Applications of Calculus of Variations to General Relativity

  • A. Masiello


We present some global results on Lorentzian geometry obtained by using global variational methods. In particular some results on the geodesic connectedness of Lorentzian manifolds and on the multiplicity of lightlike geodesies joining a point with a timelike curve are presented. Such results allow to give a mathematical description of the gravitational lens effect.


Morse Theory Critical Point Theory Lorentzian Manifold Timelike Curve Timelike Geodesic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold V.I. (1989): Mathematical Methods of Classical Mechanics. Springer, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  2. 2.
    Morse M. (1934): The Calculus of Variations in the Large. Coll. Lect. Am. Math. Soc. 18 Google Scholar
  3. 3.
    Lusternik L.A., Schnirelmann L. (1934): Methodes Topologiques dans les Problèmes Variationelles. Gautier-Villars, ParisGoogle Scholar
  4. 4.
    Struwe M. (1996): Variational Methods. Springer, Berlin Heidelberg New YorkMATHCrossRefGoogle Scholar
  5. 5.
    Palais R. (1963): Lusternik-Schnirelmann theory on Banach manifolds. Topology 5, 115–132MathSciNetCrossRefGoogle Scholar
  6. 6.
    Klingenberg W. (1978): Lecture on Closed Geodesies. Springer, Berlin Heidelberg New YorkCrossRefGoogle Scholar
  7. 7.
    Bott R. (1982): Lectures on Morse Theory Old and New. Bull. Am. Math. Soc. 7, 331–358MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Spanier E.H. (1966): Algebraic Topology. McGraw Hill, New YorkMATHGoogle Scholar
  9. 9.
    Milnor J. (1963): Morse Theory. Ann. of Math. Studies 51, Princeton University Press, PrincetonMATHGoogle Scholar
  10. 10.
    Masiello A. (1994): Variational Methods in Lorentzian Geometry. Pitman Research Notes in Mathematics 309, Longman, LondonMATHGoogle Scholar
  11. 11.
    Palais R. (1963): Morse theory on Hilbert manifolds. Topology 2, 299–340MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Palais R. (1970): Critical Point Theory and the Min-Max Principle. Proc. Symp. Pure Appl. Math. 15, 185–212MathSciNetCrossRefGoogle Scholar
  13. 13.
    Palais R., Smale S. (1964): A Generalized Morse theory. Bull. Am. Math. Soc. 70, 165–171MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Rabinowitz P.H. (1984): Min-Max Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Soc. in Math. 65 Amer. Math. Soc, ProvidenceGoogle Scholar
  15. 15.
    Mawhin J., Willem M. (1989): Critical Point Theory and Hamiltonian Systems. Springer, Berlin Heidelberg New YorkMATHCrossRefGoogle Scholar
  16. 16.
    Adams R. (1975): Sobolev spaces. Acad. Press, New YorkMATHGoogle Scholar
  17. 17.
    Fadell E., Susseini S. (1991): Category of Loop Spaces of Open Subsets in Euclidean Space. Nonlinear Analysis T.M.A. 17, 1153–1161MATHCrossRefGoogle Scholar
  18. 18.
    Benci V., Fortunato D. (1990): Existence of Geodesies for the Lorentz Metric of a Stationary Gravitational Field. Ann. H. Poincaré Analyse Non Linéaire 7, 27–35MathSciNetMATHGoogle Scholar
  19. 19.
    Beem J.K., Ehrlich P.E., Easley K. (1996): Global Lorentzian Geometry. Marcel Dekker, New YorkMATHGoogle Scholar
  20. 20.
    Hawking S.W., Ellis G.F. (1973): The Large Scale Structure of Space-Time. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  21. 21.
    O’NeilL B. (1983): Semi-riemannian Geometry with Applications to Relativity. Academic Press, New YorkMATHGoogle Scholar
  22. 22.
    Beem J.K., Parker R.E. (1989): Pseudoconvexity and Geodesic Connectedness. Ann. Mat. Pura Appl. 155(4), 137–142MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Penrose R. (1972): Techniques of Differential Topology in Relativity. Conf. Board Math. Sci 7. SIAM, PhiladelphiaGoogle Scholar
  24. 24.
    Giannoni F., Masiello A. (1991): On the Existence of Geodesies on Stationary Lorentz Manifolds with Convex Boundary. J. Funct. Anal. 101, 340–369MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Benci V., Fortunato D., Giannoni F. (1992): On the existence of geodesies in static Lorentz Manifolds with Singular Boundary. Ann. Sc. Norm. Sup. Pisa 19(4), 255–289MathSciNetMATHGoogle Scholar
  26. 26.
