SERIES ⎜ ARTICLE Snippets of Physics 8. Foucault Meets Thomas

T Padmanabhan

T he Foucaultpendulum isan elegantdevice that dem onstrates the rotation of the Earth. A fter describingit,w e w illelaborateon an interesting geom etrical relationship betw een the dynam ics of the Foucaultpendulum and T hom as preces- T Padmanabhan works at sion d iscu ssed in th e last installm en t1 . T h is w ill IUCAA, Pune and is interested in all areas helpus to understand both phenom ena better. of theoretical physics, T h e ¯ rst titular p residen t of th e F ren ch rep u b lic, L ou is- especially those which have something to do with N ap oleon B on ap arte,p erm itted F ou cau lt to u se th e P an - gravity. th eon in P aris to give a d em on stration of h is p en d u lum (with a 67 m eterw ireand a 28 kg pendulum bob)on 31 M arch 1851. In th is im p ressive ex p erim en t, on e cou ld see the plane of oscillation of the pendulum rotating in a clockw ise direction (w hen view ed from the top) w ith afrequency! = − cosμ,where − isthe angular fre- 1 Thomas Precession, Reso- nance, Vol.13, No.7, pp.610– quen cy of E arth 's rotation and μ is th e co-latitud e of 618, July 2008. P aris. (T h at is, μ isthe standard polarangle in spher- ical p olar co ord inates w ith th e z-axisbeingtheaxisof rotation of E arth . S o ¼=2 μ is th e g e o g ra p h ic a l la ti- tude).Foucaultclaimed,quitecorrectly,that¡ thise®ect arises d u e to th e rotation of th e E arth an d th u s sh ow ed th at on e can d em on strate th e rotation of th e E arth b y an in situ ex p erim ent w ithou t look ing at celestial ob jects. T h is resu lt is q u ite easy to u n d erstan d if th e exp erim en t w as p erform ed at th e p oles or eq u ator (instead of P aris!). T h e situation at th e N orth P ole is as sh ow n in Figure1. H ere w e see th e E arth as rotating (from w est to east, in th e cou n ter-clock w ise d irection w h en v iew ed from th e Keywords top) underneath the pendulum, m aking one fullturn Spin, Thomas precession, Earth's rotation. in 24 hours. It seem s reasonableto deduce from this

706 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE

Figure 1.

th at, as v iew ed from E arth , th e p lan e of oscillation of the pendulum w illm ake one fullrotation in 24 hours; so the angular frequency ! ofthe rotation of the plane of the Foucaultpendulum isjust ! = − . (Through- ou t th e d iscu ssion w e are con cern ed w ith th e rotation ofthe plane ofoscillation ofthe pendulum;not the pe- riod ofthependulum 2¼=º,which{ofcourse{isgiven b y th e stan d ard form u la involving th e len gth of th e su s- pension w ire,etc.).A t the equator,on the other hand, th e p lan e of oscillation d o es n ot rotate. S o th e form u la, Jean Bernard Lèon ! = − cosμ, cap tu res b oth th e resu lts correctly. Foucault (1819 –1868) was a French It is trivial to w rite d ow n th e equ ation s of m otion for th e physicist, famous for pendulum bob in the rotating fram e of the Earth and the demonstration of solve them to obtainthisresult[1,2]at thelinear order Earth's rotation with in − . Essentially,the Foucaultpendulum e®ect arises his pendulum. due to the C oriolis force in the rotating fram e of the Although Earth's Earthwhichleadstoanacceleration2v − ,where v , rotation was not thevelocity ofthependulum bob,isdirected£ tangential unknown then, but this to th e E arth 's su rface to a goo d ap p rox im ation . If w e easy-to-see ch oose a local coord inate system w ith th e Z -axispoint- experiment caught ing n orm al to th e su rface of th e E arth an d th e X;Y everyone's coordinates in the tangent plane at the location, then it is easy to sh ow th at th e eq u ation s of m otion for th e imagination.

