The geometry of null rotation identifications

TL;DR

This work analyzes Minkowski space quotiented by a null rotation, creating a Lorentzian orbifold that preserves half of the supersymmetry while avoiding closed timelike curves away from a lightlike singular locus. It establishes a geometric picture of the null-rotation quotient, shows that closed causal curves are absent for $x^- eq 0$, and reveals a deep link to the BTZ black hole via a double-scaling limit. In string/M-theory, the quotient can be embedded, yielding dilatonic waves and offering a resolution of the singularity through nullbranes, with a broader extension to curved backgrounds possessing SO$(1,2)$ isometries. The results point to a universal, supersymmetric sector of time-dependent backgrounds with potential applications to cosmological and brane dynamics in string theory.

Abstract

The geometry of flat spacetime modded out by a null rotation (boost+rotation) is analysed. When embedding this quotient spacetime in String/M-theory, it still preserves one half of the original supersymmetries. Its connection with the BTZ black hole, supersymmetric dilatonic waves and one possible resolution of its singularity in terms of nullbranes are also discussed.

Figures (5)

• Figure 1: Geometry of the $x^- >0$ sections of flat spacetime after null rotation identifications. The length of the strip is $L=x^-\cdot\beta$. Points P and Q are identified. The dashed line stands for the original orbit of the Killing vector $\xi_{\text{null}}$.
• Figure 2: Geometry of the $x^- < 0$ section of flat spacetime after null rotation identifications. The length of the strip is $L=|x^-|\cdot\beta$. Points P and Q are identified. The dashed line stands for the original orbit of the Killing vector $\xi_{\text{null}}$.
• Figure 3: Geometry of the $x^-=0$ section of flat spacetime after null rotation identifications. The bold line at $x^1=0$ stands for the line of fixed points. Points P and Q are identified. The angle $a$ is determined by $\tan\,a=2\beta$. The dashed line stands for a closed null curve.
• Figure 4: Resolution of the singularity at $x^-=0$ by identifying points in spacetime through a compact translation plus a null rotation. Bold lines are identified, and in particular, points P and Q are identified.
• Figure 5: Duality relations among nullbranes, dilatonic waves and flat vacua with non-trivial identifications. R stands for Kaluza-Klein reduction, T for T-duality and S for an $\mathrm{SL}(2\,,\hbox{Z})$ transformation. $a$ and $b$ stand for the two different circles as explained in the text.