Geometry and topology of the quasiplane Szekeres model

Geometrical and topological properties of the quasiplane Szekeres model and of the plane-symmetric dust model are discussed. Some related comments on the quasihyperbolic model are made. These properties include: (1) The pattern of expansion in the plane-symmetric case, and the Newtonian model that imitates it; (2) The possibility of toroidal topology of the $t=\mathrm{const}$ sections in the plane-symmetric model; (3) The absence of apparent horizons in the quasiplane and quasihyperbolic models (they are globally trapped); (4) Description of the toroidal topology in the Szekeres coordinates; (5) Consequences of toroidal topology in the nonsymmetric quasiplane model; (6) Avoidance of shell crossings in the toroidal model; (7) Interpretation of the mass function in the quasiplane model, with the toroidal and with the infinite space.

Figure 1

The cones where $u$ is constant are orthogonal to the cones where $v$ is constant. The coordinates in space are $u$, $v$, and the angle around the axis of symmetry. The figure shows an axial cross section through the setup. A gravitational potential which depends only on $u$ in these coordinates gives an expansion pattern that is qualitatively similar to the one in a plane-symmetric dust spacetime.

Figure 2

Left panel: The function $Z_0(R)$ (from 0 to 1, solid line) and the function $Z_1(R)$ (from 1 to 3, dotted line), given by Eqs. (5.4, 5.6), respectively. In the graph we chose the $-$ sign for $Z_0(R)$ and the $+$ sign for $Z_1(R)$. The graph is the cross section of the $(R,p)$-surface in the spaces (5.3) (left part of the curve) and (5.5) (right part of the curve). Right panel: The $(R,p)$ (or $(R,q)$) surface obtained by rotating the graph from the left panel around the $R=0$ axis. The lower end of the funnel is where the embedding in the Euclidean space breaks down, i.e., where the solid line meets the dotted line in the left panel. Upwards, the funnel goes infinitely far and becomes infinitely thin.

Figure 3

A sketch of the full 3-space of the plane-symmetric toroidal model (a faithful picture cannot be drawn because the two-dimensional flat torus cannot be embedded in a Euclidean space, and the 3-space of a planar Szekeres model cannot be made flat). Each square section of the funnel represents a two-dimensional flat torus, so its front edge coincides with the back edge, and the left edge coincides with the right edge. Each torus is an orbit of the symmetry group of the model. Each smaller torus is contained within all the larger ones, but the inclusion relation cannot be depicted in the Euclidean space of the picture. Also, in the curved 3-space, the 2-tori surround the asymptotic origin, which is the tip of the funnel, situated infinitely high above the plane drawn in the picture.

Figure 4

A map of the elementary torus in the $(p,q)$ coordinates (the central dotted square). Its left edge coincides in space with the right edge, the lower edge coincides with the upper edge. Sometimes it is convenient to consider the torus as a subset of the ${\mathbb{R}}^2$ plane, in which case the plane should be imagined as covered with an infinite number of copies of the elementary square. It will later turn out that the identifications have to be done with a twist, i.e., with reflections in $p=0$ and $q=0$. The open circles and the full circles show the pairs of points to be identified.

Figure 5

The image of the torus $\{p,q\}\in [-\pi ,\pi ]\times [-\pi ,\pi ]$ in the Szekeres coordinates. The values of $P$ and $Q$ were chosen arbitrarily, but other elements of the figure are drawn to scale. The small empty circles and the small solid circles mark pairs of points to be identified in the general, nonsymmetric case (see Sec. 8). More explanation in the text.

Figure 6

Four identical copies of a projective plane patched together to form a two-sided surface (the $4\Sigma$ of Eq. (11.8)). Numbers in circles identify the edges of the elementary square; roman numbers label the copies of the elementary square. Copy II is attached to copy I so that it mimics the twist in the projective plane. Similarly, copy III is attached to copy I in the same way. Copy IV is attached to copies II and III to close the picture. The right edge of the large square is then identified with the left edge without a twist; its upper edge is identified with the lower edge also without a twist. The surface thus obtained is two-sided, so orientable, and it makes sense to speak about the volume inside it.