The Weyl-Cartan theorem and the naturalness of general relativity

European Journal of Physics

Volumn 24, Issue 3, 2003, Pages 261-266

  Pesic, Peter ^a   Boughn, Stephen P ^b  

^a ST JOHN S COLLEGE   (United States)

^b HAVERFORD COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL GEOMETRY; LINEAR EQUATIONS; TENSORS; THEOREM PROVING;

WEYL-CARTAN THEOREM;

QUANTUM THEORY;

EID: 0038187775     PISSN: 01430807     EISSN: None     Source Type: Journal    
DOI: 10.1088/0143-0807/24/3/305     Document Type: Article

References (21)

1. Einstein A 1979 On the method of theoretical physics (1933) Einstein: a Centenary Volume ed A P French (Cambridge, MA: Harvard University Press) p 312
2. Einstein A 1952 The foundation of the general theory of relativity (1916) The Principle of Relativity ed H A Lorentz et al (New York: Dover) p 144
3. Weyl H 1952 Space-Time-Matter (New York: Dover) pp 315-17 (transl. H L Brose) Note that this edition is based on the fourth German edition (1921) and includes the relevant appendix II, which was not present in the first (1918) edition. We thank Anne Battis (Burndy Library, Dibner Institute for the History of Science and Technology) for checking the early editions for us.
4. Cartan E 1922 Sur les équations de la gravitation d'Einstein J. Math. Pure. Appl. 1 141-203 Cartan E 1955 Œuvres Complètes vol 1, partie III (Paris: Gauthier-Villars) pp 549-611 (reprint) This paper refers to Weyl's generalized theory of varieties, but never to Weyl's relativity book. Cartan notes the difficulties of scientific communication during the Great War, which may well have been a factor in the independent appearance of Weyl and Cartan's arguments. There is no direct discussion of the matter in Debever R (ed) 1979 Élie Cartan-Albert Einstein: Letters on Absolute Parallelism 1929-1932 (Princeton, NJ: Princeton University Press) The correspondence between them began after 1922.
5. Cartan E 1922 Sur les équations de la gravitation d'Einstein J. Math. Pure. Appl. 1 141-203 Cartan E 1955 Œuvres Complètes vol 1, partie III (Paris: Gauthier-Villars) pp 549-611 (reprint) This paper refers to Weyl's generalized theory of varieties, but never to Weyl's relativity book. Cartan notes the difficulties of scientific communication during the Great War, which may well have been a factor in the independent appearance of Weyl and Cartan's arguments. There is no direct discussion of the matter in Debever R (ed) 1979 Élie Cartan-Albert Einstein: Letters on Absolute Parallelism 1929-1932 (Princeton, NJ: Princeton University Press) The correspondence between them began after 1922.
6. Cartan E 1922 Sur les équations de la gravitation d'Einstein J. Math. Pure. Appl. 1 141-203 Cartan E 1955 Œuvres Complètes vol 1, partie III (Paris: Gauthier-Villars) pp 549-611 (reprint) This paper refers to Weyl's generalized theory of varieties, but never to Weyl's relativity book. Cartan notes the difficulties of scientific communication during the Great War, which may well have been a factor in the independent appearance of Weyl and Cartan's arguments. There is no direct discussion of the matter in Debever R (ed) 1979 Élie Cartan-Albert Einstein: Letters on Absolute Parallelism 1929-1932 (Princeton, NJ: Princeton University Press) The correspondence between them began after 1922.
7. Misner C W, Thorne K S and Wheeler J A 1973 Gravitation (San Francisco, CA: Freeman) p 198, 408 exercise 17.3
8. Weinberg S 1972 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley) pp 133-5
9. Adler R, Bazin M and Schiffer M 1975 Introduction to General Relativity 2nd edn (New York: McGraw-Hill) p 344
10. For example, see Cartan E 1986 On Manifolds with an Affine Connection and the Theory of General Relativity (Naples: Bibliopolis) (transl. A Magnon and A Ashtekar) and Cartan's classic textbook Cartan E 1983 Geometry of Riemann Spaces ed R Hermann (Brookline, MA: Mathematical Sciences Press) (transl. J Glazebrook)
11. For example, see Cartan E 1986 On Manifolds with an Affine Connection and the Theory of General Relativity (Naples: Bibliopolis) (transl. A Magnon and A Ashtekar) and Cartan's classic textbook Cartan E 1983 Geometry of Riemann Spaces ed R Hermann (Brookline, MA: Mathematical Sciences Press) (transl. J Glazebrook)
12. This is the celebrated 'Raumproblem', discussed in detail in Scholz E 2001 Weyls Infintesimalgeometrie, 1917-1925 Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 85-95 and especially Coleman R and Korté H 2001 Hermann Weyl: Mathematician, Physicist, Philosopher Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 215-50 Weyl's proof that physical space - time must have a Riemannian metric goes back to the arguments of Hermann von Helmholtz (1868), helpfully summarized in [7], pp 7-16. Cartan was not satisfied that Weyl's arguments extended to the n-dimensional case and provided his own proof in Cartan E 1923 Sur un théorème fondamental de M H Weyl dans la théorie de l'espace métrique C. R. Acad. Sci., Paris 175 82-5 and Cartan E 1923 Sur un théorème fondamental M H de Weyl J. Math. Pure. Appl. 2 167-92 both included in Cartan's Œuvres Complètes [4] pp 629-58. Here Cartan refers to the French translation of the fourth German edition of Weyl's book, implying that he had only read it in 1922. One infers that Cartan would also not have been satisfied with Weyl's proof of the uniqueness of the curvature scalar, though he makes no explicit comment on it.
