Non-singular extension of the Kerr-NUT-(anti) de Sitter spacetimes
aa r X i v : . [ g r- q c ] J a n Non-singular extension of the Kerr-NUT- (anti) de Sitter spacetimes
Jerzy Lewandowski ∗ Maciej Ossowski † Faculty of Physics, University of Warsaw,ul. Pasteura 5, 02-093 Warsaw, PolandJanuary 15, 2021
Abstract
The outstanding issue of a non-singular extension of the Kerr-NUT- (anti) de Sitter solutions to Einstein’sequations is solved completely. The Misner’s method of obtaining the extension for Taub-NUT spacetimeis generalized in a non-singular manner. The Killing vectors that define non-singular spaces of non-nullorbits are derived and applied. The global structure of spacetime is discussed. The non-singular conformalgeometry of the null infinities is derived. The Killing horizons are present.
Contents
The Kerr solution to the vacuum Einstein equations may be modified by adding a so called NUT (named afterthe discoverers: Newman, Unti, Tamburino) parameter l . The Kerr-NUT spacetime is still Ricci flat, but itstopology is considerably different than that of Kerr. Due to the Misner’s method of compacting the symmetrygroup [1], the global structure of those spacetimes is obtained as R × S and it contains timelike loops. On theother hand, for sufficiently large value of l , the Kerr singularity is smoothed out, even though the spacetimestill contains horizons. Due to the unquestionably growing relevance of the cosmological constant in physics, itis natural to generalize the family of the Kerr-NUT solutions by adding a constant Λ. That has been done longtime ago, the resulting solutions to the vacuum Einstein equations with a cosmological constant are referred toas the Kerr-NUT-(anti) de Sitter spacetimes and set a 4-dimensional family and since then has been an interest[2, 3, 4, 5, 6].In the original case of the Taub-NUT case the recipe for Misner gluing consisted of connecting two patchesTaub-NUT spacetime into a non-singular one. This had the consequence of compactifying the orbits of ∂ t Killingvector to cirles. Howecer, in the case of Kerr-NUT-(anti-) de Sitter, a generalized Misner’s gluing does not workproperly - there still persists a conical singularity irremovable at least from one of the axis of the rotationalsymmetry. We completely solve that problem in the current paper, thus our approach generalises the Misner’scompactification both in used methods and the class of applicable spacetimes.We recognize a geometric mechanism of the problem - it is hidden in the spaces of non-null orbits of Killingvectors of Kerr-NUT-(Anti) de Sitter spacetimes. Only some, distinguished Killing vectors define non-singulargeometry. We find them all. Next, one of them is used to perform a non-singular generalization of Misner’s ∗ [email protected] † [email protected] I − to the plusone I + , as well as the contained Killing horizons.This paper is the third in the series concerning the non-singular interpretation of Kerr-NUT-de Sitter space-times. However, it is a completely self contained continuation. In the previous papers we focused on thegeometry of the Killing horizons contained in that family of spacetimes. We introduced a notion of projec-tively non-singular horizon, (the horizons is said to be projectively non-singular if its space of null generatesin non-singular) and derived a 3-dimensional subfamily of the Kerr-NUT-de Sitter spacetimes, each of whichcontains a projectively non-singular horizon. For those Kerr-NUT-de Sitter spacetimes (of a well tuned valueof the cosmological constant) we were able to introduce a generalized Misner construction in a non-singularmanner. This is a special case of the generalization derived in the current paper that is valid for all the valuesof Λ independently of the remaining three parameters, regardless of the projective properties of the horizons.If a Kerr-NUT-de Sitter spacetime happens to contain a projectively non-singular horizon, then the currentconstruction of non-singular spacetimes may be reduced to the one presented in [7]. Thus the previous resultsfit neatly into the new ones. The Kerr-NUT-(anti-) de Sitter metric in the simplified form first derived in by Griffith and Podolsky [4] canbe expressed in the Boyer–Lindquist-like coordinates as ds = − Q Σ ( dt − Adφ ) + Σ Q dr + Σ P dθ + P Σ sin θ ( adt − ρdφ ) , (1)where Σ = r + ( l + a cos θ ) ,A = a sin θ + 4 l sin θ,ρ = r + ( l + a ) = Σ + aA, Q = ( a − l ) − mr + r − Λ (cid:0) ( a − l ) l + ( a + 2 l ) r + r (cid:1) ,P = 1 + 43 Λ al cos θ + Λ3 