Regularity of a double null coordinate system for Kerr-Newman-de Sitter spacetimes

We construct a double null coordinate system $(u,v,\theta_\star,\phi_\star)$ for Kerr-Newman-de Sitter spacetimes and prove that the two-spheres given by the intersection of the hypersurfaces $u=\mbox{constant}$ and $v=\mbox{constant}$ are $C^\infty$ in Boyer-Lindquist coordinates (including at the"poles"). The null coordinates allow one to immediately extend some results previously proven for Kerr. As an example, we illustrate how Sbierski's result, for the wave equation on the black hole interior, for Reissner-Nordstr\"{o}m and Kerr spacetimes, applies to Kerr-Newman-de Sitter spacetimes.


Introduction
The Kerr-Newman-de Sitter (KNdS) metric is a solution of the Einstein-Maxwell equations with a positive cosmological constant Λ: Rg µν + Λg µν = 2T µν , dF = d ⋆ F = 0, Here R µν are the components of the Ricci tensor of the spacetime metric g, R is the scalar curvature, ⋆ is the Hodge star operator, and F µν is the Faraday electromagnetic 2-form. So, the KNdS metric is an electrovacuum solution of the Einstein field equations, i.e. it is a solution in which the only nongravitational mass-energy present is an electromagnetic field. The spacetime is the four-dimensional manifold R 2 × S 2 with metric given by g = ρ 2 ∆ r dr 2 + ρ 2 ∆ θ dθ 2 + 1 ρ 2 (a 2 sin 2 θ∆ θ − ∆ r ) dt 2 + sin 2 θ Ξ 2 ρ 2 ((r 2 + a 2 ) 2 ∆ θ − a 2 sin 2 θ∆ r ) dφ 2 − 2a sin 2 θ Ξρ 2 ((r 2 + a 2 )∆ θ − ∆ r ) dφ dt (1) in Boyer-Lindquist coordinates, where ∆ r = (r 2 + a 2 ) 1 − Λ 3 r 2 − 2M r + e 2 , ∆ θ = 1 + Λ 3 a 2 cos 2 θ, ρ 2 = r 2 + a 2 cos 2 θ, θ is the colatitude, with 0 ≤ θ ≤ π, and φ is the longitude, with φ ∈ S 1 , (see Carter [6], and Akcay and Matzner [1] and Kraniotis [23], for example). Here M and a are mass and angular momentum parameters, respectively, both nonzero, and e is a charge parameter (which may be zero). Without loss of generality, we assume that the magnetic charge is zero and a is positive. This metric is supposed to represent a rotating black hole, with charge, in a universe which is expanding at an accelerated rate (as ours is). We refer to Appendix C, where we calculate Komar integrals over the event horizon, for the relation between these parameters and the physical quantities of the black hole. We wish to consider subextremal metrics, meaning that ∆ r has four distinct real roots r n < 0 < r − < r + < r c .
The event horizon H corresponds to the hypersurface where r = r + , the Cauchy horizon CH corresponds to the hypersurface where r = r − , and the cosmological horizon corresponds to the hypersurface r = r c . We are interested in studying solutions of the wave equation in the black hole region, r − < r < r + , and we wish to work in double null coordinates. This article follows closely the strategy and tools developed in [9]. Double null coordinate systems were constructed by Pretorius and Israel [28] for Kerr spacetimes (in the local setting), by Balushi and Mann [3] for Kerr-(anti) de Sitter spacetimes, and by Imseis, Balushi and Mann [19] for Kerr-Newman-(anti) de Sitter spacetimes. In [3] and [19] the authors also study the formation of caustics. In Section 2 we construct a double null coordinate system for Kerr-Newman-de Sitter spacetimes. This construction only differs from the one in [3] and [19] (that we were unaware of until the completion of this work) in the choice of λ (our choice λ = sin 2 θ ⋆ is identical to the one in [9] and [28]). Consider the transformation (t, r, θ, φ) → (t, r ⋆ , θ ⋆ , φ) in the black hole region r − < r < r + , where ∆ r < 0. The coordinate θ ⋆ is defined implicitly as the solution of F (r, θ, θ ⋆ ) = 0, where F is given by The coordinate r ⋆ is defined by r ⋆ = ̺(r, θ, sin 2 θ ⋆ (r, θ)), where ̺ is given by for some fixed r 0 ∈ (r − , r + ), and where f is the function which satisfies f (0) = 0 and (1)). The regularity of transformation of coordinates (r ⋆ , θ ⋆ ) → (r, θ) for Kerr spacetimes, namely at θ ⋆ = 0, was shown by Dafermos and Luk [9]. We adapt their work to the setting of Kerr-Newman-de Sitter spacetimes. We check that θ ⋆ is well defined and continuous at θ = 0 and θ = π 2 with sin θ ⋆ sin θ + cos θ cos θ ⋆ 1.
The proof of (2) requires that we use conditions characterizing subextremal black holes which are deduced in Appendix B. Namely, we use where l is given by (137). This implies the inequality Ξ < csc 2 arctan r + r − + arctan r − r + , which in turn implies (2). We use the fact that Λ is nonnegative so that our computations are not immediately applicable to the setting of Kerr-AdS. We also show that r and θ are smooth functions of r ⋆ and θ ⋆ . When the cosmological constant is equal to zero and e = 0, ∆ θ is equal to 1 and our formulas reduce to the ones for the Kerr spacetime in [9]. The trigonometric identity sin 2 (2θ ⋆ )∆ θ = − sin(2θ) ∂ θ D(θ, θ ⋆ ) + 2(cos(2θ) + cos(2θ ⋆ ))D(θ, θ ⋆ ), for D(θ, θ ⋆ ) = sin 2 θ ⋆ ∆ θ −sin 2 θ is the key to the calculation of ∂ r θ ⋆ and ∂ θ θ ⋆ , as well as the successful completion of some new identities, such as (53) and (58), which we need in order to calculate the derivatives of r and θ with respect to r ⋆ and θ ⋆ . We would like to emphasize that our calculations are successful because the dependence of ∆ θ on θ occurs through sin 2 θ, and not through sin θ, or on any other non-smooth function of sin 2 θ. (This is a reflection of the fact that the Kerr-Newman-de Sitter metric is regular on a manifold diffeomorphic to R 2 ×S 2 θ,φ , i.e. it is regular on the full Boyer-Lindquist spheres of constant time coordinate t and radial coordinate r. ) We also obtain bounds on the derivatives of r and θ which we need later on. These bounds parallel the ones in [9]. Using with h given by ∂ r⋆ h(r ⋆ , θ ⋆ ) = − Ξa((r 2 + a 2 )∆ θ − ∆ r ) (r 2 + a 2 ) 2 ∆ θ − a 2 sin 2 θ∆ r , with h(0, θ ⋆ ) = 0, allows one to bring the metric to the double null form which is particularly adequate for carrying out energy estimates.
