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Riemann and Theoretical Physics Joseph Samuel
S e v e r a l m a th e m a tic a l d isc ip lin e s su c h a s d i® e re n tia l g e o m e try , to p o lo g y a n d R ie m a n n g e o m e tr y o n w h ic h R ie m a n n le ft h is m a r k h a v e a m a jo r im p a c t o n p h y sic s to d a y . T h is a r tic le d e sc r ib e s so m e o f th e se c o n tr ib u tio n s o f R ie m a n n to p h y sic s. Joseph Samuel is a theoretical physicist and by natural inclination a classical mechanic. Over the years he has strayed into other fields like optics, general relativity and very recently DNA elasticity. A unifying theme in his work is differential geometry and topology in physics. He keeps moderately fit by raising and lowering indices and relaxes by playing semiclassical guitar.
Keywords Mathematical physics, geometry, toplogy, relativity.
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P la n e g eo m etry h a s b een stu d ied sin ce th e a n cien t G reek s. T h e su b ject w a s co n so lid a ted in E u clid 's E lem en ts, w h ich p u t d ow n th e m a in id ea s in a x io m a tic fo rm . G en era tio n s o f m a th em a ticia n s h av e m a rv elled a t th e a rch itectu ra l b ea u ty o f E u clid ea n g eo m etry. A few y ea rs b efo re, R iem a n n th ere b eg a n ex p lo ra tio n s o f cu rv ed g eo m etry. T h e su b ject o f n o n -E u clid ea n g eo m etry w a s a rriv ed a t in d ep en d en tly b y G a u ss, L o b a ch ev sk y a n d B o ly a i. T h is n ew g eo m etry sa tis¯ ed a ll o f E u clid 's a x io m s sav e o n e: th e ¯ fth p o stu la te. O n e v ersio n o f th e ¯ fth p o stu la te sta tes th a t g iv en a n y ¯ g u re th ere is a sim ila r ¯ g u re o f a n y size. T h is p o stu la te a sserts th a t th e E u clid ea n p la n e d o es n o t h av e a n in trin sic sca le. S ca le in d ep en d en ce is n o t tru e o f cu rv ed g eo m etries: th e sp h ere h a s a ra d iu s. N o n -E u clid ea n g eo m etry w a s a n im p o rta n t in tellectu a l d ev elo p m en t: it esta b lish ed th e lo g ica l in d ep en d en ce o f E u clid 's ¯ fth p o stu la te a n d a lso in tro d u ced th e id ea o f cu rv ed g eo m etry. T h e sp h ere is a tw o d im en sio n a l su rfa ce o f co n sta n t p o sitiv e cu rva tu re a n d th e sa d d le (o r L o b a ch ev sk y sp a ce) is a tw o d im en sio n a l sp a ce o f co n sta n t n eg a tiv e cu rva tu re. B o th h av e a n in trin sic sca le rela ted to th e cu rva tu re. G a u ss stu d ied cu rv ed tw o d im en sio n a l su rfa ces em b ed d ed in th ree d im en sio n a l ° a t sp a ce. H e reco g n ised th e d i® eren ce b etw een p ro p erties in trin sic to th e su rfa ce a n d ex trin ic o n es, w h ich d ep en d ed o n th e em b ed d in g .
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A cco rd in g ly, G a u ss in tro d u ced tw o n o tio n s o f cu rva tu re, th e m ea n cu rv a tu re, w h ich d ep en d ed o n th e em b ed d in g sp a ce a n d w h a t is n ow ca lled \ G a u ssia n cu rva tu re" . T h e m ea n cu rva tu re m ea su res h ow th e su rfa ce va ries in th e em b ed d in g sp a ce a n d is a lso ca lled th e ex trin sic cu rva tu re. F o r in sta n ce, th e cy lin d er (th e su rfa ce o f a p ip e) h a s ex trin sic cu rva tu re, b u t it is n o t in trin sica lly cu rv ed (it ca n b e cu t a n d ° a tten ed o n a p iece o f p a p er). H e em p h a sised th e in trin sic p ro p erties a s w o rth y o f stu d y a n d p rov ed th a t th e \ G a u ssia n cu rv a tu re" is a n in trin sic q u a n tity : it ca n b e d eterm in ed b y m ea su rem en ts m a d e en tirely w ith in th e su rfa ce. G a u ss w a s so im p ressed b y th is th eo rem th a t h e ca lled it th e T heorem a E gregiu m . T h is la tin p h ra se ca n b e lo o sely a n d co llo q u ia lly tra n sla ted in to m o d ern E n g lish a s \ H ea p B ig T h eo rem " .
