Comment on "Zero mass limit of Kerr spacetime is a wormhole"

TL;DR

This paper critiques a previous study by Gibbons and Volkov, arguing that their conclusions about the Kerr spacetime's zero mass limit are based on misinterpretations of the surface at r=0.

Contribution

It clarifies the correct geometric interpretation of the r=0 surface in Kerr spacetimes, challenging prior claims about the zero mass limit being a wormhole.

Findings

The surface r=0 in Kerr spacetime is not a wormhole as previously claimed.

Erroneous interpretation of the r=0 surface led to incorrect conclusions.

Correct geometric analysis refutes the wormhole interpretation in the zero mass limit.

Abstract

It is argued that the main results of the paper of Gibbons and Volkov (Phys. Rev. D {\bf 96}, 024053 (2017)) come from erroneous interpretations of the surface $r=0$ in the Kerr and Kerr-(anti)de Sitter spacetimes.

Full text

In a recent paper of Gibbons and Volkov GVo it has been claimed that a specific configuration involving two copies of a Minkowski space arises as the zero mass limit of the Kerr solution Ker extended to the negative values of the radial coordinate $r$ in the manner of Boyer and Lindquist BLi (according to the book HEl , for instance, the latter extension has two asymptotic regions – of positive and negative mass – connected on the disks $r=0$). A similar “wormhole” interpretation has also been given in GVo to the zero mass limit of the extended Kerr–(anti)-de Sitter solution Car whose asymptotic regions are glued by means of the disks $r=0$ too.

Apparently, the authors of GVo overlooked some novel results on the non-disk geometry of the surface $r=0$ in the Kerr and Kerr–de Sitter spacetimes obtained and discussed recently in the papers GMa ; MGa ; GMa2 . In the first of these papers the surface $r=0$ of the Kerr solution was shown to be a dicone (see Fig. 1), while in the second paper the surface $r=0$ of the Kerr–(anti)-de Sitter solution was found to be a concave or a convex dicone of constant Gaussian curvature $K=\Lambda/3$, $\Lambda$ being a cosmological constant (see Fig. 2, and also figures 2 and 3 of MGa ); in the third paper the specific role of quasi-Cartesian coordinates in creating a misleading image of the surface $r=0$ in the original papers on the analytical extensions of the black-hole spacetimes was clarified. This shows in particular that the maximal analytic extension of the Kerr metric considered by Boyer and Lindquist in BLi , which treats the surface $r=0$ as a disk, is actually wrong. The correct extension of the Kerr or Kerr–(anti)-de Sitter solution consists in combining the interior region of the dicone with the main manifold $r>0$, thus getting in each case a topologically trivial spacetime with a single asymptotic region depicted in Fig. 3. Of course, the zero mass limit of such extended solutions with one asymptotic region will be a Minkowski or an (A)dS space, respectively.

It follows from the above said that the extensions of the Kerr and Kerr–(anti)-de Sitter spacetimes with two asymptotic regions of positive and negative mass considered in the paper GVo actually do not exist as legitimate mathematical or physical models, and hence there is no any sense in considering their zero mass limit. It should be also noted that the analytical cross-gluing on the disks considered in the paper GVo is physically unrealizable, being just a mathematical abstraction, as the readers themselves can easily check for instance by making identical cuts on two sheets of paper and then trying to cross-glue them. At the same time, the abstract locally Minkowskian (or (A)dS) spacetime of nontrivial topology with two asymptotic regions certainly does exist as a purely mathematical construction independently of the work GVo , and everyone can readily construct it as a mental experiment from two copies of a Minkowski (or (A)dS) space. Moreover, using the results of the paper GMa , it is trivial to establish the true relation of such spacetimes to the Kerr and Kerr–(anti)-de Sitter solutions. Indeed, as was observed in GMa , several identical copies of the analytically extended Kerr spacetime (i.e., of the one shown in Fig. 3) can give rise to the more sophisticated analytic extensions of the Kerr solution characterized by several asymptotic regions and nontrivial topology (such mathematical constructions however will have a similar intrinsic “gluing” problem as the one arising during the construction of the Riemann surface of the analytical function $\sqrt[n]{z}$, $n\geq 2$). In the case of two identical copies, after making the cuts in the equatorial plane inside the singular rings (these cuts differ from the $r=0$ surface!) and gluing mentally different sides of the cuts of different copies, one readily gets a mathematical construction whose zero mass limit is exactly the locally Minkowskian space with two asymptotic regions considered in GVo . (The Kerr–(anti)-de Sitter case is treated similarly.)

Acknowledgments

The author is grateful to Eduardo Ruiz for interesting discussions. This work was supported by the CONACyT of Mexico.