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Black hole Near Horizons through the Looking Glass

Arjun Bagchi, Arkachur Bhattacharya, Sharang Rajesh Iyer, K. Narayan

Abstract

We show that the near horizon of a generic non-extremal black hole (BH) can be understood in terms of a Carrollian geometry with two null directions, also called a String-Carroll (SC) geometry. The base space of this fibre-bundle structure is a sphere (or a plane for a black brane) and the fibre is the two-dimensional Rindler spacetime. We launch a detailed study of probes in this geometry. We study particle geodesics and scalar fields. The first part of the paper constructs geodesics and probe scalar fields directly in the SC geometry. We then look at a wide class of examples, including the Schwarzschild BH and the Kerr BH in asymptotically flat spacetimes, the BTZ BH and the black brane in AdS spacetimes, as well as Lifshitz black holes and construct the explicit maps to the SC geometry to obtain results specific to each case. These results are reproduced by considering the probe particles and fields in the original BH background and taking the near-horizon limit of the solutions. Our encyclopedia of examples establishes the notion of SC geometries as near-horizon geometries of non-extremal black objects, paving the way for a detailed, intricate future analysis of the quantum aspects of this geometry.

Black hole Near Horizons through the Looking Glass

Abstract

We show that the near horizon of a generic non-extremal black hole (BH) can be understood in terms of a Carrollian geometry with two null directions, also called a String-Carroll (SC) geometry. The base space of this fibre-bundle structure is a sphere (or a plane for a black brane) and the fibre is the two-dimensional Rindler spacetime. We launch a detailed study of probes in this geometry. We study particle geodesics and scalar fields. The first part of the paper constructs geodesics and probe scalar fields directly in the SC geometry. We then look at a wide class of examples, including the Schwarzschild BH and the Kerr BH in asymptotically flat spacetimes, the BTZ BH and the black brane in AdS spacetimes, as well as Lifshitz black holes and construct the explicit maps to the SC geometry to obtain results specific to each case. These results are reproduced by considering the probe particles and fields in the original BH background and taking the near-horizon limit of the solutions. Our encyclopedia of examples establishes the notion of SC geometries as near-horizon geometries of non-extremal black objects, paving the way for a detailed, intricate future analysis of the quantum aspects of this geometry.
Paper Structure (43 sections, 220 equations, 4 figures)

This paper contains 43 sections, 220 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Carroll near the horizon
  3. Structures near the horizon
  4. Carroll symmetries
  5. Multiple null directions: Carrollian expansions
  6. Identifying the expansions
  7. Probes in String-Carroll
  8. Point particles
  9. Scalar field
  10. Checking our formulation: The algorithm
  11. Schwarzschild Black hole
  12. Explicit map
  13. Geodesics using the map
  14. Geodesics from NH limit of Schwarzschild geodesics
  15. Scalar field using the map
  16. ...and 28 more sections

Figures (4)

  • Figure 1: Image of the core of Messier 87 (M87), housing the supermassive Black hole M87* EventHorizonTelescope:2019ths.
  • Figure 2: Closing of the square -- consistency of our work is ensured by reproducing answers in two different ways of getting from Block A to D.
  • Figure 3: A radial cross-section of a point particle just outside the horizon. According to an asymptotic observer, the particle takes an infinite coordinate time to reach the horizon.
  • Figure 4: The NH region (blue) of the Schwarzschild BH in a Penrose diagram and the process of reaching the BH horizon from the NH in the inset.