Relating follicly-challenged compact stars to bald black holes: A link between two no-hair properties

Compact stars satisfy certain no-hair relations through which their multipole moments are given by their mass, spin and quadrupole moment. These relations are approximately independent of their equation of state, relating pressure to density. Such relations are similar to the black hole no-hair theorems, but these possess event horizons inside which information that led to their formation can hide. Compact stars do not possess horizons, so whether their no-hair relations are related to the black hole ones is unclear. We investigate how the two relations are related by studying relations among multipole moments for compact stars with anisotropic pressure as a toy model, which allows such stars to be more compact than those with isotropic pressure. We here show numerically that the compact star no-hair relations approach the black hole ones as the compactness approaches that of a black hole. We also prove analytically that the current dipole moment exactly reaches the black hole limit quadratically in compactness as strongly anisotropic stars approach the black hole limit. We moreover show that compact stars become progressively oblate in this limit, even if prolate at low compactness due to strong anisotropies.

Figure 1

Compactness dependence of ${\overline{M}}_2$ for incompressible strongly anisotropic compact stars. The black hole value of ${\overline{M}}_2$ is shown with a cross on the top right corner. Although the compactness of anisotropic stars with ${\lambda }_{\mathrm{BL}}$ chosen here does not actually reach the black hole limit, as one would find by zooming to the $C\sim 0.5$ region, it does get very close to it.

Figure 2

${\overline{S}}_1\text{−}{\overline{M}}_2$ (top left) and ${\overline{S}}_3\text{−}{\overline{M}}_2$ (bottom left) relations for incompressible anisotropic stars. The right panel zooms in to the black hole limit, shown with crosses. The arrows indicate the direction of increasing compactness, with the solid ones corresponding to stars with ${\lambda }_{\mathrm{BL}}\ge -0.8\pi$ and the dashed one for stars with ${\lambda }_{\mathrm{BL}}<-0.8\pi$. The solid line in the bottom left panel represents the analytic Newtonian relation, which is independent of ${\lambda }_{\mathrm{BL}}$.