Skip to main content
SpringerLink
Log in

Einstein, Hilbert, and the Theory of Gravitation

  • Chapter
The Physicist’s Conception of Nature

Abstract

From 1907 to 1925, Albert Einstein dominated the development of the general relativity theory of gravitation. His own work on the theory of gravitation from 1912 to 1916 had the drama of high adventure. David Hilbert, fascinated by Einstein’s work on relativity and Gustav Mie’s work on electrodynamics, decided to construct a unified field theory of matter. In two communications to the Gottingen Academy (on 20 November 1915 and 23 December 1916), Hilbert developed his theory of the foundations of physics. In the first of these communications, he derived the field equations of gravitation and the conditions governing them. Hilbert’s work, although inspired by Einstein, was independent of and simultaneous with Einstein’s derivation of the field equations and, in certain essential respects, went beyond Einstein’s. Einstein, at that time, was very critical of the efforts of Mie, Hilbert and Weyl to construct a unified field theory of gravitation and electromagnetism. After 1925, such a programme became his primary mission.

In this paper, we have given an account of the intellectual struggles of this fascinating period when the modern theory of gravitation was created.

This investigation developed from a question which Professor Eugene Wigner asked me in November 1971 about Hilbert’s contribution to the equations of general relativity, and my reply to him. See Acknowledgements and Appendix.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References and Notes

  1. From Einstein-Sommerfeld Briefwechsel, edited by A. Hermann, Schwabe and Co., Basel/ Stuttgart 1968, p. 26. Writing to Hilbert on 1 November 1912 Sommerfeld said, ‘My writing to Einstein was in vain…. Einstein is obviously so deeply immersed in [the theory of] gravitation, that he is deaf against everything else.’ (p. 27).

    Google Scholar 

  2. A. Einstein, Jahrbuch der Radioaktivität und Elektronik 4 (1907), pp. 443–444.

    Google Scholar 

  3. A. Einstein, Jahrbuch 4 (1907), Ref. (2), p. 454.

    Google Scholar 

  4. A. Einstein, Jahrbuch 4 (1907), Ref. (2), p. 454.

    Google Scholar 

  5. A. Einstein, Jahrbuch 4 (1907), Ref. (2), pp. 458–459.

    Google Scholar 

  6. A. Einstein, Jahrbuch 4 (1907), Ref. (2), p. 459.

    Google Scholar 

  7. A. Einstein, Jahrbuch 4 (1907), Ref. (2), p. 461.

    Google Scholar 

  8. A. Einstein, ‘Die Relativitätstheorie’, Vierteljahresschr. Naturforsch. Ges. Zürich, 56, 1–14 (1911).

    Google Scholar 

  9. A. Einstein, Ref. (12), p. 906.

    Google Scholar 

  10. A. Einstein, Ref. (12), p. 908.

    Google Scholar 

  11. They actually do so on bilinear expressions in the gradients. (See M. Abraham, Phys. Z. 13, 3.)

    Google Scholar 

  12. M. Abraham, Ref. (16), p. 3.

    Google Scholar 

  13. M. Abraham, Ref. (16), p. 4.

    Google Scholar 

  14. M. Abraham, Phys. Z. 13, p. 796.

    Google Scholar 

  15. M. Abraham, Phys. Z. 13, p. 797.

    Google Scholar 

  16. Abraham derived from it an upper limit for the mass of a star which, however, turned out to be rather large (m003C 108 m 0 ). (Phys. Z. 13, 311.)

    Google Scholar 

  17. A. Einstein, Ann. Phys. (Leipzig) 38, 355–369 (1912).

    Google Scholar 

  18. A. Einstein, Ref. (25), p. 355.

    Google Scholar 

  19. A. Einstein, Ref. (25), p. 360.

    Google Scholar 

  20. A. Einstein, ‘Zur Theorie des statischen Gravitationfeldes’, Ann. Phys. (Leipzig) 38, 443–458 (1912), in particular p. 457.

    Google Scholar 

  21. A. Einstein, Ref. 25, p. 362.

    Google Scholar 

  22. A. Einstein, Ref. 25, pp. 368 - 369.

    Google Scholar 

  23. M. Abraham, Ann. Phys. (Leipzig) 38, 1056–1058 (1912).

    ADS  Google Scholar 

  24. R. v. Eötvös, Math. u. Natw. Ber. Ungarn, 8, 65 (1890); improved later on by R. Eötvös, D. Pekär, and E. Fekete, Trans. XVI. Allgem. Konf. der int. Erdvermessung (1909), and D. Pekär, Naturwiss. 7, 327 (1919).

