The rest frame in stationary space−times with axial symmetry

Abstract

A harmonic space approach to quantum gravity of stationary space-times with SO(3) symmetry A rest frame in a stationary, axially symmetric space-time is defined as a synchronizable frame which is as nearly Killing as possible. This is a possible generalization of the Newtonian rest frame. A kinematical theorem giving the condition for the existence of a rest frame whose time vector is a linear combination of the Killing vectors is proved. The frame is also unique. The condition is shown to be weaker than the assumption of orthogonal transitivity. The surfaces of simultaneity of the rest frame are the surfaces of constancy of a particular Killing time coordinate, and its time vector is the component of the time Killing vector orthogonal to the angular Killing vector. Some properties of the rest frame are then discussed; it is shown that the frame is well-behaved down to the event horizon, where its time vector becomes null. Under a suitable condition on the event horizon, the time vector coincides with a Killing vector there. The gravitational redshift relation in the rest frame is derived. There is a dependence on the angular momentum of the geodesic. Furthermore, the event horizon is shown to be an infinite redshift surface for the rest frame observers. Finally, the three-vector potential of Landau and Lifshitz is interpreted and shown to be closely related to the rest frame, and a corresponding four-vector potential is invariantly defined.Lemma: If the vector field w a is never null in some region, then w a = (w B W B)-l w a satisfies W[a,Bl = 0if and only if

Einstein equations and inertial forces in axially symmetric stationary spacetimes

“…As has been shown by Greene et al [20], axially symmetric stationary spacetimes with orthogonal transitivity admit a globally hyper surface orthogonal timelike vector field,…”

“…Physically, these parameters represent the convective flows in the spacetime geometry. The necessary and sufficient condition for the infinitesimal two-surfaces orthogonal to the infinitesimal two-surface formed by the Killing vectors are surface forming is given by C 0 = C 1 = 0, and the Killing vector fields are said to satisfy the condition of orthogonal transitivity [18,20]. Equations (13) and (14) are equivalent to the Einstein field equations for spacetimes with two Killing vectors.…”

“…This vector filed defines the global rest frame, a generalisation of global inertial frame [20]. The inertial forces are define with respect to this global rest frame.…”

Carter constant and angular momentum

“…In SASS, the time like-Killing vector ξ a and rotational Killing vector η a are surface forming [20]. However, they are not orthogonal, i.e., ξ a η a = 0.…”

“…As ξ a and η a are not orthogonal, we span the same surface with the vector field χ and η. The vector field χ a is not a Killing vector field and referred as quasi-Killing vector field [20]. It has several interesting properties, it is hypersurface orthogonal and can be taken as frame closest to the Newtonian global rest frame in SASS [21].…”