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    Arch. Hist. Exact Sci. 51 (1997) 273-314. @ Springer-Verlag 1997

    erm ann M ink owsk i an d the Postu l a te o f R ela t i v i t y

    LEo CORRY

    ommunicated by

    J. Norton

    1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

    2. The Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

    3. The Basic Equations of Electromagnetic Processes in Moving Bodies . . . . . . . . . . . . . . . . 279

    4. Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

    5. Max Born, Relativity, and the Theories of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

    6. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

    1 I n t r o d u c t i o n

    In the history of two of Einstein s chief scientific contributions -b oth the special and

    the general theories of relati vity- two of the leading G6ttingen mathematicians o f the be-

    ginning of this century each plays a significant role: Hermann Minkowski (1864-1909)

    and David Hilbert (1862-1943). Einstein published his famous paper on the electrody-

    namics of moving bodies in 1905. Beginning in 1907, Hermann Minkowski erected the

    new theory of relativity on what was to become its standard mathematical formulation

    and devised the language in which it was further investigated. In particular, Einstein s

    adoption of Minkowski s formulation - which he had initially rejected - proved es-

    sential to his own attempts to generalize his theory to cover gravitation and arbitrarily

    accelerated systems of reference. After a long and winding process that spanned at least

    three years of intense work and included the publication of several versions he later

    deemed incorrect, Einstein presented to the Prussian Academy of Sciences in Berlin

    his generally-covariant field equations of gravitation on November 25, 1915. But, as it

    happened, David Hilbert - the undisputed, foremost living mathematician in the world

    and the lifelong close friend and collaborator of the by then deceased Minkowski - had

    already presented to the G6ttingen Academy his own version of the same equations a

    few days earlier, on November 20. Although Minkowski and Hilbert accomplished their

    most important achievements in pure mathematical fields, their respective contributions

    to relativity should in no sense be seen as merely occasional excursions into the field

    of theoretical physics. Minkowski and Hilbert were motivated by much more than a

    desire to apply their exceptional mathematical abilities opportunistically, jumping onto

    the bandwagon of ongoing physical research by solving mathematical problems that

    physicists were unable to. On the contrary, Minkowski s and Hilbert s contributions to

    relativity are best understood as an organic part of their overall scientific careers. It

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    274 L. Comfy

    is remarkable that although the close professional and personal relationship between

    Minkowski and Hilbert is well-known, no direct connection between their respective

    contributions in these fields has hitherto been established or even suggested. 1 The his-

    tory of the special and the general theories of relativity has more often than not been told

    from the perspective of Einstein's work and achievements, and the roots and true moti-

    vations of Minkowski's and Hilbert's contributions to this field have therefore remained

    only partially and incorrectly analyzed.

    A detailed examination of their careers makes it evident that a keen interest in

    physics was hardly ever distant form either Hilbert's or Minkowski's main focus of ac-

    tivity in pure mathematics. Minkowski's interest in physics dates back at least to his

    Bonn years (1885-1894), during which he was in close contact with Heinrich Hertz. 2

    In 1888 he published an article on hydrodynamics in the proceedings of the Berlin

    Academy (Minkowski

    1888).

    From his correspondence with Hilbert, 3 we know that

    during his Ztirich years Minkowski kept alive his interest in mathematical physics, and

    in particular in thermodynamics. In 1902 he moved to G6ttingen, following Hilbert's

    strong pressure on Felix Klein (1849-1925) to create a professorship for his friend.

    It is well known that during his last years there, Minkowski's efforts were intensively

    dedicated to electrodynamics. But this was not the only field of physics to which his

    attention was attracted. Minkowski was commissioned to write an article on capillarity

    for the physics volume of the

    Encyclopiidie der mathematischen Wissenschaften,

    edited

    by Arnold Sommerfeld (Minkowski

    1906).

    At several meetings of the G6ttingen Math-

    ematical Society he lectured on this, as well as on other physical issues such as Euler's

    equations of hydrodynamics and Nernst's work on thermodynamics.4 He also taught

    advanced seminars on physical topics and more basic courses on continuum mechanics,

    and gave exercises in mechanics and heat radiation. 5

    Perhaps under Minkowski's influence, Hilbert also developed a strong attraction to

    physics from very early on. He followed the latest developments in physics closely and

    taught courses and seminars on almost every current physical topic. Hilbert elaborated the

    principles of his axiomatic method between 1894 and 1899 as part of his current interest

    in problems related to the foundations of geometry; but to a considerable extent, he also

    reflected throughout these years on the relevance of the method for improving the current

    state of physical theories. Influenced by his reading of Hertz's

    Principles of Mechanics,

    Hilbert believed that physicists often tended to solve disagreements between existing

    theories and newly found facts o f experience by adding new hypotheses, often without

    thoroughly examining whether such hypotheses accorded with the logical structure of the

    1 For example, no such connection is considered in oft-cited accounts of Minkowski's work:

    Galison

    1979;

    Pyenson

    1977;

    Miller

    1981,

    238-244. Neither is it discussed in accounts of Hilbert's

    contribution to general relativity: Earman and Glymour

    1978;

    Mehra

    1974;

    Pais

    1982,

    257-261;

    Vizgin

    1994,

    54-69.

    2 See Rtidenberg and Zassenhaus (eds.)

    1973,

    39-42, and Hilbert

    1909,

    355.

    3 See Rtidenberg and Zassenhaus (eds.)

    1973,

    110-114.

    4 As registered in the

    Jahresbericht der Deutschen Mathematiker-Vereinigung JDMV).

    See

    Vol.12 (1903), 445 447; Vol.15 (1906), 407.

    5 See the announcement of his courses in

    JDMVVol.13

    (1904), 492; Vol.16 (1907), 171;

    Vol.17 (1908), 116.

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    Hermann Minkow ski and the Postulate of Relativity 27 5

    e x i s ti n g t h e o r ie s t h e y w e r e m e a n t t o i m p r o v e . I n m a n y c a s e s , h e t h o u g h t , t h i s h a d l e d t o

    p r o b l e m a t i c s it u a ti o n s i n s c i e n c e w h i c h c o u l d b e c o r r e c t e d w i t h t h e h e l p o f a n a x i o m a t i c

    a n a l y si s o f t h e k i n d h e h a d m a s t e r f u l l y p e r f o r m e d f o r g e o m e t r y . I n a c o u r s e in G 6 t t in g e n

    i n 1 90 5 o n t h e l o g i c a l p r i n c i p l e s o f m a t h e m a t i c s , H i l b e r t g a v e a q u i te d e t a i l e d o v e rv i e w

    o f h o w s u c h a n a x i o m a t i c a n a l y s i s w o u l d p ro c e e d i n th e c a s e o f se v e ra l s p e c if i c t h e o r i e s,

    i n c lu d i n g m e c h a n i c s , t h e r m o d y n a m i c s , t h e k in e t i c t h e o r y o f g a s e s, e l e c t r o d y n a m i c s ,

    p r o b a b i li t ie s , i n s u r a n c e m a t h e m a t i c s a n d p s y c h o p h y s i c s . 6

    A f t e r h is a r r i v a l i n G 6 t t in g e n , M i n k o w s k i w a s d e e p l y i n v o l v e d in a l l th e s c i e n t if i c

    a c t i v i ti e s o f H i l b e r t , in c l u d i n g h i s c u r r e n t i n t e r e st s i n t h e a x i o m a t i z a t i o n o f p h y s i c s . A n

    o n g o i n g i n t e r c h a n g e o f i d ea s b e t w e e n t h e m - i f n o t a c t u al c o l l ab o r a t i o n - s h o u l d b e

    t a k e n i n t o a c c o u n t b y t h e h i st o r i a n a s i m p o r t a n t i n th e e v o l u t i o n o f th e c o n c e p t i o n s o f

    e a c h t h ro u g h o u t t h e i r c a r e e r s , a n d e s p e c i a l l y d u r i n g t h e i r s h a r e d y e a r s a t G 6 t t i n g e n .

    M o r e s p e c i fi c a ll y f o r o u r p r e s e n t c o n ce r n s , i n 1 90 5 H i l b e r t a n d M i n k o w s k i , t o g e t h e r

    w i t h o t h e r G 6 t t i n g e n p r o f e s s o r s , o r g a n i z e d a n a d v a n c e d s e m i n a r t h a t s t u d i e d r e c e n t

    p ro g re s s i n t h e th e o r i e s o f t h e e le c t ro n . I n 1 9 0 7, t h e t w o c o n d u c t e d a j o i n t s e m i n a r o n

    t h e e q u a t i o n s o f e l e c t ro d y n a m i c s . B e g i n n i n g a t l e a s t i n 1 9 0 7 a n d u n ti l h is d e a t h i n 1 9 0 9 ,

    M i n k o w s k i d e v o t e d a l l h i s e f f o rt s to t h e s t u d y o f th e e q u a t i o n s o f e l e c t r o d y n a m i c s a n d

    t h e p o s t u l a t e o f r e l at i v it y . H i l b e r t c e r t a i n l y fo l l o w e d M i n k o w s k i s w o rk i n t h i s f ie l d w i t h

    g re a t i n t e r e s t . I n h i s s t u d y o f e l e c t ro d y n a m i c s , M i n k o w s k i a l s o a d d re s s e d t h e q u e s t i o n

    o f g r a v i ta t io n , a n d f o r m u l a t e d s o m e p r e l i m i n a r y i d e a s c o n c e r n i n g t h e p o s s i b i li t y o f a

    L o r e n t z c o v a r i a n t t h e o r y t o a c c o u n t f o r it . A n a c c o u n t o f H i l b e r t s w a y t o h i s l a t e r w o r k

    o n g e n e r a l r e l a ti v i ty o b v i o u s l y c a l ls f o r a n e x p l o r a t io n o f M i n k o w s k i s w o r k b e t w e e n

    1907 and 1909.

    T o w h a t e x t e n t H i l b e r t a c ti v e l y co n t r ib u t e d t o t h e c o n s o l i d a ti o n o f M i n k o w s k i s

    s p e c i f i c i d e a s o n e l e c t ro d y n a m i c s a n d t h e p r i n c i p l e o f r e la t iv i t y , a n d t o w h a t e x t e n t

    M i n k o w s k i i n f l u e n c ed H i l b e r t s c o n c e p t i o n s o n p h y s i c a l is s u es , i s h a r d t o d e t e r m i n e

    w i t h e x a c t i t u d e , b u t i t s e e m s s a f e t o a s s u m e t h a t t h e t w o s h a r e d m a n y b a s i c c o n c e p -

    t i o n s c o n c e rn i n g t h e s e m a t t e r s . I n t h e p r e s e n t a r t ic l e I c l a i m t h a t a p ro p e r u n d e r s t a n d i n g

    o f M i n k o w s k i s i n c u r si o n i n t o t h e f i e ld o f e l e c t r o d y n a m i c s a n d r e l a ti v i ty m u s t t a k e i nt o

    a c c o u n t i ts p r o x i m i t y to t h e k i n d o f i d e as p u t f o r w a r d i n H i l b e r t s p r o g r a m f o r th e a x i o m -

    a t iz a t io n o f p h y s i c s. M i n k o w s k i u n d e r t o o k a s y s t e m a t i c e x a m i n a t i o n - l i k e t h o s e f o u n d

    i n H i l b e r t s 1905 l e c tu r e s o n t h e a x i o m a t i c m e t h o d - o f th e l o g ic a l, m a t h e m a t i c a l a n d

    p h y s i c a l i m p l i c a t io n s o f a d d i n g t o t h e e x i s t in g e d if i ce o f p h y s i c s t h e n e w l y f o r m u l a t e d

    h y p o t h e s i s k n o w n a s th e p r i n c i p le o f r e la t iv i ty . G i v e n M i n k o w s k i s o w n p h y s i c a l b a c k -

    g r o u n d a n d m a t h e m a t i c a l in t e re s ts - w h i c h d i f f e re d in s e v e r al r e s p e c t s f r o m H i l b e r t s -

    a n d g i v e n t h e l a te s t d e v e l o p m e n t s in p h y s i c s , M i n k o w s k i s a n a l y s i s i m p l i e d a d i r e ct i o n

    o f t h in k i n g t h a t H i l b e r t d i d n o t c o v e r - a n d p e r h a p s c o u l d n o t e v e n i m a g i n e p o s s i b l e -

    w h e n t e a c h i n g h i s 1 90 5 c o u r s e . Y e t t h e v e ry m o t i v a t i o n s fo r s u c h a n a n a l y s i s , a s w e l l a s

    m a n y o f th e q u e s t i o n s a d d r e s s e d i n it, a r e c l e a r l y r e m i n i s c e n t o f H i l b e r t s o w n a n d a r e

