Strong deflection limit-analysis using Picard-Fuchs equation in Einstein-Maxwell-Dilaton spacetime
Abstract
We consider the deflection of light by a spherically symmetric and electrically charged black hole solution in the Einstein-Maxwell-Dilaton theory with specific values of the dilaton coupling constant where the deflection angle can be represented as elliptic integrals. We show that the deflection angle as a function of two dimensionless variables (s,z), which are related to the background charge and the impact parameter, respectively, satisfies a system of 2nd-order linear partial differential equations called the Picard-Fuchs (PF) equations. For each case of the dilaton coupling, the PF equations lead to 1st-order ordinary differential equations with respect to the variable s for the constants a̅ and b̅ in the log-formula for the strong deflection limit. Using the Hamiltonian system associated with Painlevé VI equations, which holds as a result of the integrability of the PF equations, we solve the equations for a̅ and b̅. By requiring consistency with the Schwarzschild case in the zero-charge limit, a̅ and b̅ for nonzero charge are uniquely determined.
Versions
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BibTeX
@misc{sasaki2026strong,
title = {{Strong deflection limit-analysis using Picard-Fuchs equation in Einstein-Maxwell-Dilaton spacetime}},
author = {Tadashi Sasaki},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.12819},
eprint = {2606.12819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12819},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strong deflection limit-analysis using Picard-Fuchs equation in Einstein-Maxwell-Dilaton spacetime
AU - Tadashi Sasaki
PY - 2026
DA - 2026/06/25
PB - arXiv
DO - 10.48550/arxiv.2606.12819
UR - https://arxiv.org/abs/2606.12819
AB - We consider the deflection of light by a spherically symmetric and electrically charged black hole solution in the Einstein-Maxwell-Dilaton theory with specific values of the dilaton coupling constant where the deflection angle can be represented as elliptic integrals. We show that the deflection angle as a function of two dimensionless variables (s,z), which are related to the background charge and the impact parameter, respectively, satisfies a system of 2nd-order linear partial differential equations called the Picard-Fuchs (PF) equations. For each case of the dilaton coupling, the PF equations lead to 1st-order ordinary differential equations with respect to the variable s for the constants a̅ and b̅ in the log-formula for the strong deflection limit. Using the Hamiltonian system associated with Painlevé VI equations, which holds as a result of the integrability of the PF equations, we solve the equations for a̅ and b̅. By requiring consistency with the Schwarzschild case in the zero-charge limit, a̅ and b̅ for nonzero charge are uniquely determined.
ER -