a cos θ. (2)Throughout this paper we use a generalisation of the Eddington-Finkelstein coordinates adopted to the rotatingspacetime dv := dt + ρ Q dr, d ˜ φ := dφ + a Q dr. (3)This provides an extension of the metric (1) that covers the roots of the function Q . Then the metric tensortakes the following form: ds = − Q Σ ( dv − Ad ˜ φ ) + 2 dr ( dv − Ad ˜ φ ) + Σ P dθ + P Σ sin θ ( adv − ρd ˜ φ ) . (4)That metric shows singularities (apparent or true) familiar from the analysis of the standard Kerr and Kerr-(anti) de Sitter solutions, which are special cases of the considered metrics.We emphasise now the consequences of the presence of the NUT parameter l . A helpful consequence of | l | > | a | is that the function Σ never vanishes. Otherwise, if | l | ≤ | a | the function Σ takes the value zero at r = 0 and θ = θ c such thatcos θ c = − la . That is a source of a non-removable curvature singularity [5]. The singularity has a similar structure to that ofKerr, in particular, there are continues curves that pass from the r > r < r >
0, and r <
0, by the analogy to the Kerr singularity.In the current paper we admit all the values of | a/l | , hence the vanishing Σ singularity either appears or not.If the NUT parameter l is large enough while a and Λ are kept constant the function P ( θ ) changes the signfor some θ d ∈ [0 , π ]. That is accompanied by a change of the signature of the metric tensor. To avoid thispathology we allow in the current paper only those values of l, a and Λ that ensure P ( θ ) > , for every θ ∈ [0 , π ] . (5)It is conceivable, that this assumption could be carefully relaxed by making the inequality non-sharp, but thislies beyond the scope of this paper.The NUT parameter l = 0introduces a notable difficulty that eventually has topological consequences. What is peculiar about this caseis the singularity of the differential 1-form Ad ˜ φ that is caused by the term4 l sin θd ˜ φ. (6)Indeed, when considered on a sphere parametrised by ( θ, ˜ φ ) ∈ [0 , π ] × [0 , π ), that term is discontinues at thepole θ = π . That singularity can be cured by introducing another chart that covers the pole θ = π . It wasdefined first by Misner in the Taub-NUT case, i.e. a = 0 = Λand can be easily generalised to arbitrary values of the parameters a and Λ. The Misner charts give rise to thetopology R × S of spacetime, and an action of the O (2) group generated by the Killing vector ∂ τ of the metrictensor (1) that induces a principal fibre bundle structure R × S → R × S , (7)that reduces to each sphere S , S → S . (8)defined by a fixed value of the variable r .For Λ = 0the glueing solves the problem at θ = π . However, at a generic case of al Λ = 0 , one more obstacle appears. As long as the fibres of the bundle (8) contained in the spacetime R × S are not nulland the spacetime is twice differentiable, the geometry induced on S should be continues and differentiable.The latter one is defined by the angular part of the spacetime metric tensor (1)Σ P dθ + P Σ sin θρ d ˜ φ (9)while the remaining parts are differentiable on S on their own and the term (4 l sin θd ˜ φ ) cured by the Misnerglueing). It is easy to see [7], that the tensor (9) gives rise to a well defined and differentiable metric tensor onentire S including the poles θ = 0 , π by a suitable rescaling of the variable ˜ φ , if and only if P (0) = P ( π ) , (10)that is the case (see 1), if and only if al Λ = 0 . Otherwise, the metric tensor (9) has an irremovable conical singularity at least at one of the poles. In thespacetime R × S , the corresponding singularity takes the form of a 2-dimensional surface on which the variable r takes all the values in R . Hence, in the case al Λ = 0, the metric tensor (1) can not be a local form of a C metric tensor defined on R × S such that the Killing vector ∂ t generates the fibres of the projection (7).The above naive approach to the solution of the problem is further justified in the following chapters using thebroader geometrical picture of the spaces with NUT parameters.In the current paper we solve the problem of a non-singular generalisation of Misner’s gluing to a generalcase of the Kerr-NUT-(anti)-de-Sitter spacetime (1). The resulting spacetime still has the O(2)-bundle structure(7) and the only singularity is the one corresponding to possible zeros of the function Σ if | l | < | a | , or otherwisecompletely non-singular.The metric tensor is well defined and analytic in the following range of the variables −∞ < v, r < ∞ , ≤ θ < π, ≤ φ < πP (0) , while we still have to take care of the half-axis θ = π . 