    6 I have presented a detailed account of the origins and early stages of Hilbert s prog ram for

    the axiomatization of physics f ro m 1894 to 1905 , including his 1905 course, in Corry 1997 The

    present article should ideally be read as a follow-up of that earlier one. For an overview of Hilb ert s

    wo rk on physical issues until 1915, see Corry 1998 For H ilbert s work on G eneral Relativity see

    Corry 1998a

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    2 7 6 L . C o a x

    c l a r if i e d b y a s s o c i a t i o n w i t h t h e la t te r . I n f a c t , o n e o f t h e i m p o r t a n t i n s i g h t s a f f o r d e d b y

    t h is r e a d i n g o f M i n k o w s k i i s t h a t it al s o s tr e ss e s t h e k i n d o f q u e s t i o n s t h a t M i n k o w s k i

    w a s

    not

    p u r s u i n g i n h is w o r k . I n p a r ti c u la r , t h e p o i n t o f v i e w a d o p t e d h e r e s u g g e s t s a

    r e i n te r p r e t a ti o n o f t h e r6 1e o f M i n k o w s k i ' s w o r k i n th e d e b a t e s o f t h e f ir st d e c a d e o f th e

    c e n t u r y - m u c h d i s c u s s e d i n t h e s e c o n d a r y li te r a tu r e - c o n c e r n i n g t h e u l t im a t e n a t u r e

    o f p h y s ic a l p h e n o m e n a .

    B e t w e e n 1 9 0 7 a n d 1 9 1 0 , th e y e ar s in w h i c h M i n k o w s k i w a s v i g o r o u s l y p u r s u i n g

    h i s i d e a s o n e l e c t r o d y n a m i c s a n d r e la t iv i ty , H i l b e rt h i m s e l f d i d n o t p u b l i s h o r l e c t u r e o n

    p h y s i c a l is s u e s a t a ll . I n fa c t , a f t e r h i s 1 9 0 5 c o u r s e o n a x i o m a t i z a t i o n a n d t h e j o i n t s e m i n a r

    o f 1 9 0 7 w i t h M i n k o w s k i , H i l b e r t t a u g h t a c o u r s e o n p h y s i c s a g a i n o n l y in 1 9 1 0 , w h e n h e

    l e c t u r e d o n m e c h a n i c s . 7 I n a s e c t i o n o f t h a t c o u r s e d e a l i n g w i t h t h e n e w m e c h a n i c s , w e

    f in d t h e f ir st e v i d e n c e o f H i l b e r t ' s r e f e r ri n g t o M i n k o w s k i ' s c o n t r ib u t i o n s . H i l b e r t s ta t e d

    t h a t t h o s e c o n t r i b u t io n s w e r e t h e s t a rt i n g p o i n t f o r h i s o w n p r e s e n t a t io n i n th a t c o u r s e .

    T h e r e f o r e , i n th e a b s e n c e o f d ir e c t e v i d e n c e t o th e c o n t r a r y , m y d e f a u l t a s s u m p t i o n w i ll

    b e t h a t M i n k o w s k i ' s p u b l i s h e d w o r k c a n b e t a k e n a s a f a it h f u l e x p r e s s i o n o f H i l b e r t ' s

    o w n v i e w s b e t w e e n 1 9 0 7 a n d , a n d a s th e s t a rt in g p o i n t f o r h i s o w n s tu d y o f p h y s ic a l

    t o p i c s a f te r M i n k o w s k i ' s d e a t h . T h i s w i l l b e i m p o r t a n t i n t r a c i n g H i l b e r t ' s o w n w a y t o

    g e n e r a l r e la t i v it y , a t a s k w h i c h I i n t e n d t o u n d e r t a k e i n t h e n e a r f u t u r e .

    2 T h e P r i n c i p l e o f R e l a t i v i ty

    M i n k o w s k i ' s i d e a s c o n c e r n i n g t h e p o s t u la t e o f r e l at iv i ty h a v e b e e n p r e s e r v e d i n t h e

    m a n u s c r i p t a n d p u b l i s h e d v e r s i o n s o f t h r e e p u b l i c ta l ks , a s w e l l a s t h r o u g h a n a r ti c le

    p o s t h u m o u s l y p u b l i sh e d b y M a x B o r n , b a s e d o n M i n k o w s k i ' s p a p er s a n d o n co n v e r s a -

    t i o n s b e t w e e n t h e t w o . M i n k o w s k i p r e s e n t e d h i s id e a s o n e l e c t r o d y n a m i c s a n d r e l a ti v i ty

    i n p u b l i c f o r t h e f i r s t t i m e i n N o v e m b e r 5 , 1 9 0 7 , i n a t a l k d e l i v e r e d t o t h e G S t t i n g e n

    M a t h e m a t i c a l S o c i e t y u n d e r t h e n a m e o f T h e P r in c i p le o f R e l at iv i ty . ' 8 O n e m o n t h b e -

    f o r e t h e ta lk , M i n k o w s k i h a d w r i t te n t o E i n s te i n a s k i n g f o r a re p r i n t o f h i s 1 9 0 5 p a p e r ,

    i n o r d e r t o s t u d y i t i n h i s j o i n t s e m i n a r w i t h H i l b e rt . 9

    R e c e n t d e v e l o p m e n t s i n t h e e l e c t r o m a g n e t i c t h e o r y o f l i g h t - M i n k o w s k i s a i d i n

    o p e n i n g h i s t a l k - h a v e g i v e n r is e t o a c o m p l e t e l y n e w c o n c e p t i o n o f s p a c e a n d t im e

    7 See the appendix to C orry

    1998.

    s Publ i shed as Minkowstd 1915. Fo r detai ls on the printed and ma nuscript versions of

    M i nkow sk i ' s w or k s ee G a i i son

    1979,

    119 -121 . T he or iginal typesc r ipt of this lecture wa s edi ted

    for publ ica t ion by Arno ld Somm erfe ld . A f ter compar ing the publ i shed vers ion wi th the or ig ina l

    typescript, L ew is Py ens on

    1977,

    82) has remarked tha t Som me rfe ld in t roduced a few changes ,

    am on g them a significant one conce rning the role of Einstein: So mm erfeld was unable to resist

    rewrit ing M inko w ski 's jud gem ent o f Ein stein 's form ulat ion of the pr inciple of relat ivi ty. H e in-

    t roduced a c lause inappropr ia te ly pra is ing Eins te in fo r having used the M ichelson exper iment to

    demo nst ra te tha t the concept of absolu te space d id not express a proper ty of phenom ena. S omm er-

    feld also suppressed M inko w ski 's conclu sion, where Einstein wa s portray ed as the clar if ier , but

    by no me ans as the principal expositor, o f the pr inciple of relat ivi ty Th e added clause is quo ted

    in Gal i son

    1979,

    93.

    9 See Stachel et ai (eds.)

    1989,

    267.

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    Hermann Minkow ski and the Postulate of Relativity 27 7

    a s a f o u r -d i m e n s i o n a l , n o n -E u c l i d e a n m a n i fo l d . W h e re a s p h y s i c i s t s a r e s t i l l s t r u g g l i n g

    w i t h t h e n e w c o n c e p t s o f t h e th e o ry , p a i n fu l l y t r y i n g t o fi n d th e i r w a y t h ro u g h t h e

    " p r i m e v a l f o r e s t o f o b s c u ri t ie s ," M i n k o w s k i a d d e d , m a t h e m a t i c i a n s h a v e l o n g p o s s e s s e d

    t h e c o n c e p t s w i t h w h i c h t o c l a r i f y th i s n e w p i c tu r e . T h e p h y s i c i st s M i n k o w s k i a s s o c i a te d

    w i t h t h i s t r e n d w e r e L o r e n t z , F i t z G e r a l d , P o i n c a r r , P l a n c k a n d E i n s t e i n . M i n k o w s k i

    t h o u g h t t h a t a p r o p e r e l a b o r a t i o n o f th e i r id e a s c o u l d b e c o m e o n e o f t h e m o s t s i g n if i ca n t

    t r iu m p h s in a p p l y i n g m a t h e m a t i c s t o u n d e r s t a n d in g t h e w o r l d , p r o v i d e d - h e i m m e d i a t e l y

    q u a l i f ie d h i s a s s e r t i o n - " t h e y a c t u a l l y d e s c r i b e th e o b s e rv a b l e p h e n o m e n a . " 10 T h i s l a t te r ,

    b r i e f r e m a r k c h a r a c te r i z es v e r y a p t l y t h e n a t u re o f M i n k o w s k i ' s i n c u r si o n i n t o t h e s t u d y

    o f t h e e l e c t r o d y n a m i c s o f m o v i n g b o d i es : a l o n g t h e l in e s o f H i l b e r t ' s a n a ly s i s o f t h e

    a x i o m s o f o t h e r p h y s i c a l d i sc i p li n e s, h e w o u l d a t t e m p t t o u n d e r s ta n d a n d s i m p l i f y t h e

    c o n c e p t u a l s t ru c t u re s o f e l e c tr o d y n a m i c s a n d m e c h a n i c s - p r e s e n t ly i n a s t a t e o f g r e a t

    c o n f u s i o n , i n v i e w o f t h e l a te s t d i sc o v e r i e s o f p h y si c s . H e w o u l d s o r t o u t t h e f u n d a m e n t a l

    s t a t e m e n t s t h a t l i e a t t h e b a s i s o f t h o s e s t r u c tu r e s , s t a t e m e n t s t h a t m u s t b e c o n f ro n t e d b y

    e x p e r i m e n t i n o rd e r t o v a l i d a t e o r r e fu t e t h e r e l e v a n t t h e o r i e s. T h e p r i n c i p l e o f r e l a t i v i t y

    w o u l d t h e n b e s h o w n t o p l a y a f u n d a m e n t a l r o l e in t h e s e n e w d e v e l o p m e n t s o f p h y s i c s.

