Schwarzschild-like AdS black holes: holographic imaging and LQG influences on Einstein rings

Abstract Based on AdS/CFT correspondence, our study delineates a crucial correlation between the dynamics of Einstein ring and LQG parameter a , horizon $$\rho _{h},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> observational positions and the wave source $$\omega ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which aptly reflects the influence of space-time geometry on the Einstein ring. The augmentation of a and $$\rho _{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> leads to the growth of the peak of absolute amplitude of the response function, which is generated due to diffraction of the scalar wave on the AdS boundary. The special optical system helps us to construct holographic images on the screen. The results indicate that the smaller $$\rho _{h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> shows a series of concentric striped patterns for an observer, and the brightest ring is close to the boundary and gradually shrinks with the aid of $$\rho _{h}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Our analysis reveals that the Einstein ring, which corresponds to the photon ring in geometric optics, constantly exhibits high levels of observational intensity at the north pole. With changes in observation positions, the ring gradually transforms into a bright arc-like shape and ultimately changes into a light spot on the screen’s left. The impact of variations of a on the Einstein ring is also presented in the bright curves, where the peak of the curves slightly moves towards the boundary with the increase of a . The influence of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> reflects on the ring; the width of the ring sharply reduces as $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> grows, which can also be confirmed from the corresponding brightness profiles. Finally, the comparison between wave optics and geometric optics outcomes is examined in depth.