RESONANCE ⎜ August 2008 707 SERIES ⎜ ARTICLE

pendulum bob arew ellapproximated by XÄ + º 2 X =2− Y_ ; YÄ + º 2 Y = 2− X;_ (1) z ¡ z whereº istheperiod ofoscillation ofthependulum and − z = − cosμ is th e n orm al com p on en t of th e E arth 's angular velocity. In arriving at these equations w e have ignored the term s quadraticin − and the vertical dis- placem ent ofthependulum. T he solution to thisequa- tion is ob tained easily b y intro d u cing th e variab le q(t) ´ X (t)+ iY (t): T h is satis¯es th e eq u ation

2 qÄ +2i− z q_+ º q =0: (2) T h e solution , to th e ord er of accu racy w e are w ork ing with,isgivenby q = X (t)+ iY (t)= (X (t)+ iY (t)) ex p ( i− t); (3) 0 0 ¡ z

whereX 0 (t);Y 0 (t) isthe trajectory ofthe pendulum in th e ab sen ce of E arth 's rotation . It is clear th at th e n et e® ect of rotation is to cau se a sh ift in th e p lan e of rota- tion at the rate − z = − cosμ. B ased on thisknow ledge and the resultsfor the poleand the equator one can give a `plainEnglish'derivation oftheresultforinterm ediate la titu d e s b y sa y in g so m e th in g lik e : \ O b v io u sly , it is th e com ponent of − n orm al to th e E arth at th e location of thependulum w hich m attersand hence ! = − cosμ." T h e ¯ rst-p rinciple ap p roach , b ased on (1), of cou rse h as the advantage ofbeing rigorous and algorithm ic;forex- am ple, ifyou w ant to take into account the e®ects of ellipticity of E arth , you can d o th at if you w ork w ith theequationsofm otion. B utitdoesnotgive you an in- tuitive understan d ing of w h at is going on , an d m u ch less a uni¯ed view of allrelated problem s havingthe sam e stru ctu re. W e sh all n ow d escrib e an ap p roach to th is problem w hich has the advantage of providing a clear geom etrical p ictu re an d con n ecting it u p { som ew h at q u ite su rp risingly { w ith T h om as p recession d iscu ssed in th e last installm en t.

708 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE

O ne point w hich causes som e confusion as regards the Foucaultpendulum isthe follow ing. W hileanalyzing the behavior ofthe pendulum at the pole,one assum es th at th e p lan e of rotation rem ains ¯ x ed w h ile th e E arth rotates underneath it.Ifw e m ake thesam e claim for a pendulum experiment done at an interm ediatelatitude, { i.e ., if w e sa y th a t th e p la n e o f o sc illa tio n re m a in s invariant w ith resp ect to, say, th e \¯ x ed stars" an d th e A Earth rotatesunderneath it{ itseem s naturalthat the period ofrotation ofthependulum plane shouldalways b e 24 h ou rs irresp ective of th e location ! T h is, of cou rse, is n o t tru e a n d it is a lso in tu itiv e ly o b v io u s th a t n o th in g h ap p en s to th e p lan e of rotation at th e eq u ator. In th is w ay ofapproachingtheproblem ,itisnotvery clearhow exactlytheEarth'srotation in°uencesthem otion ofthe pendulum. Figure 2. T o p rov ide a geom etrical ap p roach to th is p rob lem , w e w ill re p h ra se it a s fo llo w s [3 , 4 ]. T h e p la n e o f o sc illa - tion of the pendulum can be characterized by a vector norm alto itor equivalently by a vector w hich islying in th e p la n e and tan gen tial to th e E arth 's su rface. L et us now introduce a cone w hich iscoaxialw ith the axis of rotation of th e E arth an d w ith its su rface tan gen tial to the Earth at the latitude of the pendulum (see Fig- ure 2). T he base radiusof such a cone w illbe R sin μ , whereR is th e rad ius of th e E arth an d th e slant h eigh t of the cone w illbe R tan μ. Such a cone can be built out of a sector ofa circle(as show n in Figure 3) having Figure 3. the circum ference 2¼R sin μ and radius R tan μ by iden- tifyingthelinesO A and O B .T he `de¯citangles'ofthe cone, ® and ¯ 2¼ ® , sa tisfy th e re la tio n s: C ´ ¡ (2¼ ® )R tan μ =2¼R sin μ (4) ¡ O whichgives ® =2¼ (1 cosμ); ¯ =2¼ cosμ: (5) ¡ A B T he behavior of the plane of the Foucaultpendulum

