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Home , Time in physics

GENERAL  ARTICLE and in and

Vasant Natarajan, V Balakrishnan and N Mukunda

Space and tim e are conceptsthatseem to be em - bedded in our very consciousness. A s we grow up, our `intuitive understanding' of these con- cepts seem s to grow as well. A nd yet the fact is that our understanding of space-tim e in the deepest scienti¯c sense is far from com plete, al- though w e have covered a considerable distance along the route. There m ay stillbe m any sur- prises aw aiting us on the road ahead. 1.It Began w ith G eom etry A ll of us have som e intuitive ideas about the natures (left) Vasant Natarajan is at the Department of , of space and tim e in which we are em bedded. Space IISc. His research interests appears to be the stage on which all events, experi- are in high-resolution ences and phenom ena take place, w hile tim e is like a spectroscopy, optical background against w hich this happens. A llobjects,in- frequency , and cluding ourselves,exist in space and change w ith tim e. use of laser-cooled atoms to search for time-reversal In this article, w e shalldescribe in sim ple and qualita- violation in the laws of tive show ourunderstanding ofspace and tim e has physics. developed over the ,and w hat we have learnt about their properties. W e w illsee that m any strands (right) V Balakrishnan is at com e togetherin thisstory { notonly from m athem atics the Department of Physics, IIT- Madras, Chennai. His and physics,but also,in som e im portant respects,from research interests are in b iology. dynamical , stochastics and statistical physics. In one ofhis essays,SchrÄodinger1 saysthisabouttheem - inent philosopher Im m anuel K ant (1724{1804): \K an t (centre) N Mukunda is at the ... term ed space and tim e,as he knew them ,the form s Centre for High Energy ofour m entalintuition { space being the form ofexter- Physics, IISc, Bangalore. His interests are classical and n al, tim e that of intern al, intuition ." W e begin with , this quotation both because it is so profound and be- theoretical optics and cause som e of K ant's other ideas w illappear later on. . A llofus have at least a prelim inary or com m on-sense

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1 The great physicist Erwin understanding ofthe nature ofspace through oursenses, Schrödinger(1887–1961; Nobel m ainly sightand touch.2 W e see objectsarranged in var- PrizeinPhysics, 1933),whodis- iousw aysrelative to one another;w e touch them ifthey covered wave mechanics and was oneofthefoundersofquan- are nearby;w e develop a feeling for shape,distance,per- tum mechanics,wasalsoawon- spective;and so on. T hus,through direct sensory expe- derfullylucidwriter on manypro- riences, space seem s to have a de¯nite character, even found subjects. His famous though the idea ofspace by itselfisquite abstract. T im e, books include Science and Hu- on the other hand,is m ore subtle. W e cannot reach it manism, Expanding , Statistical Thermodynamics, through our senses directly;w e can experience it and be What is Life?, Mind and Matter, aw are of its passage only internally through the m ind My View of the World, Space- in a non-sensory way. Because we cannot see or touch Time Structure, and Nature and tim e, extension in space seem s a little easier to grasp the Greeks. than in tim e. T hus,space hasto be abstracted from externalexperience,tim e from internalexperience. 2 It is an interesting fact that about 80% of our sensory input O ver and above this subtlety, space and tim e are re- comes from oureyes, andabout ally not sim ple concepts at all. A n obvious fundam ental 40% of our brain capacity is de- di®erence between the two is that space appears to be voted to processing visual infor- `controllable'{ in the sense that w e can m ove from one mation – we are very ‘visual’ point to another,w ith seem ingly arbitrary freedom . O n creatures. the other hand,tim e seem s to be som ething over w hich we have no control{ it just m arches inexorably on. W e are caught up in this m arch with no apparent choice. The old adage { `tim e and tide waitforno m an'{ cap- turesthisnotion perfectly. T hisalso m eansthatw e have m em oriesofw hathashappened up tillnow ,butno idea ofw hat the holds (contrary to w hat soothsayers and horoscope-w riters m ight tellyou!) Let us begin with space. The word geom etry, as you know ,m eans `m easuring the (or land)'. T he ori- gins ofgeom etry go back severalm illennia to the great E gyptian civilization (see also B ox 1). G eom etry arose in ancient Egypt from the repeated need to survey and re-establish boundaries ofland holdings after the annual °oodsofthe riverN ile. H ere isa passage from the G reek traveller and historian H erodotus (482?{434? B C ): Keywords Space, time, geometry, relativ- \T he K ing m oreover (so they said) divided the coun try ity. am ong allthe Egyptians by giving each an equalsquare

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Box 1. The O riginsofG eom etry Ancient Babylon and Egypt are well known as the civilizations in which the subject of geometry originated. However, there were other nodal centres as well. You are undoubt- edly familiar with some ofthe great and unique achievements ofancient Indian mathemat- ics, such as the invention ofthe decimal place-value ofnumbers, negative numbers, and zero. It is important to note that these remarkable mathematical advances (which took place from about 500 AD onwards) were preceded by equally signi¯cant achieve- ments in the Indian sub- in even earlier . In recent , much new light has been shed on the scope and advanced nature of truly ancient Indian mathematics. It is being gradually recognized that ancient India (especially the Indus Valley Civilization) was in many respects (including mathematics) fully the peer of the ancient Egyptian and Babylonian civilizations. The historians ofscience J J O'Connor and E F Robertson (see http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian mathematics.html) write, \... the study of mathematical astronomy in India goes back to at least the third mil- lennium BC and mathematics and geometry must have existed to support this study in ancient times." They go on to quote V G Childe in N ew Lighton the M ostAncientEast (Routledge and Kegan Paul Ltd., , 1952): \India confronts Egypt and Babylonia by the 3rd with a thoroughly individual and independent civilization of her own, technically the peer of the rest." What is clear is that astronomy was a prime mo- tivating factor behind the development of geometry, and related topics in ancient India. parcelofland,and m ade this his source ofrevenue,ap- pointing the paym entofa yearly tax.And any m an who was robbed by the river ofa partofhis land would com e to Sesostris and declare what had befallen him ;then the K ing would send m en to look into it and m easure the space by which the land was dim inished, so that there- after he shou ld pay the appointed tax in proportion to the loss. F rom this, to m y thinking, the G reeks learn ed the art ofm easuring land." Geometry was the So geom etry was the product ofpracticalhum an needs product of practical to m easure space,m uch as trade and com m erce led to needs to arithm etic. Sim ilarly,the practicalneed to m easure tim e measure space, (for exam ple,to keep track ofdaily sunrises or the an- nual°oods) gave rise to and . H um an much as trade and scienti¯c and technological progress has required ever- commerce led to m ore accurate clocks. A brief of tim ekeeping arithmetic. over the centuries is presented in B ox 2.