3 .2 Our approach to the problem We will construct in this paper a spacetime metric ¯ g defined on R × S , such that ¯ g : • is locally isometric to (4) with l = 0, • admits an isometric action of the group O(2) that induces the principal fibre structureΠ : R × S → R × S . (11)We start with addressing necessary conditions for a choice of a Killing vector ξ of (4) whose orbits will becompactified. If the desired metric ¯ g exists, then the generator ¯ ξ of the O(2) action has to be a nowherevanishing Killing vector field. The projection (11) induces a metric tensor ¯ q on an open subset U ⊂ R × S , of the non-null (with respect to ¯ g ) orbits of the action of O(2) in R × S . In every point of U the metric tensor ¯ q should be well defined and (at least) differentiable. Therefore, in Sec. 3 we study the spaces of orbits of Killingvectors ∂ v + f ∂ ˜ φ in the spacetime (4). We determine those Killing vectors that define a non-singular metric tensor q on U . Itturns out, that at the presence of l = 0 there are allowed exactly two values of the parameter f for each tripleof values of a, Λ and l . That result is a new in the literature consequence of the presence of the NUT parameter l = 0. Indeed, in the l = 0 case, for every value of the parameter f , the corresponding Killing vector field definesa non-singular geometry q on the space of non-null orbits.The element of the desired metric tensor ¯ g on R × S encoding the non-trivial structure of the bundle (11)is the 1-form of the rotation-connection of the Killing vector ¯ ξ , namely¯ ω := ¯ ξ µ dx µ ¯ ξ α ¯ ξ α , (12)valid wherever ¯ ξ α ¯ ξ α = 0 . If the bundle extension (11) of the spacetime (4) exists, then the part of the spacetime R × S described by(4) is a trivialisation of (11) that covers the pole θ = 0 of S . Therefore, in Sec. 4, for each of the Killingvectors ξ derived in Sec. 3 we derive the rotation-connection 1-form ω . The analysis of the discontinuity of ω as θ → π leads to a complementary trivialisation of (11) that covers the pole θ = π . Remarkably, the key limitproperties of ω at θ = π are r independent. Hence, the second trivialisation covers also the null orbits. Thetransformation law between the trivialisations becomes a recipe for bundle reconstruction implemented in Sec.5. The trivialisations come with metric tensors g and g ′ , respectively. The former one is the original metrictensor (4), and g ′ is a new, transformed metric. On the overlap of the trivialisations the metric tensors g and g ′ are consistent with each other according to the trivialisation transformations. In that way they consistentlymake up a uniquely defined metric tensor ¯ g on the entire manifold R × S that satisfies all of the desiredproperties. In this section we consider the geometries of the spaces of orbits of the Killing vector fields [8] in the spacetime(4). In a generic case of (4), the most general form of nowhere vanishing Killing vector field is ξ = ∂ v + f ∂ ˜ φ , f = const . (13)In an adapted coordinates system, that is( x µ ) = ( τ, x i ) = ( v, r, θ, ˆ φ := − f v + ˜ φ ) (14)the Killing vector field ξ takes a simple form ξ = ∂ τ . The three coordinates ( x i ) = ( r, θ, ˆ φ )4atisfy ξ ( x i ) = 0 , hence they set a coordinate system at the space of the orbits. To find the metric tensor induced thereon, weuse the rotation-connection 1-form ω := ξ µ dx µ ξ α ξ α , (15)and decompose the spacetime metric (4) in the following manner g = ξ α ξ α ω + q. (16)Then, the part q of the spacetime metric satisfies ξ µ q µν = 0 = L ξ q µν , hence, it is expressed purely in terms of the three coordinates ( x i ), q = q ij ( r, θ, ˆ φ ) dx i dx j . (17)As a matter of fact, q is the pullback to the spacetime of the metric tensor induced on the space of the non-nullorbits of ξ , that is also given exactly by (17), when the variables r, θ , and ˆ φ are used both, in the space of orbitsand as a part of the spacetime coordinate system. We calculated q for the metric tensor (4) transformed to theadapted coordinate system (14), and the result reads: q = (1 − Af ) Σ Q (1 − Af ) − P sin θ ( a − f ρ ) dr + P sin θ Σ( f ρ − a ) Q (1 − Af ) − P sin θ ( a − f ρ ) drd ˆ φ ++ Σ P dθ + P Q sin θ Σ Q (1 − Af ) − P sin θ ( a − f ρ ) d ˆ φ . (18)The above metric is well defined for ( r, θ, ˆ φ ) ∈ R × ]0 , π [ × [0 , πc [, as long as the denominators do not vanish.The parameter c represents the rescaling freedom that will be used to fix the