    Article  Google Scholar 

  25. See his reply, in A. Einstein, Ann. Phys. (Leipzig) 39, 704 (1912).

    Google Scholar 

  26. Nordström had gone to Zurich to meet Einstein (and Laue), and he submitted his third paper (Ann. Phys. 42, 533 (1913)) from there on 24 July 1913. At that time, therefore, he was in close contact with Einstein.

    Google Scholar 

  27. A. Einstein, Phys. Z. 14, 1249–1266 (1913), in particular p. 1250.

    Google Scholar 

  28. The postulate (ii), which states the equivalence of masses, is also satisfied if the gravitational potential does not change significantly in the system considered.

    Google Scholar 

  29. A. Einstein, Phys. Z. 14, 1254 (1913).

    Google Scholar 

  30. Nordström’s theory has been revived during the past few years by the work of Brans and Dicke in an attempt to give a certain scalar admixture to Einstein’s tensor theory, which seems to be consistent with recent experiments on the bending of light from the sun. C. A. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).

    Google Scholar 

  31. A. Einstein, Ref. (44), p. 328.

    Google Scholar 

  32. A. Einstein, ‘Die formale Grundlage der allgemeinen Relativitätstheorie’, Sitz. Ber. Kgl. Preuss. Akad. Wiss., Math.-Phys. Kl. (1914), 1030–1085, submitted on 29 October 1914, and read on 19 November 1914.

    Google Scholar 

  33. Einstein’s lectures at the Naturforschende Gesellschaft, Zurich, were given on the occasion of the annual meeting of the Schweizerische naturforschende Gesellschaft, at Frauenfeld on 9 September 1913. The lecture at the Congress of Natural Scientists in Vienna was given in November 1913. The final version of the ‘Entwurf’ paper was written after his lectures at both of these conferences.

    Google Scholar 

  34. See the Frauenfeld lecture, 1913, the second paper in Ref. 51, p. 286.

    Google Scholar 

  35. M. Grossmann, first paper in Ref. 51, Mathematischer Teil, p. 257.

    Google Scholar 

  36. A. Einstein, first paper in Ref. 51, p. 260.

    Google Scholar 

  37. A. Einstein, S. B. Preuss. Akad. Wiss. (1914), see pp. 1054-1057.

    Google Scholar 

  38. A. Einstein, Phys. Z. 14, 1258 (1913).

    Google Scholar 

  39. A. Einstein, the first paper in Ref. (51), Z. Math. u. Phys. 62 (1913), pp. 233–234.

    Google Scholar 

  40. First paper in Ref. (51).

    Google Scholar 

  41. A. Einstein and M. Grossmann, Z. Math. u. Phys. 63, 221, (1914).

    Google Scholar 

  42. A. Einstein and M. Grossmann, Z. Math. u. Phys. 63 (1914), see p. 225.

    Google Scholar 

  43. With considerable confidence, Einstein wrote in this paper: [Ref. (52)]: The laws governing these differential forms have been obtained by Christoffel, Ricci and Levi-Civita. I would like to present here an especially simple derivation of these, which [also] appears to me to be new.

    Google Scholar 

  44. C. Seelig, Ref. (35), p. 178.

    Google Scholar 

  45. Letter to Besso, dated Zurich, March 1914. See Correspondance, Ref. (73).].

    Google Scholar 

  46. Albert Einstein, Michele Besso, Correspondance, 1903–1955 (edited by Pierre Speziali), Hermann, Paris (1972).

    Google Scholar 

  47. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. pp. 778–786 (1915).

    Google Scholar 

  48. Einstein remarked explicitly: ‘Mathematics teaches us that all of these covariants can be derived from the Riemann-Christoffel tensor of the fourth rank…. In the problem of gravitation, we are particularly interested in the tensors of the second rank, which can be constructed from these tensors of the fourth rank and the gμv by inner multiplication.’ (Ref. (74), p. 781. Compare with our earlier quotations from Grossmann’s paper.)

