×

Trajectories in Gödel-type space-times under the action of a vector field. (English) Zbl 1062.53061

A semi-Riemmanian manifold \((M,g_L)\) is a Gödel-type Lorentz manifold if \(M=M_0 \times \mathbb R^2\), where \((M_0, g_R)\) is a complete connected Riemannian manifold with metric \(g_R\) and \(g_L=g_R \oplus \alpha (x) \text{d}y^2+2\beta (x) \text{d}y \text{d}t - \gamma (x) \text{d}t^2\) where \(x \in M_0\), the variables \((y,t)\) are the natural coordinates on \(\mathbb R^2\) and \(\alpha, \beta, \gamma\) are scalar fields on \(M_0\) such that \(\beta^2(x) + \alpha(x) \gamma(x)>0\) for all \(x \in M_0\). Let \(A\) be a \(C^1\) vector field \(A : M \rightarrow \text{T}M\), dependent only on \(x \in M_0\).
The purpose of this article is to look for curves joining two fixed points of a Gödel-type Lorentz manifold \(M\) or for curves having prescribed period \((Y,T)\) and that satisfy the following system: \(D_s \dot \gamma(s) = {1 \over 2} \text{curl\,}A(\gamma(s))[\cdot , \dot \gamma(s)]\). By a \((Y,T)\)-periodic trajectory the authors mean a geodesic \(z=(x,y,t)\colon [a,b]\to M\) such that \(\big(x(a),\dot x(a)\big)=\big(x(b),\dot x(b)\big)\), \(y(b)=y(a)+T\), \(\dot y(b)=\dot y(a)\) and \(t(b)=t(a)+T\), \(\dot t(b)=\dot t(a)\).
The authors prove the existence of such curves under some assumptions on the scalar fields \(\alpha, \beta, \gamma\) and conditions on the Riemanian manifold \(M_0\). Assuming the topology of \(M_0\) is not trivial, they also prove a multiplicity result for time-like solutions of the system given above. The proofs use a variational approach. The system \(D_s \dot \gamma(s) = {1 \over 2} \text{curl\,} A(\gamma(s))[\cdot,\dot\gamma(s)]\) represents the equation of trajectories of relativistic particles under the action of an external force field, and generalizes the geodesic equation, which corresponds to the case \(\text{curl\,}A = 0\).

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
83C15 Exact solutions to problems in general relativity and gravitational theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bardi, M.; Capuzzo Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (1997), Birkhäuser: Birkhäuser Boston · Zbl 0890.49011
[2] Bardi, M.; Falcone, M., An approximation scheme for the minimum time function, SIAM J. Control Optim, 28, 950-965 (1990) · Zbl 0723.49024
[3] M. Bardi, M. Falcone, Discrete approximation of the minimal time function for system with regular optimal trajectories, Analysis and Optimization of Systems, Antibes, 1990, Lecture Notes in Control and Information Sciences, Vol. 144, Springer, Berlin, 1990, pp. 103-112.; M. Bardi, M. Falcone, Discrete approximation of the minimal time function for system with regular optimal trajectories, Analysis and Optimization of Systems, Antibes, 1990, Lecture Notes in Control and Information Sciences, Vol. 144, Springer, Berlin, 1990, pp. 103-112. · Zbl 0708.49012
[4] G. Barles, Remark on a flame propagation model, Report INRIA (1985) #464.; G. Barles, Remark on a flame propagation model, Report INRIA (1985) #464.
[5] Barles, G.; Roquejoffre, J. M., Large time behaviour of fronts governed by eikonal equations, Interfaces Free Bound, 5, 83-102 (2003) · Zbl 1038.35173
[6] Barles, G.; Soner, H. M.; Souganidis, T., Front propagations and phase field theory, SIAM J. Control Optim, 31, 439-469 (1993) · Zbl 0785.35049
[7] Caffarelli, L.; Crandall, M. C.; Kocan, M.; Swiech, A., On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math, 49, 365-397 (1996) · Zbl 0854.35032
[8] Camilli, F.; Grüne, L., Numerical approximation of the maximal solution for a class of degenerate Hamilton-Jacobi equations, SIAM J. Numer. Anal, 38, 1540-1560 (2000) · Zbl 0988.65077
[9] Camilli, F.; Siconolfi, A., Discontinuous solutions of an Hamilton-Jacobi equation with infinite speed of propagation, SIAM J. Math. Anal, 28, 1421-1447 (1997)
[10] Camilli, F.; Siconolfi, A., Hamilton-Jacobi equations with measurable dependence on the state variable, Adv. Differential Equations, 8, 733-768 (2003) · Zbl 1036.35052
[11] I. Capuzzo Dolcetta, The Hopf solution of Hamilton-Jacobi equations, in: Elliptic and Parabolic Problems, Rolduc/Gaeta, 2001, World Scientific Publishing, River Edge, NJ, 2002, pp. 343-351.; I. Capuzzo Dolcetta, The Hopf solution of Hamilton-Jacobi equations, in: Elliptic and Parabolic Problems, Rolduc/Gaeta, 2001, World Scientific Publishing, River Edge, NJ, 2002, pp. 343-351. · Zbl 1033.35021
[12] De Cecco, G.; Palmieri, G., LIP manifoldsfrom metric to Finslerian structure, Math. Z, 218, 223-237 (1995) · Zbl 0819.53014
[13] Falcone, M.; Giorgi, T.; Loreti, P., Level sets of viscosity solutionssome applications to fronts and rendezvous problems, SIAM J. Appl. Math, 54, 1335-1354 (1994) · Zbl 0808.49029
[14] Y. Giga, Surface evolution equations—a level set method, Hokkaido Univ. Technical Report Series in Mathematics #71, 2002.; Y. Giga, Surface evolution equations—a level set method, Hokkaido Univ. Technical Report Series in Mathematics #71, 2002.
[15] Lions, P. L., Generalized Solutions of Hamilton-Jacobi Equations (1982), Pitman: Pitman London · Zbl 1194.35459
[16] Newcomb II, R. T.; Su, J., Eikonal equations with discontinuities, Differential Integral Equations, 8, 8, 1947-1960 (1995) · Zbl 0854.35022
[17] Ostrov, D., Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependance, J. Differential Equations, 182, 51-77 (2002) · Zbl 1009.35015
[18] Siconolfi, A., A first order Hamilton-Jacobi equation with singularity and the evolution of level sets, Comm. Partial Differential Equations, 20, 277-308 (1995) · Zbl 0814.35012
[19] A. Siconolfi, Representation formulae and comparison results for geometric Hamilton-Jacobi equations, preprint, 2001.; A. Siconolfi, Representation formulae and comparison results for geometric Hamilton-Jacobi equations, preprint, 2001.
[20] Soravia, P., Generalized motion of front along its normal directiona differential game approach, Nonlinear Anal. TMA, 22, 1247-1262 (1994) · Zbl 0814.35140
[21] Soravia, P., Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian, Indiana Univ. Math. J, 51, 451-477 (2002) · Zbl 1032.35055
[22] Sethian, J., Curvature and the evolution of fronts, Comm. Math. Phys, 101, 487-499 (1985) · Zbl 0619.76087
[23] Ziemer, W., Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, Vol. 120 (1989), Springer: Springer New York · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.