metric at the poles. The function P > ξ µ ξ µ , more precisely g ( ξ, ξ ) = g ττ = Σ − (cid:0) P sin θ ( a − f ρ ) − Q (1 − Af ) (cid:1) . (19)For general values of the parameters m, a, Λ , l, f it does vanish for some values of ( r, θ ), but then we do notexpect the metric q to be well defined, hence we consider the metric q only where P sin ( θ )( a − f ρ ) − Q (1 − Af ) = 0 . (20)The degeneracies of q that we do worry about are the half-axis p and p π corresponding to θ = 0 and θ = π ,respectively. The term proportional to dr is manifestly regular, so is the term proportional to dr d ˆ φ becauseit can be viewed as a regular 1-form sin θd ˆ φ times an analytic function times dr . Now we turn to the purelyradial part and consider the pullbacks of q to the surfaces of the variable r =const, that is (2) q = Σ P dθ + P Q sin θ Σ Q (1 − Af ) − P sin θ ( a − f ρ ) d ˆ φ . (21)One of the tools we use for the analysis are closed curves of θ = θ , which can be view as circles around eitherpole (notice, that the ˆ φ =const curves are geodesic with respect to (2) q ). The radii as seen from either pole( R or R π , respectively) and circumference ( L ) are defined as R ( θ ) = Z θ p (2) q H θθ dθ, R π ( θ ) = Z πθ p (2) q H θθ dθ, L ( θ ) = Z πc q q (2) H ˆ φ ˆ φ d ˆ φ. (22)Then the condition for removing the conical singularity is recovering the expected limit of 2 π of ratio of thecircumference to radius of said cures as we tend to the poleslim θ → L ( θ ) R ( θ ) = 2 π = lim θ → π L ( θ ) R π ( θ ) . (23)The above amounts to P (0) = P ( π ) | − lf | , c = 1 /P (0) . (24) One may also consider an extension of the Kerr-NUT-(anti-) de Sitter spacetimes to the case with the acceleration parameter.Then the condition (23) is formally the same, although with more complicated function P . See [9] for a discussion of the non-singularity if ξ develops the horizon.
5e would be in trouble, if this condition involved the coordinate r , however, this is not the case because thefunction P depends on θ , only.Due of the absolute value in the denominator above, for l = 0, there are 2 possible branches of solutions,each depending on the parameters of the spacetime. For the further convenience let us denote σ := sgn (1 − lf ) . Either we have σ = 1 and then we find the solution f = f + , such that0 < − lf + = P ( π ) P (0) , f + = 2 a Λ3 + a Λ + 4 al Λ (25)or σ = − f = f − , satisfies0 < lf − − P ( π ) P (0) , f − = 3 + a Λ2 l (3 + a Λ + 4 al Λ) (26)We note, that the assumed inequalities rewritten in (25) and (26) are consistent with the overall assumptionthat the function P does not vanish in [0 , π ].In either case, the rescaled angle variable ranging from 0 to 2 π is ϕ = P (0) ˆ φ. (27)It is instructing to test our results on the special cases that are encountered in the literature. A very specialcase when the Killing vector field ξ develops a horizon was studied extensively in [9] and [7]. Then the coefficient f is related to the value r taken by the coordinate r at the horizon, namely f = ar +( a + l ) . (28)In that case 1 − lf = 1 − lar +( a + l ) = r + ( a − l ) r + ( a + l ) > , (29)hence it falls in the very case (25). The conditions (28) and (25) determine the value of Λ, namelyΛ = 3 a + 2 l + 2 r . (30)That is exactly the value found in [9] when the horizon can be made projectively non-singular, i.e. its space ofthe null generators is non-singular. This can be done using the same rescaled coordinate as for the surroundingspacetime. The horizon is then necessarily cosmological (more precisely: the outermost, possibly with a negativemass parameter) and non-extremal.Another compelling choice of the Killing vector is simply ξ = ∂ v , meaning f = 0 , resembling the original Misner’s choice is his non-singular interpretation of the Taub-NUT metric tensor. Uponthis choice, the condition (24) amounts to the constraint P (0) = P ( π ) , which is met iff Λ al = 0. We have discussed that case in Sec 2.1. However now, in view of our general resultderived in this section, the value f = 0 falls into the (25) case, while there is yet another solution, the one ofthe type (26), namely f = 12 l . The corresponding the Killing vector field is ξ = ∂ v + 12 l ∂ ˜ φ . Finally, when the NUT parameter is switched off, that is when l = 0, then every Killing vector field ξ = ∂ v + f ∂ ˜ φ , f = constdefines a non-singular geometry on the orbit space wherever ξ µ ξ µ = 0. That is why we never encounter thatissue while considering spacetimes without the NUT parameter. Remark.