    Google Scholar 

  49. Einstein said that on the basis of his considerations ‘the field equations of gravitation may be written in the form Rμv= — y Tμv, since we already know that these equations are covariant with respect to arbitrary transforma-tions of the determinant 1.’ (Ref. (74), p. 783.)

    Google Scholar 

  50. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. p. 785 (1915).

    Google Scholar 

  51. A. Einstein, ‘Zur allgemeinen Relativitätstheorie (Nachtrag)’, Sitz. Ber. Preuss. Akad. Wiss. pp. 799–801 (1915).

    Google Scholar 

  52. A. Einstein, Ref. (79), pp. 799–800.

    Google Scholar 

  53. A. Einstein, ‘Die Feldgleichungen der Gravitation’, Sitz. Ber. Preuss. Akad. Wiss. pp. 844–847 (1915).

    Google Scholar 

  54. Einstein had explained this result, obtained empirically in his second note, submitted on 18 November 1915, entitled ‘Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie’, Sitz. Ber. Preuss. Akad. Wiss. pp. 831–839 (1915).

    Google Scholar 

  55. Ref. (81), p. 847.

    Google Scholar 

  56. He wrote to Besso again on 21 December 1915: ‘Read the papers Einstein’s! The final release from misery has been obtained. What pleases me most is the agreement with the perihelion motion of Mercury….Even Planck now begins to take the thing seriously, although he still resists it a bit. But he is a splendid human being.’ [See Correspondance, Ref. 73.]

    Google Scholar 

  57. Writing again to Besso on 3 January 1916, he said: ‘The great success with [the theory of] gravitation pleases me extraordinarily. 1 have seriously in mind to write a book in the near future on the special and general theory of relativity….’ [See Correspondance, Ref. 73.]

    Google Scholar 

  58. See R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York (1953).

    Google Scholar 

  59. See Vorträge über die Kinetische Theorie der Materie und der Elektrizität, B. G. Teubner, Berlin (1914); Hilbert wrote the foreword to these lectures.

    Google Scholar 

  60. O. Blumenthal, Lebensgeschichte, I.E., p. 417 [see Ref. 88].

    Google Scholar 

  61. C. Caratheodory, ‘Untersuchungen über die Grundlagen der Thermodynamik’, Math. Ann. 67, 355 (1909).

    Article  MathSciNet  Google Scholar 

  62. P. Hertz, ‘Über die mechanischen Grundlagen der Thermodynamik’, Ann. Phys. (Leipzig) 33, 225–274, 537–552 (1910). See also Hertz’ article on statistical mechanics in Weber and Gans Repertorium der Physik, Vol. 1/2, Leipzig (1916). Hertz was close to Hilbert in 1912.

    Google Scholar 

  63. D. Hilbert, ‘Begründung der Kinetischen Gastheorie’, Math. Ann. 71, 562 (1912).

    Article  MathSciNet  Google Scholar 

  64. D. Hilbert, ‘Die Grundlagen der Physik I, II’, Nachr. Ges. Wiss. Gott. (1915), p. 395; (1917), p. 201. He later on united these two communications into a single article with the same title, Math. Ann. 92, 1 (1924).

    MathSciNet  Google Scholar 

  65. W. Heisenberg, Z. Phys. 33, 879–893 (1925); Math. Ann. 95, 683–705 (1925/26).

    MathSciNet  Google Scholar 

  66. P. Jordan, ‘Über kanonische Transformationen in der Quantenmechanik’, Z. Phys. 37, 383–386 (1926); 38, 513–517 (1926); ‘Über eine neue Begründung der Quantenmechanik’, Z. Phys. 40, 809–838 (1927). Jordan proved that his formal transformation theory contains not only Schrödinger’s wave mechanics, but also the Born-Wiener operator calculus and Dirac’s number algorithm as special cases.

    Google Scholar 

  67. D. Hilbert, J. von Neumann, and L. Nordheim, Math. Ann. 98, 1–30 (1928).

    Article  MathSciNet  Google Scholar 

  68. J. von Neumann, Grundlagen der Quantenmechanik, Springer, Berlin (1931) (English translation by R. T. Beyer, Princeton (1955)).

