Abstract
In this paper we argue that the well-known maximal extensions of the Kerr and Kerr-Newman spacetimes characterized by a specific gluing (on disks) of two asymptotically flat regions with ADM masses of opposite signs are physically inconsistent and actually non-analytic. We also discover a correct geometrical interpretation of the surface $r=0$, $t={\rm const}$ - a dicone in the case of the Kerr solution and a more sophisticated surface of non-zero Gaussian curvature in the case of the Kerr-Newman solution - which suggests that the problem of constructing the maximal analytic extensions for these stationary spacetimes is likely to be performed within the models with only one asymptotically flat region, in which case a smooth crossing of the ring singularity becomes possible, for instance, after carrying out an appropriate transformation of the radial coordinate.
Highlights
Unlike the maximal analytic extensions (MAEs) of the Schwarzschild [1, 2] and Reissner-Nordstrom [3,4,5,6] metrics in which the range of the radial coordinate r is restricted to non-negative values only, the MAEs of the Kerr [7, 8] and Kerr-Newman (KN) [9,10] spacetimes involve both positive and negative values of r
The procedure of the continuation of r into negative values is well described in the classical books on general relativity [11, 12] and leads to appearance of the second asymptotically flat region of negative ADM [13] mass, provided the mass of the first asymptotically flat region is positive definite
It is commonly thought that the specific gluing of the two asymptotically flat regions employed in those MAEs is smooth, a recent study of the Kerr and KN solutions endowed with negative mass [14, 15] has revealed, that the curvature singularities in the negative-mass case are massless and in addition are located outside both the symmetry axis and the mass distributions
Summary
Unlike the maximal analytic extensions (MAEs) of the Schwarzschild [1, 2] and Reissner-Nordstrom [3,4,5,6] metrics in which the range of the radial coordinate r is restricted to non-negative values only, the MAEs of the Kerr [7, 8] and Kerr-Newman (KN) [9,10] spacetimes involve both positive and negative values of r. An additional undesirable feature of the known MAEs of the Kerr and KN solutions is that in the static limit they do not reduce straightforwardly to the MAEs of static spacetimes with only one asymptotically flat region It is commonly thought (but has never been proved) that the specific gluing of the two asymptotically flat regions employed in those MAEs is smooth, a recent study of the Kerr and KN solutions endowed with negative mass [14, 15] has revealed, that the curvature singularities in the negative-mass case are massless and in addition are located outside both the symmetry axis and the mass distributions. The main result obtained by us is demonstrating that, in an appropriate coordinate system, a KN spacetime can be analytically continued from one hemisphere to another through the part of the equatorial plane encircled by the ring singularity which means that the ring singularity is traversable
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