You cannot get through Szekeres wormholes: Regularity, topology, and causality in quasispherical Szekeres models

The spherically symmetric dust model of Lemaître and Tolman can describe wormholes, but the causal communication between the two asymptotic regions through the neck is even less than in the vacuum (Schwarzschild-Kruskal-Szekeres) case. We investigate the anisotropic generalization of the wormhole topology in the Szekeres model. The function $E(r,p,q)\mathrm{}$ describes the deviation from spherical symmetry if ${\partial }_{r}E\ne 0,$ but this requires the mass to be increasing with radius, ${\partial }_{r}M>\mathrm{}0,$ i.e. nonzero density. We investigate the geometrical relations between the mass dipole and the loci of the apparent horizon and shell crossings. We present the various conditions that ensure physically reasonable quasispherical models, including a regular origin, regular maxima and minima in the spatial sections, and the absence of shell crossings. We show that physically reasonable values of ${\partial }_{r}E\ne 0$ cannot compensate for the effects of ${\partial }_{r}M>\mathrm{}0\mathrm{}$ in any direction, so that communication through the neck is still worse than in the vacuum. We also show that a handle topology cannot be created by identifying hypersufaces in the two asymptotic regions on either side of a wormhole, unless a surface layer is allowed at the junction. This impossibility includes the Schwarzschild-Kruskal-Szekeres case.