An intriguing and useful observation are the following general identities: f + + f − = 12 l , and f − − f + = P ( π )2 lP (0) . (31)6 Generalisation of Misner’s gluing
The starting point for this section is one of the Killing vector fields ξ (13) found in the previous section, that issuch that the constant f satisfies one of the conditions: either (25) or (26). In terms of the adapted coordinates(14) with the rescaled angle variable (27) the metric tensor (4) takes the following form ds = − Q Σ ((1 − f A ) dτ − AP (0) dϕ ) + 2 dr ((1 − f A ) dτ − AP (0) dϕ )+ Σ P dθ + P Σ sin θ (( a − ρf ) dτ − ρP (0) dϕ ) , (32)and the Killing vector field is just ξ = ∂ τ . (33)If this spacetime is a trivialisation of the principal fibre bundle (11) and ξ is a generator of the structure groupaction, as we want it to be, then the orbits of ξ are closed loops, and the parameter τ takes values in a finiteinterval τ ∈ [0 , τ ) . The relation of τ and the NUT parameter will follow as a consistency condition for a transformation betweenthe given one, and a new, complementary trivialisation that will cover the half-axis θ = π . It is the rotation-connection 1-form (12) that will tell us, how to construct this complementary trivialisation. The explicit formulafor ω reads, ω = dτ − (1 − Af )Σ dr + (cid:0) A Q (1 − Af ) − P sin θρ ( a − f ρ ) (cid:1) dϕ/P (0) Q (1 − Af ) − P sin θ ( a − f ρ ) (34)It is well defined at θ = 0, however it fails to be so at θ = π , ω ϕ ( r, θ = 0) = 0 ω ϕ ( r, θ = π ) = − l − lf P (0) = σ − lP ( π ) . (35)The obstruction is non-vanishing of the component ω ϕ at the second half-axis.Along with ω the metric tensor is not well defined at θ = π , what can be seen from the formula (16). Fromthe limit of ω ϕ at θ = π , we deduce a coordinate transformation that cures ω at that half-axis at the cost of θ = 0, namely τ = τ ′ + σ lP ( π ) ϕ ′ , r = r ′ , θ = θ ′ , ϕ = ϕ ′ for , θ, θ ′ = 0 , π. (36)The condition for the constant τ is hidden behind the transformation of τ . If ϕ and ϕ + 2 π correspond to asame point of spacetime for every value of τ , r , θ and φ , and the same is true for ϕ ′ and ϕ ′ + 2 π , also τ and τ ′ must parametrise circles and their period is τ = 2 π l − lf P (0) = 2 πσ lP ( π ) , (37)alternatively the period may be a n fraction of the above. Hence, the unprimed coordinates parametrise S × R × S \ { p π } , and the primed coordinates parametrise S × R × S \ { p } , where p and p π are the polesof S corresponding θ = 0 and θ = π , respectively, and the transformation (36) defines gluing, and the vectorfields ∂ τ and ∂ τ ′ give rise to a uniquely defined vector field ∂ τ = ¯ ξ = ∂ τ ′ . The manifold defined by the two charts is diffeomorphic to R × S and the flow of ¯ ξ makes it the bundle (11).The transformation (36) maps the 1-form ω into ω ′ that is extendable by the continuity to θ ′ = π and ananalytically defined 1-form in the subset of the second chart corresponding to ξ ′ µ ξ ′ ν = 0. Finally, the 2-metrictensor q is invariant with respect to the transformation (36).Applying the transformation (36) to the metric tensor (32) we obtain the metric well defined on the chartcontaining the pole θ = πds ′ = − Q Σ ((1 − f A ) dτ ′ − A ′ P ( π ) σdϕ ′ ) +2 dr ′ ((1 − f A ) dτ ′ − A ′ P ( π ) dϕ ′ )+ Σ P dθ ′ + P Σ sin θ ′ (( a − ρf ) dτ ′ − ρ ′ P ( π ) σdϕ ′ ) , (38)where A ′ ( θ ′ ) := a sin θ ′ − l cos θ ′ , ρ ′ ( r ′ ) := r ′ + ( a − l ) ,note that A ′ is dual to A is the sense that A ′ vanishes at θ = π and is singular at θ = 0Finally, we can turn to the non-singularity of the resulting