[Paper Review] Relativistic Rotation in the Large Radius, Small Angular Velocity Limit

Robert D. Klauber|ArXiv.org|Sep 9, 2002

Relativity and Gravitational Theory2 references3 citations

TL;DR

This paper challenges the traditional interpretation of relativistic rotation by comparing the local Lorentz frame approach with a non-time-orthogonal (NTO) differential geometry method in the large-radius, small-angular-velocity limit. It argues that the traditional method leads to internal inconsistencies—particularly in claiming absolute Lorentz contraction on a rotating disk—while the NTO approach, which accounts for spacetime non-orthogonality, predicts no contraction and is therefore more physically consistent. The key contribution is the demonstration that the limit case, though seemingly inertial, still exhibits non-trivial geometric structure incompatible with standard special relativity assumptions.

ABSTRACT

Relativistic rotation is considered in the limit of angular velocity approaching zero and radial distance approaching infinity, such that centrifugal acceleration is immeasurably small while tangent velocity remains close to the speed of light. For this case, the predictions of the traditional approach to relativistic rotation using local co-moving Lorentz frames are compared and contrasted with those of the differential geometry based non-time-orthogonal analysis approach. Different predictions by the two approaches imply that only the non-time-orthogonal approach is valid.

Motivation & Objective

• To examine the validity of the traditional local Lorentz frame approach to relativistic rotation in the limit of large radius and small angular velocity.
• To identify internal inconsistencies in the traditional method’s prediction of absolute Lorentz contraction on a rotating disk rim.
• To compare predictions of the traditional approach with those of the non-time-orthogonal (NTO) differential geometry approach in the same limit case.
• To argue that only the NTO approach yields consistent, experimentally plausible results in the limit where centrifugal acceleration is immeasurably small but tangent velocity approaches c.

Proposed method

• The paper analyzes the rotating frame using differential geometry to derive a metric with non-zero off-diagonal spacetime components, indicating non-time-orthogonality.
• It contrasts this NTO metric with the traditional method, which assumes local inertial frames and applies special relativity locally before integrating globally.
• The analysis focuses on how observers in different frames (lab, jet, rotating disk) perceive the length of meter sticks in other frames, using spacetime diagrams to track simultaneous events.
• It employs a thought experiment involving the sudden release of a meter stick from the rotating frame into a moving inertial frame to model a smooth transition from rotating to inertial motion.
• The paper uses event-based spacetime constructions (e.g., events O, N, M, P) to compare 4D spacetime lengths and determine perceived lengths across frames.
• It evaluates the physical consistency of predictions by checking for symmetry and relativity of simultaneity, particularly in the limit where ω → 0 and r → ∞ but v = ωr ≈ c.

Experimental results

Research questions

• RQ1 Does the traditional local Lorentz frame method correctly predict Lorentz contraction on a rotating disk rim in the large-radius, small-angular-velocity limit?
• RQ2 Can the traditional method consistently describe length contraction when both the rotating and lab observers are considered effectively inertial in this limit?
• RQ3 How do the predictions of the traditional method differ from those of the non-time-orthogonal (NTO) differential geometry approach in this limit case?
• RQ4 Is Lorentz contraction in the rotating frame absolute or relative, and does the traditional method correctly account for this?
• RQ5 What physical inconsistencies arise when the traditional method assumes absolute contraction while both observers are in Lorentz frames?

Key findings

• The traditional approach predicts absolute Lorentz contraction of the disk’s meter sticks, implying the disk surface is curved, but this contradicts the fact that the disk observer should not perceive any contraction in her own frame.
• The NTO approach predicts no Lorentz contraction on the rotating disk rim, consistent with the disk observer measuring a flat 2D surface, and resolves the contradiction in the limit case.
• In the limit case, where ω → 0 and r → ∞ but v ≈ c, the rotating frame is not truly inertial due to measurable gravitational potential effects, invalidating the assumption that it behaves like a Lorentz frame.
• The NTO method shows that the lab observer sees the disk meter stick as uncontracted, while the jet observer (moving at v = ωr) sees it as contracted, and the disk observer sees the jet stick as contracted—consistent with relativity of simultaneity.
• The transition of a released meter stick from the rotating frame to inertial motion is smooth in the lab frame, with length decreasing continuously from uncontracted to Lorentz-contracted, supporting the NTO model.
• The paper concludes that the traditional approach is inconsistent due to conflicting predictions about contraction and curvature, while the NTO approach provides a coherent, experimentally plausible description.