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Box 2. A B riefH istory ofTim ekeeping Timekeeping on earth by nature is as old as the planet itself, since all it requires is a periodic process. The periodic and revolution of the earth give rise to daily and yearly cycles, which have been used by nature as a much before human beings evolved. Many living organisms including show daily (or diurnal) rhythms reg- ulated by the , and seasonal patterns that repeat every . It is therefore natural that the earliest man-made clocks also relied on the sun. One example was the , which consisted of a pointer and a calibrated plate on which the pointer cast a moving shadow. Ofcourse this worked only when the sun was shining. The need to tell the time even when the sun was not out, such as on an overcast or at , made it necessary to invent other kinds of clocks. The earliest all-weather clocks were water clocks, which were stone vessels with sloping sides that allowed water to drip at a constant rate from a small hole near the bottom. Markings on the inside surface indicated the passage of time as the water level reached them. Other clocks were used for measuring small intervals of time. Examples included candles marked in increments, oil lamps with marked reservoirs, ¯lled with sand, and small stone or metal mazes ¯lled with incense that would burn at a constant rate. Time became signi¯cantly more accurate with the advent ofthe in the 17th . Galileo had studied the of the pendulum as early as 1582, but the ¯rst pendulum clock was built by (1629{1695) only in 1656. As we know from high-school physics, the time period of a pendulum executing small-amplitude oscillations depends only on its and the due to . Huygens' clock had an unprecedentedly small error of less than 1 per day. Later re¯nements allowed him to reduce it to less than 10 a day. While very accurate compared to previous clocks, pendulum clocks still showed signi¯cant variations, because a change of just a few degrees in the ambient temperature could change the length of the pendulum (owing to thermal expansion). Many clever schemes were therefore devised in the 18th and 19th centuries to compensate for such changes in length. In the history of clockmakers, the name of the great horologist John Harrison (1693{ 1776) stands out. He constructed many `marine chronometers'{ highly accurate clocks that were used on ships to tell the time from the start of the voyage. A comparison of local n oon (that is, the time at which the sun was at its highest point) with the time on the clock (which would give the time of the at the starting point) could be used to determine with precision the of the ship's current position. The British government had instituted the Longitude Prize so that ships could navigate on transatlantic voyages without getting lost. Harrison designed and built several prize- winning clocks based on the oscillations of a balance wheel. These maritime clocks had to maintain their accuracy over lengthy and rough sea voyages with widely-varying

Box 2. continued...

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Box 2. continued... conditions of temperature, pressure and humidity. The amazing features of Harrison's chronometer are exempli¯ed by the fact that, on a voyage from London to Jamaica, the clock only lost 5 seconds, corresponding to an error in distance of1 mile! Harrison's heroic story is brilliantly narrated by Dava Sobel in her book, Longitude: T he True Story ofa Lone G enius W ho Solved the G reatestScienti¯c Problem ofH is Tim e (Penguin Reprint Edition, 1996). However, all such mechanical clocks su®ered from unpredictable changes in timekeeping accuracy owing to wear and tear of the moving parts. Clock accuracies improved dra- matically in the ¯rst half of the 20th century with the development of the quartz crystal oscillator. These clocks rely on a property of quartz called piezoelectricity, wherein a me- chanical deformation produces an electrical signal (voltage), and vice versa. A vibrating crystal therefore produces a periodic electrical signal which acts as a precise clock. (For practical reasons, the frequency of the crystal or `resonator' is usually adjusted to be as close to 32768 or 21 5 Hz as possible. This is then stepped down by successive digital dividers to 1 Hz.) Today, we have the ubiquitous and inexpensive quartz wristwatch that is accurate to within a few seconds over a , and one does not have to worry about winding the clock every day or replacing the battery more than once a year or so. Life in our technology-driven society needs ever more precise timekeeping. Computers, manufacturing plants, electric power grids, satellite { all these depend on ultra-precise timing. The Global Positioning System (GPS), which determines the position of a receiver by triangulating with respect to the three nearest satellites, re- quires timing accuracy of 1 part in 101 2 . Modern atomic clocks, based on the oscillation frequency within an atom, have this kind of accuracy. But scienti¯c and technological needs keep pushing the requirement ever higher. Today's best atomic clocks have an accuracy of 1 part in 101 4 , or an error of 1 in 3 million years. We have indeed come a long way from the pendulum clock of just a few centuries ago.

T he science ofgeom etry m ust have developed gradually. From the E gyptians this know ledge oflength,direction, area and shape passed into the hands of the G reeks, w ho seem to have had a specialgift for this subject and for abstract thinking in general. A round 300 B C ,this m athem aticalknow ledge was codi¯ed and presented in a beautifully organized form by Euclid of A lexandria (circa 325{265 BC).H is E lem en ts becam e the suprem e pattern { or paradigm { for m athem aticalprecision and rigour, indeed for all intellectual w ork, for m any cen- turies. Starting from a few de¯nitions and axiom s (or postulates), w hich seem ed self-evident and consistent, and proceeding by a series of logical steps,m any

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Results which theorem s w ere derived as consequences. T hus, results were implicit in the w hich were im plicit in the starting axiom s were m ade explicit,or brought out into the open, by logicalargu- starting axioms m ents.From the econom y ofthe axiom sone could pass were made explicit, to the w ealth of derived consequences. A chieving this or brought out into style of thinking w as a great intellectualstep forw ard. the open, by logical arguments. 2. The N ew tonian W orldview W enow passoverseveralcenturiesand com eto thebirth ofm odern science. G alileo G alilei(1564{1642) is gener- ally regarded as the ¯rst m odern scientist. In his w rit- ings he says clearly that m athem atics is the language of N atu re: \It is w ritten in m athem atical lan guage." Soon after, in the latter part of the 17th century, the ¯rst clear statem ents about space and tim e w ere given by Isaac N ew ton (1643{1727) in his m onum entalthree- volum e w ork on the m athem aticalprinciples ofnatural philosophy,usually referred to as the P rincipia for sh ort. T he style ofthisw ork is also `Euclidean'{ a sm allnum - ber ofde¯nitions and law s are given at the beginning, and m any theorem sare then proved asconsequences. In fact,N ew ton w entso far as to allhis derivations using geom etricalargum ents (\con m ore geom etrico"). A bout space and tim e,he says that although they are com m only thought of in relation to m aterial objects placed in them ,hewould de¯nethem on theirow n.H ere are hisfam ousstatem entsfrom the opening pagesofthe P rincipia: \A bsolute, true, an d m athem atical tim e, of itself, an d from its ow n n ature, ° ow s equably w ithout relation to anything external,and by another nam e is called dura- tion ." Newton presented \A bsolute space, in its ow n n ature, w ithout relation to all his derivations anything external, rem ains always sim ilar and im m ov- using geometrical able." arguments.

