates this dilemma. The discrepancy between the measured circumference in the rotating frame and the Euclidean expectation is a manifestation of coordinate freedom; the physical observable (the ratio of circumference to radius) remains invariant, just as the Hole Argument stresses that only coordinate‑independent coincidences are physically meaningful.

In 1916 Einstein published the review “The Foundation of the General Theory of Relativity,” where he replaced the Hole Argument with the Point‑Coincidence Argument. The new argument asserts that the only physically real entities are spacetime events themselves – the coincidences of world‑lines – and that any coordinate system is merely a bookkeeping device. The rotating‑disk problem is re‑examined in this light: the measurement of the disk’s radius and circumference can be reduced to asking whether two specific events (the meeting of a light signal with a marked point on the rim and the same point on the radius) coincide. Because such coincidences are invariant under any diffeomorphism, the apparent paradox of multiple interior solutions disappears. The paper shows that the 1916 treatment of the disk fits neatly into the Point‑Coincidence framework, emphasizing that geometry is defined by the network of event coincidences rather than by coordinate labels.

Finally, the author brings Ernst Mach’s relational view of inertia into the discussion. Mach argued that an empty region of space (“a hole”) cannot be physically meaningful because all motion must be defined relative to other masses. From this perspective, the Hole Argument’s reliance on a matter‑free region is philosophically untenable. Einstein, increasingly influenced by Mach’s ideas, eventually abandoned the Hole Argument altogether. By 1918 he had incorporated Mach’s principle into his thinking, using it to argue that the metric field must be determined by the distribution of matter and that empty holes are not admissible solutions. The paper concludes that the evolution from the 1912 rotating‑disk illustration, through the Hole Argument, to the Point‑Coincidence Argument and finally to Mach’s principle reflects a coherent trajectory in Einstein’s thought: a move from concrete physical examples to abstract philosophical principles, all aimed at securing a generally covariant, deterministic, and relational theory of gravitation.