Math 2.0

Relativity and gravitational theory Special relativity: kinematics Problem

Ehrenfest paradox

Builds on Minkowski spaceLorentz factorBorn rigidityHerglotz–Noether theorem

Problem

A disc of radius RR is first at rest in an inertial frame. It is then brought into rotation about its axis with constant angular speed ω\omega, where ωR<c\omega R < c. Each element of the rim moves along its own length with speed ωR\omega R. So it should be contracted by the factor 1/γ1/\gamma, where γ=1/1−ω2R2/c2\gamma = 1/\sqrt{1 - \omega^2 R^2/c^2} is the Lorentz factor. Each radius moves perpendicular to its own length, so it should not be contracted. How can a rigid disc, whose radius stays RR while its circumference shrinks, be set into rotation? What is the geometry of the rotating disc?

Remarks

A Born rigid body at rest cannot be brought into rotation; this follows from the Herglotz–Noether theorem. So the disc must deform while it is spun up; a real disc develops stresses. Once it rotates uniformly, it can move in a Born rigid way. Then rulers at rest on the rim measure the circumference 2πRγ2\pi R \gamma, while rulers along a radius measure RR. The ratio of measured circumference to measured diameter is πγ>π\pi \gamma > \pi, so the space of the rotating observers is not Euclidean. Einstein took this observation as a hint that gravity, viewed through acceleration, goes with non-Euclidean geometry. Paul Ehrenfest posed the problem in 1909.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Born rigidity, concept, no evidence levelBorn rigidityconceptFour-velocity, concept, no evidence levelFour-velocityconceptKilling vector field, concept, no evidence levelKilling vector fieldconceptLorentz factor, concept, no evidence levelLorentz factorconceptMinkowski space, concept, no evidence levelMinkowski spaceconceptHerglotz–Noether theorem, statement, no evidence levelHerglotz–Noether theo…statementEhrenfest paradox, problem, no evidence levelEhrenfest paradoxproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Born rigidity concept
  2. Four-velocity concept
  3. Killing vector field concept
  4. Lorentz factor concept
  5. Minkowski space concept
  6. Herglotz–Noether theorem statement
  7. Ehrenfest paradox problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Ehrenfest_paradox
About this page
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AboutBorn rigidity
UsesMinkowski space, Lorentz factor, Born rigidity, Herglotz–Noether theorem
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Changed3 November 2026

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