In Subsection 2.2.6, we prove that the two atlases A BL = {(t, r, θ, φ), (t, r,x,ỹ)} and A DN = {(u, v, θ ⋆ , φ ⋆ ), (u, v, x, y)} are compatible (which would be clear if we were to exclude the points where θ = θ ⋆ = 0 and θ = θ ⋆ = π from our manifold). This implies that the two-spheres given by the intersection of the hypersurfaces u = constant and v = constant are C ∞ with respect to A BL (see Theorem 2.23).
Following [9], we analyze the decay of Ω 2 at the future event and Cauchy horizons, H + and CH + . Finally, we give regular coordinates at H + and CH + . The Christoffel symbols of g in the double null coordinates (u, v, θ ⋆ , φ ⋆ ) are given in Appendix A, along with some covariant derivatives that are needed to carry out energy estimates.
In Section 3, using the vector field method, we study, in the black hole interior, the energy of solutions of the wave equation which have compact support on H + . We apply the form (4) of the metric to construct certain blue-shift and red-shift vector fields and to calculate their covariant derivatives. We obtain the usual inequalities relating the vector and scalar currents associated to these vector fields. This allows us to illustrate how Sbierski's result in [29] applies to Kerr-Newman-de Sitter spacetimes. Let us mention that the proofs of [14] also extend to KNdS.
This work is a first step of our broader project to generalize the results of [7], which provides a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price Law, for energy of waves to remain bounded up to CH + . The work [7] used the fact that the generators of spherical symmetry are three Killing vector fields, which is not true in the context of Kerr-Newman-de Sitter spacetimes. We expect to address this in a forthcoming paper.
An alternative approach towards extending the results of [7] to the Kerr-Newman-de Sitter setting would be to work in Boyer-Lindquist coordinates as is done in the work [25] on Kerr black hole interiors.
In Appendix B, we characterize subextremal Kerr-Newman-de Sitter black holes in terms of (r − , r + , Λa 2 , Λe 2 ), proving (3), in particular, as mentioned above. The subset of R 3 where one can choose r+ r− , Λa 2 , Λe 2 is sketched in Figure 3 on page 44. We make additional remarks concerning alternative choices of parameters, namely Λ, r+ r− , a, e or (Λ, M, a, e). Related characterizations of the parameters of subextremal Kerr-de Sitter solutions, for the case when there is no charge, can be found in Lake and Zannias [24] and Borthwick [4].
Hintz and Vasy give a uniform analysis of linear waves up to the Cauchy horizon using methods from scattering theory and microlocal analysis in [18]. Moreover, Hintz proves non-linear stability of the Kerr-Newman-de Sitter family of charged black holes in [17] We look for a function r ⋆ such that the axisymmetric hypersurface, v(t, r, θ) = t ± r ⋆ (r, θ) = constant (ingoing when the plus sign is chosen, and outgoing when when the minus sign is chosen), is lightlike. Then, the function v must satisfy the eikonal equation We follow [28] and construct particular separable solutions of the eikonal equation. We define P and Q by and so the eikonal equation becomes As P is independent of r, and Q is independent of θ, we look for special solutions r ⋆ of this equation, where Both P and Q depend on (what is so far the parameter) θ ⋆ , which arises because of the degree of freedom one has in breaking up the left-hand side of (5) to a sum. Indeed, to the left-hand side of (5) we subtracted and added the quantity a 2 sin 2 θ ⋆ (which is independent of both r and θ) and then we decomposed the resulting expression into a sum of a function depending solely on r and a function depending solely on θ. We integrate (7) and obtain where the function f accounts for an integration constant. Thus we have with for some fixed r 0 ∈ (r − , r + ), and λ = sin 2 θ ⋆ .
Also, let Then, the function r ⋆ = ̺(r, θ, sin 2 θ ⋆ (r, θ)) is another solution of (7). The functions are solutions of the eikonal equation. Just as in the case of Kerr, it turns out that θ ⋆ is an appropriate angle coordinate. This can be understood starting with the construction of [28]: when Λ = M = a = e = 0 (so that we are reduced to the Minkowski spacetime), θ ⋆ is the spherical polar angle. Moreover, for r close to r + , θ ⋆ is close to θ. The function θ ⋆ is interpreted as the spherical polar angle and the hypersurfaces where u and v are constant are called quasi-spherical light cones. From (recall that ∆ r < 0), a hypersurface where r ⋆ equals a constant is spacelike.
Remark 2.2. Because of the symmetry in (10), our statements about θ and θ ⋆ will refer to the interval 0, π 2 and it is understood that corresponding statements will hold in π 2 , π .
Differentiating both sides of (9) with respect to r and θ yields Since the differentials of r and θ are given by To write the metric in (t, r ⋆ , θ ⋆ , φ) coordinates one uses (15) and (16) in (1) and obtains