    Google Scholar 

  69. See, e.g., M. Jammer, Concepts of Space, Harvard University Press ( 1954 ); Harper Torchbooks, New York (1960).

    Google Scholar 

  70. See W. Pauli, Theory of Relativity, Pergamon Press, Oxford (1958), p. 35.

    MATH  Google Scholar 

  71. W. K. Clifford’s translation, Ref. 105, p. 69.

    Google Scholar 

  72. W. K. Clifford, On the Space Theory of Matter, read 21 February 1870; abstract printed in Camb. Phil. Soc. Proc. 2 (1876), reprinted in W. K. Clifford’s Mathematical Papers, pp. 21–22.

    Google Scholar 

  73. H. Minkowski, Nachr. Ges. Wiss. Gott., pp. 53–112 (1908).

    Google Scholar 

  74. H. Minkowski, Ref. (110), p. 55.

    Google Scholar 

  75. H. Minkowski, Ref. 110, pp. 65–66. See also his lecture at the 80th Naturforscherversammlung (Congress of Natural Scientists) in Cologne on ‘Raum und Zeit’ printed in Phys. Z. 10, 104–111 (1909), where he gave a most illuminating exposition of the concepts of space and time and introduced the notions of ‘world vectors’ and ‘world postulate’.

    Google Scholar 

  76. R. Hargreaves, ‘Integral forms and their connection with physical equations’, Trans. Camb. Phil. Soc. 21, 107–122 (1908). In fact, H. Poincaré already used ict as the fourth coordinate 4 in his memoir ‘Sur la dynamique de l’électron’, Palermo Rend, 21 (1906), which was submitted on 23 June 1905.

    Google Scholar 

  77. H. von Helmholtz, ‘Über die Tatsachen die der Geometrie zugrundeliegen’, Nachr. Ges. Wiss. Gött., pp. 193–221 (1863).

    Google Scholar 

  78. H. von Helmholtz, Ref. (115), p. 193.

    Google Scholar 

  79. H. von Helmholtz, Ref. (115), pp. 194–195. Riemann’s ‘trial’ lecture [Probevorlesung] was published in the Abhandl. kgl. Ges. Wiss. Gött. 13 (1867), although it had appeared as a pamphlet in 1854.

    Google Scholar 

  80. F. Klein, ‘Vergleichende Betrachtungen über neuere geometrische Forschungen’, Erlangen, appeared in December, 1872; reprinted in Math. Ann. 43 (1893) and in Ges. Abh., Vol. I, p. 460; See F. Klein, Vorlesungen, Ref. 103, Vol. II, p. 28.

    Google Scholar 

  81. See p. 7 of Klein’s Erlangen Programme, or Ref. 103, p. 28.

    Google Scholar 

  82. G. Hamel, ‘Die Lagrange-Eulerschen Gleichungen der Mechanik’, Z. Math. u. Phys. 50, 1-57 (1904) (inaugural lecture); See also’Über die virtuellen Verschiebungen in der Mechanik’, Math. Ann. 59, 416–434 (1904).

    MathSciNet  MATH  Google Scholar 

  83. G. Hamel, Ref. 124, p. 12.

    Google Scholar 

  84. G. Herglotz, ‘Über die Mechanik des deformierbaren Körpers von Standpunkte der Relativi-tätstheorie’, Ann. Phys. (Leipzig) 36, p. 1911.

    Google Scholar 

  85. The name ‘Poincaré group’ was first used by E. P. Wigner in: ‘On Unitary Representations of the Inhomogeneous Lorentz Group’, Annals of Math. 40, 149 (1939).

    Google Scholar 

  86. His papers on the theory of invariants had the unexpected effect of withering, as it were over-night, a “discipline which so far had stood in full bloom.” [H. Weyl, Gesammelte Abhandlungen Vol. 4, Springer-Verlag, New York 1968, p. 124.]

    Google Scholar 

  87. E. Noether, Ref. 131, pp. 238–239.

    Google Scholar 

  88. E. Bessel-Hagen, ‘Über die Erhaltungssatze der Elektrodynamik’, Math. Ann. 84, 254 (1921).

    Article  MathSciNet  Google Scholar 

  89. The generalization of Noether’s Theorem to quantum mechanics was made by E. P. Wigner in his paper ‘Über die Erhaltungssätze in der Quantenmechanik’, Nachr. Ges. Wiss. Gött., p. 375 (1927).

