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Rotation of the Swing Plane of Foucault's Pendulum and Thomas

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Rotation of the Swing Plane of Foucault's Pendulum and Thomas Rotation of the swing plane of Foucault’s pendulum and Thomas spin precession: Two faces of one coin. M. I. Krivoruchenko∗ Institut f¨ur Theoretische Physik, T¨ubingen Universit¨at, Auf der Morgenstelle 14 D-72076 T¨ubingen, Germany Using elementary geometric tools, we apply essentially the same methods to derive expressions for the rotation angle of the swing plane of Foucault’s pendulum and the rotation angle of the spin of a relativistic particle moving in a circular orbit (Thomas precession effect). PACS numbers: 01.55.+b,01.65.+g,02.40.Ky,03.30.+p I. INTRODUCTION pleted essentially after the works of Johannes Kepler at the beginning of the XVII century. Jean Bernard L´eon Foucault conducted his first pen- From the viewpoint of an observer at the Earth, re- dulum experiment in Paris in January 1851, aiming to mote stars complete one rotation in the clockwise direc- prove the rotation of the Earth by demonstrating the ro- tion during 1 sidereal day = 23 hours, 56 minutes, 4.091 seconds. tation of the plane of swings. Originally, the suspension ◦ The observed rotation rate,ϕ ˙ E ≈ −11 per hour, of length of the pendulum was 2 m. The next experiment ◦ ◦ was set up with the suspension length 11 m at the Obser- Foucault’s pendulum is not equal to −360 /24 = −15 vatory of Paris. Louis-Napol´eon Bonaparte, the first tit- per hour, nor is it zero (the minus sign indicates that the ular President of the French Republic and nephew of the swing plane rotates clockwise). famed Napoleon I of France, was informed of Foucault’s If an observer in the coordinate system of the remote work and proposed him to conduct an experiment at the stars transports the Foucault’s pendulum to the North Panth´eon. The experiment took place on 31 March 1851, Pole along the Earth meridians keeping the angle be- with the pendulum bob weight 28 kg suspended under the tween the pendulum’s plane and the meridians fixed, she Panth´eon dome by a steel wire 67 m long. will detect a uniform precession of the pendulum’s plane relative to the remote stars. The rotation angles ϕS and Aristarchus of Samos who proposed the first consis- ϕ , relative to the remote stars and the Earth meridians, tent heliocentric model around 270 B. C. explained the E respectively, are related by ϕS =2π +ϕE for one sidereal observed rotation of stars by the axial rotation of the day. In the adiabatic approximation and for small swing Earth. Similar ideas were discussed earlier by Philolaus, angles, the rotation angle ϕ is given by philosopher of the Pythagorean school, in the V-th cen- S tury B. C. and by Greek philosopher Herakleides in the ϕS =2π(1 − cos ϑ), (1) IV-th century B. C. If idea on the axial rotation of the Earth is correct, one can expect that because of inertia where ϑ is the polar angle. The adiabaticity means that the plane of swings of the Foucault’s pendulum does re- the period of the Earth’s rotation is much greater than volve delaying the Earth rotation. It, however, the Earth the period of the swings. is fixed and motionless, as was embraced by both Aris- The equation of rotation of the plane of swings of Fou- totle, Ptolemy, and most Greek philosophers, the swing cault’s pendulum, as illustration of the laws of classical plane of Foucault’s pendulum cannot rotate. mechanics (see, e.g., [1] and also Section III), is an ele- ment of program of university courses of physical facul- arXiv:0805.1136v2 [nucl-th] 19 Apr 2010 From the technical point of view, the experiment with the Foucault’s pendulum was accessible to all ancient ties. Thus, the Foucault’s pendulum, placed at the North and more recent civilizations, including Greeks, never- ◦ theless, the experiment has been done in the New Era Pole, during the day turns at an angle ϕE = −360 . At the Equator the pendulum does not rotate. The only. During one and half thousands of years it was as- ◦ sumed that the problem of rotation of the stars does not Panth´eon in Paris is located at the parallel ϑ = 41.15 (in geography one speaks about the latitude α = π/2 − ϑ). require additional attention because of the incontestable ◦ authority of Aristotle and success of the Ptolemy heo- From the equations we obtainϕ ˙ E = −11.3 per hour, centric model that described and describes until now the which is consistent with the observations on the Fou- motion of planes with high precision. The interest to the cault’s pendulum and precludes accompanying ”rotation problem and discussions recommenced again in the XVI of heavens” with high accuracy. century after the works of Nicolaus Copernicus and com- The rotation of the plane of the pendulum provided the first evidence for the rotation