Definition of φ ⋆ and the metric in double null coordinates
From (11), one gets Now introduce a new coordinate φ ⋆ , defined by For a general function f , one haŝ wheref is the function f written in the (û,v,θ ⋆ , φ) coordinate system andû = u,v = v,θ ⋆ = θ ⋆ . So, it follows that These equations help us with the geometric interpretation of the change of coordinates operated by passing from φ to φ ⋆ . Of course, for functions f that do not depend on φ, like the coefficients of our metric, Defining the expression for the metric becomes For each pair (u, v), g / is a metric defined on a two-sphere. The calculation above shows that the coefficients of this metric are The determinants of the metrics g / and g are and det g = −4Ω 4 L 2 sin 2 θ Ξ 2 , and the inverse of the metric g is with coefficients given by

Normals to hypersurfaces and volume elements
We finish this subsection by writing down the volume elements of hypersurfaces of our spacetime, corresponding to constant r ⋆ , v and u. As recalling (24) for the determinant of g /, de volume element for Σ r⋆ is Our choice for the normals to constant v and u hypersurfaces are We have that and so, since the volume element associated to the metric g is the volume elements associated to constant v and u hypersurfaces are As r ⋆ = u + v, the tangent space to a hypersurface Σ r⋆ , where r ⋆ is constant, is spanned by ∂ v − ∂ u , ∂ θ⋆ and ∂ φ⋆ , and Remark 2.3. ∂ φ⋆ is equal to zero when θ ⋆ is either 0 or π.