    M i n k o w s k i ' s 1 9 0 7 t a l k c o m p r i s e d fo u r s e c t i o n s : e l e c t r i c i t y , m a t t e r , d y n a m i c s , a n d

    g ra v i t a t io n . I n t h e f ir s t t w o s e c t io n s , M i n k o w s k i e l a b o ra t e d o n i d e a s t h a t h a d b e e n d i s -

    c u s s e d r e c e n t l y i n h is j o i n t s e m i n a r w i t h H i l b e r t. I n th i s s e m i n a r , g e o m e t r i c a l s p a c e h a d

    b e e n d e s c r i b e d a s f i l le d w i t h t h r e e d i f f e r e n t k i n d s o f c o n t i n u a - e t h e r, e l e c t r ic i t y a n d

    m a t t e r - w h o s e p ro p e r t i e s m u s t b e c h a ra c t e r i z e d b y s u i t a b l e d i f f e r e n t ia l e q u a t i o n s . 11

    T h i s p a r t i c u l a r c o n c e p t i o n w a s n o t i n i t s e l f n e w . In f a c t, th e s t u d y o f th e c o n n e c t i o n

    b e t w e e n e t h e r a n d m a t t e r i n m o t i o n h a d s h a r p l y in t e n si fi e d a f t e r t h e 1 89 8 m e e t i n g o f

    t h e S o c i e t y o f G e rm a n S c i e n t i st s a n d P h y s i c i a n s i n D t i s se l d o r f , i n w h i c h t h e s u b j e c t w a s

    d i sc u s s ed . O n t h a t o c c a s i o n L o r e n t z d e s c r i b e d t h e p r o b l e m i n t h e f o l l o w i n g t e r m s :

    E t h e r , p o n d e ra b l e m a t t e r , a n d w e m a y a d d , e l e c t r i c it y a r e t h e b u i ld i n g s t o n e s f ro m

    w h i c h w e c o m p o s e t h e m a t e r ia l w o r ld , a n d i f w e c o u l d k n o w w h e t h e r m a t te r ,

    w h e n i t m o v e s , c a rr i e s th e e t h e r w i t h i t o r n o t , t h e n th e w a y w o u l d b e o p e n e d

    b e f o r e u s b y w h i c h w e c o u l d f u r t h e r p e n e tr a t e i n t o t h e n a t u re o f th e s e b u i l d in g

    s t o n e s a n d t h e i r m u t u a l r e l a ti o n s. 1 2

    T h i s d e v e l o p m e n t c o m p r i s e d t w o d i f f e r e n t p e r s p e c t i v e s : t h e m i c r o s c o p i c t h e o r i e s

    o f t h e e l e c t r o n a n d t h e m a c r o s c o p i c t h e o r i e s o f o p t i c a l a n d e l e c t r o m a g n e t i c p h e n o m -

    e n a i n m o v i n g m e d i a . 13 W h e re a s E i n s t e i n ' s 1 90 5 r e l a t iv i s t i c k i n e m a t i c s c o n c e rn e d o n l y

    L o r e n t z ' s m i c r o s c o p i c e l ec t r o n t h eo r y , i t w a s M i n k o w s k i w h o f ir st a d d r e s s e d t h e f o r m u -

    l a t i o n o f a r e l a t iv i s t i c e l e c t ro d y n a m i c s o f m o v i n g m e d i a . T h u s h i s th r e e p u b l i c l e c t u r e s

    o n t h e p o s t u l a te o f r e la t iv i t y d ea l m a i n l y w i th t h e m a c r o s c o p i c p e r s p e c ti v e , w h i l e t h e

    p o s t h u m o u s a r t ic l e p u b l i s h e d b y B o r n f o c u s e d o n t h e m i c r o s c o p i c o n e.

    10 Minkowski 1915 927: " .. . fails sie tats~ichlich die Erscheinungen richtig wiedergeben . . . . "

    9 ~ Notes o f this seminar were taken by H ermann M ierendorff, and they are kept in Hilb ert 's

    Nachlass (Cod Ms 570/5). For more details on the seminar see Pyenson 1977 83.

    12 Lorentz 1898 101. Translation q uoted from Hirosige 1976 35.

    13 O n the development of these two perspectives before Einstein an d Minkow ski, see Stachel

    et al (eds.) 1989 503-504.

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    27 8 L. CoRRY

    T h e G d t t i n g e n s e m i n a r s o f 1 9 0 5 a n d 1 9 0 7 o n e l e c t r o d y n a m i c s w e r e o s t e n s i b ly c o n -

    d u c t e d i n th e c o n t e x t o f t h e in t e n s e a c t i v it y d e v e l o p e d b y G e r m a n - s p e a k i n g p h y s i c i s ts

    o n t h e s e q u e s t i o n s , f o l l o w i n g t h e D t i s s e l d o r f m e e t i n g . 14 B u t t h e d i f fe r e n t i a l e q u a t i o n s

    b r i e f l y d i s c u s s e d i n t h e 1 9 0 7 s e m i n a r w e r e r e f o r m u l a t e d i n M i n k o w s k i ' s t a l k i n a n i n -

    n o v a t iv e w a y : M i n k o w s k i i n t r o d u c e d h e r e f o u r - v e c to r s o f fo u r a n d o f si x c o m p o n e n t s

    ( h e c a l l e d t h e l a t te r

    T r a k t o r e n )

    a s th e m a t h e m a t i c a l t o o l n e e d e d t o b r i n g t o l ig h t a ll t h e

    s y m m e t r i e s u n d e r l y i n g t h e p h y s i c a l q u e s t io n s i n v o lv e d .1 5 M i n k o w s k i e x p l i c i t ly c l a i m e d

    t h a t it is p r e c i s e l y t h e f o u r - v e c t o r f o r m u l a t i o n t h a t m a k e s e v i d e n t t h e k i n d o f i n v a r ia n c e

    c h a r a c t e ri s ti c o f L o r e n t z ' s e q u a t i o n s f o r th e e l e c t r o n ( w h i c h a l s o d e s c r i b e t h e b e h a v i o r

    o f a n e l e c t r o m a g n e t i c f i e ld i n p u r e e t h e r a n d o f a n e l e c t r i c f i e l d fi l li n g i n f in i te s p a c e , i . e .,

    t h e fi rs t a n d s e c o n d o f t h e th r e e c o n t i n u a m e n t i o n e d a b o v e ) . M o r e o v e r - h e r e m a r k e d -

    t h e w a y i n w h i c h t h is p u r e l y f o r m a l p r o p e r t y o f t h e e q u a t i o n s i s p r e s e n t e d h e r e h a d n o t

    b e e n n o t i c e d b e f o r e e v e n b y a u t h o r s l ik e P o i n c a r & 1 6 A l t h o u g h in h i s t al k d i d n o t a c t u a l l y

    w r i t e t h e M a x w e l l e q u a t i o n s i n L o r e n t z - c o v a r i a n t f o r m , h e s h o w e d s k e t c h i ly t h a t i f t h e s e

    e q u a t i o n s a r e f o r m u l a t e d i n t e rm s o f f o u r -v e c t o r s , t h e ir i n v a r i a n c e u n d e r a n y t r a n s fo r -

    m a t i o n o f t h e f o u r c o o r d i n a t e s t h a t l e a v e s i n v a r i a n t t h e e x p r e s s i o n x 2 + x22 + x 2 + x42

    (where x42 =

    i t )

    f o l l o w s a s a si m p l e m a t h e m a t i c a l r e su l t. I n M i n k o w s k i ' s f o r m u l a t i o n ,

    t h e L o r e n t z t r a n s f o r m a t i o n s r e p r e s e n t r o t a ti o n s i n t hi s fo u r - d i m e n s i o n a l s p a c e .

    I n t h e s e c o n d p a r t o f t h e t a lk M i n k o w s k i i n v e s t i g a t e d h o w t h e e q u a t i o n s a r e a f f e c te d

    w h e n m a t t e r i s a d d e d t o p u r e e th e r. M i n k o w s k i , v e r y m u c h l i k e H i l b e rt i n h i s 1 9 0 5 l e c -

    t u r e s, s t r e s s e d t h a t h i s t h e o r y d o e s n o t a s s u m e a n y p a r t i c u l a r w o r l d v i e w : i t t r e at s f i rs t

    e l e c t r o d y n a m i c s a n d o n l y l a te r m e c h a n i c s , a n d i ts s t a rt in g p o i n t i s t h e a s s u m p t i o n t h a t

    t h e c o r r e c t e q u a t i o n s o f p h y s i c s a r e s ti ll n o t e n t i re l y k n o w n t o US 17 P e r h a p s o n e d a y a

    r e d u c t i o n o f t h e t h e o r y o f m a t t e r t o t h e t h e o r y o f e l e c t ri c it y m i g h t b e p o s s i b le , h e s a id , b u t

    a t t h i s s t a g e o n l y t h is m u c h i s c l e a r : th a t e x p e r i m e n t a l r e s u l t s , a n d e s p e c i a l l y t h e M i c h e l -

    s o n e x p e r i m e n t , h a v e s h o w n t h a t t h e c o n c e p t o f a b s o l u t e r es t c o r r e s p o n d s t o n o p r o p e r t y

    o f t h e o b s e r v e d p h e n o m e n a . T h i s s it u a ti o n , M i n k o w s k i a s s e rt e d , c a n e a s i ly b e c la r i-

    f i ed i f o n e a s s u m e s t h a t t h e e q u a t i o n s o f e l e c t r o d y n a m i c s s ti ll r e m a i n i n v a r i a n t u n d e r t h e

    L o r e n t z g r o u p a f te r m a t t e r h a s b e e n a d d e d t o th e f i el d. I t is p r e c i s e l y h e r e t h a t th e p r i n c i p l e

    o f r e l a ti v i ty e n t e r s t h e p i c t u re o f p h y s i c s . M i n k o w s k i d e c l a r e d t h e p r i n c ip l e o f r e l a t i v i t y -

    i .e ., i n v a r i a n c e u n d e r L o r e n t z t r a n s f o r m a t i o n s - t o b e a tr u l y n e w k i n d o f p h y s i c a l l a w : i t

    is n o t o n e t h a t h a s b e e n d e d u c e d f r o m o b s e r v a t i o n , b u t ra t h e r

    i t i s a d e m a n d w e i m p o s e o n

    14 On these activities, see Hirosige

    1976 ,

    36- 41 .

    15 For the p lace o f Mink ow ski ' s cont r ibut ion in the developm ent of the theory of tensors , see

    Re i ch 1 9 9 4 , 168-184.

    16 M in k w sk i1 9 5 , 929: ``Ich wi h ier, w as i ibrigens be i ke in em g enn aten Autren, sebst n ic ht

    bei Poincard, gesch ehen is t, jene Sym me tr ic vo n vornherein zur Darstel lung br ingen, w odu rch in

    der Tat die F orm der Gleichu ngen , w ie ich racine, ~iuBerst durch sicht ig wird.

    17 Passag es l ike this on e have of ten bee n quo ted in the secon dary l iterature as evidence to

    suppor t the c la im tha t Minkow ski comp le te ly s ided wi th to the e lec t romag net ic world-view. For

    instance, Galison

    1979 ,

    92, t ranslates the or iginal Hier stellen wir uns auf den Stand pun kt . . . ,

    as He re we f ind ourse lves a t a s tandpoint where the t rue phys ica l l aws are not ye t com ple te ly

    kno w n to us. I read this different ly as W e place ourselves here at the s tandpo int . . . , namely,

    this is not a s tandpoint imp osed up on us, as i t were, bu t rather one w e del iberately adop t in order

    to avo id debate on this par t icular quest ion.

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    Herm ann M inkowski and the Pos tu la te of Rela t iv i ty 27 9

    y e t t o be fou nd e qua t ions d e sc r ib ing obse rv ab le phe nome na . I S

    A p p l y i n g t h is p o s t u l a t e

    t o th e s i tu a t io n i n q u e st io n , M i n k o w s k i s h o w e d th a t a s s u m i n g L o r e n t z c o v a r i a n c e a n d

    u s i n g t h e f o u r - v e c t o r f o r m u l a t i o n , t h e a s s u m p t i o n o f t h e G a l i l e a n p r i n c i p le o f i n e rt ia

    i m p l i e s t h a t t h e s p e e d o f l i g h t m u s t b e i n fi n it e . S i m i l a r ly , h e d e r i v e d t h e e l e c t r o d y n a m i c

    e q u a t i o n s o f a m o v i n g m e d i u m , m a k i n g e v i d e n t a n d s t re s s in g t h e ir i n v a ri a n c e u n d e r t h e

    L o r e n t z g r o u p . F r o m t h e k i n d o f r e a s o n i n g a p p l i e d h e re - h e r e m a r k e d i n t h e t h i rd p a r t

    o f t h e l e c t u r e - i t f o l l o w s , t h a t i f t h e p r i n c i p l e o f r e l a t iv i t y i s a c t u a l l y v a l i d a l s o f o r m a t t e r

    i n m o t i o n , t h e n t h e b a s ic l a w s o f c l a s s ic a l m e c h a n i c s s h o u l d b e u n d e r s t o o d a s o n l y a p -

    p r o x i m a t e l y t ru e . B u t th e n , t h e a b o v e - m e n t i o n e d i m p o s s i b i li t y o f d e te c t in g t h e m o t i o n

    o f t h e e a r t h r e l a ti v e t o t h e e t h e r c o n f i r m s t h a t t h is i s i n d e e d t h e c a s e . 19 M o r e o v e r , h e

    q u o t e d s o m e e la b o r a t e t e c h n ic a l r e a s o n i n g t a k e n f r o m M a x P l a n c k ' s r e c e n t c o n t r ib u t i o n

    t o a r e l a t i v i s t i c t h e r m o d y n a m i c s ( P l a n c k

    1907 ,

    a s a d d i t i o n a l a r g u m e n t s f o r r e j e c t i n g

    t h e c l a s s i c a l p r i n c i p l e o f i n e r ti a . 2 ~

    T h e f o u r t h p a r t o f M i n k o w s k i ' s l e c tu r e c o n t a i n e d a b r i e f d is c u s s i o n o n g r a v it a ti o n .

