Traces of quantum gravitational correction at third-order curvature through the black hole shadow and particle deflection at the weak field limit

This study investigates the impact of the quantum-gravity correction at the third-order curvature ($c_6$) on the black hole's shadow and deflection angle on the weak field regime, both involving finite distances of observers. While the calculation of the photonsphere and shadow radius $R_{\rm sh}$ can easily be achieved by the standard Lagrangian for photons, the deflection angle $\alpha$ employs the finite-distance version of the Gauss-Bonnet theorem (GBT). We find that the photonsphere reduces to the classical expression $r_{\rm ph} = 3M$ for both the Planck mass and the theoretical mass limit for BH, thus concealing the information about the applicability of the metric on the quantum and astrophysical grounds. Our calculation of the shadow, however, revealed that $c_6$ is strictly negative and constrains the applicability of the metric to quantum black holes. For instance, the bounds for the mass is $M/l_{\rm Pl} \in [0.192, 4.315]$. We also derived the analytic formula for the observer-dependent shadow, which confirms $c_6$'s influence on quantum black holes even for observers in the asymptotic regions. The influence of such a parameter also strengthens near the quantum black hole. Our analytic calculation of $\alpha$ is shown to be independent of $c_6$ if the finite distance $u \rightarrow 0$, and $c_6$ is not coupled to any time-like geodesic. Finally, the effect of $c_6$ manifests in two ways: if $M^2$ is large enough to offset the small value of $l_{\rm Pl}$ (which is beyond the theoretical mass limit), or if $b$ is comparable to $l_{\rm Pl}$ for a quantum black hole.
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