Gauss emphasized that the intrinsic geometry of a surface was worthy of study.
R iem a n n 's a p p ro a ch to cu rva tu re w a s fa r m o re g en era l th a n G a u ss's. In a sm a ll reg io n o f sp a ce (sm a ll co m p a red to th e sca le o f th e cu rva tu re) o n e ca n a d o p t a lo ca l n o tio n o f p a ra llelism : a ru le fo r co m p a rin g v ecto rs a t d i® eren t p o in ts. It tu rn s o u t th a t o n a cu rv ed sp a ce th is lo ca l n o tio n d o es n o t in teg ra te to p ro d u ce a g lo b a l n o tio n o f p a ra llelism . T h is is b est illu stra ted b y u sin g th e ex a m p le o f th e su rfa ce o f th e E a rth . In ev ery d ay life, w e d o n o t b o th er a b o u t th e cu rva tu re o f th e E a rth (sin ce th e E a rth is m u ch b ig g er th a n o u r cities) a n d so m e p la n n ed cities (S a lt L a k e C ity, U ta h , U S A is a g o o d ex a m p le) h av e a C a rtesia n g rid o f streets la id o u t stra ig h t lik e a ch ess b o a rd . W e ta k e p a in s to en su re th a t th e M a in ro a d s a re d irected \ p a ra llel" to ea ch o th er. L ik e w ise, th e cro ss ro a d s a re d irected \ p a ra llel" to ea ch o th er a n d m eet th e M a in ro a d s a t rig h t a n g les. If y o u sta rt o n a M a in ro a d a n d ta k e fo u r left tu rn s o f n in ety d eg rees (w a lk in g say, ¯ v e b lo ck s b etw een tu rn s) y o u w ill b e b a ck o n a M a in ro a d . H ow ev er, if th e city g row s so th a t it cov ers a g o o d fra ctio n o f th e E a rth 's su rfa ce (say 1 = 8 ), th is p a ra llelism is n o lo n g er p o ssib le. (T ry stick in g sm a ll b its o f g ra p h p a p er o n a n o cta n t o f
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Riemann’s concept of curvature was more general and therefore to relevant to Einstein’s relativity.
a sp h ere m a in ta in in g p a ra llelism !) Y o u w ill ¯ n d th a t if y o u sta rt o n a M a in ro a d a n d ta k e fo u r left tu rn s (ea ch a fter trav ersin g a q u a rter o f th e E a rth 's circu m feren ce), y o u en d u p o n a cro ss ro a d . T h e g rid h a s ro ta ted b y a rig h t a n g le! R iem a n n ia n cu rva tu re is d e¯ n ed u sin g th e n o n -in teg ra b ility o f p a ra llelism . R iem a n n 's d e¯ n itio n o f cu rva tu re h a s co n sid era b le a d va n ta g e ov er G a u ss's: it is m a n ifestly in trin sic. T h e T heorem a E gregiu m is n o lo n g er a th eo rem b u t a n o b v io u s fa ct. R iem a n n 's id ea s g en era lise ea sily to a n y n u m b er o f d im en sio n s. T h ey a lso w o rk fo r m etrics o f L o ren tzia n sig n a tu re. T h is tu rn ed o u t to b e im p o rta n t fo r G en era l R ela tiv ity. It w a s a lrea d y clea r fro m sp ecia l rela tiv ity th a t sp a ce-tim e in th e a b sen ce o f g rav ity (M in k ow sk i sp a ce-tim e) h a d a g eo m etry o f L o ren tzia n sig n a tu re. T h is fo llow s fro m th e fa ct th a t in sp ecia l rela tiv ity it is th e in terva l x 2 + y 2 + z 2 ¡ c 2 t2 w h ich is o f p rim a ry in terest, a s o p p o sed to th e (sq u a red ) len g th x 2 + y 2 o f E u clid ea n geo m etry. W h a t E in stein n eed ed to d escrib e th e g rav ita tio n a l ¯ eld w a s cu rva tu re in fo u r d im en sio n a l sp a ce-tim e, i.e sp a ce-tim e w ith a fo u r d im en sio n a l L o ren tzia n m etric. S u ch a b stra ctio n w a s n o t ea sy to a ch iev e b a sed o n G a u ss' lin e o f a tta ck u sin g em b ed d in g in E u clid ea n sp a ce. E in stein 's G en era l T h eo ry o f R ela tiv ity w en t o n to a cq u ire a life o f its ow n , p red ictin g m a n y p h y sica l e® ects th a t co u ld b e m ea su red in th e S o la r S y stem , rev o lu tio n isin g C o sm o lo g y a n d A stro p h y sics. T h e th eo ry a lso p red icts B la ck H o les, w h ich co n tin u e to stretch o u r im a g in a tio n a