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N ew ton assum ed,ofcourse,that space obeyed the law s Newton assumed ofE uclidean geom etry. that space obeyed Itm ustbe em phasized that,forthe furtherdevelopm ent the laws of ofscience,it w as very im portant that N ew ton explained Euclidean so clearly hisview saboutspaceand tim e. T hey could be geometry. exam ined in a carefulw ay,and m ade the basisforfurther progress. N ew ton's view s w ere already criticized in his ow n lifetim e by the philosopher G eorge (B ishop) B erke- ley (1685{1753) and the great m athem atician,physicist and philosopher G ottfried W ilhelm von Leibnitz (1646{ 1716). T he latter felt that there was no need to think of space apart from the relation of m aterial objects to one anotherand declared: \I hold space to be som ething purely relative as tim e is." M uch later,in the late 1800's, the A ustrian physicist-philosopher E rnst M ach (1838{ 1916) also critically exam ined N ew ton's view s,not only aboutspace and tim e,butaboutm echanicsaswell.A ll this only em phasizes the crucial role and im portance of N ew ton's unam biguous statem ents. A lbert Einstein (1879{1955; N obel P rize, 1921) him self acknow ledged this fact w hen he stated in one ofhis essays,\... w hat we have gained up tillnow would have been im possible w ithout N ew ton's clear system ." T hisN ew tonian picture ofspace and tim e served physics 3 extrem ely well for over two centuries. T he word `ab- Thus, Newton treated space and time as quantities that ap- solute'in his statem ents is im portant: it strongly sug- peared in his laws of motion as gests that space and tim e them selves are una®ected by independent variables, while all the physical processes that occur in them .3 O n ce other physical quantities were again,E instein conveysthe essence ofthisidea very w ell: functions of these. We have he says `absolute'm eans \... physically real ... indepen - already noted the ‘controllabil- ity’ of space, in contrast to the den t in its physical properties, having a physical e® ect, ‘inevitability’ of time. These ideas but n ot itself in° u en ced by physical con dition s." F ol- are embodied in Newton’s con- low ing his statem ents on the absolute natures of space ception by supposing that ob- and tim e,N ew ton gives his three law s ofm otion,w hich jects exist in space, but evolve we learn at school. M uch later,towards the end ofthe in time. The in time was determined by his theory of P rincipia,he introduces his Law of U niversalG ravita- fluxions, which we study today tion.W e do notgo into Newtonian m echanics orthe as differential calculus.

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General relativity inverse square law ofgravitation,asourfocusis on space changed precisely the and tim e. But we m ention here that general relativ- Newtonian ity (E instein's theory ofgravitation,unveiled by him in assumption of the 1915{16) changed precisely the N ew tonian assum ption ofthe absoluteness ofspace and tim e. W e w illreturn to absoluteness of this point later on. space and time. 3. Space,Tim e and H um an Intuition Building on the foundations provided by G alileo and N ewton,both m echanics and astronom y m ade rem ark- able advancesduring the 18th century. Tow ardsthe end, the law sofelectricity and m agnetism also seem ed to fol- low the generalG alilean{N ew tonian pattern. K ant w as so im pressed by these successes that he tried to present a philosophicaljusti¯cation forthem . H ism ain idea w as th at of th e syn thetic a priori principles. T hese are non- trivialstatem ents about the properties of nature w hich are necessarily true or binding,but they are regarded as notderived from experience. T hatisthe m eaning ofthe p h rase `a priori',nam ely,som ething know n even before experience. A s far as we are concerned here,K ant re- garded the concepts ofabsolute and separate space and tim e,as N ewton had viewed them ,as synthetic a pri- ori principles;so also the uniform °ow oftim e and the 4 As it happens, he was wrong E uclidean geom etry ofspace.4 on both counts (for reasons he could not have foreseen). A pro- K ant claim ed that allthese ideas are already present in found lesson about nature is to our m inds before w e have contact w ith,and experience be learnt here. See Box 3. of,nature;and that they m ust be necessarily valid be- cause science would be im possible w ithout them . T he m odern understanding of this m atter com es, surpris- ingly enough, from evolutionary biology. It is largely due to the w ork ofthe zoologist and ethologist K onrad Lorenz (1903{89; N obelPrize,1973) around 1940,fur- ther developed and described by the m olecular biologist M ax D elbrÄuck (1906{81;N obelPrize,1969). T he basic idea is that as species evolve biologically over long peri- ods oftim e,governed by naturalselection,they acquire

