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RELATIVISTIC ROTATIONAL ANGULAR MOMENTUM
Dr. Raymond HV Gallucci, PE (ret)
Abstract. This is an exercise without a conclusion, intended to examine a phenomenon that, in the
relativistic world, seems most difficult to comprehend, let alone explain – relativistic rotational angular
momentum. Many have tackled the problem, first cited by Ehrenfest in 1909, and known since then as the
“Ehrenfest Paradox,” primarily in terms of analyzing effects on the size and geometry of a relativistically
rotating disk. Length contraction and time dilation interplay with aspects of general relativity and
hyperbolic geometry according to the experts. Tackling this aspect is beyond me, so I try to examine a
somewhat more predictable phenomenon, relativistic angular momentum as a result of relativistic mass
increase in both a rotating rod and disk. No conclusions are drawn, but observations are offered as food for
thought for the reader.
1. INTRODUCTION
Rotational motion has always posed intriguing
phenomena that sometime seem to defy the laws of
physics. For example, consider the stability of a
bicycle in motion or a spinning gyroscope. Or the
underwater rotating cylinders of Ionel Dinu that
seem to mimic the behavior of bar magnets,
whereby cylinders rotating in the same direction
(clockwise-clockwise or counterclockwise-
counterclockwise) will “repel” each other, while
those rotating in opposite directions (clockwise-
counterclockwise) will “attract.” In fact, if left to
spin on their own in either direction, they will align
just as the north and south poles of a pair of bar
magnets will.
(https://sciencewoke.org/scientist/ionel-dinu/) Or
Eric Laithwaite’s famous 1983 video
demonstration of how a spinning disk at the end of
rod, too heavy to easily lift when stationary,
becomes surprisingly “lightweight” when set to
rotating rapidly.
(https://www.youtube.com/watch?v=JRPC7a_Ac
Qo) Standard explanations can be complex and
seemingly contrary to common sense, but are
relatively straightforward when compared to
explaining disk rotational behavior in the
relativistic realm, the subject of what is known as
the “Ehrenfest Paradox.”
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1 “Born rigidity is satisfied … if the length of the rigid
body in momentary co-moving inertial frames measured
by standard measuring rods (i.e., the proper length) is
constant and is therefore subjected to Lorentz
contraction in relatively moving frames. Born rigidity is
a constraint on the motion of an extended body, achieved
2. THE EHRENFEST PARADOX
“The Ehrenfest paradox [see Figure 1] concerns
the rotation of a ‘rigid’ disc in the theory of
relativity. … as presented by Paul Ehrenfest [in]
1909 in relation to the concept of Born
rigidity within special relativity1 … [I]t discusses
an ideally rigid cylinder that is made to rotate about
its axis of symmetry. The radius R as seen in the
laboratory frame is always perpendicular to its
motion and should therefore be equal to its value
R0 when stationary. However, the circumference
(2πR) should appear Lorentz-contracted to a
smaller value than at rest, by the usual factor γ. This
leads to the contradiction that R = R0 and R < R0.
“The paradox has been deepened further
by Albert Einstein, who showed that … the
circumference … would … measure greater than
2πR [contrary to Ehrenfest]. This indicates that
geometry is non-Euclidean for rotating observers,
and was important for Einstein's development
of general relativity. Any rigid object made from
real materials that is rotating with a
transverse velocity close to the speed of sound in
the material must exceed the point of rupture due
to centrifugal force, because centrifugal
pressure cannot exceed the shear modulus of
material ... Therefore, when considering velocities
close to the speed of light, it is only a thought
by careful application of forces to different parts of the
body. A body rigid in itself would violate special
relativity, as its speed of sound would be infinite.”
(https://en.wikipedia.org/wiki/Born_rigidity)
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