metric tensor ¯ g . This issue amounts to showingthe non-singularity of the metric tensors (32) and (38) in their charts. By construction, each of the metrictensors is automatically non-singular as long as ¯ ξ µ ¯ ξ µ = 0 (39)7s true, owing to the decomposition g = ξ α ξ α ω + q, g ′ = ξ ′ α ξ ′ α ω ′ + q (40)and the non-singularity of ξ α ξ α , ξ ′ α ξ ′ α , ω, ω ′ , q in the corresponding charts. Notice, that the missing prime atthe second q is intentional - indeed, in that stage of the construction we use the single 2-metric tensor q , thesame for each chart. But given the metric tensors in the form (32) and (38) we relax the assumption (39) anddecompose the formulae in non-singular elements. First of all, except for the half-axes θ = 0 and θ ′ = π , all thecoefficients are non-singular. To analyse the metrics at the poles we decompose them into the following way.First consider the purely angular partsΣ P dθ + P Σ sin θ (cid:18) ρP (0) (cid:19) dϕ = Σ P dθ + P Σ sin θ (cid:18) ρP (0) (cid:19) dϕ ! , at θ = 0 , (41)Σ P dθ ′ + P Σ sin θ ′ (cid:18) ρ ′ P ( π ) (cid:19) dϕ ′ = Σ P dθ ′ + P Σ sin θ ′ (cid:18) ρ ′ P ( π ) (cid:19) dϕ ′ ! at θ ′ = π. (42)One can easily see, that in the parentheses, the coefficients at sin θdϕ and sin θ ′ dϕ ′ are smooth (analytic)and tend to 1 at θ = 0 and, θ ′ = π , respectively.Next, consider the differential 1-forms α = Adϕ, α = sin θdϕ, (43) α ′ = A ′ dϕ ′ , α ′ = sin θ ′ dϕ ′ . (44)Clearly, they are smooth (analytic) in their domains including the poles θ = 0, and θ ′ = π respectively. Theremaining elements used in the definitions of the metric tensors g and g ′ are functions smooth (analytic) intheir domains, also at the respective half-axes.In conclusion, the metric tensors g and g ′ give rise to a metric tensor ˆ g uniquely defined on the manifoldconstructed above, diffeomorphic to R × S , and admitting the O(2) bundle structure induced by the flow ofthe Killing vector field ˆ ξ .For the construction above we have used one of the two possible choices (25) or (26) of the parameter f .Does the other vector have any special meaning in the resulting spacetime? Does the outcome of this sectiondepend on that choice? To answer those questions, suppose that f = f + above, and consider the other vector field ξ − corresponding to f − . We transform it to the coordinates adaptedto ξ , ξ − = ∂ τ + ( f − − f + ) ∂ ˆ φ = ∂ τ + P ( π )2 l ∂ ϕ It is convenient to consider to consider a rescaled version, ∂ ϕ + 2 lP ( π ) ∂ τ Upon the triviality transformation (36), it changes the form to ∂ ϕ ′ − lP ( π ) ∂ τ ′ . That symmetry indicates a special character of this vector field. Indeed, we can introduce on every surface r = const an auxiliary structure of the group SU(2), such that the vector field ξ generates a left invariantvector field, the vector field ξ − a right invariant vector field, and the vector fields coincide at the unity elementof the group. An important consequence of that symmetry between ξ + and ξ − is that the glued spacetime isindependent of whether we chose ξ + or ξ − in order to define the generalised Misner gluing. The spacetime manifold is entire R × S provided | l | > | a | . Otherwise, if | l | ≤ | a | , (45)8he vanishing of Σ produces an non-removable singularity at( r, θ ) = (0 , θ c ) = ( r ′ , θ ′ )where the critical value θ c is defined by cos θ c = − la . However, as in the Kerr spacetime, the surface r = 0 , θ, θ ′ = θ c which a curve has to cross in order to get from the r > r < l = ± a, where the ring is shrunk to a point.The vector field