New spacetimes for rotating dust in ($2+1$)-dimensional general relativity

Multiparameter solutions to the Einstein equations in $2+1$ dimensions are presented, with stress-energy given by a rotating dust with negative cosmological constant. The matter density is uniform in the corotating frame, and the ratio of the density to the vacuum energy may be freely chosen. The rotation profile of the dust is controlled by two parameters, and the circumference of a circle of a given radius is controlled by two additional parameters. Though locally related to known metrics, the global properties of this class of spacetimes are nontrivial and allow for new and interesting structure, including apparent horizons and closed timelike curves, which can be censored by a certain parameter choice. General members of this class of metrics have two Killing vectors, but parameters can be chosen to enhance the symmetry to four Killing vectors. The causal structure of these geometries, interesting limits, and relationship to the Gödel metric are discussed. An additional solution, with nonuniform dust density in a Gaussian profile and zero cosmological constant, is also presented, and its relation to the uniform-density solutions in a certain limit is discussed.

Figure 1

Illustration of the null congruences orthogonal to circles of fixed $r$ (gray arcs) on some slice of constant $t$ (green surface). The outgoing and ingoing orthogonal null congruences have tangents given in Eq. (9) by $k^a$ and ${\ell }^a$ and are illustrated by red and blue arrows, respectively. For this illustration, $j_1{j}_2$ was chosen to be negative, so the frame dragging vanishes on the surface at radius $r_0$ (dotted line) where $w(r_0)=0$ and goes in the $+\varphi$ or $-\varphi$ direction inside and outside, respectively. As the Cauchy horizon $r_{\mathrm{C}}$ (dashed line) is approached, the orthogonal null congruences are dragged around the angular direction. For $r<r_{\mathrm{C}}$, the congruence is not defined.

Figure 2

Cases in Eq. (17) for geometry with metric given in Eqs. (1) and (2). The singularities in the $rr$ component of the metric [the zeros of $N(r)$] are $r_{\pm }$, while the Cauchy horizon $r_{\mathrm{C}}$ is determined by the zero of the $\varphi \varphi$ component of the metric $D(r)$. In Cases I and II, $0<M<1$, while in Cases III and IV, $M\ge 1$. In Cases I and III, $j_1{j}_2+c_1>0$, while in Cases II and IV, $j_1{j}_2+c_1<0$. If $c_2\ge 0$ and $c_1\le 0$, then $r_{\mathrm{C}}\ge r_{\pm }$.

Figure 3

Geodesics bouncing off of a boundary at $r=r_{\pm }$. Depicted is a family of geodesics with different initial data $r_{\mathrm{i}}$ and $v_{\mathrm{i}}$ chosen to all intersect the boundary at some fixed spacetime location (black dot). The bounce occurs for geodesics originating on either side of the boundary.