Spin precession in a black hole and naked singularity spacetimes

Abstract

We propose here a specific criterion to address the existence or otherwise of Kerr naked singularities, in terms of the precession of the spin of a test gyroscope due to the frame dragging by the central spinning body. We show that there is indeed an important characteristic difference in the behavior of gyro spin precession frequency in the limit of approach to these compact objects, and this can be used, in principle, to differentiate the naked singularity from black hole. Specifically, if gyroscopes are fixed all along the polar axis upto the horizon of a Kerr black hole, the precession frequency becomes arbitrarily high, blowing up as the event horizon is approached. On the other hand, in the case of naked singularity, this frequency remains always finite and well-behaved. Interestingly, this behavior is intimately related to and is governed by the geometry of the ergoregion in each of these cases which we analyze here. One intriguing behavior that emerges is, in the Kerr naked singularity case, the Lense-Thirring precession frequency (ΩLT) of the gyroscope due to frame-dragging effect decreases as (ΩLT ∝ r) after reaching a maximum, in the limit of r = 0, as opposed to r −3 dependence in all other known astrophysical cases.

Distinguishing Kerr naked singularities and black holes using the spin precession of a test gyro in strong gravitational fields

“…We find that the divergence of the spin precession frequency on the ergosurface reported in Ref. [3] can be avoided if the test gyro moves with a non-zero angular velocity Ω. In Ref.…”

“…We now consider the full timelike Killing vector of the Kerr space-time and study spin precession for observers with u, that is stationary observers. This is in contrast to [3], where the velocity was chosen to be proportional ∂ 0 and these described static observers. Therefore, for a general stationary spacetime which also possess a spacelike Killing vector we can write the most general timelike Killing vector as,…”

“…For BHs, the dimensionless Kerr parameter satisfies a * = J/M 2 ≤ 1. In another work [3], the spin precession frequency was discussed in detail in the case of NS (a * > 1). For NS, the entire region between the inner and outer ergo radii is defined to be the ergoregion.…”

“…In Ref. [3], the motivation was to study the precession of spins of gyroscopes attached to stationary observers, that is, to answer physical questions like 'how the gyroscope of an astronaut holding his spaceship at a constant distance from a Kerr compact object will behave.' The four-velocity (u) of 'static gyros' on the ergosurface satisfies u.u = 0, that is, it becomes null.…”

“…For NS, a divergence in the precession frequency occurs at the singularity itself, which is present in the equitorial plane. In [3], the precession frequency of gyroscopes attached to static observers was considered, namely observers that do not change their spatial position with time. The velocity vectors of these observers are proportional to ∂ 0 and we found that the precession frequency for static observers located infinitesimally close to the ergosurface became arbitrarily large, in all coordinates, thus invariantly indicating the location of the ergosurface.…”