Periodic Trajectories for the Lorentz-Metric of a Static Gravitational Field

  • Vieri Benci
  • Donato Fortunato
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)


In General Relativity a gravitational field is described by a symmetric, second order tensor
$$ g \equiv g(z)\left[ {.,.} \right]z = ({z_0},...,{z_3})\varepsilon {^4} $$
on the space-time manifold R 4The tensor g is assumed to have the signature +, −, −, −; namely for all zR 4 the bilinear form g(z)[.,.] possesses one positive and three negative eigenvalues. The “pseudo-metric” induced by g is called Lorentz-metric.


Gravitational Field Critical Point Theory Periodic Trajectory Finite Dimensional Approximation Satisfy Assumption 
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]
    P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. T.M.A. 7 (1983), 981–1012.MathSciNetzbMATHGoogle Scholar
  2. [2]
    V. Benci, A geometrical index for the group S’ and some applications to the study of periodic solutions of ordinary differential equations, Comm. Pure Appl. Math. 34 (1981), 393–432.Google Scholar
  3. [3]
    V. Benci, On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), 533–572.MathSciNetCrossRefGoogle Scholar
  4. [4]
    V. Benci, A. Capozzi, D. Fortunato, Periodic solutions of Hamiltonian systems with superquadratic potential, Ann. Mat. Pura Appl. 143 (1986), 1–46.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    V. Benci, D. Fortunato, Existence of geodesics for the Lorentz metric of a stationary gravitational field,to appear in Ann. Inst. H. Poincaré Analyse Non-Linéaire.Google Scholar
  6. [6]
    H. Berestycki, J. M. Lasry, G. Mancini, B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surface, Comm. Pure Appl. Math. 38 (1985), 253–289.Google Scholar
  7. [7]
    E. R. Fadell, P. H. Rabinowitz, Generalized cohomological index theories for Lie group action with an application to bifurcation questions for Hamiltonian systems, Inv. Math. 45 (1978), 139–174.MathSciNetzbMATHGoogle Scholar
  8. [8]
    L. Landau, E. Lifchitz, Théorie des champs, Editions Mir 1970.Google Scholar
  9. [9]
    P. H. Rabinowitz, Minim ax methods in critical point theory with applications to differential equations, Conf. Board Math. Sc. A.M.S. 65 (1986).Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Vieri Benci
    • 1
  • Donato Fortunato
    • 2
  1. 1.Istituto di MatematicheApplicate — UniversitàPisaItaly
  2. 2.Dipartimento di MatematicaUniversitàBariItaly

Personalised recommendations