Gravitational field theory for the continuum: second order field equations

 Access to full text

 Full (PDF)

 Full (DjVu)

1. See for example WHEELER, J.A., in The Physicist's Conception of Nature, Dirac 70th anniversary volume (Dordrecht and Boston). 
2. THIRRING, W. (1961) - «Ann. Phys. (N.Y.)», 16, 96. See also SEXL, R. U. (1967) - «Fortschr. Phys.», 15, 269. MR135564DOI10.1016/0003-4916(61)90182-8
3. DESER, S. (1970) - «Gen. Relativ. Gravit.», 1, 9. MR391862
4. CAVALIERI, G. and SPINELLI, G. (1975) - «Phys. Rev.», 12 D, 2203. MR475612DOI10.1103/PhysRevD.12.2203
5. CAVALIERI, G. and SPINELLI, G. (1977) - «Nuovo Cimento», 39 B, 93. 
6. CATTANEO, C. (1973) - «Boll. U.M.I.», 8 Suppl, fasc. 2, 49. 
7. We employ here point transformations, not to be confused with coordinate transformations. See for instance, PLYBON, F. (1971) - «Journ. Math. Phys.», 12, 57. 
8. CAVALIERI, G. and SPINELLI, G. (1977) - «Nuovo Cimento», 39 B, 87. 
9. LANDAU, L. D. and LIFSHITZ, E. M. (1962) - The Classical Theory of Fields, second edition (Oxford, 1962), Sect. 94. Zbl0178.28704MR143451
10. CAVALIERI, G. and SPINELLI, G. (1975) - «Phys. Rev.», 12 D, 2200. MR475612DOI10.1103/PhysRevD.12.2203
11. Directly by the definition of the deformation tensor. See for example LANDAU, L. D. and LIFSHITZ, E. M. (1959) - Theory of Elasticity, (London) Chapt. 1. MR106584
12. Parentheses containing two indices, denote symmetrization, e.g. ${\psi }_{\alpha (\beta ;\gamma )}={\psi }_{\alpha \beta ;\gamma }+{\psi }_{\alpha \gamma ;\beta )}$. The traces of tensor are written by suppresing the repeated indices e.g. ${\psi }_{\sigma }^{\sigma }=\psi$. Finally $\mathrm{\square }$ is the d'Alembertian operator i.e. $\mathrm{\square }{\psi }_{\alpha \beta }={\psi }_{\alpha \beta ;{\lambda }^{\lambda }}$. 
13. WISS, W. (1965) - «Helv. Phys. Acta», 38, 469. 
14. DICKE, R. H. (1964) - The Theoretical Significance of Experimental Relativity, (New York, N.Y.). Zbl0148.46006MR189749
15. As shown in Ref. [2] an atom put in the gravitational field, undergoes, in the linear approximation, a deformation given by a tensor $f{\psi }_{\alpha \beta }$. It is the same deformation to which real rods and clocks (made out of atoms) are subjected, so that a real observer does not measure a pseudo-Euclidean but a Riemannian space-time. Taking into account that the matter is made out of atoms, all the objects are deformed by gravity in the unrenormalized picture. Hence, in such space-time a variation $\delta {\psi }_{\alpha \beta }$ causes an increase of the deformation tensor equal to $f\delta {\psi }_{\alpha \beta }$.