[Paper Review] Gravitational lensing by two photon spheres in a black-bounce spacetime in strong deflection limits

Naoki Tsukamoto|arXiv (Cornell University)|May 29, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy171 references93 citations

TL;DR

This paper investigates gravitational lensing in a black-bounce spacetime featuring two photon spheres—one primary near the black hole and one secondary at the wormhole throat—using strong deflection limit analysis. It reveals that observers on the same side of the throat can detect an infinite sequence of faint images formed by light rays deflected near an antiphoton sphere between the two photon spheres, in addition to images near each photon sphere, offering a distinctive lensing signature for distinguishing such ultracompact objects from standard black holes.

ABSTRACT

We investigate gravitational lensing by a primary photon sphere which is a sphere filled with unstable circular light orbits, and by a secondary photon sphere on a wormhole throat in a black-bounce spacetime which is suggested in [F. S. N. Lobo, M. E. Rodrigues, M. V. d. S. Silva, A. Simpson, and M. Visser, Phys. Rev. D 103, 084052 (2021)] in strong deflection limits. There is an antiphoton sphere between the primary photon sphere and the secondary photon sphere. If a light source and an observer are on the same side of the wormhole throat, in addition to an infinite number of images slightly outside of both the primary and secondary photon spheres, an infinite number of images formed by light rays reflected by the potential barrier near the antiphoton sphere, slightly inside the primary photon sphere, might be observed.

Motivation & Objective

• To investigate gravitational lensing in a black-bounce spacetime with two photon spheres under strong deflection limits.
• To determine the observable lensing signatures from light rays deflected near both the primary photon sphere and the secondary photon sphere at the wormhole throat.
• To explore the existence and characteristics of images formed by light rays reflected near the antiphoton sphere located between the two photon spheres.
• To distinguish lensing features of such ultracompact objects from those of standard black holes using deflection angle analysis.
• To extend strong deflection limit techniques to a spacetime with non-trivial topology and multiple photon spheres.

Proposed method

• Applies strong deflection limit analysis to the black-bounce metric with parameters m (mass) and a (regularization), focusing on K=2, N=1.
• Derives the deflection angle using the effective potential V(r) = E²(Ab²/Σ² − 1), with b as the impact parameter.
• Uses the trajectory integral I(r₀) = 2∫∞r₀ dr / [Σ√(Σ²/b² − A)] to compute the deflection angle α = I(r₀) − π.
• Applies the strong deflection limit expansion for b → bm+0 (near primary photon sphere) and b → bm−0 (near antiphoton sphere), yielding logarithmic deflection expressions.
• Considers light rays originating and terminating at spatial infinity on the same side of the wormhole throat, assuming far-source and far-observer conditions.
• Employs Buchdahl coordinates and assumes equatorial (ϑ=π/2) propagation to simplify the spherically symmetric, static spacetime geometry.

Experimental results

Research questions

• RQ1 Can gravitational lensing produce observable images near both the primary and secondary photon spheres in a black-bounce spacetime?
• RQ2 What lensing signatures emerge from light rays deflected near the antiphoton sphere located between the primary and secondary photon spheres?
• RQ3 How do the deflection angles and image magnitudes differ between the primary photon sphere, the antiphoton sphere, and the secondary photon sphere?
• RQ4 Can the presence of multiple photon spheres be distinguished from standard black hole lensing via strong deflection limit analysis?
• RQ5 What are the conditions under which infinite sequences of faint images form near each photon sphere in this spacetime?

Key findings

• An infinite sequence of faint images forms slightly outside the primary photon sphere (b → bm+0), with deflection angle α ≈ −¯a log((b/bm)−1) + ¯b.
• An infinite sequence of faint images forms slightly inside the primary photon sphere, near the antiphoton sphere (b → bm−0), with deflection angle α ≈ −¯c log((bm/b)−1) + ¯d.
• A second infinite sequence of images forms slightly outside the secondary photon sphere (at the wormhole throat), due to strong deflection near the throat's photon sphere.
• The presence of the antiphoton sphere leads to observable image sequences not present in Schwarzschild or standard black hole lensing.
• For 4√3/9 < a/m ≤ 2√5/5, the spacetime supports two distinct photon spheres and an antiphoton sphere, enabling multiple image sequences.
• The lensing observables depend on the spacetime parameters m and a, with distinct signatures for black hole-like and wormhole-like regimes.