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B ox 3. C aution: N ature C annot be Second-G uessed! If a single loud message comes across from the developments in the physical and the life sciences during the last one hundred years or so, it is surely this: D o n ot try to second-guess nature! As Sherlock Holmes put it so pithily, in A Study in Scarlet,\It is a capitalm istake to theorize before you have allthe evidence. It biases the judgm ent." This dictum can hardly be over-emphasized when it comes to unlocking the secrets of nature. Time and again, nature has corroborated the great geneticist and biologist J B S Haldane (1892{1964) who said, \Now m y own suspicion is that the is not only queerer than we suppose, but queerer than we c a n su ppose". Both relativity and quantum mechanics, the two cornerstones of our current understanding of the physical universe, show us in dramatic fashion how misleading our `intuition' can be, with re- gard to the most fundamental aspects of nature. We know now that (i) space is n ot Euclidean, (ii) the Newtonian assumptions of space and time as separate absolutes are not strictly correct, and (iii) space-time is quite complex even at the level up to which we understand it currently (and is likely to turn out to be even more so at the fundamental level). Now, philosophy may lead to some insights into, and systematization of, scienti¯c knowledge. B u t it is n ot, in itself, a reliable w a y to discover scien ti¯ c facts. This is why it is not appropriate to begin scienti¯c enquiry with a pre-conceived set of notions that are postulated to be `absolute truths' simply because they appear to be self-evident or based on `common sense'. As Einstein pointed out, \Com m on sense [in this context] is the collection of prejudices acquired by eighteen ". As we have explained subsequently in the main text, what is meant by all this is the fol- lowing: the `intuition' we acquire with regard to the natural world around us is based on the behaviour ofobjects and phenomena we encounter in it in everyday life, and perceive with our senses. For both these reasons, the information thus gathered is restricted to a rather narrow range of variation (or window) of the values of physical quantities such as , length, time, velocity, momentum, force, and so on. This is the so-called `world of middle dimensions'. It informs our beliefs and our expectations (\intuition") regard- ing the nature of the physical world. Broadly speaking, comprehension of the world of middle dimensions at a certain level, and the ability to deal with it, has been hard-wired into our brains for reasons based on survival in the evolutionary sense. However, there is absolutely no reason to expect that nature itself would be restricted to this narrow range of values of physical quantities, and indeed it is not { in a most spectacular way. It turns out that our own windows of mass, length and time only encompass about seven or eight orders ofmagnitude, while our present state ofknowledge shows that nature and natural phenomena extend over at least ten tim es thatnum ber oforders ofm agnitude. It is hardly surprising, then, that our naive, essentially hard-wired, intuition and common sense are totally inadequate to comprehend the universe unaided. We need the assistance of better, ¯ner and more powerful tools to do so. These are provided by our instruments (to probe nature) and by our mathematics (to analyze our ¯ndings).

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Our intuition does not and retain those capabilities w hich are m ost useful for survival. A m ong these are the abilities to recognize the apply either to the m ost im portant physical features of the w orld around sub-microscopic world us, corresponding to our ow n scales of size and tim e. of atoms or, at the T hese include the properties ofspace and tim e as stated other end, to the by N ew ton,features ofnature that are valid (to a high cosmos of galaxies, degree ofapproxim ation) at our ow n level. T hus,these since these remote abilities are the result of slow experience of nature by parts of nature do not the species over im m ense stretches of tim e, but to the appear to be individualm em ber of the species they are available at immediately relevant and soon afterbirth. T his is why they seem to be given to our day-to-day a priori, availab le before any actualexperiences during . life. From the present perspective, the m ain point is that allthis relates only to our w orld ofnorm alevery- day experiences in life. It does not apply either to the sub-m icroscopic w orld ofatom s or,at the other end,to the cosm os of galaxies,since these rem ote parts ofna- ture do not appear to be im m ediately relevant to our day-to-day existence. W e referred earlier to our intuitive ideas about space and tim e. N ow this intuition,and the w ay itisbuiltup, also has a fascinating explanation. It is not som ething available ready-m ade at birth. R ather,it is acquired in the early years of infancy out of experience, using the abilities and apparatus of sensory perception. T his is w hat biologicalevolution gives to each ofus at birth { n ot kn ow ledge about the im portant features of nature around us,but the capacity to learn about them . The experiments of the child T he experim ents of the child psychologist Jean Piaget psychologist Jean (1896{1980) in the 1950's led to rem arkable insightsinto Piaget in the 1950’s the learning process in infancy and childhood, in par- led to remarkable ticular into learning about space and tim e. It seem s that the regularity ofevents occurring soon after one an- insights into the other in tim e is recognized and rem em bered very early learning process in on; w hile the idea of space com es a little later. Sev- infancy and eral years later com e the concepts of universal space childhood. and tim e com m on to, and containing, all things and

852 RESONANCE  September 2008 GENERAL  ARTICLE events. Piaget suggested that the order in w hich chil- We are born with dren acquire concepts of space is opposite to w hat is the capacity to taught at schooland college. B road concepts that are acquire knowledge topological in character { the idea of nearness, of the about nature, not the deform ability of shapes into one another, and so on { are acquired ¯rst;next,concepts ofprojective geom etry knowledge itself. such as direction and perspective;and only at the end, the underlying concepts ofdistance,shape,size,congru- ence,etc.,on w hich Euclidean geom etry is based. B ut in form alteaching the sequence follow ed is exactly the opposite. E uclidean geom etry is alw ays taught ¯rst,at an elem entary level. T his is follow ed,at a m ore m ature level of instruction in m athem atics (or applications of m athem atics), by the concepts of projective geom etry. Topology is introduced at an even m ore advanced level of instruction! To sum m arize:W e are born w ith the capacity to acquire know ledge about nature,not the know ledge itself. N at- ural selection has tuned these capacities to the actual properties ofthe w orld ofthose dim ensions (or sizes,or m agnitudes) that are directly and im m ediately relevant to our survivalin the naturalenvironm ent. B ut we re- tain no later m em ories ofthis learning process that w e go through in ourinfancy;and w hen w e reach adulthood w e im agine thatw e w ere born w ith an `intuitive'know l- edge ofnature in this dom ain. R eturning to Lorenz,his ideas are w ellcaptured in these rem arks by D elbrÄuck: \... there can be little doubt that our spatial con cepts develop in childhood by way ofan adaptation to the world in w hich w e live ... the cogn itive capacity that perm its m an to an alyze space in geom etric term s m ust, to a large exten t, be evolution arily derived." In an appendix titled `Physics and Perception' in his b o ok S pecial R elativity,the physicist-philosopher D avid B ohm (1917{94) describes how a child gradually builds concepts of space and tim e outside of herself, and the