Regularity of the change of coordinates
The regularity of transformation of coordinates (r ⋆ , θ ⋆ ) → (r, θ) for Kerr spacetimes, namely at θ ⋆ = 0 and θ ⋆ = π 2 , was shown by Dafermos and Luk [9]. In this subsection we adapt their work to the setting of Kerr-Newman-de Sitter spacetimes.
Proof. The second integral in (9) is bounded above by for some positive c Λ,M,a,e . On the other hand, using the substitution sin θ ′ = sin θ ⋆ sinθ we see that the negative of the first integral in (9) is bounded below by Since F (r, θ, θ ⋆ ) is equal zero, we obtain the estimate provided the right-hand side is positive, i.e.
We claim that for all α greater than 1. To show this, we define the function which we wish to check is always greater than one. Since So certainlyρ(α) is greater than one for α close to 1 and α sufficiently large. Estimating, one can determine 1 < α m < α M such thatρ(α) > 36 for all 1 < α < α m , andρ(α) > 9 for all α > α M . Then one can plot the graph ofρ in the interval [α m , α M ] and check that it is also greater than one in that interval. This is illustrated in Figure 1. The claim is proven. PSfrag replacements α ρ(α) Figure 1: The fact thatρ > 1 implies that we can guarantee (33) for all subextremal black holes.
The previous paragraph implies that (32) is satisfied, and so the right-hand side of (30) is positive. Then, we have sin θ sin θ ⋆ ≥ c > 0.
If we use the substitution cos θ ′ = cos θ ⋆ secθ in (29) we get The result follows since (28) implies that the last expression is bounded above. ✷
Proof. We use expression (35). Note that For θ ⋆ close to zero, the right-hand side is bounded above by π/2 a cos θ ⋆ , whereas for θ ⋆ close to π 2 , the right-hand side is bounded above by Recalling (27), we conclude that the second term on the right-hand side of (35) is uniformly bounded below by a negative constant. Clearly, the same is true for the third term on the right-hand side of (35). Since both terms are negative, csc(2θ ⋆ ) sin(2θ) is bounded, and 1/ sin 2 θ ⋆ ∆ θ − sin 2 θ is bounded below by a positive constant, we obtain (37). ✷

First partial derivatives
Lemma 2.10. The partial derivatives of (r, θ) with respect to (r ⋆ , θ ⋆ ) are given by In the region r − < r < r + , we have Proof. We start from (15) and (16). We use (35) to obtain (39) and (41). The estimates for the derivatives of r and θ follow from the estimates which together imply that