corresponding to ∂ r in unprimed chart, and to ∂ r ′ in the primed chart is globally definedand everywhere null ¯ g ( ∂ r , ∂ r ) = 0 . A time orientation of spacetime can be defined by declaring • either ∂ r and ∂ r ′ to be future directed, • or − ∂ r and - ∂ r ′ to be future directed.The coordinate transformation ( τ ” , r ” := − τ, − r )maps the first case into the second (and vice versa), hence, without lack of generality we can assume that − ∂ r is future pointing.The Killing orbits are two dimensional (generically) surfaces endowed with the induced geometry ds = − Q Σ ((1 − f A ) dτ − AP (0) dϕ ) + P Σ sin θ (( a − ρf ) dτ − ρP (0) dϕ ) , (46)The signature of the above is • ( − , +), if Q > • (+ , +), if Q < • (0 , +), i.e. null, if Q = 0If an orbit is timelike (or null), at a given point, than a timelike (or null), vector is ξ i = ∂ τ + P (0) ρ ( r c ) ( a − ρ ( r c ) f ) ∂ ϕ , where r = r c is constant. Its time orientation is encoded in the scalar product g ( ξ c , − ∂ r ) = − Σ ρ < r i = r , r , r , r of the polynomial Q determines a Killing horizon (see [7])developed by the Killing vector ξ i , As in the Kerr-(anti-) de Sitter space time, all roots cannot have the samesign. Also would r i = 0 corresponds to singularity and not a Killing horizon [7]. It follows, that this Killingvector is always future pointing.In the very special case when Λ = 3 a + 2 l + 2 r i , (47)9he coefficient at ∂ ϕ vanishes. Then the horizon is developed by our Killing vector ¯ ξ itself. Hence, the nullgenerators are closed. Then the space of the null generators is diffeomorphic to S , and the geometry inducedthereon is the limit of the metric tensor (2) q (21), it in non-singular, and smooth. That case was discovered anddescribed in detail in [7] along with their relation to solutions of Type D equation on Hopf bundle and isolatedhorizons [10, 11]Another non-generic case is when the ratio of the coefficient at ∂ ϕ to 4 l/P ( π ) is rational. Then the nullgenerators will be finite. Each such a case requires individual characterisation. For a non-rational value of theratio, all the null generators are infinite and each of them is dense in a 2-manifold contained in S . The quotientspace of those null generators has non-Hausdorff topology and lacks a differential structure.We can introduce a coordinate Ω := 1 r valid for either r < r >
0. The the metric tensor g can be written as g = 1Ω Λ3 (cid:18) (1 − f A ) dτ − AP (0) dϕ (cid:19) + dθ P + P sin θ (cid:18) f dτ + dϕP (0) (cid:19) − d Ω (cid:18) (1 − f A ) dτ − AP (0) dϕ (cid:19) + O (Ω) ! , where O (Ω = 0) = 0. The surfaces Ω = 0, corresponding to r = ±∞ define the future / past infinity equippedwith an induced geometryΛ3 (cid:18) (1 − f A ) dτ − AP (0) dϕ (cid:19) + dθ P + P sin θ (cid:18) f dτ + dϕP (0) (cid:19) of the signature depending on Λ in the known way. The result of this paper is a 4 dimensional family (parametrised by ( m, a, l,
Λ)) of globally defined spacetimesthat are locally isometric to the Kerr-NUT-(anti) de Sitter metric tensors (1), however, they do not suffer thesingularities at none of the axis θ = 0 , π . The spacetime manifold is obtained by gluing the manifolds S × R × (cid:0) S \ { p π } (cid:1) parametrised by ( τ, r, θ, ϕ ), and S × R × (cid:0) S \ { p } (cid:1) (48)parametrised by ( τ ′ , r ′ , θ ′ , ϕ ′ ), respectively, with the transformation (36). The coordinates ( θ, ϕ ) and ( θ ′ , ϕ ′ )are the standard spherical coordinates on S , while the variables τ and τ ′ parametrise circles. The points p and p π are the poles of S corresponding θ = 0 