3. Potential for the Kerr field
Relativistic gravitational fields can be interpreted as consisting of three parts: the Newtonian (also Coulomb or gravitoelectric) component, the dragging (gravitomagnetic) component, and the space-geometry component. The Newtonian component is generated by mass-density and is given essentially by the gradient of the component of the spacetime metric. The dragging component is generated by mass-currents and determined by curl of . The space-geometry component (determined by ) has no classical analog. This view and, in particular, the analogy with electromagnetism are most straightforward within the linearized approximation, but can be given a precise and invariant meaning even in strong fields (Thorne et al. 1986; Jantzen et al. 1992, and references cited therein). Let us compose the potential for the Kerr spacetime in a way that acknowledges this approach.
3.1. The Newtonian component
Cutting the Kerr manifold at , one can obtain the spacetime which is free of causality-violating regions. The cut leads to jumps in derivatives of the metric at which are interpreted as a thin massive layer. This induced mass spreads over the hypersurface and consists of the attractive singular ring (, ) of infinite positive mass density, spanned by the repulsive disc (, ) of negative mass density. Constructing the causally maximal extension of the Kerr metric, Keres (1967) and Israel (1970) showed that the Newtonian field generated by the massive layer corresponds to the potential 2
The structure of this field was shown in Semerák (1995; cf. Fig. 2 there) by depicting its lines in the ()-plane. We will use the Keres-Israel potential as a scalar "seed" of our model. With given by Eq. (10), Eqs. (6)-(8) read
Separated first integrals of these equations (analogy of Carter's equations for the Kerr spacetime) were found by Israel (1970). It was illustrated in Semerák (1996) that Eqs. (11)-(13) often yield trajectories indiscernible from their exact-Kerr counterparts computed from Eqs. (1)-(3). However, they do not contain the terms linear in velocities which embody the very characteristic feature of the Kerr geometry - the frame-dragging effects.
3.2. The dragging component
In this section, we are led by analogy with classical electrodynamics and by observation that the Kerr dipole-like gravitomagnetic field resembles the Kerr-Newman magnetic field. One thus arrives at the potential which incorporates dragging in the form
where and the vector potential is given by that of the Kerr-Newman electromagnetic field, , with Q (electric charge of the Kerr-Newman centre) replaced by (the extra factor of 2 is ascribed to the tensorial character of gravity and the minus sign to its attractive nature) - i.e., . Similar (just half) correction for the dragging was considered by Dadhich (1985) for a special case of particles with zero axial angular momentum. The added term really introduces the desired dragging terms into the Eqs. (11)-(13): they read now
Note that the contravariant 3-potential obtained by using the inversion of the Kerr 3-metric, , is
where and ; equals the shift vector of Thorne et al. (1986) which stands for the potential of the gravitomagnetic field in this reference.
3.3. The space-geometry component
The only point in which Eqs. (20) and (21) differ from the relativistic Eqs. (1)-(2) is that the relativistic variable - given by (E and stand for the particle's energy and axial angular momentum at infinity) - is replaced by m in the classical model. This distinction reflects, however, the conflict between the very roots of classical physics (where time parameter is universal) and relativity (where proper time and some coordinate time occur, related to each other in a specific way at each point). Notice that Eq. (22) differs from (3) also in several other points.
3.4. Specific features of motion in the pseudo-Kerr potential
where . The Kerr axial angular momentum at infinity contains (again, as in Eqs. (20)-(21)) instead of m ( is the angular velocity with which the field rotates relative to an observer standing at infinity). For large r one obtains the usual forms of the energy and axial angular momentum in the Newtonian central gravitational field:
Let us determine the acceleration of an observer orbiting uniformly () at , . We will define acceleration as a specific force necessary to keep the observer in the orbit, i.e. as the minus acceleration of a free particle having at a given point. According to Eqs. (20)-(22), we find
This 3-acceleration differs from the space part of 4-acceleration of the Kerr stationary observer only by the absence of the multiplicative factor (square of the time-component of the observer's 4-velocity) [cf. Semerák 1993, Eqs. (37)-(39)].
Hence, the field is repulsive at in the sense that the radial component of (30) is negative there.
According to (27)-(29), the stationary observer needs no thrust (i) at on the axis (), and (ii) in the equatorial plane if his angular velocity is . The latter agrees exactly with the Keplerian angular velocity of an equatorial observer orbiting along a circular geodesic in the Kerr spacetime.
Important circular geodesics in the equatorial plane, on the other hand, are not reproduced properly by the potential (19), viz. the equation for marginally stable orbits reads
and for marginally bound orbits
The correct equations have respectively the forms
In the Schwarzschild case, for instance, both our equations imply .
In any pseudo-Newtonian description of a rotating black hole, the main difficulty is to simulate the presence of the horizon and of the dragging effects with an acceptable precision. Both these phenomena are outside the scope of Newtonian physics. Our potential accounts for the frame-dragging effects, and especially in the intermediate and large distances provides a good fit, whereas it does not describe correctly the innermost region where the horizon and important circular orbits lie. In order to account for the horizon, one would have to start from the scalar potential which diverges to there. The Keres-Israel potential (10), instead, reaches only at the very singularity . The potential proposed by Artemova et al. (1996) is a better alternative in this respect, being the simplest generalization of the Paczyski-Wiita potential to the Kerr case which reproduces the horizon and approximates the important orbits. Also, the epicyclic frequency of small radial oscillation , important in the theory of discoseismology, is not reproduced with good accuracy by potentials (10) and (19) although it does show a maximum, typical for a relativistic . Apparently, trying to incorporate different manifestations of the frame-dragging accurately, one may end with a long expression without practical sense.
To summarize, each particular relativistic effect can be well simulated within classical physics, but pseudo-Newtonian modelling of the complete relativistic situation has only a restricted validity. In order to clarify the value of our potential, we have carried out an extensive computation of trajectories and introduced a criterion which determines the accuracy of the pseudo-Newtonian model.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999