Application of the Thomas precession to the deformations of a rotating disk
Within the framework of Special Relativity, the recent paper [1] describes the shrinking determined by the rotation of a planar disk, and an apparent paradox emerging from the anisotropic outcoming contraction. In this paper, using the Thomas precession, we obtain the same result in a different way.
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irk-123456789-1243672017-09-25T03:02:47Z Application of the Thomas precession to the deformations of a rotating disk Celakoska, E.G. Trecevski, K. Balan, V. Within the framework of Special Relativity, the recent paper [1] describes the shrinking determined by the rotation of a planar disk, and an apparent paradox emerging from the anisotropic outcoming contraction. In this paper, using the Thomas precession, we obtain the same result in a different way. 2009 Article Application of the Thomas precession to the deformations of a rotating disk / E.G. Celakoska, K. Trecevski, V. Balan // Український математичний вісник. — 2009. — Т. 6, № 4. — С. 429-435. — Бібліогр.: 9 назв. — англ. 1810-3200 2000 MSC. 83A05, 83B05. http://dspace.nbuv.gov.ua/handle/123456789/124367 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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Within the framework of Special Relativity, the recent paper [1] describes the shrinking determined by the rotation of a planar disk, and an apparent paradox emerging from the anisotropic outcoming contraction. In this paper, using the Thomas precession, we obtain the same result in a different way. |
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Celakoska, E.G. Trecevski, K. Balan, V. |
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Celakoska, E.G. Trecevski, K. Balan, V. Application of the Thomas precession to the deformations of a rotating disk Український математичний вісник |
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Celakoska, E.G. Trecevski, K. Balan, V. |
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Celakoska, E.G. |
| title |
Application of the Thomas precession to the deformations of a rotating disk |
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Application of the Thomas precession to the deformations of a rotating disk |
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Application of the Thomas precession to the deformations of a rotating disk |
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Application of the Thomas precession to the deformations of a rotating disk |
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Application of the Thomas precession to the deformations of a rotating disk |
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application of the thomas precession to the deformations of a rotating disk |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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Application of the Thomas precession to the deformations of a rotating disk / E.G. Celakoska, K. Trecevski, V. Balan // Український математичний вісник. — 2009. — Т. 6, № 4. — С. 429-435. — Бібліогр.: 9 назв. — англ. |
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Український математичний вісник |
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AT celakoskaeg applicationofthethomasprecessiontothedeformationsofarotatingdisk AT trecevskik applicationofthethomasprecessiontothedeformationsofarotatingdisk AT balanv applicationofthethomasprecessiontothedeformationsofarotatingdisk |
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Український математичний вiсник
Том 6 (2009), № 4, 429 – 435
Application of the Thomas precession to the
deformations of a rotating disk
Emilija G. Celakoska, Kostadin Trenčevski,
Vladimir Balan
(Presented by A. E. Shishkov)
Abstract. Within the framework of Special Relativity, the recent
paper [1] describes the shrinking determined by the rotation of a planar
disk, and an apparent paradox emerging from the anisotropic outcoming
contraction. In this paper, using the Thomas precession, we obtain the
same result in a different way.
2000 MSC. 83A05, 83B05.
Key words and phrases. Special Relativity, radial velocity, circum-
ference, Lorentz contraction.
Introduction
This year, 2009, marks the 100th anniversary of the rotating disk
paradox. Namely, the problem known as “rotating disk paradox” origi-
nates from 1909 paper [2] where an ideally rigid cylinder rotating about
its axis of symmetry is considered. It is discussed that the radius R as
seen in the laboratory frame is always perpendicular to its motion and
should be equal to its value R0 when stationary. But the circumference
2πR should appear Lorentz-contracted to a smaller value than at rest,
by the Lorentz factor γ. So, a contradiction arises: R = R0 and R < R0.
Ehrenfest concludes that motions of extended bodies cannot be Born
rigid.
From 1910 many mathematicians and physicists worked on the reso-
lution of this problem, with different approaches. Plank [3] suggests that
elastic contraction as a result of the acquired spin velocity of the disk
should not be confused with the measurement differences between a sta-
tionary and rotating observer. Einstein, evolving from the discussions of
Received 24.12.2008
ISSN 1810 – 3200. c© Iнститут математики НАН України
430 Application of the Thomas precession...
simultaneity [4] to General Relativity [5] introduced the Equivalence Prin-
ciple via rotating disk, apparently noticing the underlying non-Euclidean
geometry.
Much later, Langevin introduces a frame field corresponding to the
family of rotating observers [6]. The concept of moving frames applied to
the rotating disk paradox was improved by Weyssenhoff and Rosen. On
the other side, in [7] is given a pure kinematical explanation the problem.
The history and an overview of the paradox is given in [8] up to that
date, but the relativistically rotating reference frame still continues to be
a research topic of interest. It is widely accepted that this topic should
be described by tools of differential geometry and therefore the most
accepted paradox solution is described by the curved Landau–Lifschitz
metric.
