We investigate gravitational lensing in the strong deflection regime by loop quantum gravity (LQG)-motivated rotating black hole (LMRBH) metrics with an additional parameter l besides mass M and rotation a. The LMRBH spacetimes are regular everywhere, asymptotically encompassing the Kerr black hole as a particular case and, depending on the parameters, describe black holes with one horizon only (BH-I), black holes with an event horizon and a Cauchy horizon (BH-II), black holes with three horizons (BH-III), or black holes with no horizons (NH) spacetime. It turns out that as the LQG parameter l increases, the unstable photon orbit radius  $$x_{ps}$$  <math> <msub> <mi>x</mi> <mrow> <mi>ps</mi> </mrow> </msub> </math>  , the critical impact parameter  $$u_{ps}$$  <math> <msub> <mi>u</mi> <mrow> <mi>ps</mi> </mrow> </msub> </math>  , the deflection angle  $$\alpha _D(\theta )$$  <math> <mrow> <msub> <mi>α</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </mrow> </math>   and angular position  $$\theta _{\infty }$$  <math> <msub> <mi>θ</mi> <mi>∞</mi> </msub> </math>   also increases. Meanwhile, the angular separation s decreases, and relative magnitude  $$r_{mag}$$  <math> <msub> <mi>r</mi> <mrow> <mi>mag</mi> </mrow> </msub> </math>   increases with increasing l for prograde motion but they show opposite behaviour for the retrograde motion. Using supermassive black holes (SMBH) Sgr A* and M87* as lenses, we compare the observable signatures of LMRBH with those of Kerr black holes. For Sgr A*, the angular position  $$\theta _{\infty }$$  <math> <msub> <mi>θ</mi> <mi>∞</mi> </msub> </math>    $$\in $$  <math> <mo>∈</mo> </math>   (16.4, 39.8)  $$\upmu $$  <math> <mi>μ</mi> </math>  as, while for M87*  $$\in $$  <math> <mo>∈</mo> </math>   (12.33, 29.9)  $$\upmu $$  <math> <mi>μ</mi> </math>  as. The angular separation s, for SMBHs Sgr A* and M87*, differs significantly, with values ranging  $$\in $$  <math> <mo>∈</mo> </math>   (0.008–0.376)  $$\upmu $$  <math> <mi>μ</mi> </math>  as for Sgr A* and  $$\in $$  <math> <mo>∈</mo> </math>   (0.006–0.282)  $$\upmu $$  <math> <mi>μ</mi> </math>  as for M87*. The deviations of the lensing observables  $$\Delta \theta _{\infty }$$  <math> <mrow> <mi>Δ</mi> <msub> <mi>θ</mi> <mi>∞</mi> </msub> </mrow> </math>   and  $$\Delta s$$  <math> <mrow> <mi>Δ</mi> <mi>s</mi> </mrow> </math>   for LMRBH ( $$a=0.80,l=2.0$$  <math> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.80</mn> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </math>  ) from Kerr black holes can reach up to  $$10.22\,\upmu $$  <math> <mrow> <mn>10.22</mn> <mspace width="0.166667em"></mspace> <mi>μ</mi> </mrow> </math>  as and  $$0.241~\upmu $$  <math> <mrow> <mn>0.241</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as for Sgr A*, and  $$7.683~\upmu $$  <math> <mrow> <mn>7.683</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as and  $$0.181~\upmu $$  <math> <mrow> <mn>0.181</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as for M87*. The relative magnitude  $$r_{mag}$$  <math> <msub> <mi>r</mi> <mrow> <mi>mag</mi> </mrow> </msub> </math>    $$\in $$  <math> <mo>∈</mo> </math>   (0.047, 1.54). We estimate the time delay between the first and second relativistic images using twenty supermassive galactic centre black holes as lenses to find, for example, the time delay for Sgr A* and M87* can reach approximately 23.26 min and 33,261.8 min, respectively. Our analysis concludes that, within the  $$1 \sigma $$  <math> <mrow> <mn>1</mn> <mi>σ</mi> </mrow> </math>   region, a significant portion of the BH-I and BH-II parameter space agrees with the EHT results of M87* and Sgr A*. The possibility of LMRBH being a BH-III with three horizons has been almost ruled out, except for a small portion of parameter space, by  $$\theta _{sh}$$  <math> <msub> <mi>θ</mi> <mrow> <mi>sh</mi> </mrow> </msub> </math>   bounds of Sgr A* and M87* measured by EHT. In contrast, NH without a horizon is completely ruled out. We discover that the EHT results of Sgr A* place more stringent limits on the parameter space of LMRBH black holes than those established by the EHT results of M87*.

