AHES-Corry-Hermann Minkowski and the postulate of

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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