In this paper we show that the coefficient of shrinking from [1] can
be derived in a different way, using the Thomas precession as presented
in [9].
1. Rotating disk and the shrinking coefficient
Consider the disk C = {(x, y) | x2 + y2 = R2} of radius R in the xOy
Euclidean plane. We assume that the disk rotates around the origin in
trigonometric sense, with constant angular velocity ω. In this section, we
remind of the recent results [1]. In [1] is determined the maximal radius ρ
of the moving disk, observed from the inertial system in which the center
of the disk rests. The velocity of the periphery points of C is ωρ, and its
periphery of length 2πρ is observed under contraction with the coefficient
√
1 − (ωρ
c )2. Hence we obtain the equality
2πR =
2πρ
√
1 − (ωρ/c)2
,
which implies the dependence R = ρ (1 − (ωρ/c)2)−1/2,
ρ =
R
√
1 + (ωR/c)2
. (1.1)
We notice that ρ < c
ω = ρmax (see Fig. 1), hence the boundary radius
c/ω can never be achieved. Note also that the dependence (1.1) is true
not only for the points of the periphery of the disk, but also for the points
which are interior to the disk.
The basic idea of this paper is to deduce the previous equality (1.1)
(up to c−2), in a different way, as a consequence of the Thomas precession.
E. G. Celakoska, K. Trenčevski, V. Balan 431
ñ
O
Figure 1: The disk during rotation
This will be made in Section 2. Further we shall present the consequences
of (1.1), omitting the technical details.
From Special relativity (SR), it is well known that we observe a con-
traction of lengths only in the direction of motion, but not orthogonal
to the motion. Since the radial direction is orthogonal to the direction
of motion, we have apparently a paradox: we still observe a contraction
in the radial direction. To explain this paradox we choose two infinitesi-
mally close points A and B in the radial direction (before rotation) and
examine their places A′ and B′ at a chosen moment during the rotation.
If O, A′ and B′ are collinear, then we have a paradox in SR (the rotation
is transversal to OB, hence it is supposed to affect differently A and B).
So these points should be non-collinear (Fig. 2).
Assume that the coordinates of A′ in the xOy plane are (x, y) and
the coordinates of B′ are (x + ∆x, y + ∆y). Let C ′ be a point on OB′,
such that the angle A′C ′B′ is right. Since A′ and B′ are infinitesimally
close, without loss of generality we assume OA′ ∼= OC ′. Let us denote
ρ = d(O, A′) =
√
x2 + y2. Using elementary geometrical calculation, it
is easy to find that
|A′C ′| = |x∆y − y∆x|/ρ, |B′C ′| = |x∆x + y∆y|/ρ. (1.2)
Since the distance |A′C ′| is observed under contraction with factor
√
1 − (ωρ/c)2, and there is no contraction in the direction B′C ′, we ob-
tain the following equality
|AB|2 = |B′C ′|2 +
|A′C ′|2
1 − (ωρ/c)2
. (1.3)
432 Application of the Thomas precession...
´´
´
Figure 2: The disk during rotation
On the other side,
y − xy′
x + yy′
=
±
√
3ω2ρ2
c2
− 3ω4ρ4
c4
+ ω6ρ6
c6
1 − (ωρ/c)2
, (1.4)
where y′ = lim∆x→0
∆y
∆x , i.e. y′ is the usual derivative of y = y(x). In
polar coordinates (ρ, φ), we have
dφ
dρ
=
y′x − y
ρ(x + yy′)
,
where φ = atan y
x . Now,
ρ
dφ
dρ
=
±
√
3ω2ρ2
c2
− 3ω4ρ4
c4
+ ω6ρ6
c6
1 − (ωρ/c)2
. (1.5)
Hence for the function φ(ρ) we get
φ(ρ) = φ0 −
ρ
∫
0
√
3ω2
c2
− 3ω4z2
c4
+ ω6z4
c6
1 − (ωρ/c)2
dz, (1.6)
where φ0 is the initial value of φ, i.e. φ0 = φ(0).
Notice that if ρω/c ≈ 0, then we have the following approximation
φ(ρ) ≈ φ0 −
√
3
ω̺
c
. (1.7)
E. G. Celakoska, K. Trenčevski, V. Balan 433
The previous results do not contradict the SR fact that there is no
shrinking in the direction orthogonal to the moving direction, because
the rotating systems are not inertial systems. Notice that in this case:
• the effect of shrinking in radial direction appears because the points
in the radial direction have different velocities, and
• the same coefficient of shrinking is observed from both the inertial
system and the rotating system, and it essentially differs from the
Fitzgerald contraction.
Note that shrinking in the radial direction depends on the distance
to the center, and it can be calculated as follows:
From (1.1) we get
dρ =
dR
(
√
1 + (ωR
c )2
)3
= dR
(
√
1 −
(ωρ
c
)2
)3
,
and the required coefficient k of local shrinking on distance R (before
rotation), or distance r (during rotating) is equal to
k =
1
(
√
1 + (ωR/c)2
)3
=
(
√
1 −
(ωρ
c
)2
)3
.