of the Earth by terrestrial means. In the process of movement of the pendulum along the ∗On leave of absence from Institute for Theoretical and Experi- surface of the Earth the plane of swings, as a consequence mental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia of the laws of classical mechanics, remains parallel to 2 itself [2–6]. This remarkable fact allows to investigate Group-theory aspects of the spin rotation under the the evolution of the Foucault’s pendulum using geometric Lorentz transformations were introduced to physicists by methods. Eugene Paul Wigner [14]. The term ”Wigner rotation” is The bob’s velocity in the equilibrium point is tangent used as a synonym of the rotation of a spinning particle to the surface of the Earth, and therefore belongs to the under the coordinate transformations and specifically as tangent space of the surface. The plane of swings of the a synonym of ”Thomas precession”, too. pendulum can be described by a vector orthogonal to The history of early study of the relativistic precession it. Such a vector lies in a plane tangent to the Earth’s effect is described in Ref. [15]. As claimed by Robert surface also. In the absence of external forces and/or Merton, multiple independent discoveries represent the torques tangent vectors under the displacement experi- common pattern in science [16]. ence parallel transport. For example, vectors tangent to Recently, the geometric nature of the spin precession 1,3 Minkowski spacetime, M = R , like four-velocity, re- effect of a relativistic particle has again attracted atten- main fixed under displacement, whereas tangent vectors 2 tion. It was shown in [17] that the rotation angle φS is of curved spaces like the surface of a sphere, S , or the determined by an integral over the surface limited by the space of physical relativistic velocities, rotate under dis- closed trajectory of the particle in the physical relativis- placement in order to remain tangent to the correspond- tic velocity space. This property characterizes parallel ing surface. This kind of evolution is associated with transport of vectors in a Riemannian space (see, e.g., inertial motion. In the first case the displacements occur [18]). Parallel transport in the relativistic velocity space in Minkowski spacetime, in the second and third cases and Thomas precession were discussed in [19] in detail. the displacements occur on the surface of a sphere and in Rotation angles ϕS and φS correspond to a geometrical the space of physical relativistic velocities, respectively. phase that occurs in many areas of physics [4, 5, 18]. The geometric basis of the relativistic effect known to- The analogy between rotations and Lorentz transfor- day as Thomas precession has been discovered in 1913 ´ mations is of the high heuristic value. In particular, year by French mathematician Emile Borel [7]. He we recall that the relativistic velocity addition theorem pointed out noncommutativity of non-collinear Lorentz can be obtained as the composition law for arcs of the transformations, described an analogy between transfor- great circles on a sphere of imaginary radius in a four- mations of vectors on a spherical surface and in the phys- dimensional Euclidean space with one imaginary coordi- ical relativistic velocity space, and provided the lowest nate (time). By introducing the imaginary coordinate order estimate of kinematic precession of axes of a rigid (time), it is possible to transform a hyperboloid of physi- body on a circular orbit. cal relativistic velocities into a sphere of imaginary radius The same year two young mathematicians Ludwig in a four-dimensional Euclidean space. Both for Fou- F¨oppl and Percy John Daniell derived exact expression cault’s pendulum and for Thomas precession, the surface for the rotation angle due to Thomas precession, corre- along which a displacement occurs can be considered a sponding to one period of a uniform circular motion [8] sphere. This suggests that the rotation effects of Fou- φ =2π(1 − cosh θ), (2) cault’s swinging pendulum and Thomas precession, ob- S viously, are geometrically identical; this does not neces- where θ is rapidity, cosh θ = γ = (1 − v2/c2)−1/2 is sarily contradict to the different physical natures of these Lorentz factor, v is velocity of the body, and c is speed systems. of light. The work of F¨oppl and Daniell has been recom- The aim of this methodological note is to show that mended for publication by David Hilbert. expressions for the rotation angles ϕS and φS can be ob- The relativistic precession effect has been known at tained by the same method using elementary geometrical about the same time to Ludwik Silberstein [9]. tools for parallel transport of vectors over corresponding In the early 1920s, Enrico Fermi [10], and later Arthur surfaces.
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