RESONANCE ⎜ August 2008 709 SERIES ⎜ ARTICLE

can be understo od very easily in term s of th is con e. In i- The plane of tially,the Foucaultpendulum isstarted out oscillating oscillation of the in som e arb itrary d irection at th e p oint A , say. T h e d i- pendulum will rotate rection of oscillation can b e ind icated b y som e straigh t with respect to a linedraw n along thesurfaceofthecone (like A C in Fig- coordinate system ure 3). W hilethe plane ofoscillation ofthe pendulum fixed on the Earth, w ill rotate w ith resp ect to a coord inate sy stem ¯ x ed on but it will always th e E arth , it w illalw ays coincide w ith th e lines d raw n on coincide with the lines th e con e w h ich rem ain ¯ xed relative to th e ¯ x ed stars. drawn on the cone W h en th e E arth m akes on e rotation , w e m ove from A which remain fixed toB inthe°attened outcone in Figure3 . P h y sic a lly , o f relative to the fixed cou rse, w e iden tify th e tw o p oints A an d B w ith th e sam e stars. (Figures 2,3) location on th e su rface of th e E arth . B u t w h en a vector is m oved arou n d a cu rve alon g th e lines d escrib ed ab ove, on the curved surface of E arth,its orientation does not retu rn to th e original value. It is ob v iou s from Figure 3 th at th e orientation of th e p lan e of rotation (ind icated by a vector in the plane of rotation and tangentialto th e E arth 's su rface at B ) w ill b e d i® eren t from th e cor- resp on d ing vector at A . (T h is p ro cess is called p arallel transport and the fact that a vector changes on parallel tran sp ort arou n d an arb itrary closed cu rve on a cu rved su rface is a w ell-k n ow n resu lt in d i® eren tial geom etry and general relativity.) C learly, th e orientation of th e vector ch an ges b y an an gle ¯ =2¼ cosμ d u ring on e rotation of E arth w ith p eriod T . Since therateofchange isuniform throughout because of th e stead y state n atu re of th e p rob lem , th e an gu lar velocity ofthe rotation ofthe pendulum plane isgiven by ¯ 2¼ ! = = cosμ = − cosμ: (6) T T T h is is p re c ise ly th e re su lt w e w e re a fte r. T h e k e y g e o - m etrical idea w as to relate th e rotation of th e p lan e of the Foucaultpendulum to the parallel transport of a vector ch aracterizing th e p lan e, arou n d a closed cu rve on th e su rface of E arth . W h en th is closed cu rve is n ot

710 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE a geo d esic { an d w e k n ow th at a cu rve of con stan t lat- This derivation also itu d e is n ot a geo d esic { th e orien tation of th is vector allows one to changes w hen it com pletes one loop. T here are m ore understand the sophisticated w ays of calculating how m uch the orien- tation ch an ges for a given cu rve on a cu rved su rface. Thomas B ut in the case of a sphere,the trick of an enveloping precession of the cone providesa simpleprocedure.(W hen thependulum spin of a particle. is located in th e equ ator, th e closed cu rve is th e equ a- tor itself; th is, b eing a great circle is a geod esic on th e sphereand thevectordoesnotget`disoriented'on going around it.So theplane ofthependulum doesnotrotate in th is case.) T hisisgood, but as I said, things get better. O ne can show that an alm ost identical approach allow s one to determ inetheT hom asprecession ofthespinofa particle (say, an electron) m oving in a circular orbit around a nucleus [5]. W e saw in th e last installm ent [6]th at th e rate ofT h om as precession isgiven,in general,by an expression of the fo rm

! d t =(cosh 1)(dn^ n^ ) ; (7) ¡ £ wheretanh = v=c and v is th e velocity of th e p article. In th e case of a p article m ov ing on a circu lar tra jectory, the m agnitude of the velocity rem ains constant and w e can integrate th is ex p ression to ob tain th e n et an gle of precession duringone orbit. For a circular orbit,dn^ is always perpendicular ton ^ so th atn ^ d^n is e sse n tia lly £ d μ w h ich integrates to give a factor 2¼ . H ence the net angleofT hom as precession duringone orbitisgiven by

©=2¼ (cosh  1): (8) ¡ ThesimilaritybetweenthenetangleofturnoftheFou- caultpendulum and thenetT hom as precession angleis now obvious w hen we com pare (8) w ith (5). W e know th at in th e case of L orentz tran sform ation s, on e rep laces