13. This is the celebrated 'Raumproblem', discussed in detail in Scholz E 2001 Weyls Infintesimalgeometrie, 1917-1925 Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 85-95 and especially Coleman R and Korté H 2001 Hermann Weyl: Mathematician, Physicist, Philosopher Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 215-50 Weyl's proof that physical space - time must have a Riemannian metric goes back to the arguments of Hermann von Helmholtz (1868), helpfully summarized in [7], pp 7-16. Cartan was not satisfied that Weyl's arguments extended to the n-dimensional case and provided his own proof in Cartan E 1923 Sur un théorème fondamental de M H Weyl dans la théorie de l'espace métrique C. R. Acad. Sci., Paris 175 82-5 and Cartan E 1923 Sur un théorème fondamental M H de Weyl J. Math. Pure. Appl. 2 167-92 both included in Cartan's Œuvres Complètes [4] pp 629-58. Here Cartan refers to the French translation of the fourth German edition of Weyl's book, implying that he had only read it in 1922. One infers that Cartan would also not have been satisfied with Weyl's proof of the uniqueness of the curvature scalar, though he makes no explicit comment on it.
14. This is the celebrated 'Raumproblem', discussed in detail in Scholz E 2001 Weyls Infintesimalgeometrie, 1917-1925 Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 85-95 and especially Coleman R and Korté H 2001 Hermann Weyl: Mathematician, Physicist, Philosopher Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 215-50 Weyl's proof that physical space - time must have a Riemannian metric goes back to the arguments of Hermann von Helmholtz (1868), helpfully summarized in [7], pp 7-16. Cartan was not satisfied that Weyl's arguments extended to the n-dimensional case and provided his own proof in Cartan E 1923 Sur un théorème fondamental de M H Weyl dans la théorie de l'espace métrique C. R. Acad. Sci., Paris 175 82-5 and Cartan E 1923 Sur un théorème fondamental M H de Weyl J. Math. Pure. Appl. 2 167-92 both included in Cartan's Œuvres Complètes [4] pp 629-58. Here Cartan refers to the French translation of the fourth German edition of Weyl's book, implying that he had only read it in 1922. One infers that Cartan would also not have been satisfied with Weyl's proof of the uniqueness of the curvature scalar, though he makes no explicit comment on it.
15. This is the celebrated 'Raumproblem', discussed in detail in Scholz E 2001 Weyls Infintesimalgeometrie, 1917-1925 Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 85-95 and especially Coleman R and Korté H 2001 Hermann Weyl: Mathematician, Physicist, Philosopher Hermann Weyl's Raum-Zeit-Materie and a General Introduction to His Scientific Work ed E Scholz (Basel: Birkhäuser) pp 215-50 Weyl's proof that physical space - time must have a Riemannian metric goes back to the arguments of Hermann von Helmholtz (1868), helpfully summarized in [7], pp 7-16. Cartan was not satisfied that Weyl's arguments extended to the n-dimensional case and provided his own proof in Cartan E 1923 Sur un théorème fondamental de M H Weyl dans la théorie de l'espace métrique C. R. Acad. Sci., Paris 175 82-5 and Cartan E 1923 Sur un théorème fondamental M H de Weyl J. Math. Pure. Appl. 2 167-92 both included in Cartan's Œuvres Complètes [4] pp 629-58. Here Cartan refers to the French translation of the fourth German edition of Weyl's book, implying that he had only read it in 1922. One infers that Cartan would also not have been satisfied with Weyl's proof of the uniqueness of the curvature scalar, though he makes no explicit comment on it.
16. See Minkowski H 1952 Space and time (1908) The Principle of Relativity ed H A Lorentz et al (New York: Dover) pp 75-91 at 88
17. lkαβ must depend only on the metric tensor, not its derivatives. He notes that he did not use this stipulation in his proof and also refers to the earlier work of Vermeil H 1917 Notiz über das mittlere Krümmungmass einer n-fach augedehnten Riemann'schen Mannigfaltigkeit Nachrichten von der Königliche Gesellschaft der Wissenschaften zu Göttingen (Mathematisch-Physikalische Klasse) pp 334-44 Vermeil shows the relation between a simple line-element and the curvature scalar; his methods are close to those that Weyl adopts, but Vermeil does not take the seemingly obvious step of considering the implications of quasi-linearity.
18. Mehra J 1974 Einstein, Hilbert, and the Theory of Relativity (Dordrecht: Reidel)
19. For instance, see Adler R, Bazin M and Schiffer M 1975 Introduction to General Relativity 2nd edn (New York: McGraw-Hill) pp 344-8
20. See Adler R, Bazin M and Schiffer M 1975 Introduction to General Relativity 2nd edn (New York: McGraw-Hill) pp 358-61, 376
21. Einstein A 1979 Notes on the origin of the general theory of relativity (1934) Einstein: a Centenary Volume ed A P French (Cambridge, MA: Harvard University Press) pp 308-9