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role ofm em ory in the construction ofthe ideas ofpast, presentand future tim e.O nly by the age often orm ore does a child develop the m entalcapacity to conceive of a universalspace and a universaltim e com m on to all objects and events that are perceived. T his just goes to show how deep and genuinely di± cultthese concepts are,and thatwe are notborn with them . 4. B eyond Euclidean G eom etry A fter this briefexcursion into biology,let us com e back to m athem atics. Euclid's geom etry has been adm ired and studied for over two thousand years. A s we have already said,the approach startsw ith a very sm allnum - beroffundam entalpostulates,and buildsupon them by logicalargum ent. B utfrom very early tim espeople w ere puzzled by the statusofthe ¯fth postulate ofE uclid,also 5 Let L beastraightlineofinfinite know n as the `parallelpostulate'.5 W asitindependentof extent at both ends, and let P be a point that does not lie on L. the ¯rstfourpostulates,orcould itbe derived from them The parallel postulate asserts (in w hich case there would be no need foran additional that there is one – and only one ¯fth postulate)? Em inent thinkers over the centuries – straight line passing through P pondered over this problem , from the astronom er and that is parallel to L. geographer C laudius Ptolem y (circa 85{165 AD)to the See C R Pranesachar, Euclid and ‘The Elements’, Resonance, fam ous m athem atician A drian-M arie Legendre (1752{ Vol.12, No.4, 2007. 1833). M any equivalent reform ulations of this postu- late w ere found,but the unchanging beliefw as that the geom etry ofspace had to be Euclidean,for the sake of 6 Like so many fundamental ad- consistency. A long the w ay,R en¶eD escartes(1596{1650) vances, the idea of coordinate invented the m ethod of coordinates to represent points geometry first appeared in the in space, so that the pow erful m athem atics of algebra appendix to a book! The title of becam e available for tackling problem s in geom etry.6 Descartes’ book, published in 1637, translates to Discourse A s w e have seen,K ant subsequently expressed the view on the Method of Properly Con- that physicalspace had necessarily to obey the law s of ducting One’s Reason and of E uclidean geom etry,and thatthere w asno otheroption. Seeking the Truth in the Sci- ences. Nearlyfourhundred years A few after K ant, in the early 1800's, the ¯- later, thetitles ofscientific books nal resolution of this long-standing problem occurred. and papers today are somewhat T he ¯fth postulate was indeed independent ofthe oth- more specialized, if less impos- ing! ers,since it could be replaced in a self-consistent m an-

854 RESONANCE  September 2008 GENERAL  ARTICLE ner by other assum ptions, and one could think of al- 7 There is evidence that Gauss had conceived the idea of non- ternatives to E uclid's geom etry. T his breakthrough w as Euclidean geometry already in achieved independently by three m athem aticians{the in- 1792, as a precocious teenager. com parable C arl Friedrich G auss (1777{1855), around But his first mention of the sub- 1824,building on his theory ofcurved surfaces;N ikolai ject is in a letter written in 1824. Ivanovich Lobachevsky (1792{1856),in 1829;and J¶anos Lobachevsky made his discov- ery in 1826, and published his Bolyai(1802{60)in 1832. T huswasborn the subjectof work in 1829. This was the first 7 non-E uclidean geom etry. formal publication of non-Euclid- ean geometry. Bolyai arrived at A round the m iddle of the 19th century, G auss' gifted the idea in 1823, but the first studentand one ofthe greatestofm athem aticians,B ern- publication of his results was hard R iem ann (1826{66), took the next big step: the dated 1832. All three mathema- creation ofdi®erentialgeom etry. H e presented his ideas ticians dealt with what we now call hyperbolic geometry, appli- in his fam ous probationary lecture titled O n the hy- cable to of negative cur- potheses thatlie atthe foundations ofgeom etry, given at vature. Subsequently, Riemann G Äottingen on June 10,1854. T he fundam entalinsight completed the picture with w as to determ ine the geom etry ofa space starting from spherical geometry, applicable the de¯nition of(the square of) the intervalor distance to spaces of positive curvature. betw een nearby points in the space. From this,the con- cepts of tensors, parallel transport, intrinsic di®erentia- tion,connection,curvature,etc.,could allbe developed. This was a stupendous achievem ent, and m any great contributions from a galaxy of geom eters follow ed.8 In 8 The list includes Bruno particular,R iem ann's m ethods w ere pow erfulenough to Christoffel (1829–1900), GregorioRicci-Curbastro (1853– dealw ith spaces ofany num ber ofdim ensions. 1925), his son-in-lawTullioLevi- 5. M axw elland the R oad to SpecialR elativity Civita (1873–1941), and Luigi Bianchi (1856–1928), among W e now com e back to physics. In 1865,not long af- others. Like the Soviet tradition in the theory of probability, and ter R iem ann developed di®erentialgeom etry,one ofthe the later Indian tradition in sta- greatest scientists ofalltim e,the brilliant physicist Jam es tistics, Italy has had a fine tradi- C lerk M axwell (1831{79), put together his system of tion in geometry. coupled partialdi®erentialequations fortim e-dependent electric and m agnetic ¯elds{the basis for allelectrom ag- netic phenom ena. H e then discovered the possibility of freely propagating electrom agnetic w aves, identi¯ed them w ith light,and thus uni¯ed three ¯elds of classi- cal physics{, m agnetism and optics. M axw ell assum ed,tacitly,the N ew tonian picture ofseparate and

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absolute space and tim e,and also believed in the exis- tence ofa m edium ,the `ether',to carry electrom agnetic 9 For a brief but excellent over- w aves.9 view of Maxwell’s ideas and dis- coveries, see the article titled H owever,it soon becam e clear that there was a deep- : a force seated contradiction betw een N ew ton's m echanics and for physics by F Everitt, at the the set ofequations encapsulating this new understand- website http://physicsworld. com/cws/article/print/26527.It is ing of electrom agnetism . T he equations of m echanics interestingtonotethat theterms w ere unchanged in form (or `form -invariant') under a and relativity, both of which setoftransform ations ofthe space and tim e coordinates are so basic to contemporary called G alilean transform ations. T hese transform ations physics, were first introduced in allow for the arbitrariness in the origin of the coordi- their current senses by Maxwell. nate axes and of tim e, in the alignm ent or orientation ofthe coordinate axes,and in the choice ofthe inertial fram e of reference used to analyze the m echanicalsys- tem . T im e, of course, is assum ed to `°ow inexorably' at exactly the sam e rate in allm utually inertialfram es ofreference. A stonishingly enough,M axw ell's equations did not share the property of form -invariance{they did n ot rem ain unchanged in form under G alilean trans- form ations! T his, in turn, im plied that the ether had a special feature that could be established experim en- tally: M axw ell proposed electrom agnetic experim ents w hich could detect the m otion ofthe earth through the ether,the m edium thathe believed to be at,and in fact d e¯ n ed , absolute rest. (R em em ber that, in G alilean{ N ew tonian m echanics,there isno such thing as absolute rest.) H ow ever,the fam ous and crucialexperim ents of 10 Albert Abraham Michelson M ichelson and M orley,1 0 done in 1887, failed to detect (1852–1931; Nobel Prize, 1907) the expected e®ects, and this led to a crisis. M any and Edward Morley (1838– great¯guresin physicsand m athem atics,including H en- 1923). drik A ntoon Lorentz (1853{1928;N obelP rize,1902)and H enriPoincar¶e(1854{1912) w orked on attem pts to set- tle this serious issue. T he ¯nalresolution ofthe problem cam e in 1905,w ith E instein's Special T heory of R elativity. It turned out that N ewton's m echanics had to be m odi¯ed to fallin line w ith M axw ell's electrom agnetism ! It w as the lat-

