Twisted Black Hole Is TaubNUT
Abstract
Recently a purportedly novel solution of the vacuum Einstein field equations was discovered: it supposedly describes an asymptotically flat twisted black hole in 4dimensions whose exterior spacetime rotates in a peculiar manner – the frame dragging in the northern hemisphere is opposite from that of the southern hemisphere, which results in a globally vanishing angular momentum. Furthermore it was shown that the spacetime has no curvature singularity. We show that the geometry of this black hole spacetime is nevertheless not free of pathological features. In particular, it harbors a rather drastic conical singularity along the axis of rotation. In addition, there exist closed timelike curves due to the fact that the constant and constant surfaces are not globally Riemannian. In fact, none of these are that surprising since the solution is just the TaubNUT geometry. As such, despite the original claim that the twisted black hole might have observational consequences, it cannot be.
I Twisted Black Holes: What Happens to the Singularity Theorem?
Recently, Zhang zhang showed that general relativity admits the following vacuum solution in 4dimensions (see also a followup work 1610.00886 ):
(1)  
(2) 
This should be compared to the wellknown rotating solution, namely the Kerr metric:
(3)  
(4) 
In the Kerr metric, denotes the rotation parameter where is the (ADM) angular momentum of the black hole. (We use the usual convention that the speed of light is set to unity: .) In the twisted metric, is just a constant; it cannot be interpreted as a rotation parameter in the same manner because is identically zero (see below).
One obvious difference is that some occurrences of in the Kerr metric have been replaced simply with in Zhang’s twisted metric. In particular, the metric coefficient is now independent of . Furthermore, unlike the Kerr black hole event horizon, with is situated at , the twisted solution has an event horizon at instead. Since the latter expression is always real, there is no upper bound on how large the parameter can be. This is not a problem since, as we mentioned, is not a rotation parameter in the usual sense. Also unlike the Kerr black hole which has complicated structures: an event horizon, an inner (Cauchy) horizon, and an ergosphere, the twisted black hole appears to lack all of these but the event horizon. In addition to the event horizon , the denominator of has another zero , which is negative for any nonzero real . In the case of Kerr black hole, analytic continuation to negative is not considered physically realistic. See, e.g., the comment on p.13 in visser . The geometry here is quite different from that of the Kerr solution, but for simplicity let us not consider any further in this work. (See 1610.06135 for further discussion.)
It was shown in zhang that there is no frame dragging on the horizon, i.e., the horizon itself is static. However, the exterior spacetime does experience framedragging (except on the equatorial plane), and the effect is maximum at some finite distance away from the black hole. The upper half spacetime above the equatorial plane rotates in the opposite direction than that of the lower half. Consequently, the ADM angular momentum vanishes identically, though the Komar angular momentum does not zhang .
Unlike the Kerr black hole, the twisted black hole remains a black hole solution even if the ADM mass, , is set to zero. Whether or not, the Kretschmann scalar can be computed explicitly and shown to be finite everywhere zhang . It is, surprisingly, independent of the angles and :
(5)  
in contrast to the Kretschmann scalar of the Kerr black hole 9912320 :
(6) 
As mentioned in zhang , it can been seen from Eq.(5) that the twisted black hole is regular everywhere, in the sense that there is no curvature singularity. This is unlike other vacuum black hole solutions in general relativity that harbors curvature singularity (either timelike or spacelike).
The very absence of a curvature singularity, however, raises a question: how does the gravitational collapse that resulted in the twisted black hole evade the singularity theorem of Penrose penrose ? (See jose for a recent review). Essentially, Penrose proved that if a spacetime contains a noncompact Cauchy hypersurface (“causality condition”) and a closed futuretrapped surface (“trapping condition”), and if the null Ricci condition holds for all null vector , then there exist future incomplete null geodesics. That is to say, light rays hit a singularity in finite affine time.
(The null Ricci condition is better known by the name “null energy condition”, but it is really a geometric condition. See jose , and also parikh .) Note that a vacuum solution such as the twisted black hole satisfies the null Ricci condition trivially because .
