Advertisement

Annali di Matematica Pura ed Applicata

, Volume 110, Issue 1, pp 29–136 | Cite as

A memoir on the projective geometry of spinors

  • J. D. Zund
Article

Summary

The classical methods of projective geometry are applied to a number of questions in general relativity, by using the Van der Waerden spinor analysis. These include a new geometric theory of spinors, refinements in the spinor calculus, the classification of electromagnetic and gravitational fields, Weyl-Maxwell fields, a classification of the Einstein spinor, and the projective geometry of the Bel-Petrov types.

Keywords

General Relativity Classical Method Gravitational Field Geometric Theory Projective Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. [1]
    E. H. Askwith,The analytical geometry of the conic sections, A & C. Black, London, 3rd edition (1938).Google Scholar
  2. [2]
    H. F. Baker,Principles of geometry, vol. II, Cambridge University Press, 2nd edition (1930).Google Scholar
  3. [3]
    A. B. Basset,A treatise on the geometry of surfaces, Deighton-Bell, Cambridge (1910).Google Scholar
  4. [4]
    H. Bateman,Kummer's quartic surface as a wave surface, Proc. London Math. Soc., Ser. 2,8 (1910), pp. 375–385.zbMATHGoogle Scholar
  5. [5]
    A. Bessel,Über die Invarianten der einfachsten system simultaner binärer Formen, Math. Ann.,1 (1869), pp. 173–194.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    W. S. Burnside,On the invariants and covariants α, a of a binary quartic considered geometrically as a system of two ternary quadrics, Quart. J. Math.,10 (1869–1870), pp. 211–218.zbMATHGoogle Scholar
  7. [7]
    W. S. Burnside -A. W. Panton,The theory of equations with an introduction to the theory of binary algebraic forms, vol. II, Longmans, Green & Co., London, 7th edition (1928), reprinted by Dover Publications (1960).Google Scholar
  8. [8]
    J. Casey,A treatise on the analytical geometry of the point, line, circle and conie sections containing an account of its most recent extensions with numerous examples, Hodges Figgis & Co. Ltd., Dublin, 2nd edition (1893).Google Scholar
  9. [9]
    A. Clebsch,Über die Bedeutung einer simultanen Invariante einer binären quadratischen und einer binären biquadratischen Form, Math. Ann.,3 (1871), pp. 263–264.CrossRefMathSciNetGoogle Scholar
  10. [10]
    A. Clebsch,Theorie der Binären Algebraischen Forms, B. G. Teubner, Leipzig (1872).Google Scholar
  11. [11]
    A. Clebsch,Vorlesungen über Geometrie, B. Teubner, Leipzig (1875), or French edition:Leçons sur la géométrie, Tome Premier, Gauthier-Villars, Paris (1903).Google Scholar
  12. [12]
    G. C. Debney -J. D. Zund,A note on the classification of electromagnetic fields, Tensor N. S.,22 (1971), pp. 333–40.MathSciNetGoogle Scholar
  13. [13]
    G. C. Debney -J. D. Zund,Electromagnetic theory in general relativity - I:The geometry of congruences, Tensor N. S.,25 (1972), pp. 47–52.MathSciNetGoogle Scholar
  14. [14]
    G. C. Debney -J. D. Zund,Electromagnetic theory in general relativity - II:Non-singular fields, Tensor N. S.,25 (1972), pp. 53–62.MathSciNetGoogle Scholar
  15. [15]
    Sir ArthurEddington,New pathways in science, Cambridge University Press (1934), re-print by Ann Arbor Paperbacks (1959).Google Scholar
  16. [16]
    A. Einstein,Über die formale Beziehung des Riemannschen Krümmungstensor zu den Feldgleichungen der Gravitation, Math. Ann.,97 (1926), pp. 99–103.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    J. Géhéniau -R. Debever,Les invariants de l'espace de Riemann à quatres dimensions, Acad. Roy. Belg. Bull. Cl. Sci.,47 (1956), pp. 114–123.Google Scholar
  18. [18]
    J. Géhéniau,Une classification des espaces riemanniens, C. R. Acad. Sci. (Paris),244 (1957), pp. 723–724.zbMATHGoogle Scholar
  19. [19]
    P. Gordan,Über Büschel von Kegelschnitten, Math. Ann.,19 (1882), pp. 529–552.CrossRefMathSciNetGoogle Scholar
  20. [20]
    P. Gordan -G. Kerschensteiner,Dr. Paul Gordan's Vorlesungen über Invarianten theorie, Bd. I & II, B. G. Teubner, Leipzig (1885 & 1887).Google Scholar
  21. [21]
    P. Gordan,Das erweiterte Formensystem, Math. Ann.,33 (1889), pp. 372–379.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    J. H. Grace - A. Young,Algebra of invariants, Cambridge University Press (1903), re-printed by Chelsea (1967).Google Scholar
  23. [23]
    F. Harbordt,Das simultane System einer biquadratischen und einer quadratischen binären Form, Math. Ann.,1 (1869), pp. 210–224.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    C. N. Haskins,On the invariants of quadratic differential forms, Trans. Amer. Math. Soc.,3 (1902), pp. 71–91.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    Sir WilliamHodge - D. Pedoe,Methods of algebraic geometry, vol. I, Cambridge University Press (1947).Google Scholar
  26. [26]
    R. W. H. T. Hudson,Kummer's quartic surface, Cambridge University Press (1903).Google Scholar
