A Rindler Road to Carrollian Worldsheets

TL;DR

The paper investigates how strings behave in background Rindler spacetimes, showing that approaching a horizon induces a tensionless, Carrollian worldsheet with a Rindler-induced geometry. It demonstrates the emergence of BMS$_3$ symmetry on the worldsheet via contractions of the Virasoro algebra and reveals novel features such as folds and modified periodicity that partition a closed string into gluable open segments. Through Bogoliubov transformations and boundary/gluing conditions, it constructs global worldsheet modes and boundary states, linking accelerated-string dynamics to boundary phenomena and potential open-string degrees of freedom. These results illuminate the connection between spacetime horizons and worldsheet Carrollian physics, offering a framework to study strings near black hole horizons and suggesting future extensions to more general near-horizon geometries and supersymmetric cases.

Abstract

The tensionless limit of string theory has recently been formulated in terms of worldsheet Rindler physics. In this paper, by considering closed strings moving in background Rindler spacetimes, we provide a concrete exemplification of this phenomenon. We first show that strings probing the near-horizon region of a generic non-extremal blackhole become tensionless thereby linking a spacetime Carroll limit to a worldsheet Carroll limit. Then, considering strings in $d$-dimensional Rindler spacetime we find a Rindler structure induced on the worldsheet. Novelties, including folds, appear on the closed string worldsheet pertaining to the formation of the worldsheet horizon. The closed string becomes segmented at these folding points and different segments go into the formation of closed strings in the different Rindler wedges. The Bondi-Metzner-Sachs (BMS) or the Conformal Carroll algebra emerges from the closed string Virasoro algebra as the horizon is hit. Quantum states on these accelerated worldsheets are discussed and we show the formation of boundary states from gluing conditions of the different segments of the accelerated closed string.

Figures (8)

• Figure 1: Illustration of how the string worldsheet deforms gradually under increasing acceleration. We represent accelerated worldsheets as hyperboloids, which are a natural generalization of Rindler particle worldlines.
• Figure 2: Two dimensional Rindler space spanned by $(\eta,\xi)$ coordinates, with the inertial set of coordinates being $(t,x)$. The timelike vector flows in opposite directions for two causally disconnected wedges (Right/Left), and the constant $\eta$ timeslices $\eta_1<\eta_2<\eta_3$ are shown spanning both wedges. Constant $\xi$ surfaces are families of hyperbolae corresponding to progressively accelerated observers. Light cones in Minkowski $(t,x)$ coordinates become Rindler horizons for accelerated observers.
• Figure 3: Illustration of a spinning folded string in the AdS cylinder. At the small spin limit the string is circular and stays near the center, as spin is increased folds start to appear at marked points on the worldsheet, and finally the folding points touch the boundary.
• Figure 4: Illustration of how acclereation affects the string shape. As the acceleration increases from zero, the string "folds" at the special points and gradually gets more and more elongated. One can think of the special points as boundary of two 'open' strings.
• Figure 5: Effective periodicity