856 RESONANCE  September 2008 GENERAL  ARTICLE ter that led to a new and m ore correct view of space The earlier and tim e, and m echanics had to be m ade consistent Galilean w ith this view . It becam e clear that the equations of transformations of both m echanics and electrom agnetism would have to space and time be form -invariant under a com m on set of transform a- gave way to new tions ofthe space and tim e coordinates,com prising the transformations so-called Poincar¶egroup (also know n as the inhom oge- named after neous Lorentz group). N ew ton's separate and individ- ually absolute space and tim e were replaced by a new Lorentz, and these uni¯ed space-tim e w hich alone w as absolute, the sam e form the basis of for allobservers. B ut each observerseparates space-tim e . into her own space and herown tim e in her own way. T he earlier G alilean transform ations of space and tim e gave way to new transform ations nam ed after Lorentz, and these form the basis of specialrelativity. N ew ton had posited thattim e w as universal,the sam e forevery- one,°ow ing inexorably from the into the future. If tw o events occurred sim ultaneously,allobservers w ould agree on it. B ut now ,w ith the specialtheory ofrelativ- ity, that was show n to be untrue. Sim ultaneity is not absolute,but depends on the fram e of reference. T wo events that occur sim ultaneously according to one ob- server need not do so for another observer in a di®erent fram e of reference. A ccording to the Lorentz transfor- m ations, not only the coordinates of points in space, butalso the tim es ofevents change in a speci¯c m anner from one reference fram e to another. A s a consequence, the distances betw een objects or events,as w ellas the tim e intervals betw een events,also vary from one fram e of reference to another. Even though space and tim e The distances rem ain profoundly di®erent in their physicalcharacter- between objects or istics,the new rules oftransform ation unite them m uch events, as well as m ore closely than is the case in the G alilean{N ew tonian w orld v iew . the time intervals between events, E ven in specialrelativity,though,com bined space-tim e also vary from one rem ains absolute. Its quantitative characterization by di®erent observers (using m etre rods, clocks and their to another.

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equivalents) is encoded in the Lorentz transform ation rules. It serves as the stage for allphysicalphenom ena, but is una®ected by them . 6. G eneralR elativity: D ynam icalSpace-T im e T he essentially ¯nalstep (w ithin the fram ew ork ofclas- sicalphysics)in the developm entofourunderstanding of space and tim e occurred w ith E instein'sG eneralT heory ofR elativity,w hich he com pleted in 1915. T his is a the- 11 11 “A field theory par excellence” ory ofsuperlative beauty. In it,¯nally,space and tim e – L D Landau and E M Lifshitz, no longer rem ain a stage on w hich physical processes The Classical Theory of Fields, take place,but them selves becom e actors or active par- 4th edition, Butterworth– ticipants in the proceedings. T he spirit behind this idea Heinemann, 1980. is beautifully expressed by E instein: \It is con trary to the m ode of thinking in scien ce to con - ceive of a thing ... w hich acts itself, bu t w hich can n ot be acted upon ." T he very geom etry ofspace-tim e,w hich E instein identi- ¯ed w ith gravitation,now becom es changeable and dy- nam ical. G eom etry acts upon m atter and electrom ag- netism , and in turn is acted upon by them . In the 12 12 The eminent physicist John expressive w ords ofW heeler: \S pace-tim e tells m atter ArchibaldWheeler (1907–2008) how to m ove,m atter tells space-tim e how to curve." In was one of the world’s leading generalrelativity, E instein used extensively the m athe- experts on gravitation. As you m atics initiated by R iem ann and developed further by may know, he coined the term ‘black hole’. C hristo®el,R icci,Levi-C ivita and others. B ut the fun- dam entalphysicalsense in w hich he went beyond R ie- m ann w asthat,from the geom etricalpointofview ,tim e w as treated on the sam e footing asspace,in spite ofthe profound di®erencesbetw een them . T hiswassom ething The very geometry of R iem ann had not foreseen. It becam e possible only be- space-time, which cause specialrelativity had already been developed (by Einstein identified with E instein him self). A m ore detailed discussion ofgeneral gravitation, now relativity requires m athem atics,and w e do not go into becomes changeable this here. H owever, we describe som e of the relevant and dynamical. salient features of in B ox 4.

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B ox 4. A Paean to G eneralR elativity There were three unique features about general relativity that Einstein found appealing. They also represented part of the reason for his lifelong opposition to quantum mechanics as an approach to understanding nature. He believed that quantum mechanics was indeed successful in describing nature within its domain ofapplicability, and that a future uni¯ed ¯eld theory would have to reproduce the results of quantum mechanics, perhaps as a linear approximation to a deeper nonlinear theory. If true, this would be similar to how the relativistic gravitational ¯eld of general relativity (with a ¯nite propagation of the gravitational force) led to Newton's law of gravitation (with its -at-a-distance force) in the non-relativistic limit. But Einstein was convinced that quantum mechanics was not the correct approach to deducing the fundamental laws of physics. The three important features of general relativity were as follows. With the equations of general relativity, Einstein found that space-time was no longer ² just a passive stage on which particles and ¯elds performed their acts, but was an active participant in the performance. Thus, the geometric structure of space-time was deter- mined by the matter in it, and ofcourse the matter responded to this geometry and was constrained by its structure. General relativity was the ¯rst theory in physics that was inherently nonlinear. An ² important consequence of this was that the equation of motion of an in¯nitesimal test particle in the gravitational ¯eld of a massive object was contained in the ¯eld equations themselves. This equation speci¯ed the geodesic path followed by the particle in the curved space-time in which it moved. In other words, one did not have separate equations for the gravitational interactions between the matter present, and for the response of the matter to these interactions. Everything was contained in one set of ¯eld equations. In contrast, a linear theory like Maxwell's equations for the electromagnetic ¯eld could only describe the (electromagnetic) ¯eld manifestation of(charged) matter, while the response of the matter (or inertial manifestation) was contained in separate `'. Indeed, this is true of any linear theory { hence Einstein's opposition to regarding the linear formalism of quantum mechanics as the ¯nal or underlying theory. For the ¯rst time in physics, a theory predicted that the inertia of a system depended ² on its surroundings. In keeping with Mach's idea that inertia is a consequence ofa body's interactions with the rest of the universe, the equations of general relativity showed that the inertia of a system increases when it is placed in the vicinity of other ponderable . Inertia was no longer some inherent, given property of a system, but was at least partly determined by the environment. Einstein's unful¯lled dream was to ¯nd a fully uni¯ed ¯eld theory which would show that all of the inertia (and not just part of it) was due to the interactions with the environment.