  27. [27]
    L. Infeld - B. L. Van der Waerden,Die Wellengleichung des Elektrons in der allgemeinen Relativitätstheorie, Sitz. Ber. Preuss. Akad. Wiss. Phys. Math. Kl. (1933), pp. 380–402.Google Scholar
  28. [28]
    C. M. Jessop,A treatise on the line complex, Cambridge University Press (1903), re-printed by Chelsea (1968).Google Scholar
  29. [29]
    E. Kasner,The invariant theory of the inversion group geometry upon a quadric surface, Trans. Amer. Math. Soc.,1 (1900), pp. 430–498.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    O. Laporte -G. E. Uhlenbeck,Application of spinor analysis to the Maxwell and Dirac equations, Phys. Rev.,37 (1931), pp. 1380–1397 and p. 1552.CrossRefGoogle Scholar
  31. [31]
    G. Ludwig,Geometrodynamics of electromagnetic fields in the Newman-Penrose formalism, Comm. Math. Phys.,17 (1970), pp. 98–108.CrossRefMathSciNetGoogle Scholar
  32. [32]
    G. Ludwig -G. Scanlan,Classification of the Ricci tensor, Comm. Math. Phys.,20 (1971), pp. 291–300.CrossRefMathSciNetGoogle Scholar
  33. [33]
    E. T. Newman -R. Penrose,An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys.,3 (1962), pp. 566–578.CrossRefMathSciNetGoogle Scholar
  34. [34]
    G. Peano,Formazioni invariantive delle corrispondenze, Gior. di Mat. (Battaglini),20 (1882), pp. 97–100, orOpere Scelte, vol. III, pp. 8–30, Cremonese, Roma (1959).Google Scholar
  35. [35]
    R. Penrose,A spinor approach to general relativity, Ann. Phys.,10 (1960), pp. 171–201.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [36]
    A. Z. Petrov,Einstein spaces, English translation Pergamon Press, London (1969).Google Scholar
  37. [37]
    H. S. Ruse,On the geometry of the electromagnetic field, Proc. London Math. Soc.,41 (1936), pp. 302–322.zbMATHGoogle Scholar
  38. [38]
    H. S. Ruse,On the geometry of Dirac's equations and their expression in tensor form, Proc. Roy. Soc. (Edinburgh),57 (1937), pp. 97–127.zbMATHGoogle Scholar
  39. [39]
    H. S. Ruse,On the line geometry of the Riemann tensor, Proc. Roy. Soc. (Edinburgh),62 (1944), pp. 64–73.zbMATHMathSciNetGoogle Scholar
  40. [40]
    R. K. Sachs,The outgoing radiation condition, Proc. Roy. Soc. (London), A264 (1961), pp. 309–338.zbMATHMathSciNetGoogle Scholar
  41. [41]
    D. M. Y. Sommerville,Analytical conics, G. Bell & Sons Ltd., London, 3rd edition (1933).Google Scholar
  42. [42]
    J. J. Sylvester,Sketch of a memoir on elimination, transformation and canonical forms, Cambridge & Dublin Math. J.,6 (1851), pp. 186–200, orCollected Mathematical Papers, vol. I, pp. 184–197, Cambridge University Press (1904).Google Scholar
  43. [43]
    J. L. Synge,Principal null-directions defined in space-time by an electromagnetic field, Univ. Toronto Studies in Applied Math.,1 (1935), pp. 1–50.MathSciNetGoogle Scholar
  44. [44]
    J. A. Todd,Projective and analytical geometry, Sir Isaac Pitman & Sons, London (1960).Google Scholar
  45. [45]
    H. W. Turnbull,Double binary forms, Proc. Roy. Soc. (Edinburgh),43 (1922–1923), pp. 43–50.Google Scholar
  46. [46]
    H. W. Turnbull,A geometrical interpretation of the complete system of the double binary (2.2) form, Proc. Roy. Soc. (Edinburgh),44 (1923–1924), pp. 23–50.Google Scholar
  47. [47]
    B. L. Van der Waerden,Spinoranalyse, Gött. Nachr., (1929), pp. 100–109.Google Scholar
  48. [48]
    O. Veblen,Geometry of two-component spinors, Proc. Nat. Acad. Sci.,19 (1933), pp. 462–474.zbMATHGoogle Scholar
  49. [49]
    O. Veblen -von Neumann,Geometry of complex domains, lecture notes Institute for Advanced Studies, Princeton (1935).Google Scholar
  50. [50]
    O. Veblen,Spinors and projective geometry, C. R. du Cong. int. des Math. Oslo 1936, pp. 117–127, Brøggers, Oslo (1937).Google Scholar
  51. [51]
    W. G. Welchman,Introduction to algebraic geometry, Cambridge University Press (1950).Google Scholar
  52. [52]
    E. T. Whittaker,On the relations of the tensor calculus to the spinor calculus, Proc. Roy. Soc. (London), A158 (1937), pp. 38–46.zbMATHGoogle Scholar
  53. [53]
    L. Witten,Invariants of general relativity and the classification of spaces, Phys. Rev.,113 (1959), pp. 357–362.CrossRefzbMATHMathSciNetGoogle Scholar
  54. [54]
    J. D. Zund,Algebraic invariants and the projective geometry of spinors, Ann. di Matem. Pura ed Appl., Ser. IV,82 (1969), pp. 381–412.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [55]
    J. D. Zund -W. F. Maher,A spinor approach to some problems in Lorentzian geometry, Tensor N. S.,21 (1970), pp. 70–74.MathSciNetGoogle Scholar
  56. [56]
    J. D. Zund,A note on the Bel-Robinson spinor, Tensor N. S.,21 (1970), pp. 250–254.zbMATHMathSciNetGoogle Scholar
  57. [57]
    J. D. Zund -E. A. Brown,The theory of bivectors, Tensor N.S.,22 (1971), pp. 179–185.MathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1975

Authors and Affiliations

  • J. D. Zund
    • 1
  1. 1.Las CrucesU.S.A.

Personalised recommendations