Higher order derivatives
Lemma 2.11. The functions r and ∂θ ∂θ⋆ are C ∞ . For every k ≥ 2, we have that Moreover, the derivatives of the function L given in (13) are bounded as follows: To prove Lemma 2.11 we need to know the derivatives of Note that the dependence of ∂r ∂r⋆ , 1 sin(2θ⋆) ∂r ∂θ⋆ , ∂θ ∂θ⋆ and L on θ and θ ⋆ is done through sin 2 θ, sin 2 θ ⋆ , ∆ θ (which is a function of sin 2 θ, and this is crucial for our argument to work), sin 2 (2θ ⋆ ), and R 1 to R 4 . The same is true for ∂θ ∂r⋆ , which however also has a factor sin(2θ ⋆ ). Moreover, note that This is clearly bounded for θ ⋆ close to zero. It is also bouded for θ ⋆ close to π 2 . Indeed, in this situation, we have because cos θ cos θ⋆ is bounded. So the function is bounded for θ ⋆ ∈ 0, π 2 . Remark 2.13. The reader will notice, using the expressions below, that for each i between 1 and 4.
Derivatives of R 1 . For any integer n ≥ 1, the following identities hold: Proof. The proof of (52) is immediate using (40), as By differentiation we obtain We keep the two first integrals unchanged and use (34) on the integral marked (54) to obtain Integrating the last integral by parts and using the fact that From equality (41), it follows that Using the last equality in (55) and dividing by sin(2θ ⋆ ), we obtain (53). ✷ Derivatives of R 2 . The following identities hold: Proof. From (40), we get (57). To start the proof of (58), note that For any A, the last expression is equal to because the terms with A cancel out and the terms with sin 2 (2θ) cancel out. The value of A will be chosen taking (41) into account, that is we choose The reason we made a term with A − 1 appear is that such a term is proportional to ∆ r . In fact, According to (34), we have that Hence, the expression above is equal to The final expression (58) is obtained using (41). ✷ Derivatives of R 3 . The following identities hold: Proof. From (40), we get (59). Arguing as in the proof of (58), we obtain One concludes by once again applying (41). ✷ Derivatives of R 4 . The following identities hold: Proof. From (40), we immediately get (60). Moreover Equality (61) is obtained using (56).
Regularity of the metric. We check the regularity of the metric at θ ⋆ = 0 using coordinates Lemma 2.17. The metric is smooth at θ ⋆ = 0 and Proof. The relations Using (12), (21), (23) The trigonometric functions of φ ⋆ , which would make the metric discontinuous at θ ⋆ = 0, have disappeared. Moreover, since lim we conclude that (64) holds. This shows that the metric is continuous at θ ⋆ = 0. Lemma 2.15 implies that the extension of the metric is smooth. ✷ Calculation of ∂ θ⋆ θ(r ⋆ , 0).
Proof. Using the definition of F in (9), the equation F (r, θ, θ ⋆ (r, θ)) = 0 can be written as We will take the limit of both sides of (66) as θ goes to 0. As lim θ→0 θ ⋆ (r, θ) = 0, uniformly in r, Using the substitution s = sin θ ′ sin θ⋆ , with cos θ ′ = 1 − s 2 sin 2 θ ⋆ . The last integrand converges uniformly to 1 √ Ξ − s 2 as θ goes to zero. Thus, we get . So, from (66) we conclude that We know that the right-hand side of this equality is positive because we guaranteed that (31) holds. And the right-hand side is obviously smaller than π 2 . Therefore, we have This is strictly greater than 1 (and goes to 1 as r ր r + ). According to (6) and (14), the derivative of the map (r, θ) → (r ⋆ , θ ⋆ ) at (r, 0) is represented by the matrix Regularity of functions at the poles. Using the change of coordinates (63), the variable θ ⋆ is written in terms of x and y as θ ⋆ = arcsin x 2 + y 2 .
Given a function f that transforms pairs (r ⋆ , θ ⋆ ), we want to study the differentiability of (r ⋆ , x, y)f −→ f r ⋆ , arcsin x 2 + y 2 .
Proof. The derivative off with respect to x is We observe that, although the map (x, y) → x 2 + y 2 is not differentiable at the origin, when x = y = 0 the quotient (67) is +1 or −1, according to the calculation of a right or a left derivative. If we assume that 1 sin(2θ⋆) (∂ θ⋆ f ) has a limit at ((r ⋆ ) 0 , 0), then ∂ θ⋆ f ((r ⋆ ) 0 , 0) = 0, so there is no indetermination in (67), and The functionf has continuous partial derivatives with respect to r ⋆ , x and y in a neighborhood of each point ((r ⋆ ) 0 , 0, 0), and so it is differentiable. ✷ Note that we can write (67) as (This is the derivative of f with respect to sin 2 θ ⋆ = x 2 + y 2 multiplied by the derivative of sin 2 θ ⋆ with respect to x.) Expression (68) shows that if the quotient 1 sin(2θ⋆) (∂ θ⋆ f ) were unbounded, then one might have problems with the differentiability when (x, y) = (0, 0). For example, if this quotient were to behave like 1 sin θ⋆ around ((r ⋆ ) 0 , 0), then ∂ xf would behave like cos φ ⋆ , so that ∂ xf ((r ⋆ ) 0 , 0, 0) would not exist. Proof. It is clear that ∂ 2 r⋆f , ∂ x ∂ r⋆f and ∂ y ∂ r⋆f are continuous. The continuity of ∂ 2 xf is a consequence of The continuity of ∂ x ∂ yf and ∂ 2 yf follow in the same way. Sof is C 2 . One proves by induction thatf is C ∞ . ✷ Remark 2.21. A differentiable function such that f (r ⋆ , θ ⋆ ) = f (r ⋆ , π − θ ⋆ ) satisfies ∂ θ⋆ f r ⋆ , π 2 = 0. The quotient 1 sin(2θ⋆) ∂ θ⋆ f has a finite limit at r ⋆ , π 2 , provided that the numerator is analytic. Indeed, both the numerator and the denominator vanish at r ⋆ , π 2 and the denominator has a first order zero there.
Thus r,x andỹ are C ∞ functions of r ⋆ ,x andŷ.

Regularity of the spheres given by the intersection of hypersurfaces u = constant and v = constant.
It is obvious that the two atlases

The decay of Ω 2 at the horizons
Recall that the surface gravities of the horizons are given by (confirm the formula for κ − with Example A.3).
Lemma 2.24. Given C R ∈ R, there exist c, C > 0 such that Proof. The formula shows that ∂r ∂r⋆ (r = r − , θ) = 0. To write the linear approximation for this function, we start by calculating According to Remark 2.1, there exists C > 0 and (r ⋆ ) 0 such that for r ⋆ ≥ (r ⋆ ) 0 we have Remembering that κ − < 0, it follows that Integrating from (r ⋆ ) 0 to r ⋆ yields These inequalities can be rearranged to This shows that there exists D > 0 and (r ⋆ ) 0 such that Let C R be a real number. Decreasing c and increasing C if necessary, one sees that Combining the previous two inequalities, they hold for r ⋆ ≥ C R . Moreover, in this region, for other appropriate constants c, C > 0. Hence, given C R ∈ R, there exist c, C > 0 such that for r ⋆ ≥ C R . As Ω 2 is comparable to |∆ r |, we conclude that Ω 2 is comparable to e 2κ−r⋆ in the region r ⋆ ≥ C R . This proves (73). The proof of (74) is analogous. ✷ Remark 2.25. The function Ω 2 given in (18) is obviously smooth on the spheres where u and v are simultaneously constant.