    Google Scholar 

  90. Gustav Mie, ‘Grundlagen einer Theorie der Materie (I)’, Ann. Phys. (Leipzig) 37, 511–534 (1912); (II), 39, 1-40 (1912); (III), 40, 1–66 (1913).

    Google Scholar 

  91. G. Mie, I.E., (I), p. 511.

    Google Scholar 

  92. G. Mie, I.E., (I), p. 513.

    Google Scholar 

  93. G. Mie, I.E., (I), p. 512.

    Google Scholar 

  94. G. Mie, Section 35 of (III), in Ref. 135, entitled ‘Das Plancksche Wirkungsquantum’ does not represent a step in this direction.

    Google Scholar 

  95. Constance Reid, Hilbert, Springer, New York, (1970), pp. 140–141.

    Google Scholar 

  96. Einstein-Sommerfeld Briefwechsel (see Ref. 1), p. 30.

    Google Scholar 

  97. A. Einstein, S. B. Preuss. Akad. Wiss. (1915), p. 844.

    Google Scholar 

  98. C. Reid, Hilbert, Springer, New York (1970), p. 142.

    MATH  Google Scholar 

  99. Einstein once remarked, ‘The people in Göttingen sometimes strike me, not as if they want to help one formulate something clearly, but as if they only want to show us physicists how much brighter they are than we.’ (Reid, I.E., p. 142.)

    Google Scholar 

  100. C. Reid, Hilbert, p. 142.

    Google Scholar 

  101. Einstein has posed immense problems, has brought forth profound thoughts and unique con-ceptions, and has invented ingenious methods for dealing with them. Mie was able to construct his electrodynamics [on the basis of these], and they have opened up new avenues for the investigation of the foundations of physics.’ (D. Hilbert, see Ref. 97, ‘Die Grundlagen der Physik’, p. 395.)

    Google Scholar 

  102. D. Hilbert, Ref. (97), I, p. 395.

    Google Scholar 

  103. D. Hilbert, Ref. (97), I, p. 396.

    Google Scholar 

  104. D. Hilbert, Ref. (97), I, p. 396.

    Google Scholar 

  105. D. Hilbert, Ref. (97), I, p. 397.

    Google Scholar 

  106. See the remarks in Section HI.3. Emmy Noether had given the proof in its most general form. Hilbert gave a complete proof for a special case (which he called Theorem 2) of the general theorem.

    Google Scholar 

  107. D. Hilbert, Ref. (97), I, pp. 397 - 398.

    Google Scholar 

  108. A. Einstein, Ann. Phys. (Leipzig) 49, 769 (1916), p. 810.

    Google Scholar 

  109. D. Hilbert, Ref. (97), I, p. 403.

    Google Scholar 

  110. D. Hilbert, Ref. (97), I, p. 404.

    Google Scholar 

  111. D. Hilbert, Ref. (97), I, p. 406.

    Google Scholar 

  112. D. Hilbert, Ref. (97), I, p. 407.

    Google Scholar 

  113. C. Seelig, Albert Einstein [Ref. 35], p. 188.

    Google Scholar 

  114. A. Einstein, Ann. Phys. (Leipzig) 49, 769–822 (1916).

    ADS  Google Scholar 

  115. A. Einstein, Ref. (165), p. 772.

    Google Scholar 

  116. A. Einstein, Ref. (165), p. 775.

    Google Scholar 

  117. A. Einstein, Ref. (165), p. 779.

    Google Scholar 

  118. F. Kottler, Ann. Phys. (Leipzig) 50, 955 (1916). Kottler had already given, as early as 1912, the expression for the electromagnetic field equations in a generally covariant form (S. B. Akad. Wiss. Wien 121, 1659 (1912)), and Einstein recognized his contribution. ‘Among the papers which deal critically with the general theory of relativity, those of Kottler are particularly remarkable, because this expert has really penetrated into the spirit of the theory.’ [A. Einstein, ‘Über Friedrich Kottiers Abhandlung “Über Einsteins Äquivalenzhypothese und die Gravitation” ’, Ann. Phys. (Leipzig) 51, 639–642 (1916).]