Of course, strictly speaking, the Penrose singularity theorem is about singularities in the sense of geodesic completeness; it does not have anything to say about curvature singularities. However, in the case of black holes in general relativity, these two notions of singularity usually coincide (see also Section 5.1.5 of jose and the references therein). In fact, if there exists incomplete timelike or null geodesics, we should ask where and why does the geodesic end physically. For example, a Minkowski spacetime with one point artificially removed is not geodesically complete, but this is not very physical. On the contrary, it is quite natural that geodesics end when something very drastic happens to the geometry, such as when curvature blows up. Therefore, the fact that the twisted black hole has no curvature singularity at least suggests that the geometry should also be geodesically complete. This is therefore in tension with the singularity theorem, unless the geometry of this black hole spacetime contains other pathologies, such as a conical singularity. (A famous example of a spacetime that contains a conical singularity is that of a cosmic string. However, a cosmic string induces a conical defect due to its energy density. The twisted black hole is quite different as it is a vacuum solution.) Indeed,
an important premise in the singularity theorem is that the metric has to be “sufficiently nice”. Technically this means that the metric has to be twice differentiable, i.e., of class . An extension to metric of class , i.e., the first derivative is Lipschitz continuous, has been achieved recently 1502.00287 . Despite the discussion in this paragraph, which motivated our investigation, we will see later on that the twisted spacetime is indeed geodesically incomplete despite it not satisfying the conditions for the singularity theorem to apply – this geodesic incompleteness arises from the spacetime pathology itself.
We will show that the twisted black hole harbors some kind of conical singularity, which means that the metric tensor is not differentiable everywhere, which would explain why the singularity theorem does not apply. Furthermore, we will show that the exterior geometry of the twisted black hole is rather peculiar: the 2dimensional surfaces of constant and constant are not globally Riemannian, with metric signature changing from to sufficiently near the poles. This change of signature is due to the change of sign in the metric coefficient . This indicates that the twisted spacetime harbors closed timelike curves (CTCs), which is yet another reason why the singularity theorem does not apply (the “causality condition” is not satisfied).
If the twisted black hole solution is in fact new to general relativity, then it is a remarkable discovery and we should indeed try to find observational evidences for such a black hole. However, the presence of CTC means that this is probably not a physical solution that one might observe in the real universe. As it turned out, the solution is in fact not new either. If one computes the Petrov type of the twisted black hole, one finds that it is of Petrov type D. However, all asymptotically flat vacuum solutions of Petrov type D are already known kinnersley – there are only 10 of them – and therefore the twisted black hole cannot be a new solution. We show below that it is just the TaubNUT (TaubNewmanUntiTambourino) solution nut1 ; nut2 ; nut3 , which is known to exhibit the aforementioned pathological properties, and belongs to Petrov type D.
As such, the claim of zhang that the twisted black hole might have observational consequences is very unlikely.
Note that the TaubNUT solution is indeed known to be geodesically incomplete (see, e.g., MKG ; 1002.4342 . However, see 1508.07622 ), although it does not have a curvature singularity.
Ii The Peculiar Geometry of Twisted Black Holes
It is easier to focus our discussion on the massless case. As we shall see later, the inclusion of the mass term does not qualitatively affect our conclusions.
For , the event horizon is located at zhang . Let us consider constant and constant surfaces. For the massless black hole, we thus obtain the following 2dimensional geometry:
(7) 
Let us call this surface . The indices emphasize that for a fixed , we have a family of surfaces indexed by .
We start by examining the Gaussian curvature of this 2geometry. As we shall see, the Gaussian curvature across the horizon at the poles is not continuous. The expression for the Gaussian curvature, , of is rather complicated in general and is not particularly illuminating to include here. At the poles, and assuming , the expression simplifies to
(8) 
We provide a plot in Fig.(1). On the horizon itself, i.e., when , the Gaussian curvature is , this is just the Gaussian curvature of a 2sphere of radius . This is as expected because the 2geometry defined by the metric 7 reduces to precisely such a sphere when .
Outside the horizon we find that the Gaussian curvature is positive, and as one approaches the horizon, it diverges: . Inside the horizon, is initially positive near , but becomes negative, and eventually tends to as one approaches the horizon. This shows that is not continuous at the horizon in a very extreme manner, despite being finite on the horizon. This in turn implies that does not have a “nice” geometry at the poles, when .
Let us now examine in details the 2geometry in two regions: the interior spacetime and the exterior spacetime, i.e., inside and outside the event horizon, respectively.
ii.1 Interior Geometry
A conical singularity at some point means that the total angle going around is not equal to . If it is less than , we say that the conical singularity has an angle deficit, whereas if it is more than , then we say that the conical singularity has an angle excess.