    Giannoni F, Piccione P. (1999): An Intrinsic Approach to the Geodesical Connectedness of Stationary Lorentzian Manifolds. Comm. Anal. Geom. 7, 157–197MathSciNetMATHGoogle Scholar
  27. 27.
    Geroch R. (1970): Domains of Dependence. J. Math. Phys. 11, 437–449MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Benci V., Fortunato D., Masiello A. (1994): On the geodesic connectedness of Lorentzian manifolds. Math. Z. 217, 74–94MathSciNetCrossRefGoogle Scholar
  29. 29.
    Giannoni F, Masiello A. (1995): Geodesies on Product Lorentzian Manifolds. Ann. Inst. H. Poincaré Analyse non linéaire 12, 27–60MathSciNetMATHGoogle Scholar
  30. 30.
    Fadell E. (1985): Lectures in Cohomological Index Theories of G-Spaces with Applications to Critical Point Theory. Sem. Dip. Mat. Université della Calabria, CosenzaGoogle Scholar
  31. 31.
    Fadell E., Husseini S. (1994): Relative category and coproducts. Rend. Sem. Mat. Fis. Milano 64, 99–117MathSciNetCrossRefGoogle Scholar
  32. 32.
    Masiello A. (1994): Convex regions of Lorentzian manifolds. Ann. Mat. Pura Appl. 167(4), 299–322MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Giannoni F., Masiello A. (1993): Geodesies on Lorentzian Manifolds with Quasi-Convex Boundary. Man. Math. 78, 381–396MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Schneider P., Ehlers J., Falco E. (1992): Gravitational Lensing. Springer, Berlin Heidelberg New YorkGoogle Scholar
  35. 35.
    Weyl H. (1917): Zur Gravitationstheorie. Ann. Phys. 54, 117–145MATHCrossRefGoogle Scholar
  36. 36.
    Levi-Civita T. (1928): Fondamenti di Meccanica Relativistica. Zanichelli, BolognaMATHGoogle Scholar
  37. 37.
    Uhlenbeck K. (1975): A Morse Theory for Geodesies on a Lorentz Manifold. Topology 14, 69–90MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Fortunato D., Giannoni F, Masiello A. (1995): A Fermat Principle for Stationary Space-Times with Applications to Light Rays. J. Geom. Phys. 15, 159–188MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    Giannoni F, Masiello A. (1998): On a Fermat Principle in General Relativity. A Lusternik-Schnirelmann Theory of Light Rays. Ann. Mat. Pura Appl. 174(4), 161–207MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Antonacci F., Piccione P. (1996): A Fermat principle on Lorentzian manifolds and applications. Appl. Math. Lett. 9, 91–96MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Perlick V. (1995): Infinite dimensional Morse theory and Fermat’s principle in general relativity. I. J. Math. Phys. 36, 6915–6928MathSciNetADSMATHCrossRefGoogle Scholar
  42. 42.
    Perlick V. (1990): On Fermat’s principle in general relativity: I. The general case. Class. Quant. Grav. 7, 1319–1331MathSciNetADSMATHCrossRefGoogle Scholar
  43. 43.
    Giannoni F., Masiello A., Piccione P. (1998): A Morse Theory for Light Rays on Stably Causal Lorentzian Manifolds. Ann. Inst. H. Poincaré Physique Théorique 69, 359–412MathSciNetMATHGoogle Scholar
  44. 44.
    Giannoni F., Masiello A. (1996): On a Fermat principle in General Relativity. A Morse Theory for light rays. Gen. Rel. Grav. 28, 855–897MathSciNetADSMATHCrossRefGoogle Scholar
  45. 45.
    Giannoni F., Masiello A., Piccione P. (1997): A Variational Theory for Light Rays on Stably Causal Lorentzian Manifolds. Existence, Regularity and Multiplicity Results, Commun. Math. Phys 187, 375–415MathSciNetADSMATHCrossRefGoogle Scholar
  46. 46.
    Giannoni F., Masiello A., Piccione P.: The Fermat Principle in General Relativity and Applications, in preparationGoogle Scholar
  47. 47.
    Giannoni F., Masiello A., Piccione P. (1999): Convexity and the Finiteness of the Number of Geodesies. Applications to the Multiple-Image Effect. Class. Quant. Grav. 16,731–748MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    Giannoni F., Masiello A., Piccione P. (1998): A Timelike Extension of Fermat’s Principle in General Relativity and Applications. Cale. Var. P.D.E. 6, 263–283MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Giannoni F., Masiello A., Piccione P.: A Morse Theory for Massive Particles and Photons. J. Geom. Phys. in pressGoogle Scholar

Copyright information

© Springer-Verlag Italia 2000

Authors and Affiliations

  • A. Masiello

There are no affiliations available

Personalised recommendations