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Arthur Stanley Eddington MAVEN BOOKS
The Mathematical Theory of Relativity
M MAVEN BOOKS
The Mathematical Theory of Relativity
Arthur Stanley Eddington M.A., M.Sc., F.R.S Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge
M MAVEN BOOKS
Chennai
Trichy
New Delhi
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dra t this was in as a athe atical supple ent to the uring the ensuing eigh teen onths I have pursued intention developing it into a ore syste atic and o prehensive treatise the athe atical theory elativi ty The atter has been re ritten the sequ ence the argu ent rearranged any places and nu erous additions ade throughou t so that the work is e panded to three ti es or er si e It is hoped that as now enlarged it eet the nee s those wish to enter ully into these proble s reconstructio theoretical physics Th e reader is e pected to have a general acquaintan e with th e less technical discussion the th eory given in although there not o ten oc asion to ake direct re erence it i t is e inently desirable to have a general grasp the revolu tion though t asso iated wi h the theory elativity be ore approaching it along the narrow lines strict athe atical deduction In the or er work we plained the older conceptions physics had beco e untenable traced gradual ascent to the ideas which ust supplant the Here our task is to or ulate athe atically this new conception the world and ollo the consequences to the ullest e tent present widespread i terest in theory arose ro the veri3 ation certain inute deviations ro ew o ian la s To those who are still hesitating and reluctant to leave th e aith these deviations will re ain the hie centre interes bu t those who have caught the spirit the new ideas the observational predictions or only a inor part the subj ect I t is clai e the theory that it leads to an unders anding the world physics learer and re pene rating than that previously a tained and it has been develop the theory in a or which thro s ost light on the origin and signi3can e the great laws physics It is hoped that di 3culties which are erely analyti al have been i i by giving rather ully the inter ediate steps in all the proo s ith abundant cross re erences the au iliary or u lae used those do not read the book consecutively attention be called to the ollowing points in the no ation su ation convention p is used er an letters always denote the product the corresponding p nglish letter by is the sy bol Ha iltonian di eren introduced on p asterisk is pre3 ed to sy bols generalise to be i epen en ovariant ith the gauge p
P REF A C E selected list original papers on the subj ect is given in the iblio graphy at the end and any these are sources either directly or at second hand the de elop ents here orth To 3t these into a con chain deduction has invol ved considerable odi3 ations ro their original or that it has not generally been ound practicable indicate the sources the separate sections requent cause deviation in treat ent is the act that in the V iew ost conte porary riters the rinciple tationary ction is the nal governing law the world reasons e plained in the te t I unwilling to ac ord it e al ed a position ter the original papers instein and those de itter ro which I 3rst acquired an interest in the theory I ost indebted to Weyl s Z Weyl s in4uence will be especially traced in as well as in the sections re erring to own theory I under great obligations to the o 3cers and sta the niversity ress their help and care in the intricate printing
IN
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TION I
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E L E M E N AR
RI N C I LE
In et r inaten s s pa ti e ra e u n a nt l qu a rati or e ur ent i nte r al s e t ngula r oor i nat ti e Lorent t ran or at ion lo i t ligh t inter al s I e iate on i ou sne s t i e i nsi onal worl ont ra ti on i u ltaneit at i e rent pla es o ent u as ne rg te p e rat u e D en i t G ene ral t ran o r ati o n oor i nates Fi el or e ri n ipl e qui al en e et ro p e t II
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C o nt ra ar ant o ar ant e t or s ath at i a l n o ti o n a t o r ph i al no t io n a e tor s u ati o n on enti o n ens or s Inner u tipli at io n ontra t ion un a enta l t ns or s o i ate t n or s C h r t o el s i n e b ol s qu ation ge o e ic C o a riant ri at i e a e t or C o a ri ant e r i at i e a t n or te r nati e i u s i o n o ariant u r a e l e ents t o e s th eor i gn i( an e o ari nt i er ntiat ion Ri e ann ten or i e ll an ou s o r u ae
q o tient
i at i e
CO N T EN T S
SE
T I ON
el e t ro ag et i qu at ion le t ro agn ti a es Lorent t ran or ati o n el t ro g et i or e e h an i al e e ts le t ro agneti (el el e t ro ag eti en rg ten or e l e t ro n gra i tat ional ( el l t ro agnet i a t io n pl natio n e h anical or e l t ro gneti olu e a ro opi qu ati ons