Let us take to be the north pole of this geometry (by symmetry everything we discuss below also applies to the south pole). Consider a circle centered at . Then we can compute the quantity “circumference/radius”, which we shall denoted by :
(9) 
In the absence of any conical singularity at the pole, this quantity should reduce to in the limit . However, one finds that
(10) 
At the pole, and only the first term remains. However, as is evident from the expression, the first term is not wellbehaved at the pole:

Inside the horizon, , the term is real and divergent.

Outside the horizon, , the term is imaginary and divergent.
In fact the only place where is welldefined at the pole, is on the horizon itself: there Eq.(9) reduces to , which, in the limit , is exactly . This is, again, as expected, since the 2geometry defined by the metric 7 reduces to a 2sphere of radius precisely when .
Therefore, although the twisted black hole has a regular event horizon without any conical singularity, any surface with has a conical singularity at the pole with an infinite angle excess. One way to visualize this geometry is by plotting its embedding diagram. We equate the metric 7 with
(11) 
so that the surface is isometrically embedded in . We need to solve the differential equation
(12) 
where
(13) 
For the case , one obtains, of course, a sphere embedded in . For however, the integration can only be solved numerically. Fig.(2) compares the region near the pole for (in the regime that is small) to that of case. We see that indeed the geometry is drastically different for the two cases. Note that despite the similarity with the Flamm paraboloid of an asymptotically flat Schwarzschild black hole (with constant and ), the left plot of Fig.(2) is a plot for a constant and constant surface, and only in a small neighborhood of the north pole.
It is interesting to compare this result with the socalled “superentropic black hole” 1504.07529 , which is a kind of asymptotically antide Sitter rotating black hole, whose 2geometry of constant and constant approaches that of a hyperbolic space near the pole. Here, the interior geometry of a massless twisted black hole behaves in an “opposite manner” compared to the superentropic black hole. In the latter case, the 2surface in the embedding diagram becomes that of a “horn” toward the poles. See Fig.(1) of 1504.07529 for the embedding diagram of the horizon geometry. Here, however, the 2surface in the embedding diagram “opens wider” toward the poles (left plot in Fig.(2)). Indeed, a quick calculation will reveal that vanishes at the poles for the superentropic black hole (for all values of ), whereas it diverges for the interior surfaces for the massless twisted black hole. However, the axis through the pole is actually excised from the superentropic black hole spacetime, and the horizon is therefore noncompact. Similarly, we should remove the rotation axis from the twisted geometry, to avoid the pathology of divergent conical angle excess.
One may argue that the interior of the black hole is not so important, after all, exterior observers have no access to any information behind the horizon, and so any pathological feature behind the horizon is not particularly interesting. However, for the twisted black hole, the exterior geometry is even more problematic.
ii.2 Exterior Geometry
The surfaces for cannot be globally embedded in . This in itself is not a problem. We caution the readers that one must be careful drawing any conclusion from embedding diagrams.
Consider, for example, the KerrNewman
black hole. It is wellknown that as the rotation becomes sufficiently fast, , the Gaussian curvature at
the poles becomes negative, and a global isometric embedding into
is no longer possible smarr . (A global embedding in is achievable frolov .) A marked difference between the twisted case and the KerrNewman case is that in the twisted case, for any surface outside the horizon, no global embedding exists.
A natural question to ask at this point is the following: what happens outside the horizon in the neighborhood of the north pole, where the “angle excess” becomes complex? The origin of this “angle excess” is that becomes negative near the pole. Explicitly, the numerator of
(14) 
can a priori be either positive or negative. At any fixed value of , is positive if
(15) 
This implies that for , the angle must be bounded away from 0 in order for to have the “right sign”.
One legitimate concern is that a sign change in might result in a change of the spacetime metric signature from to . We will show later that this does not happen.
For now, let us look into the change of sign in , and why it remains a concern. This for three reasons.
Firstly, for Kerr black holes, any constant and constant surfaces are just ellipsoids. This agrees with the fact that the Kerr metric is a good description of the spacetime around a rotating star. If we insist that the twisted solution should in fact describe what it purported to — two counterrotating halves that is nevertheless static on the horizon, then ideally we would like the constant and constant surfaces to be topologically 2sphere as well, or at least be globally Riemannian, that is, with signature throughout (away from the rotational axis).