    N a t u r a l l y , i f t h e p r i n c i p l e o f r e l a t iv i t y i s t o b e t r u l y u n i v e r s a l i t s h o u l d a c c o u n t a l s o f o r

    p h e n o m e n a o f t h is k in d . M i n k o w s k i m e n t i o n e d a s im i l a r d is c u s s i o n th a t h a d a p p e a r e d

    i n P o i n c a r 6 ' s r e l a t iv i ty a r ti c le , a n d e n d o r s e d P o i n c a r 6 ' s c o n c l u s i o n t h a t g r a v i t a ti o n m u s t

    p r o p a g a t e w i t h t h e v e l o c i t y o f li g h t. T h e p u r e l y m a t h e m a t i c a l t a s k th u s r e m a i n e d o p e n ,

    t o f o r m u l a t e a l a w t h a t c o m p l i e s w i t h t h e r e l a t i v i t y p r i n c i p l e , a n d a t t h e s a m e t i m e

    h a s t h e N e w t o n i a n l a w a s i t s l i m i t i n g c a s e . P o i n c a r 6 h a d i n d e e d i n t r o d u c e d o n e s u c h

    l aw , M i n k o w s k i s a id , b u t h i s la w i s o n l y o n e a m o n g m a n y p o s s i b l e o n e s , a n d P o i n c a r d ' s

    r e s u lt s h a d h i t h e rt o b e e n f a r f r o m c o n c l u s iv e . M i n k o w s k i l e f t a m o r e e l a b o r a t e t r e a t m e n t

    o f t h is p o i n t , f o r a l a te r o c c a s i o n .

    3 T h e B a s ic E q u a t i o n s o f E l e c t r o m a g n e t i c P r o c e s s e s i n M o v i n g B o d i e s

    M i n k o w s k i ' s s e c o n d t a l k o n e l e c t r o d y n a m i c s a n d r e l a ti v it y w a s g i v e n l es s th a n t w o

    m o n t h s a f t e r h i s fi rs t o n e , t h is t i m e a t th e m e e t i n g o f th e G 6 t t i n g e n S c i e n t i f ic S o c i e t y

    o n D e c e m b e r 2 1 , 1 9 0 7 . T h e p r i n te d v e r si o n , e n ti tl e d T h e B a s i c E q u a t i o n s o f E l e c t r o -

    m a g n e t i c P r o c e s s e s i n M o v i n g B o d i e s , w a s M i n k o w s k i ' s o n l y p u b li c a ti o n o n t h is t o p ic

    t o a p p e a r b e f o r e h i s d e a t h i n 1 9 0 9 . It c o n t a i n e d h i s m o s t d e t a il e d m a t h e m a t i c a l t r ea t -

    m e n t o f th e d i f f er e n ti a l e q u a t i o n s o f e l e c t ro d y n a m i c s . I t a l so p r e s e n t e d a n i l lu m i n a t i n g

    c o n c e p t u a l a n a l y s i s - o n c e a g a i n , v e r y s i m i l a r i n s p i r i t t o H i l b e r t ' s a x i o m a t i c t r e a t m e n t

    o f p h y s i c a l t h eo r i e s - o f th e m a i n i d e a s i n v o l v e d i n t h e c u r r e n t d e v e l o p m e n t s o f th e

    18 Minkowski 1915, 931: H ier trit t nun das Relativitatsprinzip als ein wirkliches neu es

    physikalisches Ge setz ein, ind em es f iber noc h gesuchte G leichunge n ff ir Ersch einung en eine

    Fordemng stel l t .

    19 Minkowski

    1915,

    93 4-9 35 : N achde m, was ich berei ts t iber das VerhNtnis der Rela-

    tivit/i tsprinzipes zu m Tr~igheitsgesetze ges agt habe, ist von vom he rein klar, dab die bishe rigen

    Gru ndge setze der M echa nik nur als eine App roxim ation an die Wirkl ichkeit gel ten k6nn en, falls

    anc h in der M ech an ik das R elativit~itsprinzip gelten soll. Da s mfiBte aber w iede r der Fall sein,

    wei l sons t doc h w ieder e ine M 6gl ichkei t vor l iegen wf irde, e ine Bew egung der E rde relat iv zum

    A the r konstatieren.

    20 Minkowski

    1915,

    935-9 37. For an accoun t of Planck 's paper, see Mi ller

    1981,

    360- 362 .

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    28 0 L . Com fy

    t h e o r ie s o f t h e e l e c t r o n a n d o f t h e r o le p l a y e d b y t h e p r in c i p l e o f r e la t iv i ty i n t h o s e

    t h e o ri e s. M i n k o w s k i d i s t in g u i s h e d t h r e e p o s s i b le d i f fe r e n t m e a n i n g s o f th i s p ri n c ip l e .

    F i rs t, t h e p la i n m a t h e m a t i c a l f a c t t h a t t h e M a x w e l l e q u a t i o n s , a s f o r m u l a t e d i n L o r e n t z ' s

    t h e o r y o f e l e c tr o d y n a m i c s , a r e i n v a ri a n t u n d e r th e L o r e n t z t r a n s fo r m a t i o n s . M i n k o w s k i

    c a l l e d t h i s f a c t t h e

    theorem

    o f r e l at i v it y . I t s e e m s n a t u r a l t o e x p e c t , M i n k o w s k i s a i d ,

    t h a t t h e d o m a i n o f v a l i d it y o f t h e t h e o r e m - a m a t h e m a t i c a l l y e v i d e n t t h e o r e m , i n h is

    o p i n i o n - m i g h t b e e x t e n d e d t o c o v e r al l l a w s g o v e r n i n g p o n d e r a b l e b o d i e s , in c l u d i n g

    l a w s t h a t a r e s ti ll u n k n o w n . T h i s i s t h e postulate o f r e l a t i v i t y ; i t e x p r e s s e s a c o n f i d e n c e

    (Zuvers icht)

    r a t h e r t h a n a n o b j e c t i v e a s s e s s m e n t c o n c e r n i n g t h e a c t u a l s ta t e o f a f fa i rs .

    O n e c a n e m b r a c e t h i s c o n f i d e n c e , M i n k o w s k i e x p l i c i t l y s t r e s s e d , wi thou t thereby com-

    mi t t ing one se l f to any par t i cu lar v iew o f the u l t imate re la tionsh ip be tween e lec t ri c i ty and

    m atte r. 21

    H e c o m p a r e d t h i s p o s t u la t e t o t h e p o s t u l a t i o n o f t h e v a l i d it y o f th e p r i n c i p le

    o f c o n s e r v a t io n o f en e r gy , w h i c h w e a s s u m e e v e n f o r fo r m s o f e n e r g y t h at a r e n o t y e t

    k n o w n . L a s t ly , i f w e c a n a s s e r t th a t th e e x p e c t e d L o r e n t z c o v a r i a n c e a c t u a l l y h o l d s a s

    a r e l a ti o n b e t w e e n d i r e c t ly o b s e r v a b l e m a g n i t u d e s r e l a t in g t o a m o v i n g b o d y , t h e n t hi s

    p a r t i c u l a r r e l a t i o n i s c a l l e d

    ' 'principle

    o f r e l a t i v i ty .

    I t i s i n t e re s t i n g t o c o m p a r e t hi s a n a l y s i s o f M i n k o w s k i ' s w i t h a s i m i la r o n e a d v a n c e d

    b y H i l b e r t in a c o u r s e o f t h e k i n e ti c t h e o r y o f g a s e s i n t h e w i n t e r s e m e s t e r o f 1 9 1 2 - 1 3 .

    F a c i n g t h e e n o r m o u s m a t h e m a t i c a l d i ff ic u lt ie s r a i s e d b y t h e t h e o ry , H i l b e r t s t re s s e d t h e

    n e e d to a p p r o a c h it u s i n g a p h y s i c a l p e r s p e c t i v e n a m e l y , t h r o u g h a t h o r o u g h a p p l i c a t io n

    o f th e a x i o m a t i c m e t h o d , i n o r d e r t o p o i n t c l e a r l y t h o s e p a r ts o f t h e t h e o r y i n w h i c h

    p h y s i c s e n t e rs i n t o m a t h e m a t i c a l d e d u c t i o n . I n t hi s w a y , H i l b e r t p r o p o s e d t o s e p a r at e

    t h r e e d i f f e re n t c o m p o n e n t s o f a p h y s i c a l t h e o r y : f i rs t, w h a t i s a r b it r a ri ly a d o p t e d a s

    d e f in i t io n o r a s s u m e d a s th e b a s i s o f al l e x p e r i e n c e ; s e c o n d , w h a t w e

    a pr ior i

    e x p e c t

    s h o u l d f o l lo w f r o m t h e s e a s s u m p t i o n s , b u t w h i c h t h e c u r r e n t st at e o f m a t h e m a t i c s d o e s

    n o t y e t a l l o w u s t o c o n c l u d e w i t h c e r t a i n t y ; a n d t h i r d , w h a t i s t r u l y p r o v e n f r o m a

    m a t h e m a t i c a l p o i n t o f v ie w . 22 T h u s , b o t h M i n k o w s k i a n d H i l b e rt s t re s s e d t h e n e e d t o

    s e p a r a te i n a c le a r w a y t h e v a r i o u s a s s u m p t i o n s , p h y s i c a l a n d m a t h e m a t i c a l , i n v o l v e d i n

    a th e o r y , a n d t h is i s p r e c i s e l y w h a t M i n k o w s k i a t t e m p t e d t o d o h er e .

    M i n k o w s k i ' s a n a l y s is a l l o w s o n e t o u n d e r st a n d m o r e c l e a r ly h is o w n v i e w s a b o u t

    t h e s p e c i fi c c o n t r ib u t i o n s o f v a r io u s p h y s i c i s t s t o t h e th e o r y o f th e e l e c t r o d y n a m i c s o f

    m o v i n g b o d i e s . L o r e n t z , M i n k o w s k i t h o u g h t , h a d d i s c o v e r e d th e t h e o r e m a n d h a d a l s o

    s e t u p t h e p o s t u l a t e i n t h e f o r m o f t h e c o n t r a c t i o n h y p o t h e s i s . E i n s t e i n ' s c o n t r i b u t i o n

    w a s , a c c o r d i n g t o M i n k o w s k i , t h a t o f h a v i n g v e r y c l e a r ly c l a i m e d t h a t t h e p o s tu l a t e i s

    21 Minkowsld

    1908,

    353: Nu n kann man, ohne noc h zu bes timmten Hy pothesen fiber den

    Zu sam me nh ang v on Elektrizit~it und M ater ie s ich zu bekennen, erwarten, jenes m athem atisch

    evidente Th eorem w erde se ine K onsequenzen so w ei t er st recken, dab dadu rch auch d ie noch n icht

    erkannten Gese tze in bezug au f ponderable K rrper i rgendw ie e ine Kovarianz be i den Lorentz-

    Tran sforma tionen tibernehmen werden.

    22 Hilbert

    1912-3,

    1: Da bei werden w ir aber s treng axioma tisch die Stel len, in denen die Ph ysik

    in die ma them atische D edu ct ion eingreif t, deut l ich hervorheben, und das vone inander trennen,

    was erstens als logisch wil lki i r l iche Defini t ion oder Annahme der Erfahrung entnomen wird,

    zw eitens das, wa s a pr ior i s ich aus diesen A nna hm en folgern liesse, ab et wegen m athem atischer

    Schw ierigkei ten zur Zei t noc h nicht sicher gefo lgert we rden kann, un d dr it tens, das, w as bewiesene

    mathem at i sche Folgerung i st .

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    Herm ann M inkowski and the Pos tu la te of Rela t iv i ty 281

    n o t a n a r ti fi c ia l h y p o t h e s i s , b u t r a th e r, t h a t th e o b s e r v a b l e p h e n o m e n a f o r c e i t u p o n u s

    a s p a r t o f a n e w c o n c e p t i o n o f t im e . M i n k o w s k i d i d n o t m e n t i o n P o i n c a r 6 t h is t i m e ,

    b u t g i v e n t h e la t te r ' s c o n c e p t i o n o f t h e g e n e r a l v a l i d it y o f th e t h e o r e m , M i n k o w s k i

    w o u l d p r e s u m a b l y h a v e c l a s s if ie d P o i n c a r r ' s c o n t r i b u t io n a s h a v i n g a l so f o r m u l a t e d t h e

    r e l a t i v it y p o s t u l a te . I n f a c t , i t w a s P o i n c a r 6 w h o h a d f i rs t s u g g e s t e d e x t e n d i n g t h e

    d o m a i n o f v a l i d it y o f L o r e n t z i n v a r i an c e t o a l l l a w s o f p h y s i c s . I n 1 9 0 4 , f o r i n s ta n c e ,

    P o i n c a r 6 f o r m u l a t e d t h e p r i n c i p l e a s a n e m p i r i c a l t r u th , s t i ll to b e C o n f i r m e d o r r e f u t e d

    b y e x p e r i m e n t , a c c o r d i n g to w h i c h t h e la w s o f p h y s i c s s h o u l d b e t h e s a m e f o r a n y t w o

    o b s e r v e r s m o v i n g w i t h r e c t il i n e ar , u n i f o r m m o t i o n r e l a t i v e t o e a c h o t h e r . 23

    M i n k o w s k i c l a i m e d t h a t t h e p r i n c ip l e h a d n e v e r b e e n f o r m u l a t e d f o r th e e l e c t r o d y -

    n a m i c s o f m o v i n g b o d i e s i n th e w a y i n w h i c h h e w a s d o i n g i t. T h e a i m o f h i s p re s e n ta t io n

    w a s t o d e d u c e a n e x a c t f o r m u l a t i o n o f th e e q u a t i o n s o f m o v i n g b o d i e s f r o m t h e p ri n c i p le

    o f re la t iv i ty , t h i s d e d u c t i o n , h e c l a i m e d , s h o u l d m a k e i t c l e a r t h at n o n e o f th e f o r m u -

    l a t i o n s h i t h e r t o g i v e n t o t h e e q u a t i o n s i s f u l l y c o m p a t i b l e w i t h t h e p r i n c i p l e . I n o t h e r

    w o r d s , M i n k o w s k i b e l i e v e d t h a t h i s a x i o m a t i c a n a l y s i s o f t h e p r in c i p l e o f r e la t iv i ty a n d

    o f t he e l e c t r o d y n a m i c t h e o ri es o f m o v i n g b o d i e s w a s t h e b e s t a p p r o a c h f o r u n e q u i v o c a ll y

    o b t a i n i n g t h e c o r r e c t e q u a t io n s .

    A s i n h is f o r m e r l e ct u re , i n t h e f ir st p a r t o f th e p r e s e n t o n e M i n k o w s k i d i s c u s s e d t h e

    e q u a t i o n s o f a p u r e e l e c t r o m a g n e t i c f ie ld , i. e ., e t h e r w i t h o u t m a t te r . A s p a r t o f h is d i s -

    c u s s i o n o f th e i n v a r i a n c e o f th e s e e q u a t i o n s u n d e r t h e L o r e n t z g r o u p o f t ra n s f o r m a t i o n s ,

    M i n k o w s k i i n t r o d u c e d t h e n e w m a t h e m a t i c a l t o o l t h a t a l lo w e d h i m t o p u t f o r w a r d h i s

    o w n v e r s i o n o f t h e p r in c i p l e o f r e la t iv i t y a n d t h a t t u rn e d i n t o t h e s t a n d a r d l a n g u a g e o f

    a ll f u tu r e d e v e l o p m e n t s o f e l e c t ro d y n a m i c s a n d r e la t iv i ty : t h e f o u r - v e c t o r s o f f o u r a n d

    s ix c o m p o n e n t s ( w h i c h h e c a l le d s p a c e - t i m e v e c t o r s o f t y p e I a n d I I , r e s p e c ti v e l y ). 24

    H e s t re s s e d t h r o u g h o u t t h e i n v a r ia n c e o f t h e m e t r i c e l e m e n t x 2 + x 2 + x 2 x 2 , w h e r e

    x4 = it,

    a n d s h o w e d t h a t th e i n v a r i a n c e o f th e e q u a t i o n s e x p r e s s e d i n t h e f o u r - v e c t o r

    l a n g u a g e f o l lo w s f r o m s i m p l e s y m m e t r y c o n s id e r a t io n s .

    M i n k o w s k i d e d i c a t e d a s e p a ra t e s e c t io n o f t h e f ir st p a r t t o a d i s c u s s io n o f th e c h a n g e s

    i n th e c o n c e p t o f t i m e b r o u g h t a b o u t b y i n t r o d u c i n g t h e L o r e n t z t r a n s f o r m a t i o n s i n t o

    k i n e m a t i c s , a n d i n p a r ti c u l a r th e i m p o s s i b i l i ty o f s p e a k i n g a b o u t t h e s i m u l t a n e i t y o f t w o

    e v e n ts . H i s e x p l a n a t i o n w a s b a s e d o n f o r m a l p r o p e r t ie s o f th e t r a n s fo r m a t i o n s , d i s c u s s e d

    i n a n e a r li e r s e c t i o n : i f i n a c e r t a i n r e f e r e n c e s y s t e m w e a r e g i v e n a s p a c e p o i n t A a t

    t i m e t o = 0 , and a s e co nd po i n t P a t a d i f f e r en t t i m e t, and i f t - t o < P A P A

    b e i n g t h e t i m e r e q u i r e d f o r l ig h t t o tr a v e r se t h e d i s t a n c e b e t w e e n t h e t w o p o i n ts ) , t h e n

    i t i s a l w a y s p o s s i b l e t o c h o o s e a L o r e n t z t r a n s f o r m a t i o n t h a t t a k e s b o t h t o a n d t , t o t h e

    v a l u e t r = 0 . T h e s a m e i s t r u e i f w e a r e g i v e n t w o p o i n t s a t to = 0 a n d a t h i r d o n e a t

    t , o r t h r e e n o n - c o l l i n e a r p o i n t s i n s p a c e a t t o = 0 a n d a f o u r t h o n e a t t ( a g a i n , t - to

    s a t i sf y i n g a s i m i l a r c o n d i t i o n l ik e t h e o n e j u s t m e n t i o n e d ) . H o w e v e r , i f w e a r e g i v e n

    f o u r n o n - c o p l a n a r e v e n t s i t i s n o l o n g e r p o s s i b l e t o f i n d t h e d e s i r e d t r a n s f o r m a t i o n .

    23 See Poincm'61905, 17 6-1 77 ; 1906, 495. An d again in 1908 Poincar6 wrote: I t i s impo ssible

    to escape the impression that the Principle of Relat ivi ty is a general law o f natu re. . . I t i s w el l in

    any case to see wh at a re the consequ ences to which th is poin t o f v iew w ould lead , and then subm i t

    these conseq uence s to the test of experiment. See Poincar~ 1908, 221. ::

    24 Vectors o f type II correspond to m od em second-rank, ant isym me tr ic tensors .

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    2 8 2 L. CORRY

    M i n k o w s k i ' s a r g u m e n t s c a n e s s e n t i a ll y b e c o n s t r u e d , i n h in d s i g h t , a s lo c a t i n g p o in t s

    o u t s i d e o r in s i d e t h e l i g h t -c o n e - a s t h e c a s e m a y b e - o f a g iv e n s p a c e - t i m e e v e n t.

    S u c h a f o r m u l a t i o n w o u l d s e e m i n d e e d t o s u g g e s t it s e lf i n t h is c o n t e x t , y e t M i n k o w s k i

    d i d n o t i n t r o d u c e t h o s e c o n c e p t s a n d a r g u m e n t s a t t h is s ta g e . I n th e c l o s i n g s e c t io n s

    o f th is l e c t u r e h e c a m e m u c h c l o s e r to t h o s e i d e a s , a n d t h e y f i n a ll y a p p e a r e d f u l ly -

    f l e d g e d o n l y in h i s b e s t - k n o w n a r ti c le o n th i s i s su e , t h e f a m o u s l e c t u r e o n S p a c e a n d

    T i m e . O n e s h o u l d a l so n o t i c e t h at , s in c e M i n k o w s k i ' s d i s c u s s i o n w a s i n t e n d e d a s a n

    a x i o m a t i c i n v e s t i g a ti o n o f th e s p e c i f ic i m p l i c a t i o n s o f t h e v a r i o u s a s s u m p t i o n s i n v o l v e d ,

    i t i s s i g n i f i c a n t t h a t h e r a i s e d t h e q u e s t i o n o f s i m u l t a n e i t y a t t h e e n d o f t h e s e c t i o n

    d e a l i n g w i t h t h e e q u a t i o n s i n e m p t y e th e r. W e l e a rn f r o m t h is t h at , f o r M i n k o w s k i , the

    relativity of simultaneity is a consequence of the Lorentz theorem for the equations in

    empty ether, and it is therefore independent of whatever conception of the nature of

    matter one may adopt.

    M i n k o w s k i c o n c l u d e d t h i s s e c t i o n w i t h a r e m a r k t h a t c l a r i f i e s

    h i s u n d e r s t a n d i n g o f th e b a s i c m o t i v a t i o n s b e h i n d E i n s t e i n ' s c o n t r i b u t io n t o t h e la t es t

    d e v e l o p m e n t s i n e l e c tr o d y n a m i c s : m a t h e m a t ic i a n s - M i n k o w s k i s a id - a c c u s t o m e d a s

    t h e y a r e t o d i s c u s s m a n y - d i m e n s i o n a l m a n i f o l d s a n d n o n - E u c l i d e a n g e o m e t r i e s , w i l l

    h a v e n o s e r io u s d i ff ic u l ti e s i n a d a p t i n g t h e ir c o n c e p t o f t i m e t o th e n e w o n e , i m p l i e d

    b y t h e a p p l i c a ti o n o f t h e L o r e n t z tr a n s f o r m a t i o n ; o n t h e o t h e r h a n d , t h e t a s k o f m a k i n g

    p h y s i c a l s e n s e o u t o f t h e e s s e n c e o f th e s e t r a n s f o r m a t i o n s h a d b e e n a d d r e s s e d b y E i n s t ei n

    i n t h e i n t r o d u c t i o n t o h i s 1 9 0 5 r e l a t i v i ty a r ti c le . 25

    A s i n h is e a r li e r 1 9 0 7 t a lk , t h e s e c o n d p a r t o f th e D e c e m b e r 1 9 0 7 p a p e r c o n s i d e r e d

    h o w t h e e q u a t i o n s c h a n g e w h e n m a t t e r i s a d d e d t o t h e e th e r. F o r t h e c a s e o f a b o d y a t r e s t

    i n t h e et h er , M i n k o w s k i s i m p l y re l i e d o n L o r e n t z ' s v e r s i o n o f M a x w e l l ' s e q u a t i o n s , a n d

    a n a l y z e d t h e s y m m e t r y p r o p e r t ie s o f th e la tt er . H e f o r m u l a t e d t h e e q u a t i o n s a s f o l l o w s :

    0 e

    cu r l m - s ( I )

    a t

    d i v e = p ( I I )

    aM

    c u r i e + - - = 0 ( I II )

    Ot

    d i v M = 0 ( I V )

    H e r e M a n d e a re c a l l e d t h e m a g n e t i c a n d e l e c t r ic i n te n s i t ie s Erregung) r e s p e c t i v e l y ,

    E a n d m a r e c a l l e d t h e e l e c t r i c a n d m a g n e t i c f o r c e s , p i s t h e e l e c t r i c d e n s i t y , s i s t h e

    e l e c t r i c c u r r e n t v e c t o r elektrischer Strom).26 T h e p r o p e r t i e s o f m a t te r , i n th e c a s e o f

    i s o t r o p i c b o d i e s, a r e c h a r a c t e r i z e d b y t h e f o l l o w i n g e q u a t i o n s :

    e : e E M = I x m s = c ~ E

    V)

    w h e r e e i s t h e d i e l e c t ri c c o n s t a n t , / z i s t h e m a g n e t i c p e r m e a b i l i t y , a n d c r i s t h e c o n d u c t i v i t y

    o f m a t t e r .

    25 Minkowski 1908, 362: D em Bedti rfnisse, s ich das W esen dieser Transfo rmatione n

    physikal isch n~her zu br ingen, kommt der in der Einlei tung zi t ier te Aufsatz yon A. Einstein

    e n t g e g e n

    26 In Einstein & Lau b 1908, 1908a,in which M inko ws ki 's ar ticle is referred to, the vector M

    in thes e equa tions is called the m ag netic induction, wh ereas e is the dielectric displacement.

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    H e r m a n n M i n k o w s k i a n d t h e P o s t u la t e o f R e l a t i v it y 2 8 3

    F r o m t h e b a si c p r o p e r ti e s o f th e e q u a t io n s f o r b o d i e s a t r es t, M i n k o w s k i d e d u c e d

    t h e f u n d a m e n t a l e q u a t i o n f o r t h e c a se o f a b o d y i n m o t i o n . T h i s d e d u c t i o n i s w h e r e

    t h e d e t a il e d a x i o m a t i c d e r i v a t io n i s r e a l iz e d : M i n k o w s k i a s s u m e d t h e v a l i d i ty o f t h e

    p r e v i o u s l y d i s c u s s e d e q u a t i o n s f o r m a t t e r a t r e s t t o w h i c h h e a d d e d t h re e a x i o m s . H e

    t h e n s o u g h t t o d e ri v e th e e q u a t i o n s f o r m a t t e r i n m o t i o n e x c l u s iv e l y f r o m t h e a x i o m s

    t o g e t h e r w i t h t h e e q u a t i o n s f o r r e st . M i n k o w s k i ' s a x i o m s a r e:

    1 . W h e n e v e r t h e v e l o c i t y v o f a p a r t i c le o f m a t t e r e q u a l s 0 a t

    x , y , z , i t

    i n s o m e

    r e f e r e n c e s y s t e m , t h e n e q u a t i o n s ( I ) - ( V ) a l s o r e p r e s e n t , i n t h a t s y s t e m , th e r e la t i o n s

    a m o n g a ll t h e m a g n i t u d e s : p , t h e v e c t o r s s , v , e , M , E , a n d t h e i r d e r i v a t iv e s w i t h

    r e s p e c t t o x , y , z , i t .

    2 . M a t t e r a l w a y s m o v e s w i t h a v e l o c i t y w h i c h i s le s s t h a n t h e v e l o c i t y o f l i g h t in

    e m p t y s p a c e ( i. e ., I v [ = v < 1 ).

    3 . I f a L o r e n t z t r a n s f o r m a t i o n a c t i n g o n t h e v a r i a b l e s

    x , y , z , i t ,

    t r a n s f o r m s b o t h

    m

    - i e

    a n d

    M , - i E

    a s s p a c e - t i m e v e c t o r s o f t y p e I I , a n d

    s , i p

    a s a s p a c e - t im e v e c t o r o f

    t y p e I , t h e n i t t r a n s f o r m s t h e o r i g i n a l e q u a t i o n s e x a c t l y i n t o th e s a m e e q u a t i o n s w r i t t e n

    f o r t he t r an s f o r m e d m a g n i t u d e s F

    M i n k o w s k i c a l le d th is l a s t a x i o m , w h i c h e x p r e s s e s i n a p r e c i s e w a y t h e r e q u i r e m e n t o f

    L o r e n t z c o v a r i a n c e f o r th e b a s ic e q u a t i o n s o f t h e e l e c t ro d y n a m i c s o f m o v i n g m a t t er , t he

    p r i n c i p l e o f r e la t iv i ty . T h a t i s to s a y : i t i s o n l y a f t e r e s t a b l i s h i n g t h e e q u a t i o n s f o r e m p t y

    e th e r, a n d p r o v i n g t h e L o r e n t z t h e o r e m o f i n v a r i a n c e , t h a t w e c a n s p e a k o f th e p r in c i p l e o f

    r e l a t i v i t y , w h i c h , t o g e t h e r w i t h t w o a d d i t i o n a l a s s u m p t i o n s , y i e l d s t h e e l e c t r o d y n a m i c s

    o f m o v i n g m a t t e r . It i s r e l e v a n t t o s e e i n s o m e d e t a i l h o w M i n k o w s k i i n t h is s e c t i o n

    a p p l i e s t h e a x i o m s t o d e r i v e t h e e q u a t io n s .

    S i n c e v < 1 ( a x i o m 2 ) , M i n k o w s k i c o u l d a p p l y a r e s u l t o b t a i n e d i n t h e f ir s t p a r t ,

    a c c o r d i n g t o w h i c h t h e v e c t o r v c a n b e p u t i n a o n e - t o - o n e r e l a t i o n w i t h t h e q u a d r u p l e

    Vx Vy v z i

    tO l - ~ , w 2 - l / - ~ v 2 , w 3 - ~ 2 ~ v 2 , w 4 = x / 1 - 1 2

    w h i c h s a t i s f i e s t h e f o l l o w i n g r e l a t i o n :

    A g a i n f r o m t h e r e s u lt s o f t h e f ir s t p a r t , i t f o l l o w s t h a t th i s q u a d r u p l e t r a n s f o r m s a s a

    s p a c e - t i m e v e c t o r o f t y p e I. M i n k o w s k i c a l l e d i t t h e v e l o c i t y s p a c e - t im e - v e c t o r . N o w ,

    i f v = 0 , b y a x i o m 1 , e q u a t i o n s ( I ) - ( V ) a r e a l s o v a l i d f o r t h i s c a s e . I f v 0 , s i n c e

    I v I < 1 , a g a i n t h e r e s u l ts o f e a r l i e r s e c ti o n s a l l o w t h e i n t r o d u c t i o n o f a t r a n s f o r m a t i o n

    f o r w h i c h

    / = 0 , l = 0 , / = 0 , l = i .

    1 to 2 ~o3 w 4

    I n th i s c a s e , w e a l s o o b t a i n a t r a n s f o r m e d v e l o c i t y v I = 0 . A c c o r d i n g t o a x i o m 3 , w h a t -

    e v e r th e b a s i c e q u a t io n s m a y b e t h a t h o l d f o r th i s c a s e m u s t r e m a i n i n v a r i an t w h e n

    w r i t t e n f o r t h e t r a n s f o r m e d v a r i a b l e s x ~, y l , z ~, t ~ a n d t h e t r a n s f o r m e d m a g n i t u d e s

    M ~, e t , E ' , m ~, y , s ' , a n d t h e d e r i v a t i v e s o f th e l a t t e r w i t h r e s p e c t t o x ' , y ' , z ~, t ' .

    27 Se e Minkwsd198 36 9. Fr the sa ke f simpici ty m y f rmuat in he re is s ighty d i f fe re nt

    but essen t ia l ly equ iva len t to the o r ig ina l one .

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    28 4 L. CoRRY

    B u t , s i n c e v / = 0 , t h e t r a n s f o r m e d e q u a t i o n s a r e ( b y a x i o m 1 ) j u s t ( I / )- ( IV ~ ) , o b t a i n e d

    f r o m ( I ) - ( I V ) b y t a g g i n g a ll v a ri a b l e s .T h e s a m e is tr u e f o r e q u a t i o n ( V ) ( a l t h o u g h t h e r e

    i s n o n e e d t o a p p l y a x i o m 3 ), b u t w i t h e , / z , a n d o- r e m a i n i n g u n c h a n g e d . F i n a l ly , o n e

    a p p l ie s t h e i n v e r s e o f t h e L o r e n t z t r a n s f o r m a t i o n o r i g i n a l l y a p p l i e d a n d , b y a x i o m 3 , i t

    f o l l o w s t h a t t h e f o r m o f t h e b a s i c e q u a t i o n s f o r t h e o r i g i n a l v a r ia b l e s i s i n f a c t p r e c i s e l y

    ( I ) -( I V ) . M i n k o w s k i t h u s c o n c l u d e d t h a t th e b a s i c e q u a t io n s o f e le c t r o d y n a m i c s f o r

    m o v i n g b o d i e s a r e th e s a m e a s t h e e q u a t i o n s f o r s t a ti o n a r y b o d i e s , a n d t h e e f f e c ts o f th e

    v e l o c i t y o f m a t t e r a r e m a n i f e s t o n l y t h r o u g h t h o s e c o n d i t io n s i n w h i c h i ts c h a r a c t e ri s ti c

    c o n s t a n t s e , / x , a n d ~r a p p e ar . A l s o , M i n k o w s k i c o n c l u d e d , t h e t r a n s f o r m e d e q u a t i o n

    ( V ' ) c a n b e t r a n s f o r m e d b a c k i n t o th e o r i g i n a l e q u a t i o n ( V ) .

    T h e p a r ti c u la r k in d s o f a rg u m e n t a d v a n c e d i n t hi s s e c ti o n b y M i n k o w s k i s e e m s o m e -

    w h a t o u t o f p l a c e a m i d s t t h e e la b o r a t e m a t h e m a t i c a l a n d p h y s i c a l a r g u m e n t s d i s p l a y e d

    t h r o u g h o u t t h e t al k . T h e y f i n d a n a t u ra l p l a c e , h o w e v e r , i n th e l i g h t o f th e k i n d o f a x -

    i o m a t i c c o n c e p t u a l c l a r i f i c a t i o n p r o m o t e d b y H i l b e r t i n h i s l e c t u r e s o n p h y s i c s , f o r ,

    l ik e H i l b er t, M i n k o w s k i w a s s t r e s si n g h e r e p r e c i s e l y t h a t k i n d o f t a sk . M i n k o w s k i , i n

    a d d i ti o n , w e n t o n t o c h e c k t o w h a t e x t e n t d i f f er e n t e x is t in g v e r s i o n s o f th e e q u a t i o n s

    s a t is f i ed t h e p r i n c i p l e a s s t a t e d i n h i s a x i o m s . S i n c e n o t h i n g s i m i l a r t o h i s a n a l y s i s h a d

    b e e n a t te m p t e d b e f o r e , M i n k o w s k i ' s i m p l i c it a s s u m p t i o n w a s t h a t o n l y e q u a t io n s w h i c h

    c o m p l y w i th h i s o w n v e r s i o n o f t h e p ri n c ip l e c a n b e a c c e p t e d a s c o rr e ct . W i t h o u t g o i n g

    a n y f u r t h e r i n t o d e ta i ls h e r e , I w i ll o n l y p o i n t o u t t h a t M i n k o w s k i f o u n d t h e m a c r o s c o p i c

    e q u a t io n s f o r m o v i n g m e d i a w h i c h w e r e f o r m u l a t e d in L o r e n t z ' s Encyc lop i id ie a r t i c l e

    ( L o r e n t z 1 9 0 4 ) t o b e i n c e r t a in c a s e s i n c o m p a t i b l e w i t h h i s p r i n c ip l e . 28 M i n k o w s k i a l s o

    d i s c u s s e d t h e e q u a t i o n s f o r m u l a t e d i n 1 9 0 2 b y E m i l C o h n , p o i n t i n g o u t t h a t t h e y a g r e e

    w i t h h i s o w n u p t o t e r m s o f f i rs t o r d e r i n t h e v e l o c i t y . 29 A f t e r h a v i n g f o r m u l a t e d t h e

    e q u a t i o n s a n d d i s c u s s e d t h e i r i n v a r i a n c e p r o p e rt i es , M i n k o w s k i d e a l t in d e ta il , i n t h re e

    a d d i t io n a l s e c t i o n s , w i t h t h e p r o p e r ti e s o f e l e c tr o m a g n e t i c p r o c e s s e s i n t h e p r e s e n c e o f

    m a t t e r .

    M i n k o w s k i ' s p a p e r h a s a n a p p e n d i x d i s c u s s in g t h e re l at io n s b e t w e e n m e c h a n i c s a n d

    t h e p o s t u l a t e ( n o t t h e p r i n c i p l e ) o f r e la t iv i t y . I t i s in t h i s a p p e n d i x t h a t t h e s i m i l a r i t y

    o f M i n k o w s k i ' s a n d H i l b e r t ' s tr e a t m e n t o f p h y s i c a l t h e o r i e s is m o s t c l e a r l y m a n i f e s t :

    28 Minkowski 1908 372 (I tal ics in the or iginal) : "Danach entsprechen die al lgemeinen

    Differential gleic hu ng en von Lo ren tz fiir belieb ig magnetisierte Ktirper nicht dem Re l a -

    tiviNtsprinzipe."

    29 Minkowski c i t ed here Cohn

    1902.

    For Cohn 's e lec t rodynamics see Dar r igol

    1993

    2 7 1 -

    276; Hi ros ige 1966 31-37; Mil ler 1981 181-182. M il ler gives a long l is t o f w orks that cr i t ical ly

    discussed Cohn's theory, but Minkowski 's ar t icle is not ment ioned in this context . On the other

    hand, M il ler descr ibes C oh n's theory in the fol lowing terms: "C oh n speculated on nei ther the

    nature o f the ether , no r the nature o f electr ici ty (his the ory w as no t bas ed u po n an atomist ic

    conce pt ion of e lec tr ic ity) , nor d id he a t t empt to reduce the l aws of e lec t romag net i sm to those of

    me chanics." Moreover, adds Mil ler , C oh n sugg ested that the ether shou ld be ut i l ized as "heuris t ic

    conce pt ," that sho uld not acquire an imp ortance relat ive to the theo ry in question." Given the view s

    of M inkow ski as presented here, these remark s su gges t a possible, direct or indirect, influence

    of Coh n ' s w or k on Mi nkow sk i ( A l t hough acco rd i ng t o Pyenson

    1979

    Cohn's ar t icles were not

    among the texts s tudied in the 1905 seminar on electron theory.) A more detai led discussion of

    this p oint m ust be lef t for a future oc casion.

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    Hermann Minkow ski and the Postulate of Relativity 28 5

    t h e a p p e n d i x i s a n e x p l o r a t io n o f t h e c o n s e q u e n c e s o f a d d i n g t h e p o s t u l a t e o f r e la t iv i t y

    t o t h e e x i s t in g e d i f i c e o f m e c h a n i c s , a n d o f t h e c o m p a t i b i l i t y o f th e p o s t u l a t e w i t h t h e

    a l r e a d y e s t a b l i s h e d p r i n c i p l e s o f t h i s d is c i p li n e . T h e e x t e n t to w h i c h t h i s a d d i t i o n c a n b e

    s u c c e s s f u l l y r e a l i z e d p r o v i d e s a s t a n d a r d f o r a s s e s s i n g t h e s t a t u s o f L o r e n t z c o v a r i a n c e

    a s a t r u l y u n i v e r s a l p o s t u l a t e o f a ll p h y s i c a l s c i e n c e .

    M i n k o ws k i s h o we d - u s i n g t h e f o r m a l i s m d e v e l o p e d in t h e e a r l i e r s e c t i o n s - t h a t

    i n o r d e r f o r t h e e q u a t io n s o f m o t i o n o f c l as s ic a l m e c h a n i c s t o r e m a i n i n v a r i an t u n d e r

    t h e L o r e n t z g r o u p i t i s n e c e s s a r y to a s s u m e t h a t c = e c . I t w o u l d b e e m b a r r a s s i n g o r

    p e r p l e x i n g ( v e r w i r r e n d ) , h e s a id , i f t h e l a ws o f t r a n s f o r m a t i o n o f th e b a s i c e x p r e s s i o n

    _ x 2 _ y 2 _ Z 2 c 2 t 2

    i n t o i t s e l f we r e t o n e c e s s i t a t e a c e rt a i n f in i te v a l u e o f c i n a c e r ta i n d o m a i n o f p h y s i c s a n d

    a d i ff e r en t , in f in i te one , in a secon d do ma in . A ccord ing ly , the pos tu la te o f r e la t iv i ty ( i .e . ,

    o u r c o n f i d e n c e i n t h e u n i v e rs a l v a l id i t y o f t h e t h e o r e m ) c o m p e l s u s t o s e e N e w t o n i a n

    m e c h a n i c s o n l y a s a t e n t at i v e a p p r o x i m a t i o n i n i ti a ll y s u g g e s t e d b y e x p e r i e n c e , w h i c h

    m u s t b e c o r r e c t e d to m a k e i t in v a r ia n t f o r a f in it e v a lu e o f c . M i n k o w s k i n o t o n l y t h o u g h t

    t h a t r e f o r m u l a t i n g m e c h a n i c s i n t h i s d i r e c t i o n wa s p o s s i b l e ; i n t e r m s v e r y s i m i l a r t o

    t h o s e t h a t c a n b e f o u n d i n H i l b e r t ' s le c t u r e n o t e s , h e a s s e r t e d t h a t s u c h a r e f o r m u l a t i o n

    s e e m s c o n s i d e r a b l y t o p e r f e c t th e a x i o m a t i c s t ru c t u r e o f m e c h a n i c s . 3 ~

    Na t u r a l l y , a ll t h e d i s c u s s i o n i n t h is s e c t i o n i s c o u c h e d i n t h e l a n g u a g e o f s p a c e - t i m e

    c o o r d i n a te s x , y , z , t . B u t M i n k o w s k i r e f e rr e d t h r o u g h o u t to t h e p r o p e r t ie s o f m a t t e r

    a t a c e r t a i n p o i n t o f

    s p a c e

    a t a g i v e n

    t i m e ,

    c l e a r l y s e p a r a t i n g t h e t h r e e e l e m e n t s , a n d

    f o c u s i n g o n t h e p a t h t r a v e rs e d b y a p a r t i cl e o f m a t t e r a l o n g a l l ti m e s t . T h e s p a c e - t i m e

    l i n e o f t h a t p i e c e o f m a t t e r is t h e c o l l e c t io n o f a ll t h e s p a c e - t i m e p o i n t s x , y , z , t

    a s s o c i a t e d w i t h t h a t p a r t i c le , a n d t h e t a s k o f s t u d y i n g t h e m o t i o n o f m a t t e r is d e f i n e d

    a s f o ll o w s : " F o r e v e r y s p a c e - t i m e p o i n t t o d e t e r m i n e t h e d i r ec t io n o f t h e s p a c e - t i m e

    l i n e t r a v e r s e d b y i t ." L i k e w i s e , t h e c o l l e c t i o n o f a ll s p a c e - t i m e l in e s a s s o c i a t e d w i t h t h e

    m a t e r i a l p o i n t s o f a n e x t e n d e d b o d y i s c a ll e d i ts s p a c e - t i m e t h r e a d ( R a u m - Z e i t f a d e n ) . O n e

    c a n a l s o d e f i n e t h e " p r o p e r t im e " o f a g i v e n m a t t e r p a r t i c l e i n t h e s e t e r m s , g e n e r a l i z i n g

    L o r e n t z ' s c o n c e p t o f lo c a l t im e . O n e c a n a l s o a s s o c i a te a p o s i t iv e m a g n i t u d e ( c al le d

    m a s s ) t o a n y w e l l -d e l i m i t e d p o r ti o n o f ( th r e e - d i m e n s i o n a l ) s p a c e a t a g i v e n ti m e . T h e s e

    l a s t t wo c o n c e p t s l e a d to t h e d e f in i t io n o f a r e s t - m a s s d e n s i t y , wh i c h M i n k o w s k i u s e d

    t o f o r m u l a t e t h e p r i n c i p l e o f c o n s e r v a t i o n o f m a s s i n v o l v i n g a l l t h e s e c o n c e p t s . T h u s ,

    M i n k o w s k i r e l ie d h e r e o n t h e f o u r d i m e n s i o n a l l a n g u a g e a s a n e f fe c t iv e m a t h e m a t i c a l

    t o o l p r o v i d in g a v e r y c o n c i s e a n d s y m m e t r i c m e a n s o f e x p r es s io n , b u t h i s a p p e a l t o t h e

    f o u r - d im e n s i o n a l g e o m e t r y d o e s n o t s e e m t o c o n v e y a t th i s s t ag e a n y d i r e c t e v id e n c e o f

    a n e w , a r t i c u la t e d c o n c e p t i o n o f t h e e s s e n c e o f t h e r e la t i o n b e t w e e n s p a c e a n d t i m e , l i k e

    t h e o n e th a t c h a r a c t e r iz e s M i n k o w s k i ' s f a m o u s 1 9 08 K6 1 n l e c t u r e ( d i sc u s s e d b e l o w) .

    U s i n g t h is l a n g u a g e , t h e n , M i n k o w s k i a n a l y z e d t h e c o m p a t i b i l it y o f th e p o s t u l a te

    o f r e la t i v it y w i t h t w o a c c e p t ed , b a s i c p r i n c ip l e s o f m e c h a n i c s : H a m i l t o n ' s p r i n c i p le a n d

    30 Minkowski 1 9 0 8 , 393 (Italics in the original.): " Ich rntc hte ausf 'tihren, dab durch eine R e -

    f o r m i e r u n g d e r M e c h a n i k , w o b e i a n S te l le d e s N e w t o n s c h e n R e l a t iv i t ii t sp o s t u l a te s m i t c = o o

    e i n s o l c h e s f i i r e i n e n d l ic h e s c tr i tt , s o g a r d e r a x i o m a t i s c h e A u f b a u d e r M e c h a n i k e r h e b l i c h a n

    V o l le n d u n g z u g e w i n n e n s c h e i n t .

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    14/42

    2 8 6 L. CORRY

    t h e p r i n c i p le o f c o n s e r v a t i o n o f e n e r g y. C o m p a t i b i l i ty w i th t h e f o r m e r h e d i s c u s s e d i n

    a w a y a n a l o g o u s t o h i s d i s c u s s i o n o n e l e c t ro d y n a m i c s i n e a r l i e r s e c t i o n s . A s fo r t h e

    c o n s e r v a t i o n o f e n e rg y , M i n k o w s k i s t re s s e d w i t h p a r ti c u l a r e m p h a s i s t h e f u l l s y m m e t r y

    o f t h e e q u a t i o n s o b t a i n e d fo r a l l f o u r v a r i a b l e s x , y , z , t . I n t e g ra t i n g th e t e rm s o f

    t h e e q u a t io n s o f m o t i o n d e r i v e d u s i n g t h e H a m i l t o n p r in c i p l e, h e d e d u c e d f o u r n e w

    d i f f e r e n t ia l e q u a t i o n s

    d d x

    m - - R x ,

    d r d r

    d d y

    m d r d r - R y ,

    d d z

    rrt - R z ,d r d T

    d d t

    m - R , .

    d r d r

    H e re m i s t h e c o n s t a n t m a s s o f a t h r e a d , r i s t h e p ro p e r ti m e , a n d R i s a v e c t o r o f t y p e

    I : the m o v i n g f o r c e o f th e m a t e r i a l p o i n t s i n v o lv e d . T h e f u l l s y m m e t r y o b t a i n e d h e r e b y

    t h e a d o p t i o n o f t h e p o s t u l a t e o f r e l a t i v i t y s t r u c k M i n k o w s k i a s v e ry t e l li n g , e s p e c i a l l y i n

    r e l a t i o n t o t h e s t a t u s o f t h e fo u r t h e q u a t i o n . A s i n t h e p r e v i o u s l y c o n s i d e re d , a n a l o g o u s

    c a s e o f e l e c t r o d y n a m i c s , h e c l a i m e d , h e r e t o o t h e r e i s a h i g h d e g r e e o f p h y s i c a l e v i d e n c e

    i n i ts f a v o r . 31 M o re o v e r , h e c o n c l u d e d - a g a i n i n t e rm s s t r i k i n g l y s i m i l a r to t h o s e fo u n d

    i n H i l b e r t ' s l e c tu r e s o n p h y s i c s - t h e d e r i v a t i o n p r e s e n t e d h e re j u s t i f ie s t h e a s s e r t i o n t h a t

    i f th e p o s t u l a te o f r e la t i v it y is p l a c e d o n t o p o f t h e b u i l d i n g o f m e c h a n i c s , t h e e q u a t i o n s

    o f m o t i o n c a n b e f u l l y d e r i v e d f r o m t h e p r i n c i p l e o f c o n s e r v a t i o n o f e n e r g y a l on e . 3 a

    S o m u c h f o r th e b a s i c p r i n c ip l e s o f m e c h a n i c s a n d t h e l a w s o f m o t i o n . B u t c l e a rl y ,

    t h e t r u l y u n i v e r s a l v a l id i t y o f t h e p o s t u l a t e o f r e l at i v R y c o u l d o n l y b e e x p e c t e d i f o n e

    c o u l d s h o w t h a t i ts a s s u m p t i o n d o e s n o t c o n t r a d i c t th e o b s e r v a b l e p h e n o m e n a r e l a t e d to

    g ra v i t a t io n . T o t h a t e n d , in t h e c l o s i n g p a s s a g e s o f th e t a l k , h e s k e t c h e d h i s p ro p o s a l f o r

    a L o r e n t z c o v a r i a n t th e o r y o f g r a v it a ti o n , m u c h m o r e e l a b o r a t e t h a n h i s e a r l ie r o n e . A s

    i n h is f o r m e r ta l k , M i n k o w s k i a g a i n m e n t i o n e d P o i n c a r 4 ' s s i m i l a r a tt e m p t , b u t d e c la r e d

    t h a t h i s o w n fo l l o w e d a d i f f e r e n t d i r e c ti o n .

    M i n k o w s k i e l a b o r a t e d h i s f o u r - d i m e n s i o n a l f o r m u l a t i o n e v e n f u r th e r h e re , i n t ro d u c -

    i n g i d e a s q u i te c l o s e to t h e n o t i o n o f a l i g h t c o n e a n d t h e k i n d o f r e a s o n i n g a s s o c i a te d

    w i t h it. I t i s p e r t i n e n t t o p r e s e n t b r i e f l y t h e b a s i c t e rm s o f h i s d e r i v a t i o n o f t h e l a w o f

    g ra v i t a t i o n , s i n c e t h e y c o n v e y a d i s t i n c t g e o m e t r i c f l a v o r ( i n t h e b a s i c , i n t u it i v e s e n s e o f

    t h e t e r m g e o m e t r i c , t h o u g h i n f o u r d im e n s i o n s i n s t e a d o f t h e u s u a l th r e e ) - a fl a v o r

    t h a t is o f t e n a d d u c e d i n c o n n e c t i o n w i t h M i n k o w s k i ' s a p p r o a c h t o r e la t iv i ty , b u t w h i c h

    a p p e a r s o n l y i n t h i s s e c ti o n , a n d n o t i n h i s p r e v i o u s o n e s o n e l e c t r o d y n a m i c s o r e v e n o n

    m e c h a n i c s .

    31 Minkowski 1 9 0 8 , 401 (Italics in the original): . . .g lei ch sa m e ine hOhere phys i ka l i sche

    E v i d e n z z u z u s c h r e i b e n i s t .

    32 Minkowski 1908 , 401 (Italics in the original): W i r d d a s R e l a t iv i t ii t s p o s tu l a t a n d i e S p i t z e

    d e r M e c h a n i k g e s t el lt , s o f o I g e n d i e v o I s ta n d i ge n B e w e g u n g s g e s e t z e a l le i n a u s d e m S a t z e v o n d e r

    Energ ie .

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    Hermann Minkow ski and the Postulate of Relativity 28 7

    I n o r d e r to a d a p t N e w t o n ' s t h e o r y o f g r a v i ta t i o n t o t h e d e m a n d o f L o r e n t z c o v a r i a n c e

    M i n k o w s k i d e s c r i b e d i n f o u r - d i m e n s i o n a l g e o m e t r i c a l t e rm s t h e f o r c e v e c t o r a c ti n g o n

    a m a s s p a r t ic l e rn a t a c e r ta i n p o i n t B . T h i s v e c t o r h a s to b e o r t h o g o n a l t o t h e wo r l d - l i n e

    o f t h e p a r t i c l e a t B , s i n c e f o u r - f o r c e v e c t o r s a r e o r t h o g o n a l t o f o u r - v e l o c i t y v e c t o r s .

    T o r e m a i n c l o s e t o N e w t o n ' s t h e o r y , M i n k o w s k i a l s o a s s u m e d t h a t t h e m a g n i t u d e o f

    t h i s v e c t o r is i n v e r s e l y p r o p o r t i o n a l t o t h e s q u a r e o f th e d i s t a n c e ( in o r d i n a r y s p a c e )

    b e t w e e n a n y t wo m a s s p a r t ic l e s . F i n a ll y , h e a l s o a s s u m e d t h a t t h e a c t u a l d i re c t i o n o f t h e

    o r t h o g o n a l v e c t o r to t h e wo r l d - l i n e o f m i s in f a c t d e t e r m i n e d b y t h e l in e c o n n e c t i n g

    t h e t wo a t t r a c ti n g p a r t ic l e s . T h e s e r e q u i r e m e n t s m u s t a l l b e s a t is f ie d b y a n y a d a p t a t i o n

    o f N e w t o n ' s l a w s t o L o r e n t z c o v a r ia n c e , b u t o f c o u r se , M i n k o w s k i s t il l h a d t o b e m o r e

    s p e c i f ic i n h i s c h o i c e o f s u c h a la w . He d i d s o i n t h e f o l lo w i n g w a y : T a k e a fi x e d s p a c e -

    t i m e p o i n t

    B*( x * , y * ,

    z * , t * ) , a n d c o n s i d e r a l l th e p o i n t s

    B ( x , y , z , t )

    s a t i s f y i n g t h e

    e q u a t i o n

    (x - x* ) 2 + (y - y ,) 2 _q_ (z - z*) 2 = ( t - t*) 2, ( t - t* => 0) .

    T h i s i s c a l l e d th e l i g h t - s t r u c t u r e o f B * , a n d B * i s a l ig h t - p o i n t in t h e s e t o f al l t h e p o i n t s

    l o c a t e d t o w a r d s t h e c o n c a v e s i d e o f th e 3 - s u r f a c e d e fi n e d b y t h e l i g h t- s t ru c t u r e . U s i n g t h e

    l a n g u a g e i n t r o d u c e d l a t er b y M i n k o w s k i h i m s e l f, o n e c a n s a y th a t B * c a n c o m m u n i c a t e

    b y l i g h t s i g n a l s w i t h a l l p o i n t s o f wh i c h i t i s a l ig h t - p o i n t. I f i n t h e a b o v e r e l a t i o n B * i s

    t a k e n a s v a r i a b l e a n d B a s f i x e d , t h e n M i n k o ws k i c l a i m e d t h a t f o r a n a r b i t r a r i l y g i v e n

    s p a c e - t i m e l i n e t h e r e e x i s t s o n l y o n e p o i n t B * wh i c h i s a l i g h t - p o i n t o f B . T h i s l a t t e r

    c o n c l u s i o n i s v a l i d o n l y i f t h e s p a c e - t i m e l in e i s ( u s in g t h e t e r m i n o l o g y i n t r o d u c e d l a t e r)

    t i m e - l i k e , wh i c h i s i m p l i c i t i n M i n k o w s k i ' s d e f i n i t io n o f s p a c e - t i m e l in e s a s wo r l d - li n e s

    o f m a t t e r. 33 G i v e n t wo m a t t e r p o i n t s F , F * w i t h m a s s e s m , m * , r e s p e c t iv e l y , a s s u m e F

    i s a t s p a c e - t i m e p o i n t B , a n d le t

    B C

    b e t h e i n f i n t e s im a l e l e m e n t o f t h e s p a c e - t i m e l i n e

    t h r o u g h F . T h i s s p a c e - t i m e l in e i s n o t h i n g b u t t h e ( m o d e r n l a n g u a g e ) w o r d - li n e s o f t h e

    p a r t ic l e s a t th o s e e v e n t s, w i t h m a s s e s m , m * . M i n k o w s k i c l a i m e d t h a t t h e m o v i n g f o r c e

    o f t h e m a s s p o i n t F a t B s h o u l d

    (mOge)

    b e g i v e n b y a s p a c e - t i m e v e c t o r o f t y p e I , w h i c h

    i s n o r m a l t o

    B C ,

    a n d w h i c h e q u a ls t h e s u m o f th e v e c t o r d e s c ri b e d b y t h e f o r m u l a

    ( O A ] 3

    ram * \ ~ 1 B D * ,

    ( N)

    and a second , su i tab le vec tor , pa r a l le l to

    B C * .

    F i g u r e 1 m a y h e l p i n u n d e r s t a n d in g

    M i n k o w s k i ' s t r ai n o f t h o u g ht .

    T h e a d d i ti o n a l s p a c e - t i m e p o i n ts t h a t a p p e a r h e r e a r e d e fi n e d b y M i n k o w s k i ( w i th o u t

    h i m s e l f u s i n g a n y f i g u r e ) a s f o ll o ws : B * i s t h e l ig h t - p o i n t o f B a l o n g t h e s p a c e - t i m e li n e

    o f F * ; O i s t h e o r ig i n o f th e c o o r d i n a te s y s t e m a n d

    O A t

    i s a s e g m e n t p a r a l l e l to

    B C *

    ( C * b e i n g t h e l ig h t - p o in t a l o n g t h e w o r l d -l i n e o f F * , o f s p a c e - t i m e p o i n t C ) w h o s e

    e n d p o i n t A t l i es o n t h e f o u r - d i m e n s i o n a l h y p e r b o l i c s u r f a c e

    _ x 2 y 2 _ z a + t 2 = 1 .

    F i n a l ly , D * i s t h e i n t e rs e c t i o n p o i n t o f th e l i n e t h r o u g h B C * a n d t h e n o r m a l t o O A I

    p a s s i n g th r o u g h B .

    33 Minkowski

    1908,

    393.

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    16/42

    2 8 8 L. CORRY

    I S p a c e o r m a l ~ C C

    to B C D ~ ~ .....

    F igu re

    Us in g F ig . 1 , s o m e fu r t h e r ex p l an a t i o n s m ay h e lp t o c l a r i fy M in k o ws k i ' s s o m e-

    wh a t o b s cu re t r ea tm en t o f g rav i t a t i o n . In d ev e lo p in g t h i s t o p i c , M in k o ws k i ad d s t h e

    as s u m p t io n t h a t t h e m a te r i a l p o in t F * m o v es u n i fo rm ly , i. e. , t h a t F * d es c r i b e s a s t r a ig h t

    l in e . T h u s , a t t h e o u t s e t M in k o ws k i h a s p re s u m ab ly a s s u m e d t h a t F * m o v es a rb i tr a r i ly

    (as descr ibe d in F ig . 1 above) . In th is mo re ge nera l case , BC an d B C* r ep re s en t t h e

    t an g en t v ec to r s t o t h e cu rv es F an d F* , an d t h ey can b e p h y s i ca l l y i n t e rp re t ed a s th e

    fo u r -v e lo c i t ie s o f th e m as s es w i th wo rd - l i n e s F an d F* , r e s p ec ti v e ly . No w , M in k o w s k i ' s

    g rav i ta t i o n a l fo rce m u s t b e o r t h o g o n a l t o t h e fo u r -v e lo c i t y o f F a t B , an d t h e re fo re o r -

    t h o g o n a l t o BC. B C*, o n t h e o th e r h an d , h