O
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EOME R E ORY
Nat ural geo et r worl geo t r nte ab ili t l eng h ran o at ion gauge te s Gauge n ari an e gene ra i e R i ann ten or i n ariant a r gio n nat ural ga ge l s a t ion prin i pl e A T EN E A I S E E ORY a rall l i pla ent D i pla e ent rou n in ( nit i al i r ui t Int ro u t io n a etri alu at io n u n a enta l tensor nat ural ga uge worl prin iple i enti ( at ion b i ur ti on geo et r ele tro na i s Gene ral re la ion st ru t ure te sor general prop er ies wo rl D na i al on e u en es g n rali e olu e N u e ri al alu es C on lu s io n I
IO
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n ar iants
subj ect this athe atical treatise is not pure athe atics but phy ics The vocabulary th e physicist co prises a nu ber words such as length a gle velocity orce work potential current etc which we shal l c l l brie4y physical quan tities o e these ter s occur in pure athe also in that subj ect they have a generalised eaning which does concern us here The pure athe atician deals with ideal quantities de3ned as having th e properties which he deliberately assigns to the in an e peri en tal science have discover properties assign the and physical quantities are de3ned pri arily according to the in which recognise the when con ronted by the observation the orld arou d us C onsider e a ple a length distance between points It is a nu eric l quantity associated with the two points and all know th e procedure ollowed in practice i n assigni ng this n u erical quantity to two points in nature de3nition distance will be obtained by stating the e act procedure that clearly u st be the pri ary de3nition we are to ake sure using th e word in the sense a iliar everybody The pure ath e atician proceeds di erently h e de3nes distance as an attribute the points which obeys certain la s the a io s the geo etry which h e happens to have chosen and he is concerned with the question this distance wou ld e hibit itsel in pra tical observation as his investigations are concerned h e takes care to use the word sel consistently but it does necessarily denote the thing which the rest ankin are ccusto ed to recog ise as the distance the two points 3nd any physical quantity per or certain pra tical operations ollowed by calcu latio s the O perations are called e peri ents or observations accordi g as the conditions are ore less closely under our control Th e physical quantity discovered is pri arily th e result the operations and alculations i t to speak an u actured by operations the physicist is not generally content to bel ieve that the quan ity he arrives at is so ething whose nature is inseparable ro the kind operations which led it he has an idea that h e cou ld beco e a god conte plating the e ter al world h e would his anu actured physical uantity or ing a distinct eature the picture 3nding that he can lay unit easuring rods in a line between two points he has anu actured the quantity hich he alls th e distance between the points but he believes that that distance is so ething already e isting in the picture the world gul hich wo uld be apprehended by a superior i ntelligen e as e isting itsel i hou t re eren e to the notion O perations with easuring rods
I N TR OD U CTIO N he akes curious and apparently illogical discri inations The paralla a star is ound by a well known series operations and calculations the distance across the roo is ound by operations with a tape easure oth paralla and distance are quantities anu actured by our operations but so e reason we do not e pect paralla to appear as a distinct ele ent in aga n the true picture nature in the sa e that distance does instead cutting short the astrono ical calculations when we reach the paralla we igh t go on to take th e cube th e resul t and so obtain another anu actured quantity a cubic paralla so e obscure reason e pect to see distance appearing plainly a gul in the true world picture paralla does appear directly though it can be e hibited as an angle by a co parati vely si ple constr ction and cubic paralla is not in the picture at all The physicist would say that he a length and a cubic paralla but it is only because he inheri ted a preconceived theory the world that he akes the distinction We shall venture to cha lenge this distinction istance paralla and cubic paralla ha e the sa e kind potential e istence even when the operations easure ent are not actually ade will ove sideways you will be able deter i e th e angular Shi t will lay easuri g rods in a line to the obj ect will be able count their nu ber one the three an indication to us so e e istent condition relation in the world outside