Note that this coefficient of shrinking as well as the change dφ given
with the differential equation (1.5) can be applied generally for noncon-
stant ω and more generally, for particles which arbitrarily move including
rotation.
2. Application of the Thomas precession in
deducing the shrinking coefficient
Thomas precession is a kinematic effect closely related to the prob-
lem of finding Lorentz transformation as a result of two successive non-
collinear Lorentz transformations. More precisely, Thomas precession is
defined as an additional spatial rotation Rt contained in the Lorentz
transformation which represents the resultant of two successive non-
collinear Lorentz transformations. This is more obvious for non-collinear
Lorentz boosts B1 and B2 giving Lorentz transformation L which is not
Lorentz boost, but it contains additional rotation Rt. Thomas precession
is explicitly given as a vector of angular velocity and is expressed as
~ωThomas = −1
2
~v × ~a
c2
434 Application of the Thomas precession...
where ~v is the velocity and ~a is the acceleration of a point as seen in the
inertial system.
Let us consider a point with small mass m on the periphery of the
rotating disk. This point moves with velocity ~v tangent to the circle of
the periphery, with centripetal acceleration ~a = −~ρω2, so the observer
from the inertial system observes also the Thomas precession
~ωThomas =
1
2
~v × ~ρ
c2
ω2. (2.1)
Notice that this angular velocity ~ωThomas has the opposite direction of
the initial angular velocity ~ω, and so the module of the total angular
velocity depends on the distance ρ to the center of the body and it is
given by
ωρ = ω − 1
2
ωρ2
c2
ω2 = ω
(
1 − (ωρ)2
2c2
)
. (2.2)
We give a physical interpretation of this fact. Assume that the disk is
not a solid body, but it is soft where cohesive forces completely disappear.
In this case it is possible for different particles to move with different
angular velocities, which happens according to (2.2). Notice that in this
case each particle still preserves its distance to the center of the body; for
example the particles on the periphery of the body are located at distance
R from the center of the body. Now, assume that the cohesive forces
become gradually stronger, until the body becomes a solid one, which
means that at the end all particles must rotate with the same angular
velocity ω. But, since the energy of each particle must be preserved in
that process, the initial velocity of the each particle of the periphery must
be equal to the final velocity of the particle, i.e.,
RωR = ρω, (2.3)
and according to (2.2)
Rω
(
1 − (ωρ)2
2c2
)
= ρω. (2.4)
Hence, the equality (1.1) just follows, namely, if the equality (1.1) is rep-
resented as the Taylor series with respect to ωρ/c and all terms (ωρ/c)n
for n > 2 are rejected, then we get (2.4). Thus, the formula for the
shrinking coefficient holds true.
References
[1] K. Trencevski, V. Balan, Shrinking of rotational configurations and associated
inertial forces, J. of Calcuta Math. Soc. 1(3) & 4 165-180 (2005).
E. G. Celakoska, K. Trenčevski, V. Balan 435
[2] P. Ehrenfest, Similar type rotation of rigid bodies and the theory of relativity,
Phys. Zeitschrift 10 918-918 (1909).
[3] M. Planck, Similar form rotation and the Lorentz-contraction. Phys. Zeitschrift
11 294-294 (1910).
[4] A. Einstein, About the Ehrenfest’s paradox. Phys. Zeitschrift 12 509-510 (1911).
[5] A. Einstein, The Principle of Relativity, Dover Publications, New York, (1952).
[6] P. Langevin, Remarques au sujet de la Note de Prunier, Comptes Rendus Acad.
Sci. Paris 200 48 (1935)
[7] V. Cantoni, What is wrong with Relativistic Kinematics?, Il Nuovo Cimento 57
B 220–223 (1968).
[8] O. Gron, Relativistic description of a rotating disc, Am. J. of Phys. 43 (10)
869-876 (1975).
[9] W. Misner, K.S. Thorne, I.A. Wheeler, Gravitation, Freeman Eds., San Francisco,
1973.
Contact information
Emilija G.
Celakoska
Department of Mathematics
and Informatics,
Faculty of Mechanical Engineering
St. Cyril and Methodius University P. O.
Box 464 1000 Skopje,
Macedonia
E-Mail: cemil@mf.edu.mk
Kostadin
Trenčevski
Institute of Mathematics,
Faculty of Natural Sciences
and Mathematics
St. Cyril and Methodius University
P. O. Box 162 1000 Skopje,
Macedonia
E-Mail: kostatre@pmf.ukim.mk
Vladimir Balan Department Mathematics-Informatics I,
University Politehnica of Bucharest
Splaiul Independenţei 313,
RO-060042 Bucharest,
Romania
E-Mail: vbalan@mathem.pub.ro
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