Centennial of General Relativity: A Celebration by Vasconcellos Cesar Augusto Zen

Author:Vasconcellos, Cesar Augusto Zen
Language: eng
Format: epub
ISBN: 9789814699679
Publisher: World Scientific Publishing Co Pte Ltd
Published: 2017-02-13T16:00:00+00:00

It is evident from the signs of the second derivatives in Eq. (37) that along +(r) the angular maxima and minima coincide with radial maxima and minima, whereas along −(r) the signs of Eq. (38) reveal the opposite pattern: radial maxima coincide with angular minima and vice versa. The sign pattern shown in Eqs. (37) and (38) goes on for maxima and minima in n intervals following the concavity pattern of the left panel of Fig. 1 (the right panel simply switches the sequence of maxima and minima). As a consequence, we can now state that:

The 2n spatial extrema of Szekeres scalars located at r > 0 that follow from the choice of initial conditions (23)–(24) are of the following type:

•an alternating pattern of n local spatial maxima and minima in the radial intervals along the curve +(r). If the extremum at r = 0 is a local maximum (central clump) then the pattern is maximum, minimum, maximum and so on. If there is a local minimum at r = 0 (central void) the pattern is the opposite.

•n spatial saddle points in the radial intervals along the curve _(r).

A rigorous classification of the spatial extrema of A is undertaken in [Sussman & Delgado Gaspar (2015)] in terms of the determinants of the Hessian matrix and its minors for all the second derivatives of A.

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