Generalized Extended Uncertainty Principle Black Holes: Shadow and Lensing in the Macro- and Microscopic Realms

Abstract

Motivated by the recent study about the extended uncertainty principle (EUP) black holes, we present in this study its extension called the generalized extended uncertainty principle (GEUP) black holes. In particular, we investigated the GEUP effects on astrophysical and quantum black holes. First, we derive the expression for the shadow radius to investigate its behavior as perceived by a static observer located near and far from the black hole. Constraints to the large fundamental length scale, L*, up to two standard deviations level were also found using the Event Horizont Telescope (EHT) data: for black hole Sgr. A*, L*=5.716×1010 m, while for M87* black hole, L*=3.264×1013 m. Under the GEUP effect, the value of the shadow radius behaves the same way as in the Schwarzschild case due to a static observer, and the effect only emerges if the mass, M, of the black hole is around the order of magnitude of L* (or the Planck length, lPl). In addition, the GEUP effect increases the shadow radius for astrophysical black holes, but the reverse happens for quantum black holes. We also explored GEUP effects to the weak and strong deflection angles as an alternative analysis. For both realms, a time-like particle gives a higher value for the weak deflection angle. Similar to the shadow, the deviation is seen when the values of L* and M are close. The strong deflection angle gives more sensitivity to GEUP deviation at smaller masses in the astrophysical scenario. However, the weak deflection angle is a better probe in the micro world.

Testing symmergent gravity through the shadow image and weak field photon deflection by a rotating black hole using the M87$$^*$$ and Sgr. $$\hbox {A}^*$$ results

“…Authors in Ref. [55] found constraints to L * that if uncertainties are ignored, L * = 7.950x10 13 m for the shadow radius of M87*, R sh = 5.5M .…”

“…For instance, the effect of dark matter halo on the black hole shadow was studied [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. The possibility of gaining information on the quantum nature of black holes was also considered [52][53][54][55][56][57][58][59]. These studies are only a few that we mention from the vast literature, proving the importance of such studies which we cannot underestimate with how fast our space technology evolves.…”

A critique on some aspects of GUP effective metric

“…( 16), but one will need more justification for such a choice. 8 Similar approaches were used in other generalized/extended version of uncertainty principles [11][12][13].…”

A Critique on Some Aspects of GUP Effective Metric

“…This result is strange because it means that as the black hole becomes very small, the shadow is in fact becoming large. 8 Similar approaches were used in other generalized/extended version of uncertainty principles [11][12][13]. 9 Ref.…”