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T he well-know n physicistC hen-N ing Yang (1922{ ;N o- 13 Paul Adrien Maurice Dirac bel Prize, 1957) was once asked by D irac13 w h at h e (1902–84; Nobel Prize, 1933), (Y ang) regarded as E instein's greatest discovery. W hen one of the founders of quantum mechanics. Y ang chose generalrelativity,D irac disagreed and opined that it had to be specialrelativity,because it show ed for the ¯rsttim e thatspace and tim e had to be treated and transform ed together. G eneralrelativity provided for the ¯rst tim e a depend- able language fordiscussing problem sofcosm ology { the behaviour ofthe universe as a w hole. It also led to the study ofexotic objects like black holes. O n the practical side,the G lobalPositioning System in current use takes into account the e®ects ofboth specialand generalrel- ativity. Ifthis is not done,the G PS would go hayw ire w ithin about an . In the last few years,a spectac- ular con¯rm ation ofgeneralrelativistic e®ects has been found in a double-pulsar system about 2000 light years aw ay from us. T he pulsars,separated by a distance ofa m illion kilom etres,orbit around each other w ith a tim e p eriod of 2:4 . T his tim e period is a®ected by the strong gravitational¯elds present (about 100,000 tim es strongerthan thatdue to the sun). A com parison ofthe observed variationsin the intervalbetw een the pulsesar- riving here from the system w ith the calculated values 14 Hermann Weyl (1885–1955) show s that the predictions ofgeneralrelativity are borne was one of the great mathema- out to an astonishing accuracy of99:95% . ticians – today, we would also rankhimasapre-eminentmath- Soon after E instein presented general relativity, H er- ematical physicist – of the 20th m an n W ey l14 in 1917{18 suggested an extension ofR ie- century. His book Raum-Zeit- m annian geom etry for space-tim e, involving w hat he Materie (Space-Time-Matter), first published in 1918, was called the gauge principle,w hich he thoughtw ould unify highly influential for several de- gravitation and M axw ell'sclassicalelectrom agnetism . A cades in spreading the ideas of few years later, T heodore K aluza (1885{1954) in 1921 relativity among physicists and and O skarK lein (1894{1977)in 1926 independently pub- mathematicians. Weyl also lished theirideason anotherw ay to unify the tw o forces, played a prominent role in ap- plying the ideas of symmetry based on increasing the num ber ofdim ensions ofspace- and group theory to physics, in- tim e from four to ¯ve { nam ely,four spatialdim ensions cluding quantum mechanics. and one tim e dim ension. K lein, in fact, pioneered the

860 RESONANCE  September 2008 GENERAL  ARTICLE idea that the extra space dim ension w as actually `curled up'into a tiny circle ofradius ofthe order ofthe P lanck len gth , l = G h =c3 10 35 m etres. H e also advanced P q » ¡ the idea that the `quantization'of could be related to a topologicalfeature such as the curling up ofa dim ension into a circle. T hese ideas w ere extrem ely original and beautiful, but did not succeed or lead to any progress at that tim e. M any decades later, how- ever, they have becom e im portant, in a m odi¯ed and extended form ,in the context ofstring theory. 7.Space and Tim e in Q uantum Physics A llthat we have discussed so far has been in the realm ofclassicalphysics. W e m ust now give at least a brief accountofthe surprising resultsthatem erge in quantum th eory vis-µa-vis space and tim e. T he initialform ofquantum m echanics,com pleted dur- ing the period 1925{27, assum ed that space and tim e w ere N ew tonian, i.e., non-relativistic, as w e w ould say today. T hissu± cesform ost ofchem istry,atom ic,m ole- 15 With the important proviso 15 cular and even nuclear physics. Soon after, in 1928, that the effects of spin are to be D irac found a description of the electron w hich com - incorporated. The origin of spin bined the (then) new quantum m echanics w ith special is often believed to lie in relativ- relativity and w as spectacularly successful. From the istic quantum physics, but is actually a logical and rigorous 1930's to the 1970's and beyond,this led to the im pres- consequence of non-relativistic sive developm ent ofquantum ¯eld theory in w hich,too, quantum mechanics. space-tim e is treated according to special relativity{so that once again it (space-tim e) rem ains a stage for phe- nom ena. T he theoriesofthe strong,electrom agnetic and weak interactions are allof this form . A long the way, in 1956,T sung-D ao Lee (1926{ ;N obelP rize,1957)and Yang found thatin the w eak processes,such asthose re- sponsible for radioactive decays,space re°ection or par- Nature does ity is not a valid sym m etry. T hat is,nature does distin- distinguish between guish betw een left-handedness and right-handedness at left-handedness and a fundam entallevel,contrary to w hat one m ight expect right-handedness at `intu itively'. a fundamental level.

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The inexorable A nalogous to space re°ection,w e have tim e re°ection or march of time tim e reversal. T he situation here is quite subtle. T he gives us the notion inexorable m arch oftim e gives us the notion ofan ar- of an row of tim e that always points to the future. A drop of ink that falls into a glass of w ater disperses sponta- that always points neously in the w ater, tillit becom es virtually invisible. to the future. It does not re-assem ble into the originaldrop,no m at- terhow long we wait. A llofusage and grow older;we never evolve `backwards'from old age to infancy. A nd yet,the law s governing alm ost allkinds of tim e evolu- tion, both classical and quantum m echanical, are { at the fundam ental level { alm ost alw ays tim e-reversalin- variant. T hat is,they rem ain unchanged w hen the sign ofthe tim e variable ischanged. T he deep question,then, is h ow th is m icroscopic reversibility in tim e leads,nev- ertheless, to m acroscopic irreversibility. The answer is a subject in itself,and w e shallnot go into it here. It is related,as you m ight guess,to the statisticalnature of the second law oftherm odynam ics and the m echanism by w hich increases spontaneously. It is also re- lated to dynam ical chaos (or exponential sensitivity to initial conditions), and the im possibility of specifying these conditions w ith in¯nite precision. W e m ust also m ention that a com plete answ er that is fully w ithin the purview ofquantum m echanics is not yet known,and that the m atter is the subject ofon-going research. But there is yet another twist to the tale. There is now indirect evidence that there is a certain violation

