E that for black holes, the ergosphere extends towards the rotation (symmetry) axis, but this is not the case for naked singularities.two 1 y 0 -1 -2 -2 -1 0 x 2 1 y 0 -1 -2 -2 -1 0 x 1 a = 0.five 2 y 2 1 0 -1 -2 -2 -1 0 x 1 a = 0.9 2 y 1 a = 0.5 2 y two 1 0 -1 -2 -2 -1 0 x 2 1 0 -1 -2 -2 -1 0 x 1 a=1 two y 1 a = 0.9 two y two 1 0 -1 -2 -2 -1 0 x two 1 0 -1 -2 -2 -1 0 x 1 a = 1.1 2 1 a=1 2 y two 1 0 -1 -2 -2 -1 0 x 1 a = 1.1(Z)-Semaxanib Purity & Documentation figure 1. The ergosphere extension, represented by the equatorial as well as the meridional sections, is provided for Kerr black holes and Kerr naked singularity.2.2. Test Particle Motion and Locally Non-Rotating Frames Motion of (uncharged) test particles possessing rest mass m is governed by the geodesic equation Dp=0 (13) D complemented by the normalization situation pp= (14)where pis the particle four-momentum and = -m2 for huge particles, when = 0 for massless particles. Two Killing vector fields of your Kerr geometry, /t, and /, imply the existence of conserved power E and axial angular momentum L. As all of the particles are dragged by the rotating spacetime, it’s helpful to figure out limits on the angular velocity = d/dt of the orbiting matter (fixed at a given radius r r )–the limits correspond towards the motion of photon within the sense of rotation and in the opposite one. We therefore discover that the angular velocity of any circulating particle has to be limited by the interval – (15) exactly where the restricting angular velocities are given by the relation = – gt g gt g-gtt . g(16)Universe 2021, 7,five ofNow we are able to directly see that we can define within the Kerr geometry the notion in the locally non-rotating frames (LNRF), related towards the zero angular momentum observers (ZAMO) with axial angular momentum L = 0, and four-velocityt uLNRF = (uLNRF , 0, 0, uLNRF ), t (uLNRF )two =(17) – gtt g gt 2ar . (18) LNRF (r, ) = – = 2 g (r a2 )2 – a2 sin22 gtg,t uLNRF = LNRF uLNRF ,The LNRFs (ZAMOs) four-velocity are well defined above the horizon (r r ) in the black hole case and for all radii within the naked singularity case and are corotating together with the Kerr spacetime at fixed coordinates r and . The ZAMOs represent a generalization of the static observers within the Schwarzschild spacetime–this is usually effectively demonstrated by the fact that the particles falling from rest at infinity remain purely radially falling relative to static observers within the Schwarzschild spacetimes and relative to LNRFs within the Kerr spacetimes ; for the principal null congruence (PNC) photons, i.e., purely radially moving photons, this property is, in Kerr spacetimes, realized in the Carter frames that differ slightly in comparison towards the LNRFs . Introducing the abbreviation A = (r2 a2 )2 – a2 sin2 the orthonormal tetrad of your LNRFs might be introduced as follows  r t r (19)= = = =/, 0, 0 , 0, 0, , 0 , /A, 0, 0, 0 , -LNRF A/ sin , 0, 0, A/ sin . 0,(20) (21) (22) (23)The three-velocity of a particle obtaining four-velocity U has inside the LNRFs the elements vi given by the relation vi = U U (i ) = (t) U (t) U (i )(24)where i = r, , . For the circular geodesic orbits, the only non-zero (axial) component reads  M1/2 (r2 2aM1/2 r1/2 a2 ) ; (25) v = 1/2 (r3/2 aM1/2 ) the upper sign determines the initial household orbits (purely corotating in the black hole spacetimes), even though the lower sign determines generally the counter-rotating orbits. Recall that the first loved ones steady circular orbits can become counter-rotating relative to the LNRFs (having L 0) about naked Alvelestat Protocol singularities having a three 3/4.