n d d efy o u r u n d ersta n d in g . M a n y id ea s w h ich n ow fo rm th e G en era l T h eo ry o f R ela tiv ity w ere a n ticip a ted b y R iem a n n . H e rea lised th a t sp a ce m ay b e cu rv ed a n d th a t th is is a q u estio n w h ich h a s to b e settled b y ex p erim en t, n o t p h ilo so p h ica l sp ecu la tio n . H e ev en co n ceiv ed o f th e p o ssib ility th a t sp a ce m ay b e d iscrete, a n id ea w h ich is o n ly n ow co m in g o f a g e in q u a n tu m g rav ity ; it is b eliev ed b y so m e th a t sp a ce-tim e is d iscrete a t th e P la n ck sca le o f 1 0 ¡ 3 3 cm . E v id en tly, R iem a n n
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th e m a th em a ticia n w a s w ell p lu g g ed in to th e p h y sica l w o rld . H e su g g ested (sev en y ea rs b efo re M a x w ell's eq u a tio n s w ere w ritten ) th a t electro m a g n etic in tera ctio n s p ro p a g a te a t th e sp eed o f lig h t! F o r h is h a b ilita tio n lectu re (a req u irem en t b y th e u n iv ersity fo r a n a sp irin g tea ch er) R iem a n n h a d to g iv e th ree p o ssib le to p ics. T w o o f th ese w ere o n electricity ! B u t it w a s th e th ird o n th e fo u n d a tio n s o f g eo m etry th a t G a u ss p ick ed . T h is w a s th e fa m o u s lectu re in w h ich R iem a n n 's id ea s o n d i® eren tia l g eo m etry lik e n -d im en sio n a l m a n ifo ld s w ere p u t fo rw a rd . It is fa ir to say th a t a p a rt fro m G a u ss (w h o h a d th e h a b it o f p u b lish in g fa r less th a n h e k n ew ) n o o n e in th e a u d ien ce a p p recia ted th e fu ll d ep th o f th e id ea s p ro p o sed th ere.
Riemann’s name appears so often in physics texts that we sometimes forget it is a proper noun!
A n o th er su b ject to w h ich R iem a n n co n trib u ted is co m p lex a n a ly sis. C o m p lex a n a ly tic tech n iq u es a re n ow w id ely u sed in p h y sics a n d en g in eerin g . T w o d im en sio n a l p ro b lem s in p o ten tia l th eo ry, ela sticity a n d ° u id d y n a m ics a re ro u tin ely a d d ressed u sin g co m p lex a n a ly tic tech n iq u es lik e co n fo rm a l m a p p in g s. It is in terestin g th a t R iem a n n ¯ rst en co u n tered co n fo rm a l m a p p in g s w h en stu d y in g a p ro b lem rela ted to th e h ea t eq u a tio n ! T h e R iem a n n zeta fu n ctio n a p p ea rs in p h y sics tex ts d ea lin g w ith th e sta tistica l d istrib u tio n o f F erm i a n d B o se p a rticles. It is a lso u sed in ren o rm a lisa tio n in q u a n tu m ¯ eld th eo ry to m a k e sen se o f d iv erg en t in ¯ n ite su m s. O n e u ses a n a ly tic co n tin u a tio n to g iv e a m ea n in g to th e su m b y eva lu a tin g th e zeta fu n ctio n o n a d i® eren t R iem a n n sh eet. R iem a n n 's a p p ro a ch to co m p lex a n a ly sis w a s g eo m etric a n d in tu itiv e. M a n y o f h is co n tem p o ra ries v iew ed co m p lex a n a ly sis fro m a n a lg eb ra ic p o in t o f v iew . R iem a n n 's g eo m etric p o in t o f v iew resu lted in a d va n ces in d i® eren tia l g eo m etry a n d to p o lo g y. T o p o lo g ica l id ea s a re ex trem ely in tu itiv e a n d a t th e sa m e tim e v ery h a rd to fo rm a lise. L eib n iz (1 6 4 6 -1 7 1 6 ) w a s o n e o f th e ¯ rst m a th em a ticia n s to stu d y \ A n alysis S itu s" , a s to p o lo g y
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Suggested Reading [1] M Monastyrsky Riemann, Topology and Physics, Birkhauser, Boston, 1987. [2] Joseph Samuel, Crossing Bridges, Resonance, Vol.4, p.23, 1999. [3] S Weinberg, Gravitation and Cosmology, John Wiley and Sons, 1972. [4] James Hartle, Gravity: An Introduction to Einstein’s General Theory of Relativity, Pearson Education, New Delhi, 2002. [5] C Nash and S Sen, Topology and Geometry for Physicists, Academic Press, 1983.