condition created by operations there see s reason conclude that this world condition distance any ore closely than it rese bles paralla or cubic paral la Indeed any notion rese blance between physical quantities and the world conditions underlying the see s be inappropriate the length double the length the paralla ro is hal the paral ro there is undoubtedly so e world relation hich di erent and but there is reason to regard the world relation as being better represented by double than by hal the world relatio The connection anu actured physical quantities with the e istent world condition can be e pressed by saying that the physical quantities are the world condition easure n u bers be assigned according to any code the only require en t being that the sa e easure nu ber always indicates the sa e world condition and that di erent orld conditions receive di erent easure nu bers ore physical quantities thus be easure nu bers the sa e world conditio paralla and distance ass and energy stellar agnitude and lu i The constant or u lae connecting these pairs physical quantities give the relation between the respective codes in ad itting that physical quantities can be used as easure nu bers world conditions e isting independently our operations we do alter their status as anu a tured quantities The sa e series operations will naturally anu acture the
I N T R OD U CTIO N sa e resul t when world conditions are the sa e and di erent results when they are di erent i erences world conditions which do not in4 uence the results e peri ent and observation are e cluded ro the do ain physical knowledge The si e to which a crystal grows be a easu re nu ber the te perature the other liquor bu t it none th e less a anu actured si e and we do conclude that the true nature si e is caloric Th e study physical quantities although they are the resu lts operations actual potential gives us so e kind knowledge the world conditions since the sa e operations will give di erent resul ts in di erent world conditions It see s that th is indirect know le ge is all that we can ever attain and that it is only through its in4uences such opera tions that can represent to ourselves a condition the world atte pt to describe a condition the world otherwise is either athe atical sy olis eaningless j argon To grasp a condition th e world as co pletely as it is i n power grasp it we ust have in ou r inds a sy bol which co prehends at th e sa e ti e its in4 uence th e resu lts all possible kinds operati ons what co es th e sa e thing we ust conte plate its easures according to all possible easu re codes course without con using the di erent codes It igh t well see i possible to realise co prehensive an outlook bu t we shall 3nd that the athe atical calculus tensors does represen t and deal with w orld conditions precisely in this way tensor e presses si u ltaneously the whole group easure nu bers associated with any world condition and achinery is provided keeping the various codes distinct this reason the so ewhat di cult tensor calculus is not to be regarded as an e vil necessity i n this subj ect which ought possible to be replaced by si pler analytical devices knowledge conditions in the e ternal world as it co es u s through observation and e peri ent is precisely the kind which can be e pressed by a tensor and not otherwise j ust in arith etic can deal reely with a billion obj ects without trying visualise the enor ous collection the tensor calculus enables u s deal with the w orld condition i n the totality its aspects withou t atte pting picture it aving regard to th is disti nc tion between physical quantities orld conditions we Shall de3ne a physical quantity as though it were a eature in the world pictu re which had to sought The tendency to this kind de nition had progressed even in pre relativity physics orce had beco e ass acceleration and was no longer an visible agent the world pictu re at least so as its de3nition conce ned ass is de3ned by e peri en s inertial properties no longer as quantity atter so e ter s the older kind de nition or lack de3ni tion has been obstinately adhered to and these the relativity
The Mathematical
Theory of Relativity OTHER BOOK TITLES • • • • • • • • • • • • • • • • •
Linear Associative Algebra Linear Algebra Linear Associative Algebra Some Problems of Geodynamics The Internal constitution of the stars The Mathematical theory of relativity Theory of Probability Theoretical Mechanics an introductory Treatise on the principles of Dynamic The Theory of Optical Instruments Functions of a complex variable Introduction to Higher Algebra Vector Calculus Vectorial Mechanics An Elementary Treatise the integral calculus An Introduction to combinatory Analysis Calculus of finite differences Elementary Mathematical Analysis