Systematic study of event horizons and pathologies of parametrically deformed Kerr spacetimes

In general relativity, all black holes in vacuum are described by the Kerr metric, which has only two independent parameters: the mass and the spin. The unique dependence on these two parameters is known as the “no-hair” theorem. This theorem may be tested observationally by using electromagnetic or gravitational-wave observations to map the spacetime around a candidate black hole and measure potential deviations from the Kerr metric. Several parametric frameworks have been constructed for tests of the no-hair theorem. Due to the uniqueness of the Kerr metric, any such parametric framework must violate at least one of the assumptions of the no-hair theorem. This can lead to pathologies in the spacetime, such as closed timelike curves or singularities, which may hamper using the metric in the strong-field regime. In this paper, I analyze in detail several parametric frameworks and show explicitly the manner in which they differ from the Kerr metric. I calculate the coordinate locations of event horizons in these metrics, if any exist, using methods adapted from the numerical relativity literature. I identify the regions where each parametric deviation is unphysical as well as the range of coordinates and parameters for which each spacetime remains a regular extension of the Kerr metric and is, therefore, suitable for observational tests of the no-hair theorem.

Figure 1

Null surface of the central object in the QK metric with a spin of $|a|=0.3M$ for several values of the parameter ${ϵ}_{\mathrm{QK}}$. If ${ϵ}_{\mathrm{QK}}\ne 0$, the null surface is bound by the singularity located at radius $r=2M$, and the central object is a naked singularity.

Figure 2

Location of the BK null surface and the singularities in the $xz$ plane for $a=0$, $|a|=0.5M$, and $|a|=M$. For $a=0$, the singularity $r_{1,+}$ coincides with the null surface, and the singularities $r_{2,+}$ and $r_{2,-}$ are located inside the BK null surface. At $\theta =\pi /2$, the null surface intersects with the singularity $r_{2,+}$. As $|a|$ increases, the singularity $r_{1,+}$ moves outside of the BK null surface. For small but positive values of the spin $|a|$, the null surface is closed if $B_2>0$. Otherwise, it terminates at the singularity $r_{2,+}$. For all values of the deviation parameter $B_2\ne 0$ the object constitutes a naked singularity.

Figure 3

Region of the parameter space of the MK metric where a null surface exists. In this and the shaded regions, for values of the parameter ${ϵ}_3\ne 0$, the central object is a naked singularity located at the Killing horizon, which is of spherical topology. If a null surface exists, the Killing horizon coincides with the null surface at the poles and in the equatorial plane and lies outside of the null surface otherwise. In the blue shaded regions, the Killing horizon is of disjoint topology. Due to numerical uncertainties in the determination of the location of the null surface, I find that a null surface may not exist in this region if ${ϵ}_3\lesssim -8$ or if ${ϵ}_3\gtrsim 0.1$. The black dashed line corresponds to a Kerr black hole.

Figure 4

Different shapes of the Killing horizon in the MK metric for values of the spin $|a|=0.9M$ and deviation parameter ${ϵ}_3$. Left panel: ${ϵ}_3=0.31<{ϵ}_3^{\max \mathrm{\text{−}}\mathrm{eq}}(a)$, where ${ϵ}_3^{\max \mathrm{\text{−}}\mathrm{eq}}(a)$, $a>0$, denotes the boundary between the regions of the parameter space with different topologies of the Killing horizon. The Killing horizon has spherical topology consisting of an inner and outer sphere. The event horizon is located outside of the outer Killing horizon, and the central object is a black hole. Center panel: ${ϵ}_3=0.32\approx {ϵ}_3^{\max \mathrm{\text{−}}\mathrm{eq}}(a)$. The inner and outer Killing horizons merge in the equatorial plane, and the event horizon vanishes. Right panel: ${ϵ}_3=0.33>{ϵ}_3^{\max \mathrm{\text{−}}\mathrm{eq}}(a)$. The Killing horizons about the origin have split into two sphere-like surfaces located above and below the equatorial plane, respectively. In the central and right panels, the central object is a naked singularity located at the Killing horizon. For all values of the parameter ${ϵ}_3$, the origin is likewise singular.

Figure 5

Different shapes of the Killing horizon in the MK metric in the case ${ϵ}_3=-1$ for several values of the spin. Left panel: $|a|=0.9M$. The Killing horizon has spherical topology consisting of an inner and outer spherical surface. Center panel: $|a|=M$. The inner and outer Killing horizons merge at the poles. Right panel: $|a|=1.1M$. The topology of the Killing horizon is toroidal. In all cases, the central object is a naked singularity located at the Killing horizon. For all values of the spin, the origin is likewise singular.

Figure 6

Null surface and regions with Lorentz violations and closed timelike curves in the QK metric for $a=0.3M$. Top panel: ${ϵ}_{\mathrm{QK}}=0.1$; Lorentz violations and closed timelike curves occur around the poles. Bottom panel: ${ϵ}_{\mathrm{QK}}=-0.1$; Lorentz violations and closed timelike curves occur near the equatorial plane.

Figure 7

Outermost radius, at which a pathology occurs in the QK metric, as a function of the deviation parameter ${ϵ}_{\mathrm{QK}}$. This radius depends only very weakly on the value of the spin. Note that I extrapolated this radius to include larger values of the deviation parameter; in this case, the location of the outermost radius is only approximate. In the range of the parameter $-0.5\le {ϵ}_{\mathrm{QK}}\le 0.5$, the outermost radius lies well inside of the ISCO radius (see Ref. 21). The dashed line corresponds to the naked singularity located at the radius $r=2M$.

Figure 8

Null surface and regions with Lorentz violations and closed timelike curves in the BK metric for $a=0.3M$ and $B_2=\pm 0.1$. The null surface is closed if $B_2=0.1$ and open if $B_2=-0.1$. In both cases, Lorentz violations or closed timelike curves occur outside of the null surface and the outermost singularity.

Figure 9

Outermost radius, at which a pathology occurs in the BK metric, as a function of the deviation parameter $B_2$ for several values of the spin $a$. Note that I extrapolated this radius to include larger negative values of the deviation parameter; in this case, the location of the outermost radius is only approximate. The dashed line corresponds to the equatorial singularity located at $r_{1,+}=2M$, which is independent of the spin.

Figure 10

Singularity and regions with Lorentz violations and closed timelike curves in the MN metric for $\chi =0.3$ and $q_{\mathrm{MN}}=\pm 0.1$. Lorentz violations lie inside and closed timelike curves occur both inside and outside of the (singular) boundary that coincides to the event horizon of a Kerr black hole with the same value of the spin. In the black shaded region, the MN metric is imaginary.

Figure 11

Outermost radius, at which a pathology occurs in the MN metric, as a function of the deviation parameter $q_{\mathrm{MN}}$ for several values of the spin $a=\chi M$. The dashed line corresponds to a Kerr black hole.

Figure 12

Inner and outer Killing horizons and regions with closed timelike curves in the MK metric for $a=0.3M$ and ${ϵ}_3=-0.1$. The metric is regular outside of the outer Killing horizon, which is singular and located just outside of the null surface. Regions with closed timelike curves occur only inside the inner Killing horizon.

Figure 13

Event horizon and regions with Lorentz violations and closed timelike curves in the MGBK metric for $a=0.3M$ and ${\gamma }_{1,2}={\gamma }_{3,1}={\gamma }_{3,3}={\gamma }_{4,2}=0.1$. The metric is regular outside of the event horizon, and regions with Lorentz violations and closed timelike curves occur only inside the event horizon.