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r -q c /0105081v 1 23 M a y 2001New variables for the Lema ˆıtre-Tolman-Bondi dust solutions.
Roberto A.Sussman †and Luis Garc´ıa Trujillo ‡†Instituto de Ciencias Nucleares,Apartado Postal 70543,UNAM,M´e xico DF,04510,M´e xico.
‡Instituto de F´ısica,Universidad de Guanajuato,Leon,Guanajuato,M´e xico.
We re-examine the Lemˆa itre-Tolman-Bondi (LTB)solutions with a dust source admitting symme-
try centers.We consider as free parameters of the solutions the initial value functions:Y i ,ρi ,(3)R i ,obtained by restricting the curvature radius,Y ≡
√g θθ,containing two arbitrary functions of the radial coordinate:M (r )and E (r ).The integration of the dynamical equation yields a third arbitrary function,t bb (r ).The first two of these functions (M and E )are usualy identified as the relativistic analogues of the newtonian mass and the local energy associated with the comoving dust shells,the third function (t bb )can be interpreted as the “bang time”,marking the proper time for the initial (or
final)singularity for each comoving oberver.The functions M,E,t bb are then the free parameters of the solutions,
though any one of them can always be used to fix the radial coordinate,thus there are realy only two free functions.Regularity conditions,related to well behaved symmetry centers,abscence of shell crossing singularities and surface layers,can be given as restrictions on these functions and their radial gradients.These functions are usualy selected by means of convenient ansatzes based on the interpretations described above and/or the compatibility with regularity conditions or observational criteria.Once the free parameters have been selected we have a completely determined LTB model.
Since the standard free functions lead to a consistent description of LTB models,these free functions are used in most literature dealing with these solutions (see [22]for a comprehensive and autoritative review of this literature).There are papers considering specific modifications of the standard free functions in order to formulate initial conditions for studying the censorship of singularities [20],or aiming at an “intial value”formulation of LTB models [9].We believe that new variables,defined along an arbitrary and regular “initial”hypersurface t =t i ,lead to a more intuitive understanding of LTB solutions.Therefore,we propose in this paper using a new set of basic free functions along the following steps
•Consider the rest mass density,ρ,the 3-dimensional Ricci scalar,(3)R ,and the curvature radius,Y ,all of them evaluated along an arbitrary Cauchy hypersurface,T i ,marked by constant cosmic time t =t i .This leads to:ρi (r ),(3)R i (r )and Y i (r ),where the subindex i indicates evaluation along at t =t i (we shall use this convention henceforth).