India

Jitendra Kumar

South Africa

Shafqat Islam

Sushant Ghosh

Strong gravitational lensing by loop quantum gravity motivated rotating black holes and EHT observations

We study gravitational lensing in strong-field limit by a static spherically symmetric black hole in quartic scalar field Horndeski gravity having additional hair parameter q, evading the no-hair theorem. We find an increase in the deflection angle  $$\alpha _D$$  <math> <msub> <mi>α</mi> <mi>D</mi> </msub> </math>  , photon sphere radius  $$x_{ps}$$  <math> <msub> <mi>x</mi> <mrow> <mi>ps</mi> </mrow> </msub> </math>  , and angular position  $$\theta _{\infty }$$  <math> <msub> <mi>θ</mi> <mi>∞</mi> </msub> </math>   that increases more quickly while angular separation s more slowly, but the ratio of the flux of the first image to all other images  $$r_{mag}$$  <math> <msub> <mi>r</mi> <mrow> <mi>mag</mi> </mrow> </msub> </math>   decreases rapidly with increasing magnitude of the hair q. We also discuss the astrophysical consequences in the supermassive black holes at the centre of several galaxies and note that the black holes in Horndeski gravity can be quantitatively distinguished from the Schwarzschild black hole. Notably, we find that the deviation  $$\Delta \theta _{\infty }$$  <math> <mrow> <mi>Δ</mi> <msub> <mi>θ</mi> <mi>∞</mi> </msub> </mrow> </math>   of black holes in Horndeski gravity from their general relativity (GR) counterpart, for supermassive black holes Sgr A* and M87*, for  $$q=-0.2$$  <math> <mrow> <mi>q</mi> <mo>=</mo> <mo>-</mo> <mn>0.2</mn> </mrow> </math>  , respectively, can reach as much as  $$2.4227~\mu $$  <math> <mrow> <mn>2.4227</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as and  $$1.82026~\mu $$  <math> <mrow> <mn>1.82026</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as while  $$\Delta s$$  <math> <mrow> <mi>Δ</mi> <mi>s</mi> </mrow> </math>   is about  $$0.04650~\mu $$  <math> <mrow> <mn>0.04650</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as for Sgr A* and  $$0.03493~\mu $$  <math> <mrow> <mn>0.03493</mn> <mspace width="3.33333pt"></mspace> <mi>μ</mi> </mrow> </math>  as for M87*. The ratio of the flux of the first image to all other images suggest that the Schwarzschild images are brighter than those of the black holes in Horndeski gravity, wherein the deviation  $$|\Delta r_{mag}|$$  <math> <mrow> <mrow> <mo>|</mo> <mi>Δ</mi> </mrow> <msub> <mi>r</mi> <mrow> <mi>mag</mi> </mrow> </msub> <mrow> <mo>|</mo> </mrow> </mrow> </math>   is as much as 0.70673. The results suggest that observational tests of hairy black holes in Horndeski gravity are indeed feasible. Taking the supermassive black holes Sgr A* and M87* as the lens, we also compare our hairy Horndeski black holes observable signatures with those of the neutral Horndeski black holes, Galileon black holes and charged Horndeski black holes. It turns out that although it is possible to detect some effects of the strong deflection lensing by the hairy Horndeski black holes and other black holes with the Event Horizon Telescope (EHT) observations, but it is unconvincing to discern these black holes as deviations are  $${\mathcal {O}}(\mu $$  <math> <mrow> <mi>O</mi> <mo>(</mo> <mi>μ</mi> </mrow> </math>  as). We also find that the shadow size is consistent with EHT observation if the deviation parameter  $$q \in (-0.281979,0)$$  <math> <mrow> <mi>q</mi> <mo>∈</mo> <mo>(</mo> <mo>-</mo> <mn>0.281979</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </math>

Investigating strong gravitational lensing effects by supermassive black holes with Horndeski gravity

Among the higher curvature gravities, the most extensively studied theory is the so-called Einstein–Gauss–Bonnet (EGB) gravity, whose Lagrangian contains Einstein term with the GB combination of quadratic curvature terms, and the GB term yields nontrivial gravitational dynamics in  $$ D\ge 5$$  <math> <mrow> <mi>D</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math>  . Recently there has been a surge of interest in regularizing, a  $$ D \rightarrow 4 $$  <math> <mrow> <mi>D</mi> <mo>→</mo> <mn>4</mn> </mrow> </math>   limit of, the EGB gravity, and the resulting regularized 4D EGB gravity valid in 4D. We consider gravitational lensing by Charged black holes in the 4D EGB gravity theory to calculate the light deflection coefficients in strong-field limits  $$\bar{a}$$  <math> <mover> <mrow> <mi>a</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </math>   and  $$\bar{b}$$  <math> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </math>  , while former increases with increasing GB parameter  $$\alpha $$  <math> <mi>α</mi> </math>   and charge q, later decrease. We also find a decrease in the deflection angle  $$\alpha _D$$  <math> <msub> <mi>α</mi> <mi>D</mi> </msub> </math>  , angular position  $$\theta _{\infty }$$  <math> <msub> <mi>θ</mi> <mi>∞</mi> </msub> </math>   decreases more slowly and impact parameter for photon orbits  $$u_{m}$$  <math> <msub> <mi>u</mi> <mi>m</mi> </msub> </math>   more quickly, but angular separation s increases more rapidly with  $$\alpha $$  <math> <mi>α</mi> </math>   and charge q. We compare our results with those for analogous black holes in General Relativity (GR) and also the formalism is applied to discuss the astrophysical consequences in the case of the supermassive black holes Sgr A* and M87*.