Gravitational lensing outside and inside of a marginally unstable photon sphere at a Z2-symmetric wormhole throat in strong deflection limits
Abstract
The deflection angles of rays diverge logarithmically near outside and inside of a photon sphere in usual strong deflection limits and the rays near the photon sphere could be detected by near-future space observations on black hole shadows. On the other hand, we show that the deflection angles of the rays near the photon sphere at a throat of a Z2-symmetric wormhole, as an example among some commonly used types of wormholes or simple wormholes, which is often studied by the Event Horizon Telescope collaborations, diverge in power law but not logarithmically. This is because the Z2-symmetric wormhole has a marginally unstable photon sphere at the throat and it has neither a common nor simple but special property. We investigate gravitational lensing of the rays nearly outside and inside of the marginally unstable photon sphere at the throat in a class of Z2-symmetric wormholes in the strong deflection limits. We use a numerical method to calculate the deflection angles diverging in power law and we apply it to the commonly-used simple wormhole spacetime, a Damour-Solodukhin wormhole spacetime, and a Simpson-Visser black-bounce spacetime. Our numerical results imply universal property of the deflection angle of the rays near the marginally unstable photon sphere on Z2-symmetric wormhole throat in the strong deflection limits as similar to the deflection angles of ray bent by a marginally unstable photon sphere off a throat in the strong deflection limits which have a different power. We also correct the coefficients of power-law divergent terms of the deflection angles of the rays bent nearly outside of the marginally unstable photon sphere at the Z2-symmetric wormhole throat in a semi-analytic approach investigated by the author and others.
Versions
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BibTeX
@misc{tsukamoto2026gravitational,
title = {{Gravitational lensing outside and inside of a marginally unstable photon sphere at a $Z_2$-symmetric wormhole throat in strong deflection limits}},
author = {Naoki Tsukamoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23269},
eprint = {2608.23269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gravitational lensing outside and inside of a marginally unstable photon sphere at a Z_2-symmetric wormhole throat in strong deflection limits
AU - Naoki Tsukamoto
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2608.23269
UR - https://arxiv.org/abs/2608.23269
AB - The deflection angles of rays diverge logarithmically near outside and inside of a photon sphere in usual strong deflection limits and the rays near the photon sphere could be detected by near-future space observations on black hole shadows. On the other hand, we show that the deflection angles of the rays near the photon sphere at a throat of a Z_2-symmetric wormhole, as an example among some commonly used types of wormholes or simple wormholes, which is often studied by the Event Horizon Telescope collaborations, diverge in power law but not logarithmically. This is because the Z_2-symmetric wormhole has a marginally unstable photon sphere at the throat and it has neither a common nor simple but special property. We investigate gravitational lensing of the rays nearly outside and inside of the marginally unstable photon sphere at the throat in a class of Z_2-symmetric wormholes in the strong deflection limits. We use a numerical method to calculate the deflection angles diverging in power law and we apply it to the commonly-used simple wormhole spacetime, a Damour-Solodukhin wormhole spacetime, and a Simpson-Visser black-bounce spacetime. Our numerical results imply universal property of the deflection angle of the rays near the marginally unstable photon sphere on Z_2-symmetric wormhole throat in the strong deflection limits as similar to the deflection angles of ray bent by a marginally unstable photon sphere off a throat in the strong deflection limits which have a different power. We also correct the coefficients of power-law divergent terms of the deflection angles of the rays bent nearly outside of the marginally unstable photon sphere at the Z_2-symmetric wormhole throat in a semi-analytic approach investigated by the author and others.
ER -