Coordinates at the Cauchy horizon
Let us recall how one may define coordinates to cover the Cauchy horizon. We consider a new smooth coordinate v CH + (v), with positive derivative, equal to v for v ≤ −1, satisfying v CH + → 0 as v → +∞, and satisfying for v ≥ 0. Moreover, we define Remark 2. 26. v CH + is also a smooth function of v, and so the change of coordinates For v ≥ 0, the differentials of φ ⋆ and φ ⋆,CH + are related by For v ≥ 0, the expression of the metric (20) in the new coordinates is Henceforth we will assume that we are working in the region v ≥ 0, our formulas will always refer to this region. .
Using (11)  for u + v ≥ C R . Thus, inequalities (73) implies the following bounds for Ω 2 CH + and b φ⋆ CH + , when u + v ≥ C R : For a general function f , we have wheref is the function f written in the coordinates (u,ṽ, θ ⋆ , φ ⋆,CH + ) andṽ = v. So This, (76) and ∂ φ⋆ = ∂ φ ⋆,CH + imply that We can write the vector field ∂ t using the coordinates at the Cauchy horizon as The vector field ∂ t is not null on the Cauchy horizon. A Killing vector field which is null on the Cauchy horizon is (80)

Coordinates at the event horizon
We consider a new smooth coordinate u H + (u), with positive derivative, equal to u for u ≥ 1, satisfying u H + → 0 as u → −∞, and satisfying du = e −2κ+u du H + for u ≤ 0. Moreover, we define v H + = v and Remark 2.27. u H + is also a smooth function of u, and the change of coordinates For u ≤ 0, we may write the metric as Henceforth we will assume that we are working in the region u ≤ 0, our formulas will always refer to this region. For a general function f , we have wheref is the function f written in the coordinates (u H + , v H + , θ ⋆ , φ ⋆,H + ). So, defining we get This, (76) and ∂ φ⋆ = ∂ φ ⋆,H + imply that A Killing vector field which is null on the event horizon is The value is the angular velocity Ω H on the event horizon.

The energy of the solutions of the wave equation
We will use the vector field method to study the energy of solutions of the wave equation which have compact support on H + . As is well known, the method, used by Morawetz [26], John [20], Klainerman [21,22], Dafermos [8,9,10,11,12] and Rodnianski [11,22], among many others, consists in applying the Divergence Theorem to some currents obtained by contracting the energy-momentum tensor T µν with appropriate vector field multipliers constructed specifically according to each region of spacetime.
We refer to the region close to the Cauchy horizon as the blue-shift region, and the region close to the event horizon as the red-shift region. We call the intermediate region the no-shift region. A very general construction of red-shift vector fields on general spacetimes which contain Killing horizons with positive surface gravity is carried out in the lecture notes [11]. Here we perform the computations explicitly in double null coordinates.
The blue-shift vector field (84) is constructed using the vector field Y in Lemma 3.1, and the red-shift vector field (95) is constructed using the vector field V in Lemma 3.2. We go on to calculate the covariant derivative of Y , ∇ µ Y ν , and the scalar current associated to Y , T µν ∇ µ Y ν . We obtain the usual inequalities for the currents associated to the blue-shift vector field, and for the currents associated to the red-shift vector field. We finish with Theorem 3.5, which is Sbierski's result, for the Reissner-Nordström and Kerr spacetimes, applied to Kerr-Newman-de Sitter spacetimes.

