Abstract
We present an apparent paradox within the special theory of relativity, involving a trolley with relativistic velocity and its rolling wheels. Two solutions are given, both making clear the physical reality of the Lorentz contraction, and that the distance on the rails between each time a specific point on the rim touches the rail is not equal to 2πR, where R is the radius of the wheel, but 2πR/1−R2Ω2/c2, where Ω is the angular velocity of the wheels. In one solution, the wheel radius is constant as the velocity of the trolley increases, and in the other the wheels contract in the radial direction. We also explain two surprising facts. First that the shape of a rolling wheel is elliptical in spite of the fact that the upper part of the wheel moves faster than the lower part, and thus is more Lorentz contracted, and second that a Lorentz contracted wheel with relativistic velocity rolls out a larger distance between two successive touches of a point of the wheel on the rails than the length of a circle with the same radius as the wheels.
Highlights
The efforts to understand the consequences of the special theory of relativity as applied to rotating systems have a long history.1 Such efforts have been useful, as analyzing seemingly paradoxical situations usually leads to a more detailed understanding of relativistic kinematics.2–13We here present a new special relativistic “paradox.” The situation involves a combination of translational and rotating motion—the translational motion of a trolley and the rotational motion of its wheels
We present an apparent paradox within the special theory of relativity, involving a trolley with relativistic velocity and its rolling wheels
Both making clear the physical reality of the Lorentz contraction, and that the distance on the rails between each time a specific point on thqe ffirffiffiiffimffiffiffiffiffiffiffitffioffiffiuffiffifficffiffihffiffieffiffisffiffi the rail is not equal to 2pR, where R is the radius of the wheel, but 2pR= 1 À R2X2=c2, where X is the angular velocity of the wheels
Summary
The efforts to understand the consequences of the special theory of relativity as applied to rotating systems have a long history. Such efforts have been useful, as analyzing seemingly paradoxical situations usually leads to a more detailed understanding of relativistic kinematics.. The efforts to understand the consequences of the special theory of relativity as applied to rotating systems have a long history.1 Such efforts have been useful, as analyzing seemingly paradoxical situations usually leads to a more detailed understanding of relativistic kinematics.. We here present a new special relativistic “paradox.” The situation involves a combination of translational and rotating motion—the translational motion of a trolley and the rotational motion of its wheels. This example makes clear the importance of taking the relativity of simultaneity into account when predicting the behavior of objects moving relativistically. It sheds light on the physical reality of the Lorentz contraction and provides a method for measuring the increase of the rest length of the circumference of a wheel with constant radius and increasing rotational motion
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