    Google Scholar 

  119. A. Einstein, Ann. Phys. (Leipzig) 51, 639 (1916).

    ADS  Google Scholar 

  120. A. Einstein, Ann. Phys. (Leipzig) 51, 639 (1916), esp. pp. 640–641.

    Google Scholar 

  121. E. Kreischmann ‘Über den physikalischen Sinn der Relativitätspostulate, A. Einsteins neue und seine ursprüngliche Relativitätstheorie’, Ann. Phys. (Leipzig) 53, 575–614 (1917).

    Google Scholar 

  122. E. Kretschmann, Ref. (172), p. 576.

    Google Scholar 

  123. E. Kretschmann, Ref. (172), p. 584.

    Google Scholar 

  124. E. Kretschmann, Ref. (172), p. 610.

    Google Scholar 

  125. A. Einstein, Ann. Phys. (Leipzig) 55, 241–244 (1918). He praised the author (Kretschmann) for his sharp wit, but did not find the sharper form of the equivalence principle either valuable or desirable.

    Google Scholar 

  126. A. Einstein, Ann. Phys. (Leipzig) 49, 769 (1916), esp. p. 781.

    Google Scholar 

  127. A. Einstein, Ref. (177), p. 789.

    Google Scholar 

  128. A. Einstein, Ref. (177), pp. 802–803.

    Google Scholar 

  129. A. Einstein, Ref. (177), pp. 803–804. The formal proof of this statement was given by H. Vermeil, Nachr. Ges. Wiss. Göttingen (1917) 334; see also H. Vermeil, Math. Ann. 79 (1918) 289.

    Article  MathSciNet  Google Scholar 

  130. A. Einstein, Ref. (177), p. 804.

    Google Scholar 

  131. A. Einstein, Ref. (177), p. 808.

    Google Scholar 

  132. A. Einstein, Ref. (177), p. 810.

    Google Scholar 

  133. A. Einstein, Ref. (177), pp. 810–811.

    Google Scholar 

  134. Einstein had already done this in a paper entitled ‘Eine neue formale Deutung der Maxwellschen Feldgleichungen der Elektrodynamik’, Sitz. Ber. Preuss. Akad. Wiss. (1916), pp. 184–187. (Submitted and read on 3 February 1916.)

    Google Scholar 

  135. C. Seelig, Ref. (35), p. 199.

    Google Scholar 

  136. This value had already been calculated in his communication of 18 November 1915, dealing with the motion of the perihelion of Mercury.

    Google Scholar 

  137. A. Einstein, Ref. (189), (1918), p. 164.

    Google Scholar 

  138. A. Einstein, S.B. Preuss. Akad. Wiss. (1917), p. 143.

    Google Scholar 

  139. A. Einstein, Ref. (193), pp. 143–144.

    Google Scholar 

  140. A. Einstein, Ref. (193), p. 145.

    Google Scholar 

  141. A. Einstein, Ref. (193), p. 148.

    Google Scholar 

  142. A. Einstein, Ref. (193), p. 151.

    Google Scholar 

  143. A. Einstein, Ann. Phys. (Leipzig) 49, 771.

    Google Scholar 

  144. A. Einstein, Ref. (200), p. 776.

    Google Scholar 

  145. Letter from Einstein to M. Besso, dated Berlin, 31 October 1916. [See Ref. (73)]

    Google Scholar 

  146. D. Hilbert, Nachr. Ges. Wiss. Gött., Math. Phys. Kl. (1917), 53–76, in particular p. 53.

    Google Scholar 

  147. D. Hilbert, ‘Die Grundlagen der Physik’, Math. Ann. 92, 1–32 (1924). This is a slightly condensed version of Hilbert’s two memoirs of 1915 and 1916, which were published in the Proceedings of the Göttingen Academy. The third axiom is the one which expresses H as a sum of a gravitational term K and an electrical term L. Axiom IV was stated in this paper (p. 11). [This paper was reprinted in Hilbert’s Gesammelte Abhandlungen, Vol. 3, pp. 258–289.]

    Google Scholar 

  148. D. Hilbert, Nachr. Gött. (1917), p. 54.

    Google Scholar 

  149. D. Hilbert, Nachr. Gött. (1917), p. 57.

    Google Scholar 

  150. W. Pauli, I.e. in Ref. 106, Section 22, ‘Reality relations’, English edition, pp. 62–64.

    Google Scholar 

  151. H. Reichenbach, Axiomatik der relativistischen Raum-Zeit -Lehre, Vieweg, Braunschweig (1965); English translation by M. Reichenbach, University of California Press, Berkeley (1969), pp. 179–181. It seems to be quite remarkable that Reichenbach ignored Hilbert’s system of axioms completely.