Secondly, the frame dragging angular velocity of the twisted black hole is zhang :
(16) 
If can change sign, then by continuity, it can also be zero. This means that the angular velocity will become divergent (the numerator only vanishes on the horizon, or on the equator, so it does not prevent the divergent behavior).
Thirdly, a change of sign in indicates that there is a closed timelike curve (CTC) in the geometry. The angular coordinate is associated to the vector field , the integral curves of which are closed (though its period might not be ). Now
(17) 
is just the inner product of the vector . Thus if this quantity becomes timelike we have a closed timelike curve. Similarly, if , then we have a closed null curve. See Fig.(3).
The situation is in fact rather similar to the wellknown example of CTC spacetime: the van Stockum dust spacetime LC ; VS , whose metric is, in the WeylPapapetrou form (in fact any stationary and axisymmetric metric can be put into such a form, with suitable metric coefficients),
(18) 
The metric coefficient vanishes when , where one has a closed null curve. Likewise for , the metric coefficient becomes negative, and the spacetime admits CTCs.
Unlike the CTCs in KerrNewman spacetime, which lie inside the black hole, the CTCs in the twisted spacetime occur outside the event horizon, and are therefore worrying. The presence of CTCs also explain why the singularity theorem does not apply.
If one includes the mass term, then the RHS of Eq.(15) becomes
(19) 
However, since the numerator is only zero on the event horizon, the angle is always bounded away from 0, so the mass term does not prevent the problem we discussed. In fact, if the twisted black hole metric were to be used to model astrophysical black holes, then is large. For supermassive black holes, the mass term will completely dominate the expression in both the numerator and denominator at some finite coordinate distance away from the black hole, so that , and so the region that admits CTCs becomes tremendously huge: the portion on the 2geometry that is less than has CTCs. By symmetry, a similar region centered at the south pole also admits CTCs. Thus, only a very thin strip around the equator has no causality problem. This shows that unlike the Kerr metric, the twisted metric is not a good model of a realistic rotating black hole.
Iii On the Signature of the Metric Tensor
As we mentioned previously, there is a concern that if becomes negative, then there is a danger that the metric signature might change from to . As we shall see below, this does not happen, although the metric signature does change to at the poles of any , i.e., on the rotational axis of the black hole, which is a further evidence that the poles harbor some kind of conical singularity.
Since the metric is not diagonal, it is not at all obvious what is the spacetime metric signature if . One should check the eigenvalues of the full metric tensor in Eq.(1). Again, the full expression is complicated and not interesting to show here. In the special case , and , however, the eigenvalues are greatly simplified:
(20) 
Note that and are always positive outside the horizon . However, vanishes outside the horizon, although it is positive inside (). On the contrary, vanishes inside the horizon, and is negative outside.
One might suspect that this pathological property of metric signature change is due to “finetuning” the mass term to zero. This is not the case. Restoring the mass term, for example, by setting , , and , one finds that are as before, is still positive inside the horizon, but zero outside. Similarly, vanishes inside the horizon but negative outside. Thus the metric at the poles, even in the presence of nonzero mass, has signature .
(A signature change in the metric, of the form from to , can in principle also happen in the case of AdSKerr black hole, unless one imposes that the rotation parameter is less than the AdS curvature length scale . Imposing such a condition is essential for holography to be consistent, see 1504.07344 . See also 1403.3258 .)
In fact, to see that the mass term does not help, we can examine the determinant of the metric tensor in Eq.(1). We have a surprisingly simple expression:
(21) 
which is independent of .
One easily sees that the determinant can become zero if and only if . This agrees with the fact that vanishes outside the horizon at the poles.
Therefore, we see that away from the axis of rotation the spacetime always have a Lorentzian signature , which means that the spacetime is welldefined despite the fact that can change sign. Again, let us recall that for the Kerr black hole, the constant and constant surfaces are just ellipsoids. In other words, the 3geometry of constant is foliated by a family of ellipsoids. The fact that can change sign means that we do not have such a foliation in the twisted case. Specifically this means that at any fixed time , the twisted geometry should not be thought of as a family of (topologically) spherical surfaces indexed by . However, since the spacetime remains Lorentzian, there should exist a new set of coordinates such that the family of surfaces of constant and constant are Riemannian.
Iv The Twisted Spacetime Is TaubNUT
It turns out that the twisted spacetime we have explored above is actually a known solution in general relativity – it is just the TaubNUT solution. The TaubNUT metric is, in coordinates ,
(22) 
where
(23) 
Here we follow, except for the coordinate , the notation in bonner .
