Emergence of Lorentzian geometry from string Carroll geometry
Abstract
The near horizon region of a generic non extremal black hole has been extensively studied in terms of a Carrollian geometry with two null directions, also called a string Carroll geometry. We investigate how this string Carroll geometry evolves to a Lorentzian throat as the four dimensional Reissner-Nordström (RN) black hole approaches extremality. We identify the quadratic temporal contribution that is suppressed in the non extremal description but is required to recover the AdS2 curvature. Approaching the outer horizon and extremality simultaneously, scaling the horizon separation together with the radial distance and correspondingly rescaling Killing time, brings this contribution into the leading longitudinal metric alongside the Rindler sector. In ingoing Eddington-Finkelstein coordinates, the string Carroll metric alone has a Lorentzian limit with a flat longitudinal sector. The quadratic temporal term arises from the bilinear of the leading and subleading longitudinal coframe fields and supplies the missing curvature, while higher radial contributions vanish. The reconstructed limiting geometry is already locally AdS2× S2 at finite positive scaled horizon separation; setting this parameter to zero makes the selected horizon extremal without changing the local curvature. The limiting metric and Maxwell field satisfy the Einstein-Maxwell equations. In static coordinates, the temporal component again follows from the quadratic term, while the radial component requires summing contributions from all radial orders. The reconstruction provides a geometric starting point for investigating which expansion data must be retained when extending calculations based on the non extremal string Carroll description into the throat regime, including their implications for string, particle and field dynamics.
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BibTeX
@misc{shinde2026emergence,
title = {{Emergence of Lorentzian geometry from string Carroll geometry}},
author = {Anirudhda Shinde and Mangesh Mandlik},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.23760},
eprint = {2607.23760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23760},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Emergence of Lorentzian geometry from string Carroll geometry
AU - Anirudhda Shinde
AU - Mangesh Mandlik
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2607.23760
UR - https://arxiv.org/abs/2607.23760
AB - The near horizon region of a generic non extremal black hole has been extensively studied in terms of a Carrollian geometry with two null directions, also called a string Carroll geometry. We investigate how this string Carroll geometry evolves to a Lorentzian throat as the four dimensional Reissner-Nordström (RN) black hole approaches extremality. We identify the quadratic temporal contribution that is suppressed in the non extremal description but is required to recover the AdS_2 curvature. Approaching the outer horizon and extremality simultaneously, scaling the horizon separation together with the radial distance and correspondingly rescaling Killing time, brings this contribution into the leading longitudinal metric alongside the Rindler sector. In ingoing Eddington-Finkelstein coordinates, the string Carroll metric alone has a Lorentzian limit with a flat longitudinal sector. The quadratic temporal term arises from the bilinear of the leading and subleading longitudinal coframe fields and supplies the missing curvature, while higher radial contributions vanish. The reconstructed limiting geometry is already locally AdS_2× S^2 at finite positive scaled horizon separation; setting this parameter to zero makes the selected horizon extremal without changing the local curvature. The limiting metric and Maxwell field satisfy the Einstein-Maxwell equations. In static coordinates, the temporal component again follows from the quadratic term, while the radial component requires summing contributions from all radial orders. The reconstruction provides a geometric starting point for investigating which expansion data must be retained when extending calculations based on the non extremal string Carroll description into the throat regime, including their implications for string, particle and field dynamics.
ER -