Particle motions and gravitational waveforms in rotating black hole spacetimes of loop quantum gravity

Abstract

In this article, we investigate the influence of the quantum gravity corrections on the horizons, timelike geodesic motions and the gravitational wave emission in two different rotating black hole spacetimes which are constructed via the revised Newman-Janis algorithm from two spherically symmetric loop quantum gravity black holes. The quantum gravity effect is encoded in the regularization parameter $ξ$ of the holonomy correction, and the constraint range of $ξ$ is provided. For the timelike geodesic motion, we find that when the spin parameter $a$ is small, $ξ$ significantly affects the orbital angular momentum $L$. In equatorial periodic orbits, as $ξ$ increases, the allowed energy range for fixed $L$ also increases, while in generic off-equatorial motion, as $ξ$ increases, the permissible range of the Carter constant which effectively confines trajectories toward the equatorial plane decreases. For the gravitational wave emission, by using a simplified extreme-mass-ratio inspiral model within the leading order post-Newtonian approximation, we compute the gravitational waveforms and show how increasing $ξ$ enhances the waveform deviations, particularly near the event horizon. To summarize, the results in this article preliminarily reveal some universal features that holonomy corrections imprint on potentially observable signatures of rotating black holes.

Figures (9)

• Figure 1: The critical values of $r_c$ and $\xi^{2}_{e}$ as functions of $a$ for given $M$ (which is set to $1$ unless specified). The red dashed line (corresponding to the right axis) shows the critical $r_c$ as a function of $a$. The black line (corresponding to the left axis) shows the critical $xi_c$ as a function of $a$, for each value of $\xi$ in the gray region the rotating LQGBH has an event horizon and a Cauchy horizon. The region below the green dotted line shows the restriction condition given by $\xi^2<-r_c^3/(r_c-2M)$ for a well-defined metric of BH-II.
• Figure 2: The angular momentum $L_{MBO}$ and the radius $r_{MBO}$ of the particles on the MBOs as functions of $\xi$ with $a=0.1,\,0.5,\,0.9$. The right superscripts $P$ and $R$ represent the prograde and retrograde orbits, respectively. The first row is calculated corresponding to BH-I, and the second row corresponds to BH-II. The dotted blue thin line shows the new restriction on $\xi$, which is constructed under the condition that the MBOs exist with given $a/M$.
• Figure 3: The angular momentum $L_{ISCO}$, the energy $E_{ISCO}$ and the radius $r_{ISCO}$ of the particles on the ISCOs as functions of $\xi$ with $a=0.1,\,0.5,\,0.9$ for BH-I. The right superscripts $P$ and $R$ represent the prograde and retrograde orbits, respectively. The dotted blue thin line shows the new restriction on $\xi$, which is determined under the requirement that the ISCOs exist for arbitrary given $a/M$.
• Figure 4: The angular momentum $L_{ISCO}$, the energy $E_{ISCO}$ and the radius $r_{ISCO}$ of the particles on the ISCOs as functions of $\xi$ with $a=0.1,\,0.5,\,0.9$ for BH-II. The right superscripts $P$ and $R$ represent the prograde and retrograde orbits, respectively. The dotted blue thin line shows the new restriction on $\xi$, which is determined under the requirement that the ISCOs exist for arbitrary given $a/M$.
• Figure 5: The allowed region of $E$ as the function of $\xi$ with selected $L$ for the prograde cases in the BH-I (Top) and BH-II (Bottom). From left to right, $a=0.1,\,0.5,\,0.9$.