RESONANCE ⎜ August 2008 711 SERIES ⎜ ARTICLE

real angles by im aginary angles w hich accounts for the It will turn out that d i® eren ce b etw een th e cos an d cosh factors. W h at w e the sphere and the n eed to d o is to m ake th is an alogy m ath em atically p re- cone we cise w h ich w ill b e ou r n ext task . It w ill tu rn ou t th at introduced in the thesphereand thecone w e introduced intherealspace, real space, to to study theFoucaultpendulum,have to be introduced study the Foucault in th e velocity sp ace to an alyze T h om as p recession . pendulum, have to be introduced in A s a w arm -u p to ex p loring th e relativistic velocity sp ace, the velocity space let u s start b y ask ing th e follow ing q u estion : C on sider to analyze Thomas tw o fram es S 1 and S 2 whichmovewithvelocitiesv 1 and precession. v 2 w ith resp ect to a th ird inertial fram e S 0 .Whatis themagnitudeoftherelativevelocitybetweenthetwo fram es? T his ism ost easily done using Lorentz invari- ance and four-vectors (and to sim plify notation w e w ill use units w ith c = 1). W e can associate w ith th e 3- velocities v 1 and v 2 , the corresponding four-velocities, i i given by u 1 =(° 1 ;°1 v 1 )and u 2 =(° 2 ;° 2 v 2 ) w ith a ll th e com p on en ts b eing m easu red in S 0 . O n th e oth er h an d , w ith resp ect to S 1 , this four-vector w illhave the com - i i ponents u 1 =(1;0) and u 2 =(°;°v ), w h ere v (by d ef- in itio n ) is th e re la tiv e v e lo c ity b e tw e e n th e fra m e s. T o determ inethem agnitude ofthisquantity,we note that i in th is fra m e S 1 w e can w rite ° = u 1 iu 2 . B ut since th is ex p ression is L oren tz invarian t,¡ w e can evaluate it i i in any inertial fram e. In S 0 ,with u 1 =(° 1 ;°1 v 1 );u 2 = (° 2 ;°2 v 2 )thishasthevalue 2 1= 2 ° =(1 v )¡ = ° ° ° ° v v : (9) ¡ 1 2 ¡ 1 2 1 ¢ 2 S im p lifying th is ex p ression w e get (1 v v )2 (1 v 2 )(1 v 2 ) v 2 = ¡ 1 ¢ 2 ¡ ¡ 1 ¡ 2 (1 v v )2 ¡ 1 ¢ 2 (v v )2 (v v )2 = 1 ¡ 2 ¡ 1 £ 2 : (10) (1 v v )2 ¡ 1 ¢ 2 Let us next consider a 3-dimensionalabstract space in w h ich each p oint rep resen ts a velocity of a L oren tz fram e

712 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE m easured w ith respect to som e ¯ducialfram e. W e are interested in d e¯ n ing th e n otion of `distan ce' b etw een two pointsin thisvelocity space. C onsider two nearby p oints w h ich corresp on d to velocities v and v +dv th at di®erby an in¯nitesimalquantity.B y analogy w ith the usual3-dimensional°at space,one w ouldhave assum ed th at th e `distan ce' b etw een th ese tw o p oints is ju st

d v 2 =dv 2 +dv 2 +dv 2 =dv 2 + v 2 (d μ 2 +sin2 μd Á 2 ); j j x y z (11) wherev = v and (μ;Á ) denote the direction of v .In n on -relativisticj j p h ysics, th is d istan ce also corresp on d s to themagnitudeoftherelativevelocitybetweenthetwo fram es. H ow ever, w e have just seen that the relative velocity betw een tw o fram es in relativistic m echanics is di®erent and given by (10).Itism orenaturalto de¯ne thedistancebetweenthetwopointsinthevelocityspace to b e the relative velocity betw een the resp ective fram es. Inthatcase,thein¯nitesimal`distance'between thetwo points in the velocity space willbe given by (10) with v 1 = v and v 2 = v +dv .So (d v )2 (v d v )2 d l2 = ¡ £ : (12) v (1 v 2 )2 ¡ U sin g th e re la tio n s

(v d v )2 = v 2 (d v )2 (v d v )2 ;(v d v )2 = v 2 (d v )2 £ ¡ ¢ ¢ (13) an d u sing (11) w h ere μ, Á are the polar and azimuthal angles of the direction of v ,we get d v 2 v 2 d l2 = + (d μ 2 +sin2 μ d Á 2 ): (14) v (1 v 2 )2 1 v 2 ¡ ¡ Ifw e use therapidity  in p la c e o f v through theequa- tion v =tanh , th e lin e e le m e n t in (1 4 ) b e c o m e s:

2 2 2 2 2 2 d lv =d +sinh (d μ +sinμ d Á ): (15)

RESONANCE ⎜ August 2008 713 SERIES ⎜ ARTICLE

T his is an exam ple of a curved space w ithin the con- text of sp ecial relativity. T h is p articu lar sp ace is called (three-dimensional)Lobachevsky space. Ifw e now change from realangles to the im aginary ones, by w riting  = i´ , the line elem ent becom es d l2 =d´ 2 +sin2 ´ (d μ 2 +sin2 μ d Á 2 ); (16) ¡ v w h ich (ex cep t for an overall sign w h ich is irrelevan t) rep resen ts th e d istan ces on a 3-sp h ere h aving th e th ree angles ´;μ and Á a s its c o o rd in a te s. O f these three angles, μ and Á denote the direction of velocity in the realspace as w ell. W hen a particle m oves in th e x y plane in the real space, its velocity vector lie s in th¡ e μ = ¼=2 plane and the relevant part of the m etric red u ces to

2 2 2 2 d L v =d´ +sin´ d Á (17) w h ich is ju st a m etric on th e 2-sp h ere. F u rth er, if th e particleism oving on a circularorbitw ith constant m ag- nitude for the velocity,then itfollow s a curve of ´ = constant on this2-sphere. T he analogy w ith the Fou- caultpendulum, w hich m oves on a constant latitude cu rve, is n ow com p lete. If th e p article carries a sp in, th e orb it w ill tran sp ort th e sp in vector alon g th is cir- cu lar orb it. A s w e h ave seen earlier, th e orien tation of the vector w illnot coincidew ith the originalone w hen th e orb it is com p leted an d w e exp ect a d i® eren ce of 2¼ (1 cos´ )= 2¼ (1 cosh  ). S o th e m agn itud e of th e T hom¡ as precession,over¡ one period isgiven preciselyby (8). I w illlet you w ork ou t th e d etails ex actly in an alogy w ith theFoucaultpendulum and convince yourselfthat th ey h ave th e sam e geom etrical interp retation . W hen one m oves along a curve in the velocity space, one is sam pling di®erent (instantaneously) co-m oving L oren tz fram es ob tained b y L oren tz b o osts alon g d i® er- ent d irection s. A s w e d escrib ed in th e last installm en t,

714 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE

L orentz boosts along di®erent directions do not, in gen- eral,com m ute. T hisleads to the resultthat ifw e m ove alon g a closed cu rve in th e velocity sp ace (treated as rep resen ting d i® eren t L oren tz b oosts) th e orientation of thespatialaxesw ouldhave changed w hen we com plete the loop. It tu rn s ou t th at th e ideas d escrib ed ab ove are actu ally offarm oregeneralvalidity.W henevera vectoristrans- p orted arou n d a closed cu rve on th e su rface of a sp h ere, th e n et ch an ge in its orien tation can b e related to th e solidanglesubtended by thearea enclosed by thecurve. In the case of the Foucault p endulum , the relevant vec- tor describes theorientation ofthe plane ofthependu- lum an d th e tran sp ort is arou n d a circle on th e su rface of th e E arth . In th e case of T h om as p recession , th e rel- evan t vector is th e sp in of th e p article an d th e tran sp ort occurs in the velocity space. U ltim ately,both the e® ects { the Foucaultpendulum and T hom asprecession { arise because the space in w hich one isparalleltransporting th e vector (su rface of E arth , relativistic velocity sp ace) is curved.

Suggested Reading Address for Correspondence T Padmanabhan [1] L D Landau and E M Lifshitz, Mechanics, Pergamon Press, p.129, 1976. IUCAA, Post Bag 4 [2] T Padmanabhan, Theoretical Astrophysics: Astrophysical Processes, Pune University Campus Cambridge University Press, Vol.1, p.50, 2000. Ganeshkhind [3] J B Hart et al., Am. J. Phys, Vol.55, No.1, p.67, 1987. Pune 411 007, India. [4] Jens von Bergmann, Gauss–Bonnet and Foucault, Email: URL: http://www.nd.edu/~mathclub/foucault.pdf, 2007. [email protected] [5] M I Krivoruchenko, arXiv:0805.1136v1 [nucl-th], 2008. [email protected] [6] T Padmanabhan, Resonance, Vol.13, No.7, p.610, July 2008.

RESONANCE ⎜ August 2008 715