Construction
Here the blue-shift vector field is defined to be where Y is given in Lemma 3.1. Let ι ∈ R + be given. The initial value problem (80)) has a unique time invariant solution, Y , defined in a neighborhood of the Cauchy horizon, i.e. defined for r ⋆ sufficiently large. Proof.
(a) (Y time invariant.) The vector field 1 In fact, we have 1 then, Y commutes with ∂ t .
(c) (An auxiliary calculation.) In the next step we will use the following identity. We mention that it implies thath is not identically equal to zero. Using (110) and (111), we have (92) (d) (The characteristics do not cross the boundary.) We now check that h(r, 0) = 0 andh r, π 2 = 0.
This implies V θ (r, 0) = 0 and V θ r, and guarantees that the characteristics of our differential equations do not leave the region, [r − , r + ]× 0, π 2 , where we want to solve our system. It also guarantees that Y is well defined when θ = 0, notwithstanding ∂ θ⋆ not being well defined when θ = 0. The vanishing ofh at θ = π 2 can also be seen as a consequence of the symmetry of our problem under the reflection θ → π − θ, which implies that Y should not have any component in the ∂ θ⋆ direction at the equators of the spheres where u and v are both constant. The right hand side of (90) consists of a sum of terms which we divide into two parts. The first part consists of sum of the eight summands that haveh as a factor. The term is proportional toh. The second part consists of the sum of the remaining eight summands, which are dθ ⋆ applied to Taking into account (92), the second part is At θ ⋆ = 0 and at θ ⋆ = π 2 this sum is zero because each of the terms is equal to zero. Indeed, all terms are a product of a differentiable function by sin(2θ ⋆ ). This is easy to check. Let us exemplify this assertion with one of the less immediate terms to analyze, the one which contains (see (105)). Since both g / θ⋆φ⋆ and g / θ⋆φ⋆ contain sin(2θ ⋆ ), so does this Christoffel symbol.
(e) (Existence and uniqueness of solution.) Using (12) and (38), we have This shows that the segment {r − } × 0, π 2 is noncharacteristic for our system of four first-order quasilinear partial differential equations.
We observe that when the Christoffel symbols Γ φ⋆ vθ⋆ and Γ φ⋆ θ⋆φ⋆ , which blow up at θ ⋆ = 0 like 1 sin θ⋆ (see Corollary A.2), appear in the system above, then they appear multiplied byh, which has to vanish to first order at (r, 0). So that the summands where these Christoffel appear are continuous functions.
By a standard existence and uniqueness theorem for non characteristic first order quasilinear partial differential equations, we know that our initial value problem has a solution for (r, θ) ∈ [r − , r − + δ] × [0, π], for some positive δ. Recall that constructing the solution involves solving the system of ordinary differential equations right-hand side of (88) g = right-hand side of (89) h = right-hand side of (90)  = right-hand side of (91) with We know from Remark 2.1 that, for r ⋆ sufficiently large, r is close to r − . So we have the existence of a solution for large r ⋆ .

✷
Here the red-shift vector field is defined to be where V is given in Lemma 3.2. Let ι ∈ R + be given. The initial value problem (82)) has a unique time invariant solution, V , defined in a neighborhood of the event horizon, i.e. defined for r ⋆ sufficiently negative (i.e. for r ⋆ < −C with C sufficiently large).

Covariant derivative
One could consider working in the frame (Z, Y, ∂ θ⋆ , ∂ φ⋆ ) but this is not a good choice because the energymomentum tensor does not have a simple expression in this frame. So, instead, we define and work in the frame (X T , X Y , X θ⋆ , X φ⋆ ) := (T , Y , ∂ θ⋆ , ∂ φ⋆ ).

Currents
The energy momentum tensor of a massless scalar field and the vector current. The energy momentum tensor is given by and one readily checks that The energy-momentum tensor is written as The vector currents associated to the blue-shift and red-shift vector fields are respectively.
The scalar current associated to the blue-shift vector field. The scalar current associated to N b is Since Z is a Killing vector field this is equal to K Y , which is We estimate K Y . Suppose we are given δ ∈ 0, 1 3 . Let c = 2(−κ − )δ. Choose ι = c + c + 1, where c = c δ is the constant below (which is independent of Y ). There exists r 0 > r − such that r ∈ (r − , r 0 ) implies that We have used the fact that g / φ⋆φ⋆ (∂ φ⋆ ψ) 2 ≤ |∇ / ψ| 2 g / .