u sed to b e ca lled , b u t h e w a s u n a b le to in terest p h y sicists o f h is tim e (lik e C h ristia a n H u y g en s) in th e su b ject. P h y sicists w ere m o re in terested in q u a n ti¯ ca tio n o f p h y sica l id ea s a n d th e tim e w a s n o t y et rip e o f th e q u a lita tiv e rea so n in g th a t m a rk s to p o lo g y. M u ch la ter E u ler (see C ro ssin g B rid g es, in S u g g ested R ea d in g b elow ) p lay ed w ith to p o lo g ica l id ea s in h is fa m o u s so lu tio n o f th e p ro b lem o f th e B rid g es o f K Äo n ig sb erg . W h en R iem a n n stu d ied co m p lex a n a ly sis, h is u n iq u e a p p ro a ch led h im to th e id ea o f a R iem a n n S u rfa ce. T h is led to co n sid era b le a d va n ces in to p o lo g y. H e w a s fo llow ed b y B etti a n d th en b y P o in ca re, w h o la id th e fo u n d a tio n s o f to p o lo g y in its m o d ern fo rm . T h e tw en tieth cen tu ry h a s seen a ra p id d ev elo p m en t o f th e su b ject, cu lm in a tin g recen tly (2 0 0 6 ) in th e so lu tio n o f th e P o in ca r¶e co n jectu re b y G rig o ri P erelm a n . P h y sics h a s g a in ed fro m th ese d ev elo p m en ts in m a th em a tics. T o p o lo g y is u sed to cla ssify d efects in liq u id cry sta ls a n d to d escrib e v o rtices in su p erco n d u cto rs a n d m a g n etic m o n o p o les. D i® eren tia l g eo m etry is u sed in th e stu d y o f g a u g e th eo ries a n d g rav ita tio n . W e m o d el sp a ce-tim e a s a rea l d i® eren tia b le m a n ifo ld . C o m p lex m a n ifo ld s (C a la b i-Y a u sp a ces) a re u sed b y S trin g th eo rists. T h eo retica l p h y sicists th ese d ay s g lib ly sp ea k o f ten d im en sio n a l sp a ces. It is im p o rta n t to rem em b er th a t in R iem a n n 's d ay, h ig h er d im en sio n s w ere v iew ed w ith su sp icio n ev en b y m a th em ticia n s. T h e id ea o f sp a ce a n d tim e fo rm in g a fo u r d im en sio n a l co n tin u u m w a s n o t th en in ex isten ce. R iem a n n 's a p p ro a ch w a s m a rk ed b y g en era lity a n d a b stra ctio n a n d p av ed th e w ay fo r th e m a jo r in tellectu a l d ev elo p m en ts o f th eo retica l p h y sics in o u r tim e.
Address for Correspondence Joseph Samuel Raman Research Institute Bangalore 560 080, India. Email:
[email protected]
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A c k n o w le d g e m e n ts: I th a n k B im a n N a th , S u p u rn a S in h a a n d S u m a ti S u ry a fo r rea d in g th ro u g h th is a rticle a n d h elp in g m e to im p rov e it.
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