Shaving off soft hairs and the black hole image memory effect

Abstract

Soft hairs of black holes are the Noether charges associated to the generalized Bondi-Metzner-Sachs symmetries. In this work, the images of soft-haired Kerr black holes were studied. For an eternal black hole, the image is rotated, dilated and drifting compared to that of the bald counterpart in the celestial plane. The rotation and the dilation are independent of the time, while the drifting is at a constant speed and in a fixed direction. These effects all depend on angular directions. The soft hair of an astronomical black hole can change due to the emission of the gravitational or electromagnetic wave from the various physical processes occurring in the vicinity of the horizon. Then, the image roams in the observer's view, causing the image memory effect, the smoking gun for the existence of the soft hair. The magnitude of the image memory effect of a huge, spinning black hole accompanied by a much smaller one was estimated. It turns out that this effect is proportional to the mass of the large black hole, increases with its spin, but descreases with the mass ratio. Due to the limited angular resolution of the current and the future detectors, this effect is hard to be detected, if the impact of the cosmological expansion is ignored.

Figures (4)

• Figure 1: The impacts of the soft hair on the image of an eternal black hole, represented by the changes in the shadow of a highly spinning black hole, illuminated by a huge planar light source. Left: the rotation ${\mathcal{R}}^{\hat{i}}_{\hat{j}}$ of the image; middle: the dilation $e^{\mathcal{W}}$ of the image; right: the shifting ${\mathcal{S}}^{\hat{i}}$ of the image at the constant velocity ${\mathcal{V}}^{\hat{i}}$.
• Figure 2: The image memory effect. The red line on the left represents the trajectory before the emission of the radiation, while the magenta line on the right represents the trajectory after the emission. During the emission, the image accelerates, and its path is schematically represented by the brown dotted curve. The "D" shaped areas stand for the black hole shadows at different places. And the blue arrows represent the components $\Delta {\mathcal{I}}^{\hat{i}}$ of the image memory effect.
• Figure 3: The estimation of the image memory effect for an intermediate mass ratio inspiral. The central black hole has a spin $a=0.8$ and the mass ratio is fixed to be $q=M_1/M_2=100$. They are at the distance $1\,\text{Gpc}$ from the earth. The inset displays the details of the image memory effect during the last $20 \frac{M_1}{10^{12}M_\odot}$ yrs, corresponding to the small red rectangle at the top right corner.
• Figure 4: The total $|\Delta\bm{\mathcal{I}}^\text{tot}_{20}|$ as the functions of $a$ and $q$ for the different values of $M_1$. The left panel shows $|\Delta\bm{\mathcal{I}}^\text{tot}_{20}|\text{ v.s. }a$ at $q=100$, while the right panel displays $|\Delta\bm{\mathcal{I}}^\text{tot}_{20}|\text{ v.s. }q$ at $a=0.8$.