Energy estimates
We are interested in solutions of the wave equation which are regular up to, and including, H + , and which have compact support on H + , i.e. we are interested in functions belonging to the space F := ψ ∈ C ∞ (M ∪ H + ) : ✷ g ψ = 0 and there exists v 0 ∈ R such that ψ| H + ∩{v≥v0} = 0 .
The following theorem, established by Sbierski in his thesis [29] for Reissner-Nordström and Kerr black holes, applies to Kerr-Newman-de Sitter black holes.
Theorem 3.5. Let 2κ + > −κ − and ψ ∈ F . Then, for any u 0 , v 0 ∈ R, we have Proof. We sketch the proof and refer to [29] for further details.
(a) Estimates in the red-shift region. For κ < κ + , define N r = e 2κv N r . Choose the δ in (102) such that κ < κ + (1 − δ). Since K N r ≥ 0, the Divergence Theorem implies that we have The left-hand side can be bounded below by where u(u H + ) denotes the u corresponding to u H + . As ψ is a regular function on H + , we have We conclude that (b) Estimates in the no-shift region. We recall [29,Lemma 4.5.6]: Given (r ⋆ ) 1 > (r ⋆ ) 0 and a smooth future directed timelike time invariant vector field N , there exists a constant C > 0 such that holds for all solutions of the wave equation. Then, for all 0 < α 2 < α 1 and t ≥ t 0 , we have By assumption 2κ + > −κ − . If we choose κ < κ + sufficiently close to κ + and and κ < κ − sufficiently close to κ − , then 2κ > −κ. Using (103), (104) and the fact that are comparable, we conclude that This finishes the proof. ✷
Using (40), we see that ∂θ ∂r⋆ contains a factor sin(2θ ⋆ ). Moreover, We recall that, according to (42), ∂θ ∂θ⋆ is bounded. Thus, recalling (27), ∂ r⋆ (A sin 2 θ) behaves like sin 2 θ ⋆ and ∂ θ⋆ (A sin 2 θ) behaves like sin(2θ ⋆ ). On the other hand, according to Lemma 2.15, g / θ⋆φ⋆ = A sin 2 θ ⋆ sin(2θ ⋆ ), for another smooth function A. Therefore, ∂ θ⋆ g / θ⋆φ⋆ behaves like sin 2 θ ⋆ . ✷ Note that the vector fields ∂u Ω 2 and 1 Ω 2 ∂ v + b φ⋆ ∂ φ⋆ are geodesic: Since we work in the frame it is also convenient to have the following covariant derivatives: A.2 (u, v CH + , θ ⋆ , φ ⋆,CH + ) coordinates If we use coordinates (u, v CH + , θ ⋆ , φ ⋆,CH + ) then, in the formulas above, the value of ς continues to be 1, but one has to replace Ω 2 by Ω 2 CH + , b φ⋆ by b φ⋆ CH + , and the value of σ has to be as in (79). Indeed, the expressions σ∂ r⋆ arise when taking derivatives with respect to v CH + . Now, using (76) and (78), we have But the coefficients of the metric do not depend either on φ ⋆ or on t. So, when computing the Christoffel symbols, and differentiating functions that depend exclusively on r and θ, the derivative ∂ v CH + may be replaced by The expressions for the innermost parenthesis are obtained from (38) and (40). Note that they are well defined on the Cauchy horizon, notwithstanding the coordinate r ⋆ not being defined there. The factor e −2κ−v ∆ r is bounded above and below according to (75). So, given C R ∈ R, there exist constants c, C > 0 such that for u + v ≥ C R . Special care has to be taken when differentiating b φ⋆ CH + and Ω 2 CH + with respect to v CH + because these functions also depend on v. This occurs when calculating Γ * v CH + v CH + : Notice that all terms are bounded except the last two summands of Γ φ⋆ v CH + v CH + that add up to We remark that no terms that blow up at the Cauchy horizon appear when calculating ∇ X † X ‡ , for X † and X ‡ elements of the frame (109). For example, we have that Example A.3. As a simple example, we calculate directly the surface gravity of the Cauchy horizon. Recall that the surface gravity κ − is given by where Z is the Killing vector field given by (80). On the Cauchy horizon ∇ ∂u ∂ φ⋆ = ∇ ∂ φ⋆ ∂ u = 0 apply, because Γ u uφ ⋆,CH + = Γ θ⋆ uφ ⋆,CH + = Γ φ ⋆,CH + uφ ⋆,CH + = 0 on the Cauchy horizon, and Γ v uφ ⋆,CH + is identically zero. Obviously, hold because the vector fields ∂ u and ∂ φ⋆ are tangent to the Cauchy horizon. Moreover, we know that So, we conclude that Z.
Remark B.7. If we start with a parameter set (Λ, M, a, e), we can regard (115) and (118) as a system for r − and α. In this case, it is natural to consider the quantities In terms of these quantities, (115) and (118) can be written as The equation (142) can be solved for A, yielding The system (143)−(144) allows us to determine A and R from √ ΛM , Λa 2 and Λe 2 . Of course, another way to do this would be by calculating the roots of the polynomial ∆ r .

C Komar integrals
The purpose of this appendix is to obtain expression (149), for the mass of the black hole at the event horizon. Besides the expected term M Ξ 2 , the mass has one correction term which is a constant multiple of the product of the cosmological constant by the Parikh volume of the black hole, and has another term which reflects the energy due to the electric field. In this way we recall the physical interpretation of the parameters of our metric. LetK := ∂ φ⋆ . Consider the 2-dimensional surface on the event horizon S = {(u H + , v H + , θ ⋆ , φ ⋆,H + ) : u H + = 0, v H + = (v H + ) 0 }, where v H + = (v H + ) 0 , for some fixed (v H + ) 0 . Let us calculate the Komar integral (JQ is not the angular momentum if e = 0. See [15,Subsection III.B] for the definition of the angular momentum.) Remark C.1. In the calculation below we use the coordinates (u, v, θ ⋆ , φ ⋆ ) of Subsection 2.1.3, notwithstanding the fact that S is contained in the event horizon, where these coordinates are not defined. Our computations may be easily justified by considering a sequence S n of surfaces which approximate S from within the black hole, which is covered by the coordinates, computing the integrals over S n , and then passing to the limit.