16 As you maybe aware, protons of tim e-reversalsym m etry in the fundam entalphysical and neutrons (via the quarks of law s them selves. T his is based on the discovery in 1964 which they are composed), as ofw hat is called C P -violation in certain elem entary par- well as electrons, have intrinsic ticledecays,forw hich Jam esW C ronin (1931{ )and Val magnetic dipole moments re- L Fitch (1923{ )were awarded the N obelPrize in 1980. lated to the spins of these par- ticles. Thisdoesnotviolatetime- A consequence of the violation of tim e-reversal invari- reversal invariance. We are now ance is that a particle such as an electron, or a neu- speaking of intrinsic electric di- tron,or a free atom w illhave an intrinsic electric d ip ole polemoments,whoseexistence m om en t.16 T he search for such an electric dipole m o- would indicate such a violation.

862 RESONANCE  September 2008 GENERAL  ARTICLE m entisan im portantendeavourin experim entalphysics, The violations of as it am ounts to a test ofthe validity ofa fundam en- parity invariance and talprinciple of nature. Severallaboratories around the time-reversal w orld are currently engaged in this pursuit (e.g.at H ar- invariance suggest vard,Yale,C olorado,W ashington,and P rinceton in the that nature has some US,and Im perialCollege and K VI G roningen in Eu- asymmetry built into rope),including that ofone ofthe authors (V N ) at the Indian Institute of Science. T he violations of parity in- it. variance and tim e-reversal invariance m entioned above suggest that nature has som e asym m etry built into it: elem entary particle interactions do not exhibit the de- gree of sym m etry w ith regard to space-tim e itself that one m ight expect a priori. T hese asym m etries,in turn, have im plicationsin cosm ology and the m annerin w hich the universe has evolved from the . A s you can see,allthese aspects go to show how deeply inter- tw ined are the apparently diverse parts ofthe subject of physics{clearly, nature is above these arti¯cial distinc- tion s. 8.W hatD oesthe Future H old? Finally, we com e to the problem of com bining general relativity and quantum m echanics. T his problem has been attacked since the early 1930's,but it has proved to be uncom m only di± cult. In a very realsense,itm ay be regarded as an `ultim ate problem 'ofsorts,atleastas far as the physics built up over the last four centuries or so is concerned. A s the jury is stillout on this question whatwe go on to say now m ustberegarded astentative and speculative. In recent decades,som e prom ising lines ofprogress m ay The problem of have appeared in the attem pts to tackle the problem . combining general T hey involve quite novel ideas and, along w ith them , relativity and com plicated m athem atics to im plem ent these ideas in a quantum consistent m anner. In essence,itappears that the ingre- mechanics is an dients or features necessary for a consistent and satisfac- ‘ultimate problem’ tory am algam ation of gravity and quantum m echanics of sorts.

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There could be an are likely to be,at the very least,the follow ing: (i)the underlying symmetry replacem ent of zero-dim ensional point-like particles as between bosons and the ultim ate fundam ental objects in the universe by fermions called extended objects, speci¯cally, one-dim ensional strings ; supersymmetry that is (ii) a space-tim e w ith a dim ensionality higherthan four{ spontaneously broken perhaps a space-tim e w ith nine spatialdim ensions and in the present-day one tim e dim ension,six ofthe spatialdim ensions being universe. curled up (or `com pacti¯ed') to a size w ithin the P lanck length, but in a far m ore (topologically) intricate and com plicated m anner than K lein anticipated; and (iii) an underlying sym m etry between bosons and ferm ions called supersym m etry that is spontaneously broken in the present-day universe. Taken together, they consti- tute the basic features of the so-called superstring the- ory. W hat is slow ly em erging is even m ore interesting,and cuts even deeper. O ver and above the features listed above, there appear to be theoretical indications that the resolution ofthe problem w illentaileven m ore dras- tic revisions ofour understanding ofboth quantum m e- chanics and space-tim e itself. For instance: this uni- verse m ay be only one ofa vast (or even in¯nite) array ofever-branching-out universes in a `m ultiverse';the di- m ensionality ofspace-tim e m ay em erge as a dynam ical property ofthis universe;there m ay be an ultim ate un- derlying granularity ordiscretenessin the very notions of length and tim e durations; and space-tim e coordinates (even including the possible higher-dim ensional ones) m ay have to be augm ented by other,`internal'variables of a di®erent kind in order to provide a com plete de- scription ofnature at the m ost fundam entallevel. T hese are but theoretical and m athem atical specula- tions,as yet. T im e w illtell(!) w hether they are correct, and to what extent. W hile there has been a great deal ofprogress on the theoreticalside over the past tw enty- ¯ve years or so, experim ental advances in elem entary particle physics have lagged behind. T his phase lag is

864 RESONANCE  September 2008 GENERAL  ARTICLE rather an anom aly,and the experience ofthe past four hundred years suggests strongly that true progress w ill Address for Correspondence Vasant Natarajan m ostlikely be m ade only w hen the gap between experi- Indian Institute of Science m ent and theory is reduced considerably. T his is one of Bangalore 560 012, India. the leading reasons w hy physicists are eagerly aw aiting Email: the ¯rst results from the Large H adron C ollider (LH C ), [email protected] which isexpected to go on stream atC ER N ,G eneva,in V Balakrishnan the very near future. Department of Physics Indian Institute of Technology B ut it is good to conclude this account here,seeing how Chennai 600 036, India. far we have com e from the Egyptian farm ers toiling on Email: the banks ofthe N ile,from whose m ud the subject of [email protected] geom etry wasborn { aided by the upward look into the sky, and the challenge posed by the splendour of the N Mukunda 103, 6th Main Road cosm os stretching out before the eyes ofhum ankind. Malleswaram Bangalore 560 003, India. Email: [email protected]

Einstein, (shown here with Niels Bohr in a photo taken by Paul Ehrenfest in 1925) was opposed all his life to the “Copenhagen interpretation’’ of quan- tum mechanics championed by Bohr. The result was several public debates with Bohr in the 1920s, and in 1926 Einstein wrote in a letter “I, at any rate, am convinced that He [God] does not throw dice.’’ In 1935, he wrote a path- breaking paper on what is now called the EPR paradox, which brought out the inherent nonlocality in quantum mechanics. From: Wikipedia

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