and θ ′ = π .For every quadruple of values of the parameters ( m, a, l, Λ), the spacetime metric tensor is defined by (32)and (38), where the functions Q , Σ , ρ, P and A were defined in (1). The only possible singularity of ourspacetime metric tensor may be caused (depending of the ratio la , see (45)) by vanishing of the function Σ, andhas similar character to the Kerr spacetime singularity - in particular it does not restrict the domain of the r coordinate −∞ < r < ∞ . What is new about our result, is the simultaneous presence of the Kerr parameter a = 0, the NUT parameter l = 0 and the cosmological constant Λ = 0.An underlying structure for the construction was the assumed isometric action of the group O(2) that makesthe spacetime a principal fiber bundle R × S → R × S (modulo the possible singularities discussed above). The key element of our method was determining a suitablecandidate Killing vector field in the the Kerr-NUT-(anti) de Sitter metric tensor (1) that could become agenerator of that non-singular action of O(2) on R × S . We have achieved that by studying the geometry of thespaces of non-null orbits of each Killing vector field of the Kerr-NUT-(Anti) de Sitter spacetime, and selectingthose that induce non-singular 3-geometry.We studied the global structure of the constructed spacetimes. Depending on the value of the cosmologicalconstant Λ, our spacetime is asymptotically de Sitter or anti-de Sitter. We derive the conformal geometry of thenull infinity and find it non-singular as well. The spacetime contains up to four Killing horizons correspondingto the roots of the function Q . In general and generically, the null generators of the horizons are infinitecurves, and each of them densely covers a 2-surface.Hence the space of the null generators is not a differentiable2-dimensional manifold. For special values of ( m, a, l, Λ) (see (30)) the null generators of one of the horizonscoincide with the fibers of the bundle (48). Then, the horizon is projectively non-singular, that is the space ofthe null generators has a non-singular geometry diffeomorphic to S [7].10 eferences [1] Charles W. Misner. The Flatter Regions of Newman, Unti, and Tamburino’s Generalized SchwarzschildSpace. Journal of Mathematical Physics , 4(7):924–937, 1963.[2] J. G. Miller. Global analysis of the Kerr-Taub-NUT metric.
Journal of Mathematical Physics , 14(4):486–494, 1973.[3] Marc Mars and Jos´e Senovilla. A Spacetime Characterization of the Kerr-NUT-(A)de Sitter and RelatedMetrics.
Annales Henri Poincar´e , 16, 07 2013.[4] J B Griffiths and J Podolsk´y. A note on the parameters of the Kerr–NUT–(anti-)de Sitter spacetime.
Classical and Quantum Gravity , 24(6):1687–1689, mar 2007.[5] Jerry B. Griffiths and Jiˇr´ı Podolsk´y.
Exact Space-Times in Einstein’s General Relativity . CambridgeMonographs on Mathematical Physics. Cambridge University Press, 2009.[6] J. B. Griffiths and J. Podolsk´y. A new look at the Pleba´nski–Demia´nski family of solutions.
InternationalJournal of Modern Physics D , 15(03):335–369, 2006.[7] Jerzy Lewandowski and Maciej Ossowski. Projectively nonsingular horizons in kerr-nut–de sitter space-times.
Phys. Rev. D , 102:124055, Dec 2020.[8] Piotr T. Chru´sciel.
Elements of General Relativity . Compact Textbooks in Mathematics. Birkh¨auser Basel,2019.[9] Jerzy Lewandowski and Maciej Ossowski. Non-singular Kerr-NUT-de Sitter spacetimes.
Classical andQuantum Gravity , 37(20):205007, sep 2020.[10] Denis Dobkowski-Ry lko, Jerzy Lewandowski, and Tomasz Paw lowski. Local version of the no-hair theorem.