It is not too difficult to show that this is the same as Zhang’s metric in Eq.(1),
after a simple coordinate transformation .
Indeed, in Eq.(1), we see that asymptotically the metric coefficient
(24) 
so that the geometry is in fact not asymptotically flat, since for asymptotically flat spacetimes, one has instead
(25) 
In fact, the parameter , which we have emphasized in the introduction that it is not a rotation parameter in the sense of Kerr black hole, is now clearly seen as the NUT charge, which has no Newtonian analog.
The “conical singularity” at the poles that we have explored in the previous section, which corresponds to where the metric tensor becomes noninvertible, is none other than the socalled “wire singularity” in the TaubNUT literature.
Not all geodesics are complete due to these wire singularities MKG ; 1002.4342 .
The wire singularities can nevertheless be removed by introducing new coordinate patches and compactifying the temporal coordinate so that the topology of the spacetime becomes nut3 .
Of course, it is also known that TaubNUT spacetime admits CTCs outside of the horizon.
One obvious way to improve the geometry is to excise the regions with CTCs, and topologically identify the resulting boundaries. More specifically, after one removes the cap near the north pole (similarly for the south pole) for which (see Fig.(3)), the leftover boundary can be topologically identified to a point. This will, however, result in more conical singularities. A similar idea was proposed by Bonner back in 1969 to remove the region plagued by CTCs bonner , but there the boundary after the excision is left as it is, thus making an incomplete Riemannian surface.
V Conclusion: The Price of Evading the Singularity Theorem
In this work we have investigated the geometry of the 4dimensional twisted black hole spacetime first proposed in zhang .
We have shown that such a twisted rotating black hole, which is free of a curvature singularity, nevertheless harbors a “conical singularity”. This conical singularity is of a rather extreme type: inside the black hole horizon, it has an infinite amount of angle excess at the poles. On the other hand, outside the black hole horizon, the “angle excess” becomes complex. The Gaussian curvature is not continuous at the pole across the event horizon: it is finite on the horizon, but diverges as one approaches the horizon from either side ( if approaches from the inside, and from the outside). Furthermore the signature of the metric along the rotational axis is .
We started this investigation by asking how the twisted solution evades the singularity theorem.
We now have an answer: The singularity theorem simply does not apply because the spacetime contains both closed timelike curves and a severe conical singularity along the rotational axis (except on the horizon itself).
However, none of these are new, since the twisted solution is actually the TaubNUT metric written in a slightly different coordinate system.
The proposal to construct a rotating black hole that nevertheless has globally vanishing ADM angular momentum is an interesting one. The question is: can one construct a new black hole spacetime that is free of pathologies, including conical singularities (regardless of whether it has a curvature singularity inside the horizon), using this approach? It may well be possible that the current uniqueness theorems of black holes (see 1205.6112v1 for a comprehensive review) can be extended to include rotating black hole spacetimes that have vanishing ADM angular momentum. (Note that there are other rotating black hole solutions that nevertheless have vanishing angular momentum mann2 , but these are neither solutions to pure Einstein’s general relativity, nor do they achieve this via two counterrotating halves.)
Remarks
The first version of the arXiv preprint of this manuscript only pointed out that the “twisted black hole” spacetime contains many pathological properties. It was subsequently pointed out to the author that the spacetime is in fact just TaubNUT. This fact was incorporated in the current, updated version, of the manuscript. On the same day version two of the manuscript appeared on the arXiv, another paper (Ref.1610.06135 ) that criticizes the twisted black hole also appeared on the arXiv.
Acknowledgement
YCO thanks Brett McInnes, Michael Good, and Keisuke Izumi for fruitful discussions and suggestions. He thanks the following colleagues for pointing out that the “twisted black hole” is none other than TaubNUT: Bahram Mashhoon, Edward Teo, Erickson Tjoa, David Chow, Cristian Stelea, Gérard Clément, and Carlos Herdeiro. He thanks Patrick Connell for further suggestion. YCO also thanks Qiuyue Miao and Zhen Yuan for their much appreciated help during his recent move to Shanghai. Last but not least, YCO thanks Bin Wang and his group in the Center of Astronomy and Astrophysics for their warm welcome. He also thanks the National Natural Science Foundation of China (NNSFC) for support.
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