    Google Scholar 

  152. D. Hilbert, Nachr. Gött. (1917), p. 61.

    Google Scholar 

  153. D. Hilbert, Nachr. Gött. (1917), pp. 63–64.

    Google Scholar 

  154. D. Hilbert, Nachr. Gött. (1917), p. 66.

    Google Scholar 

  155. D. Hilbert, Nachr. Gött. (1917), p. 70.

    Google Scholar 

  156. Einstein oscillated between the opinions whether he should take the singular solution as the one corresponding to the real particle (electron, etc.) or not.

    Google Scholar 

  157. W. Pauli, Theory of Relativity, p. 145, footnote 277 (see Ref. 106).

    Google Scholar 

  158. Einstein-Lorentz correspondence, The Hague, The Netherlands.

    Google Scholar 

  159. H.A. Lorentz, Versl. K. Akad. Wetensch. Amsterdam 23, 1073 (1915); Proc. K. Akad. Amsterdam 19, 751 (1915); reprinted in H.A. Lorentz, Collected Papers, Vol.5, The Hague (1937), pp. 229–245.

    Google Scholar 

  160. H. A. Lorentz, Ref. (217a), p. 1073, or Collected Papers, Vol. 5, p. 229.

    Google Scholar 

  161. H. A. Lorentz, Versl. K. Akad. Wetensch. Amsterdam 24, 1389, 1759; 25, 468, 1380 (1916); Proc. K. Akad. Amsterdam 19, 1341, 1354, 20, 2, 20 (1916), reprinted in Collected Papers, Vol. 5, pp. 246 - 313.

    Google Scholar 

  162. H. A. Lorentz, Collected Papers, Vol. 5, p. 246.

    Google Scholar 

  163. H. A. Lorentz, Ref. 220 ), p. 246.

    Google Scholar 

  164. This part is contained in the second paper in H. A. Lorentz, Versl. K. Akad. Wetensch. Amsterdam 24, 1759 (1916).

    Google Scholar 

  165. A. Einstein, ‘Hamiltonsches Prinzip und allgemeine Relativitätstheorie’, Sitz. Ber. Preuss. Akad. Wiss. (1916), pp. 1111–1116.

    Google Scholar 

  166. A. Einstein, Ref. (224), p. 1111.

    Google Scholar 

  167. F. Klein, Nachr. Ges. Wiss. Gött., Math. Phys. Kl. (1918), pp. 171–189.

    Google Scholar 

  168. He proudly noted: ‘As one can see, there is nothing more to be really calculated in the integrals, except to make the obvious use of the elementary formulas of the classical variational calculation.’ (F. Klein, Ref. (226), p. 172)

    Google Scholar 

  169. F. Klein, Ref. (226), p. 185.

    Google Scholar 

  170. F.Klein, Nachr. Ges. Wiss. Gott. (1918), pp. 394–423 (submitted and read on 6 December 1918 ).

    Google Scholar 

  171. W. de Sitter, Proc. Acad. Sci. Amsterdam 19, 1217; 20, 229 (1917).

    Google Scholar 

  172. F. Klein, Nacht. Gott. (1918), Ref. (226), pp. 394–395.

    Google Scholar 

  173. F. Klein, Ref. (232), p. 399.

    Google Scholar 

  174. F. Klein, ‘Zu Hilberts erster Note iiber die Grundlagen der Physik’, Nachr. Ges. Wiss. Gott (1917), pp. 469–489 (read on 25 January 1917). Klein’s aim was to simplify Hilbert’s calculations of the variational problem and obtain 4 a clearer insight into the significance of the conservation laws’. [F. Klein, I.E., p. 469].

    Google Scholar 

  175. F. Klein, Ref. (234), p. 475.

    Google Scholar 

  176. Because of all this I can hardly believe that it is appropriate to regard the arbitrarily chosen quantities tva as the energy-components of the gravitational field.’ (Ref. (234), p. 477.)

