Testing Loop Quantum Gravity from Observational Consequences of Nonsingular Rotating Black Holes

The lack of rotating black hole models, which are typically found in nature, in loop quantum gravity (LQG) substantially hinders the progress of testing LQG from observations. Starting with a nonrotating LQG black hole as a seed metric, we construct a rotating spacetime using the revised Newman-Janis algorithm. The rotating solution is nonsingular everywhere and it reduces to the Kerr black hole asymptotically. In different regions of the parameter space, the solution describes (1) a wormhole without event horizon (which, we show, is almost ruled out by observations), (2) a black hole with a spacelike transition surface inside the event horizon, or (3) a black hole with a timelike transition region inside the inner horizon. It is shown how fundamental parameters of LQG can be constrained by the observational implications of the shadow cast by this object. The causal structure of our solution depends crucially only on the spacelike transition surface of the nonrotating seed metric, while being agnostic about specific details of the latter, and therefore captures universal features of an effective rotating, nonsingular black hole in LQG.

Figure 1

The spacetime structure of the rLQGO metric (5) with respect to the parameter space $\{a,A_{\lambda }\}$. In regions I, II, and III, the transition surface ($y=0$) is located outside the outer horizon, between the inner and outer horizons, and inside the inner horizon, respectively. On the red (blue) curve, the transition surface is located on the outer (inner) horizon.

Figure 2

The embedding diagram of rLQGO. (a) A timelike wormhole without horizon (region I). (b) A spacelike transition surface inside the event horizon (region II). (c) The transition surface is inside the inner horizon (region III).

Figure 3

The Penrose diagram of rLQGO in (a) region I and (b) region II. Blue dashed lines denote the transition surface and slanted red lines, at an angle of 45°, are event horizons.

Figure 4

The Penrose diagram of rLQGO in region III. The colored regions are inside the outer event horizon (red lines). After entering inner horizons (solid blue lines), a timelike trajectory can be extended to another universe either by going upward (trajectory A) without touching the transition surface (dashed blue lines) or by crossing the transition surface to another interior patch (trajectory B). Note that the exterior regions of the two adjacent interior patches can be causally disconnected.

Figure 5

The Ricci scalar $R$ of the rLQGO spacetime expressed in the $(y,\cos \theta )$ plane. In the figure, we set $M_B=1$ and $a/M_B=0.9$. The solid red and blue lines represent the outer and the inner event horizons, respectively. The dashed curves represent the ergosurface. (a) A wormhole without horizon (region I with $A_{\lambda }=0.4$). (b) The transition surface is covered by the outer horizon (region II with $A_{\lambda }=0.1$) (c) The transition surface is inside the inner horizon (region III with $A_{\lambda }=0.01$).

Figure 6

The apparent size $R_S/M_B$ of the shadow cast by rLQGO is shown with respect to the parameter space $\{a,A_{\lambda }\}$. The 17% bound of $R_S/M_B$ (black curve) inferred from the M87* shadow disfavors parameters corresponding to the wormhole geometry (the region to the right of the red curve). Here we have taken into account the spin measurement obtained using the radio intensity data [45] (cyan lines) and the inclination angle measured by the jet direction [46].

Figure 7

The apparent size $R_S/M_B$ of the shadow cast by the rotating black hole metric, corresponding to [6, 7], in the parameter space of $\{a,\tilde{\mathrm{\Delta }}\}$. The black boundary represents a spin-dependent upper bound of the quantum parameter, above which shadow contours disappear.