Four dimensional metrics and the hamilton-jacobi equation

Four dimensional metrics and the Hamilton-Jacobi Equation

Enrique Montiel-Pi˜aa , Jos´ Guadalupe Santiago-Santiagob and Gilberto n e Silva-Ortigozab
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Facultad de Ingenier´a de la Universidad Aut´noma de Puebla, Boulevard ı o Valsequillo Esq. 14 Sur S/N, Ciudad Universitaria, Edif.124. Apartado Postal J39 C. P. 72570, Puebla, Pue., M´xico. e

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Facultad de Ciencias F´sico Matem´ticasde la Universidad Aut´noma de Puebla, ı a o Apartado Postal 1152, 72001, Puebla, Pue., M´xico. e

Abstract In the present work the issue of how a two-parameter family of solutions of the Hamilton-Jacobi equation in a four-dimensional manifold can be used to reconstruct the metric of the manifold is presented. Explicit conditions for this family and the conditions that the family must satisfy arepresented. The main goal of the present work is to present the conditions under which a two-parameter family of solutions of the Hamilton-Jacobi equation in a fourdimensional space-time can be used to En el presente trabajo In this work we show that on the space of solutions of a second-order PDE’s system ∂ss Z = Λ(s, s∗ , Z, ∂s Z, ∂s∗ Z, ∂ss∗ Z) ∂s∗ s∗ Z = Λ∗ (s, s∗ , Z, ∂s Z, ∂s∗ Z, ∂ss∗ Z, afour-dimensional metric, gab , can be reconstructed, such that the HamiltonJacobi equation, g ab (xa ) a Z(xd , s, s∗ ) b Z(xd , s, s∗ ) = A, holds. Furthermore, our general results are applied to the Minkowski space-time.

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Keys:Hamilton-Jacobi, Minkowski space-time, metric.

PACS:

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Introduction

In the decade of the nineties Newman and coworkers presented a formalreformulation of GR in terms of null surfaces [1, 2, 3], which represents a different point of view, but equivalent in content, to the Einstein’s theory of GR. That reformulation is actually known as Null Surface Formulation (NSF) of GR. In this new approach GR is formulated as equations for families of surfaces on a four-dimensional manifold, M, that is to say, starting from a four-dimensional manifold,M, (eventually to be the space-time) and no other structure, the conditions under which a family of local foliations over the manifold can be considered as family of null surfaces for an unknown conformal metric, which is possible to be determined, are founded. In that way, in the NSF emerge variables and concepts which result to have a more fundamental character than space-time geometry itself,which is usually described by the metric tensor gab (xc ). The fundamental variables in the NSF are two functions, Ω(xa , s, s∗ ) and Z(xa , s, s∗ ), where the xa are the space-time local coordinates and s is the stereographic coordinate; the former function plays the role of a conformal factor, while the later function, Z, describes on each space-time point, a set of null surfaces from which it ispossible to reconstruct any conformal Lorentzian metric. To guarantee the reconstruction of the conformal Lorentzian metric, the functions Z and Ω must satisfy a complex nonlinear coupled PDE’s system known as metricity or W¨nschmann-like conditions. It is important to mention that the Z function besides u being a complete integral of the eikonal equation on parameters s and its conjugated s∗ , isalso a solution of a two second order PDE’s system. That such space of solutions turn to be the differentiable manifold, M. On the other hand, from the studies on the structure and transformation properties about second and third order ODE’s under certain kind of transformations such as the called contact transformations carried out by Tresse, Lie, W¨nschmann, Cartan, u Chern and others [4-13], ithave been demonstrated that with any four-dimensional space it is possible to associate an equivalence class of second order PDE’s system [14, 15, 16]. It has even been showed that it is also possible associate to the same space-time, an equivalence class of fourth order ODE’s [17] using a one-parameter family of solutions of the eikonal equation instead of a complete integral of the same. Even...