    Google Scholar 

  177. D. Hilbert, Nachr. Gott. (1917), p. 477 - 480.

    Google Scholar 

  178. D. Hilbert, Ref. (238), p. 477. Hilbert also mentioned that Emmy Noether, whose help he had requested in his study of the question of energy conservation over a year previously, had, in fact, studied the question thoroughly and arrived at Klein’s conclusions. Klein had also seen Noether’s proof after writing down his own derivation. (Ref. (234), p. 476.)

    Google Scholar 

  179. D. Hilbert, Ref. (238), p. 480.

    Google Scholar 

  180. F. Klein, Ref. (234), pp. 481-482.

    Google Scholar 

  181. H. Weyl, Raum-Zeit-Materie, Leipzig (1918); the fourth edition (1921) was translated as Space-Time-Matter, Methuen, London (1922). Dover edition, New York (1952). The quotations are from the Dover edition, p. 102.

    Google Scholar 

  182. T. Levi-Civita, Rend. del. Circ. Mat. di Palermo 42, 173 (1917).

    Article  Google Scholar 

  183. G. Hessenberg, ‘Vektorielle Begründung der Differentialgeometrie’, Math. Am. 78, 187 (1917).

    MathSciNet  MATH  Google Scholar 

  184. H. Weyl, Math. Z. 2, 384 (1918). See also Weyl, Sitz. Ber. Preuss. Akad. Wiss. (1918), p. 465.

    Google Scholar 

  185. This follows by integrating the Equation (101), and noting that/is a constant throughout the space if (frßdxt* is a total differential, which means that the expressions

    Google Scholar 

  186. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (1918), p. 478.

    Google Scholar 

  187. A. S. Eddington, Proc. Roy. Soc. (London) A99, 104 (1921); A. Einstein, Sitz. Ber. Preuss. Aad. Wiss. A23, 32, 76, 137.

    Google Scholar 

  188. T. Kaluza, Sitz. Ber. Preuss. Akad. Wiss. (1921), p. 27. This theory was extended by O. Klein in several papers: Z.Phys. 37, 895 (1926); Z. Phys. 36,188 (1927), /. Phys. Rad. 8,242 (1927); Ark. Mat. Ast. Fys. 34, No. 1 (1946).

    Google Scholar 

  189. O. Klein, Z. Phys. 37, 903 (1926).

    ADS  Google Scholar 

  190. O. Veblen and D. Hoffman, ‘Projective Relativity’, Phys. Rev. 36, 810 (1930).

    Article  ADS  Google Scholar 

  191. Letter to H. Weyl, on 23 November 1916 (quoted from C. Seelig, Ref. (35), p. 200).

    Google Scholar 

  192. D. Hilbert, ‘Die Grundlagen der Physik’, Math. Ann. 92,1–32 (1924), reprinted in D. Hilbert, Gesammelte Abhandlungen, Berlin (1935), pp. 258–289.

    Google Scholar 

  193. D. Hilbert, pp. 1–2 (Ges. Abh. 3, pp. 258–259 ).

    Google Scholar 

  194. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (1919), pp. 348–356.

    Google Scholar 

  195. A. Einstein, Ref. (260), p. 350.

    Google Scholar 

  196. A. Einstein, Ref. (260), p. 351.

    Google Scholar 

  197. A. Einstein, Ref. (260), pp. 355-356.

    Google Scholar 

  198. A. Einstein to M. Besso, dated Berlin, 5 January 1929 [see Ref. 73]. The paper in question was ‘Einheitliche Feldtheorie’, Sitz. Ber. Preuss. Akad. Wiss. (1929), pp. 2–7.

    Google Scholar 

  199. P. A. M. Dirac, Lecture on ‘Methods in Theoretical Physics’, in the series, From A Life of Physics, Trieste, Italy (June, 1968 ).

    Google Scholar 

Download references

Authors
  1. Jagdish Mehra
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. The University of Texas at Austin, Austin, Texas, USA

    Jagdish Mehra

Rights and permissions

Reprints and permissions

Copyright information

© 1973 D. Reidel Publishing Company, Dordrecht-Holland

About this chapter

Cite this chapter

Mehra, J. (1973). Einstein, Hilbert, and the Theory of Gravitation. In: Mehra, J. (eds) The Physicist’s Conception of Nature. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2602-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-2602-4_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-2604-8

  • Online ISBN: 978-94-010-2602-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation