BPS Kerr-AdS Time Machines

It was recently observed that Kerr-AdS metrics with negative mass describe smooth spacetimes that have a region with naked closed time-like curves, bounded by a velocity of light surface. Such spacetimes are sometimes known as time machines. In this paper we study the BPS limit of these metrics, and find that the mass and angular momenta become discretised. The completeness of the spacetime also requires that the time coordinate be periodic, with precisely the same period as that which arises for the global AdS in which the time machine spacetime is immersed. For the case of equal angular momenta in odd dimensions, we construct the Killing spinors explicitly, and show they are consistent with the global structure. Thus in examples where the solution can be embedded in a gauged supergravity theory, they will be supersymmetric. We also compare the global structure of the BPS AdS$_3$ time machine with the BTZ black hole, and show that the global structure allows to have two different supersymmetric limits.


Introduction
The Kerr metric [1] is arguably the most important exact vacuum solution in Einstein's theory of General Relativity. Over the years, the solution has been generalised to include the cosmological constant and also to higher dimensions [2][3][4][5][6][7]. These metrics are asymptotic to Minkowski, de Sitter (dS) or anti-de Sitter (AdS) spacetimes, depending on the cosmological constant. They carry mass (M ) and angular momenta (J i ) as conserved quantities. For a given set of angular momenta, provided that the mass is sufficiently large, the metrics describe rotating black holes. Such a rotating black hole contains closed-time-like curves (CTCs), surrounded by a velocity of light surface (VLS), which is typically referred to as a time machine. 1 In a rotating black hole, the time machine is hidden inside the black hole event horizon.
If the black hole is over-rotating, the time machine can extend outside the horizon. For example, it was demonstrated, for a supersymmetric charged black hole with equal angular momenta in five dimensions [9], that in the over-rotating situation the boundary of the time machine lies outside the horizon and so it becomes naked [10]. (See also [11][12][13].) An examination of geodesics showed that they could not penetrate the horizon, and hence the spacetime configuration is called a repulson [10]. (See also [14].) In fact the "horizon" become a Euclidean Killing horizon that can induce a conical singularity unless the real time coordinate becomes periodic with some specific period, in which case the spacetime configuration is smooth and geodesically complete [15].
Recently, it was observed [16] that for general odd dimensions, Kerr and Kerr-AdS metrics can also generally extend onto smooth time machines when mass is negative, provided that all the angular momenta are non-vanishing. For Kerr-AdS metrics, there exist special points in the parameter space of the mass and charges, namely when the BPS condition holds, where 1/g is the "radius" of the AdS spacetime in which the solution is immersed.
When this BPS condition is satisfied, the spacetime admits a Killing spinor. The BPS condition was studied in [17] for the five-dimensional Kerr-AdS black hole, and the Killing spinors were constructed in the case where the two equal angular momenta were equal.
The BPS limit of the Kerr-AdS metric no longer describes a black hole, however, since the singularity is no longer shielded by an horizon. Interestingly, if one Euclideanises the spacetime and takes the cosmological constant to be positive the BPS Kerr-dS becomes an Einstein-Sasaki metric, which can smoothly extend onto a complete, compact manifold for appropriate discretised values of the metric parameters [18,19]. This generalises an earlier construction of smooth Einstein-Sasaki spaces in [20].
In this paper, we shall remain in Lorentzian signature and consider the case with a negative cosmological constant, but now we consider the BPS Kerr-AdS metrics where the mass is taken to be negative. Now, unlike the example considered in [17] where the mass was assumed to be positive, this yields a smooth time-machine spacetime. BPS time machines have been constructed previously in literature, typically having positive mass and with additional electric charges [10,15,21,22]. In this paper, we focus on the pure gravity BPS Kerr-AdS metrics. We analyse the global structure and find that the metrics can become smooth, provided that the mass is negative. The completeness of the spacetime requires that the asymptotic Lorenzian time coordinate be periodic, with period precisely equal to that of the time coordinate in the global AdS in which the spacetime is immersed. Furthermore, the mass and angular momenta become discretised, in a manner analogous to the discretisation of the parameters in the Einstein-Sasaki spaces, even though the spacetimes we are considering here are non-compact. For Kerr-AdS metrics with equal angular momenta in odd dimensions, we construct the Killing spinors in the BPS limit explicitly, and show that they are compatible with the global structure required for the completeness of the spacetime. Thus in dimensions where the solution can be embedded within a supergravity theory, it will be superymmetric.
The paper is organised as follows. In section 2, we begin by reviewing the time machine spacetimes that were obtained in [16] from D = (2n + 1)-dimensional Kerr-AdS spacetimes with equal angular momenta, by taking the mass to be negative, and we describe their BPS limits. We give an explicit construction of the Killing spinors in the BPS spacetimes, showing how they can be obtained by making use of the gauge-covariantly constant spinors that exist in the underlying CP n−1 spaces that form the bases of the (2n − 1)-dimensional spherical surfaces in the spacetimes. We also study the restrictions on the metric parameters that result from requiring completeness of the spacetimes, resulting from the compatibility conditions for periodicities at the various degenerate surfaces. These restrictions imply that mass and angular momentum must be rational multiples of a basic unit. They also imply that the time coordinate must be periodic, with exactly the periodicity of the time coordinate in the global AdS spacetime in which the time machine is immersed.
In section 3 we consider the case of even-dimensional spacetimes, showing that Kerr-AdS metrics with equal angular momenta can give rise in the BPS limit to metrics describing foliations of the previously discussed odd-dimensional time machines. In section 4 we discuss the analogous odd and even-dimensional BPS limits of Kerr-AdS metrics with general, unequal, angular momenta. Again these give rise to time machines if the mass is taken to be negative, and we analyse the restrictions on the metric parameters to ensure global completeness of the spacetime manifolds. Again, the mass and the angular momenta are discretised, in the sense that they are constrained to be certain rational multiples of a basic unit.
In section 5 we discuss the special case of three dimensions. Here, the Kerr-AdS metric is necessarily locally isomorphic to AdS 3 , and thus it is also locally isomorphic to the BTZ black hole [23]. We study the relation between the time machine and the BTZ spacetimes, and compare their Killing spinors in the respective BPS limits. Interestingly, the limits are different, but in each case the Killing spinors are compatible with the global structures.
Finally, after our conclusions, we include two appendices. Appendix A gives an explicit construction of the gauge-covariantly spinors in the complex projective spaces, employing an iterative construction of CP n in terms of CP n−1 that was given in [26]. We use these gaugecovariantly constant spinors in the construction of Killing spinors in section 2. Appendix B contains some results relating the various vectors and tensors that can be built from Killing-spinor bilinears.

Kerr black holes and time machines
We begin with the Kerr-AdS metrics in D = 2n + 1 dimensions with all n angular momenta set equal. The metric, satisfying R µν = −(D − 1)g 2 g µν , contains two integration constants (m, a), and it is given by [27] where Ξ = 1 − a 2 g 2 , and dΣ 2 n−1 is the standard Fubini-Study metric on CP n−1 . There is circle, parameterised by the coordinate ψ with period 2π, which is fibred over the CP n−1 base, and σ is the 1-form on the fibres, given by σ = dψ + A with dA = 2J where J is the Kähler form on CP n−1 . The terms (σ 2 + dΣ 2 n−1 ) in the metric are nothing but the metric on the unit round sphere S 2n−1 , with R i j = (n − 1)δ i j . The metric (2.1 is asymptotic to anti-de Sitter spacetime with radius ℓ = 1/g. The mass and the (equal) angular momenta are given by where A k is the volume of a unit round S k , given by . (2. 3) It will be helpful to make a coordinate transformation and a redefinition of the integration constants to replace (m, a) by (µ, ν), as follows: The metric (2.1) becomes [16] The mass and angular momenta become The metric (2.5) describes a rotating black hole if µ and ν are both positive, and a time machine if µ and ν are both negative [16], as we shall review later.

BPS limits
Under certain conditions the metric (2.5) will admit a Killing spinor, obeying the equation A necessary condition for this to occur is that the BPS condition on the mass and angular momentum, namely M = ngJ , (2.8) should hold. This implies that (2.9) The these two conditions correspond to ag = (and hence Ξ = 0) or ag = 2n − 1 respectively.
However, as we shall see, only the first of these cases gives a solution admitting a Killing spinor.
In AdS itself (i.e. µ = 0 and ν = 0), the Killing vectors have the property that g µν K µ ± K ν ± = −1, and in fact they can each be expressed in the form K µ ± =ǭ ± Γ µ ǫ ± , where each of ǫ ± is one of the Killing spinors of the AdS spacetime. We expect that if the BPS spacetime where µ and ν are non-zero, obeying one or other of the conditions in (2.9), does admit a Killing spinor, then it should be such that it limits to one of the aforementioned AdS Killing spinors in the limit where µ and ν go to zero. This means that if the BPS spacetime admits a Killing spinor, the norm K µ K µ should be manifestly negative (see [15] for a discussion of this). For the two cases in (2.9) we find µ = g 2 ν : where K + is defined in (2.10). This indicates that (2.11) gives rise to a true BPS limit, in the sense that the K + Killing vector (but not K − ) admits a spinorial square root, whereas for (2.12) it does not (nor does K − ).
For positive µ = g 2 ν, the metric has a curvature power-law naked singularity at r = 0.
We shall thus focus on the case when µ = g 2 ν is negative. Defining ν = −α, the metric becomes We have made the specific choice for the sign of √ µν → ν 2 g 2 = µg = −αg when sending µ = νg negative, and with this choice, the Killing vector admitting the spinorial square root is again given by (2.10) with the plus sign choice, for which we now define (2.14) The mass and angular momentum are given by (recall that we have made the sign choice that √ µν → +αg when sending µ and ν negative).
The metric has a power-law curvature singularity at r = 0, but there is a Euclidean Killing horizon at r = r 0 > 0 for which f (r 0 ) = 0. Thus we have The absence of a conical singularity at r = r 0 requires that the degenerate Killing vector must generate a 2π period. As we shall discuss later, this implies that the t coordinate must be periodically identified. Note that we have scaled the Killing vector so that the corresponding Euclidean surface gravity is precisely unity.
Defining a radius r * ≡ α 1 2n , we see that g ψψ < 0 in the region and thus ψ is the time coordinate in this region. (The VLS is located at r = r * where g ψψ = 0.) Since ψ is periodic, with perod ψ as stated earlier, it follows that there are closed timelike curves in the region defined by (2.18). This situation is commonly described as a time machine (see [15] for a more detailed discussion).
Finally, it is worth pointing out that in the case µ = g 2 ν, for which there is a Killing spinor, the corresponding metric (2.13) can be expressed, after we make a coordinate change ψ → ψ − g t, as a time bundle over a D = 2n dimensional space: The length of the time fibre is constant, and the base is a 2n-dimensional Einstein-Kähler metric. In fact this is Lorentzian version of the situation in an Einstein-Sasaki space, which can be written, at least locally, as a constant-length circle fibration over an Einstein-Kähler base space.

Killing spinors
Here, we construct the Killing spinor η in the (2n + 1)-dimensional BPS time machine with equal angular momenta, whose metric is given by (2.13), obeying We shall make use of the fact that CP n−1 admits a gauge-covariantly constant spinor ξ being the Dirac matrices andẽ i denoting a vielbein basis for CP n−1 . 2 With an appropriate choice of basis for the Dirac matrices one can easily establish that ξ obeys where Γ * denotes the chirality operator on CP n−1 . (We give an iterative construction of the gauge-covariantly constant spinor ξ in appendix A.) We introduce the vielbein basis e a for (2.13), with where The inverse vielbein E a is given by where E i is the inverse vielbein for CP n−1 . The torsion-free spin connection ω ab for the vielbein (2.23) is easily calculated, leading to the spinor-covariant exterior derivative D = d + 1 4 ω ab Γ ab given by Writing the 2n + 1)-dimensional Lorentz indices as a = (α, i) with α = 0, 1, 2, we may decompose the (2n + 1)-dimensional Dirac matrices in the form where γ α are 2 × 2 Dirac matrices, which we take to be It then follows that the spinor-covariant exterior derivative (2.26) is given by where D is the spinor-covariant exterior derivative on CP n−1 that we introduced earlier, andd denotes the standard exterior derivative in the three directions orthogonal to CP n−1 , With these preliminaries, it is now straightforward to obtain the equations for the Killing spinor η in the (2n + 1)-dimensional spacetime, satisfying (2.20). It takes the form where ξ is the gauge-covariantly constant spinor on CP n−1 that we introduced earlier. After further straightforward computations, we find that the 2-component spinor ǫ is given by We may now straightforwardly verify that the the Killing vector (2.14) may be written in terms of the Killing spinor η as K a =ηΓ a η . (2.34)

Global considerations and discretisation of parameters
The discussion in this section is closely analogous to that in [18,19], where the global structure of Einstein-Sasaki spaces was studied. We begin by defining the Killing vectors where we have included a 1/g in the definition of ℓ 0 in order to make it dimensionless. ℓ 1 generates a 2π period. It follows from (2.17) that Since ℓ and ℓ 1 both generate periodic translations by 2π, the ratio of their coefficients must be rational, since otherwise one there would be identifications in the time direction, generated by ℓ 0 , of arbitrarily close points in the spacetime manifold. Hence g 2 r 2 0 must be rational, which we shall write as g 2 r 2 0 = p/q, for coprime integers p andq. Consequently (2.36) can be written as where the integers q and q 1 are given by q = (n + 1)p + nq , Note that the set of integers {p, q, q 1 } are necessarily coprime, since p andq are coprime.
It is straightforward also to see from (2.38) that since p andq are coprime, it must also be the case that q and q 1 are coprime. It then follows from (2.37) that ℓ 0 generates a smallest translation period of 2π, and hence that gt has period 2π. Interestingly, this is precisely the same as the period of the time coordinate in a global AdS with radius Thus the periodicity of t that is required in order to eliminate the conical singularity at the Euclidean Killing horizon at r = r 0 is exactly the same as the time periodicity of the embedding AdS spacetime itself. Consequently, the Killing spinor (2.33) is consistent with the global structure of the time machine spacetime, and hence the solution would be supersymmetric if it can be embedded in a gauged supergravity.
The fact that g 2 r 2 0 = p/q is rational implies that the possible masses (and angular momenta) for the BPS time-machine spacetimes are discretised. From (2.15) and (2.16), we have (2.39)

Equal angular momenta in D = 2n
The Kerr-AdS metrics in even D = 2n dimensions with all equal angular momenta can be expressed as [27] where U = ρ 2 (r 2 + a 2 ) n−2 r , V = 1 r (1 + g 2 r 2 )(r 2 + a 2 ) n−1 ∆ θ = 1 − a 2 g 2 cos 2 θ , ρ 2 = r 2 + a 2 cos 2 θ , Ξ = 1 − a 2 g 2 . (3. 2) The mass and the (equal) angular momenta are [27] 3) The BPS limit M = ngJ implies that ag = 1 and hence Ξ → 0. This requires that so that M and J remain finite. In this limit, for the metric to be real and the coordinate θ to be spacelike, we need make the coordinate transformation After some algebra we end up where ds 2 2n−1 is the time machine metric obtained earlier for odd dimensions with all equal angular momenta. In deriving this, we need to further redefine the scaled m as The origin of this is that in the (V − 2m) factor, there is a term of 2mr.

General non-equal angular momenta
In this section, we consider the BPS limit of general Kerr-AdS black holes with general angular momenta.

D = 5
The Kerr-AdS metric in five dimensions was constructed in [5], given by where The metric satisfies R µν = −4g 2 g µν . The mass and angular momenta are [27]: And Riemann tensor squared is We can take the BPS limit by setting and sending ǫ → 0. The metric becomes (An analogous scaling procedure was used for five-dimensional Kerr-AdS with equal angular momenta in [17]. The Riemann tensor squared is The metric has a power-law curvature singularity atρ = 0. For positivem, the singularity is naked. However, whenm is negative, there exist a Euclidean Killing horizon at r = r 0 where∆ r (r 0 ) = 0. The absence of the conic singularity associated with the degenerate cycles atr = r 0 , θ = 0 and θ = π/2 requires that the Killing vectors must all generate 2π period. Here the Euclidean surface gravity κ on the Killing horizon is .

(4.12)
It is worth pointing out that the metric (4.7) is written in the asymptotically rotating frame.
We can make a coordinate transformation φ i → φ i + gt such that the metric becomes nonrotating asymptotically. This implies that (4.13) Defining ℓ 0 = g −1 ∂ t , we see that the Killing vectors must satisfy the linear relation with p = q + q 1 + q 2 . Consistency requires that (p, q, q 1 , q 2 ) are coprime integers, and consequently ∆t = 2π. The integration constants can expressed in terms of two rational numbers (p/q 1 , p/q 2 ): The mass and angular momenta are completely discretised, given by (4.17)

D = 2n + 1
The Kerr-AdS metric in D = 2n + 1 dimensions is given by [6,7] They satisfy R µν = −(D − 1)g 2 g µν . The mass and angular momenta are [27] The metric is non-rotating at asymptotic infinity. We take the following transformation, so that g tt → −1 at asymptotic infinity. We now take the BPS limit by setting 22) and sending ǫ → 0. The metric becomes The metric is again constant time bundle over D = 2n space, indicating that the solution admits a Killing spinor. The mass and angular momenta becomẽ The metric has a power-law curvature singularity at ∆ ψ = 0. The singularity is naked for positivem, but outside the Euclidean Killing horizon y 0 with ∆ y = 0. The Killing vectors associated with the degenerated null surfaces are Here the surface gravity κ on the horizon is Making a coordinate transformation φ i → φ i + gt, we find that the Killing vector ℓ becomes It follows that the Killing vectors satisfy As in the previous D = 5 case, consistency requires that ∆t = 2π.
We can now expressed the n integration constant b i as The mass and charges are completely discretised, given by
where ds 2 2p+1 is the metric for the BPS time machine in (2p + 1) dimensions.

Further comments in D = 3
The solutions we gave in section 2 specialise to D = 3 dimensions if we set n = 1. It is instructive to compare this with the BTZ black hole solution [23] since they are, of course, necessarily locally equivalent, both being locally just AdS 3 .
The spinor-covariant exterior derivative D = d + 1 4 ω ab γ ab is where the Dirac matrices are defined in (2.28). We find that the two-component Killing spinor is given by where (ζ + , ζ − ) satisfy the constraints , (5.14) and the exponent ∆ is given by This is not the only way to achieve the supersymmetry, however. We can instead impose M = gJ, corresponding to M BTZ − gJ BTZ = −1, in which case, we have In this case, the Killing vector is time-like, and the Killing spinor now has periodic dependence on φ, with the same period as that in the global AdS 3 . The resulting metric with negative mass then leads to the BPS time machine.
Killing spinors of BTZ black holes were also studied in [24,25].

Conclusions
In this paper, we studied the global structure of the Kerr-AdS metrics in general dimensions, when the mass and angular momenta satisfy the BPS condition (1.1). In odd dimensions with equal angular momenta, we construct explicitly the Killing spinors. A CP n and gauge-covariantly constant spinor Here we make use of the iterative construction of CP n in terms of CP n−1 that was obtained in [26], in order to give an explicit iterative construction of the gauge-covariantly constant spinor that we employed in the construction of the Killng spinor in the previous section.
As was shown in [26], the Fubini-Study metric dΣ 2 n on CP n can be written in terms of the Fubini-Study metric dΣ 2 n−1 on CP n−1 as follows: whereẽ i is a vielbein for CP n−1 . The inverse vielbein is then given by A straightforward calculation shows that the spinor-covariant exterior derivative D = d + 1 4 ω ab Γ ab on CP n is given by Decomposing the 2n-dimensional Dirac matrices Γ a for CP n as where Γ i are the (2n − 2)-dimensional Dirac matriaces for CP n−1 , it can be seen that the spinor-covariant exterior derivative (A.5) can be written as where D =d + 1 4ω ij Γ ij is the spinor-covariant exterior derivative on CP n−1 .

B Identities for spinorial square roots
In this appendix, we record some basic results for spinors in odd dimensions, which are related to our discussion about the Killing vector (2.14) in the time-machine spacetimes.
Only in the first two cases, in D = 3 and D = 5 dimensions, we see that N (1) is simply equal to N (0) . This means that in these two cases, and only in these cases, one has the relation (χΓ µ χ)(χΓ µ χ) = (χχ) 2 , (B.8) where χ is any commuting spinor. 4 The fact that (B.8) holds for any commuting spinor in D = 3 or D = 5 implies in particular that in these dimensions, any Killing vector K µ that has a spinorial square root, meaning that it can be written as in terms of a Killing spinor η as K µ =ηΓ µ η, will necessarily have constant (negative) norm.
The Killing vector (2.14) in the BPS time-machine spacetime has constant and negative norm K µ K µ = −1 in any odd dimension, and we saw in section 2.3 that it always has a spinorial square root, as in (2.34). In odd dimensions D ≥ 7, the fact that the norm is constant therefore depends upon special additional properties of the Killing spinor η that would, a priori, not necessarily hold for an arbitrary Killing spinor.