Point particles on the string in a three-dimensional AdS universe

TL;DR

The paper investigates 2+1 dimensional gravity on AdS, where point particles appear as conical defects and strings as domain walls, and develops a cut-and-glue framework to build static and dynamic spacetimes. It introduces additive notions of mass (local, Killing) and a dark-energy–subtraction approach using canonical spacetimes to isolate nontrivial matter content, then extends to dynamical configurations by boosting along boost-Killing vectors to model oscillating particle–string systems. The main contributions include explicit constructions of static systems (single particle on a semi-infinite string and two particles on a finite string), a dynamical two-particle oscillator on a finite string, and several interesting topologies (negative mass instantons, Möbius-strip spacetimes, wormholes) with corresponding mass accounting. These results illuminate how topology and nontrivial asymptotics interact with mass definitions in locally AdS spacetimes and lay groundwork for further study of BTZ-like objects interacting with strings and particles in 2+1 dimensions.

Abstract

BTZ spacetime is a long-known locally AdS solution to the Einstein equations in 1 timelike and 2 spacelike dimensions. Its static variant is interpreted as a black hole whose mass is related to the period of the angular coordinate. This solution can be parametrically continued into one without horizons but with a conical deficit in the center. Such a solution is interpreted as a spacetime with a massive particle. It has been shown that this particle can be in static equilibrium with a cosmic string passing through the spacetime to infinity. In this work, we explore the interaction of point particles with strings, such as a bound system of two particles connected by a string of finite length. We identify additive local mass in the static spacetimes and apply it to the case of particles and strings. Finally, using the cut and glue method, we construct the system of two particles oscillating on the string, which goes out of the scope of static systems.

Figures (20)

• Figure 1: Left: coordinate lines of $\{\chi,\varphi\}$ on a global time-slice with geometry of a hyperbolic plane. The time-slice is conformally related to a hemisphere with $\{\chi,\varphi\}$ being the standard spherical coordinates. Right: coordinate lines of $\{{\overline\chi},{\overline\varphi}\}$. These are spherical coordinates on a conformally related sphere centered on a pole rotated by $\chi_{\mathrm{o}}$ with respect to the left diagram. The curves ${\overline\varphi}=\text{const}$ are exocycles of the hyperbolic plane.
• Figure 2: Diagram of a spatial section ${\tau=\text{const}}$ of the spacetime with a conical deficit. It is drawn using the coordinates of the global AdS universe: radial coordinate ${\chi\in\left(0,\frac{\pi}{2}\right)}$ and angular coordinate ${\varphi\in(-\pi,\pi)}$. The boundary circle ${\chi=\frac{\pi}{2}}$ represents the conformal infinity. The angular coordinate is restricted to the interval ${\varphi\in(-\varphi_{\mathrm{o}},\varphi_{\mathrm{o}})}$ and radial edges ${\varphi=\pm\varphi_{\mathrm{o}}}$ represent the identified hyperplanes. The hatched angle parameterized by $\Delta\varphi$ is excluded. There is a conical deficit at the origin $\chi=0$ which represents a static point particle (represented by the black disk). Although the metric is not smooth at the conical singularity, one can estimate the acceleration of close-by worldlines and conclude that it vanishes. The point particle is thus in a free motion.
• Figure 3: Diagram of a global time-slice $T=\text{const}$. The origin of the accelerated frame and its shift by $\chi_{\mathrm{o}}$ from the origin of the unaccelerated frame is indicated. Two curves, ${\overline\varphi}=\varphi_{\mathrm{o}}$ and ${\overline\varphi}=\pi-\varphi_{\mathrm{o}}$, starting at this point are shown. These curves together form an exocycle, i.e., the equidistant line to the axis (dashed line) going through the center of gravity.
• Figure 4: Left: A particle at ${\overline\chi}=0$, positioned at $\chi=\chi_{\mathrm{o}}<0$, is in equilibrium between a pull of the semi-infinite string and gravitational attraction toward the origin. The excluded region is hatched, and the thick curves denote the hypersurfaces identified into a string $\Sigma$. The diagram is drawn in standard polar coordinates $\chi,\,\varphi$. Radial lines show the direction of the apparent gravitational force in the global static frame. Relation between the particle's acceleration, $a_{\mathrm{o}}<0$, and parameter $\chi_{\mathrm{o}}<0$ is given by \ref{['eq:aodef']}. The small diagram suggests how an observer inside this spacetime might perceive the situation. The line $\Sigma$ represents the semi-infinite string depicted as a spatially linear defect. The same 'radial' lines as in the main diagram are drawn, showing the direction of the gravitational force. Right: Setting the particle's position at $\chi=\chi_{\mathrm{o}}>0$ leads to the particle being pushed by a strut against the gravitational pull. Again, radial lines show the gravity direction. The acceleration parameter $a_{\mathrm{o}}$ is positive. Notice that the center of gravity is excluded from the spacetime in this case. The small diagram is again deformed in such a way that the strut $\Sigma$ is depicted as a one-dimensional object. In this case, gravity points everywhere in the spacetime towards the strut, which acts as its source. This is because the original center of gravity is in the removed region.
• Figure 5: Left: Two particles on a string. The thick solid curves suggest the two hypersurfaces forming the history of the finite string. The thick dashed line segments represent the flat hypersurfaces reaching to infinity. The hatched area again represents the excluded region. $\Delta\varphi$'s parameterize the conical defects at respective points. $\Delta\varphi_{{\mathrm{tot}}}$ parameterizes the mass of the whole system as observed in the far region. Notice that $\Delta\varphi_2 < 0$ and the corresponding particle thus has a negative mass. We thus have a self-accelerating system of negative and positive mass particles joined by the string. The diagram in the top right corner shows a qualitative suggestion of how an observer inside the spacetime might perceive it. The thick curve represents the string, the small discs represent particles (semi-discs are used when identification of multiple points forms a single particle), the thin and dashed semi-lines indicate flat surfaces, and the small cross denotes a location of the center of gravity. Right: The situation represents two particles separated by a finite strut. It keeps both particles in equilibrium and compensates for the AdS gravitational pull. Note that this time $\Delta\varphi_2 > 0$ and thus both particles have positive mass. Again, a qualitative representation of what an observer inside might perceive is presented in the top right corner. Notably, this time, the center of gravity is not part of the spacetime.