Universal horizons and black holes in gravitational theories with broken Lorentz symmetry

# Universal horizons and black holes in gravitational theories with broken Lorentz symmetry

## Abstract

In this paper, we first show that the definition of the universal horizons studied recently in the khronometric theory of gravity can be straightforwardly generalized to other theories that violate the Lorentz symmetry, by simply considering the khronon as a probe field and playing the same role as a Killing vector field. As an application, we study static charged ()-dimensional spacetimes in the framework of the healthy (non-projectable) Horava-Lifshitz (HL) gravity in the infrared limit, and find various solutions. Some of them represent Lifshitz space-times with hyperscaling violations, and some have black hole structures. In the latter universal horizons always exist inside the Killing horizons. The surface gravity on them can be either larger or smaller than the surface gravity on the Killing horizons, depending on the space-times considered. Although such black holes are found only in the infrared, we argue that black holes with universal horizons also exist in the full theory of the HL gravity. A simple example is the Schwarzschild solution written in the Painleve-Gullstrand coordinates, which is also a solution of the full theory of the HL gravity and has a universal horizon located inside the Schwarzschild Killing horizon.

###### pacs:
04.60.-m; 98.80.Cq; 98.80.-k; 98.80.Bp

## I Introduction

Lorentz symmetry has been the cornerstone of modern physics, and verified to such a high accuracy that any modification of it must face one of the most severe experimental constraints existing today in physics Liberati13 (), although it is arguable that such constraints in the matter sector are much stronger than those in the gravitational sector LZbreaking (). Nevertheless, if space and time emerge from some discrete substratum, as what we currently understand, this symmetry must be an accidental one at low energies.

Following this line of thinking, various gravitational theories that violate Lorentz symmetry have been proposed, ranging from ghost condensation GC (), Einstein-aether theory EA (), and more recently, to Horava-Lifshitz (HL) gravity Horava (). While the ghost condensation and Einstein-aether theory are all considered as the low energy effective theories of gravity, the HL gravity is attempted to be ultraviolet (UV) complete, and by construction is power-counting renormalizable reviews (). It is consistent with all the solar system tests carried out so far EA (); KMWZ () and binary pulsar observations Yagi (), and exhibits various remarkable features when applied to cosmology InflationA (); InflationB (); InflationC (); InflationD ().

However, when applying the HL theory to astrophysics, it seems to indicate that black holes are only low energy phenomena KKCP (); GLLSW (). This is because, in order to have the theory power-counting renormalizable, high-order spatial operators up to at least sixth-order must be included Horava (). Hence, the dispersion relation becomes nonlinear, and generically takes the form reviews (),

 E2=c2pp2(1+α1(pM∗)2+α2(pM∗)4), (1.1)

where and are the energy and momentum of the particle considered, and are coefficients, depending on the particular specie of the particle, while denotes the suppression energy scale of the higher-dimensional operators. Then, one can see that both phase and group velocities of the particles are unbounded with the increase of energy. This makes the causal structure of the spacetimes quite different from that given in general relativity (GR), where the light cone of a given point plays a fundamental role in determining its causal structure relative to other events. In the case described by Eq.(1.1), the causal structure is very much similar to the Newtonian case GLLSW (). This suggests that black holes may not exist at all in the HL theory, as any ray initially trapped inside a horizon can penetrate it and propagate to infinity, as long as the ray has sufficiently large velocity. On the other hand, in the infrared (IR) the high-order terms are negligible, and the first term in Eq.(1.1) becomes dominant, so one may still define black holes, following what was done in GR 2.

Surprisingly, in contrast to the above physical intuition, recently it was shown that there still exist absolute causal boundaries, the so-called universal horizons, and particles even with infinitely large velocities would just move around on these boundaries and cannot escape to infinity BS11 (). This has immediately attracted lot of attentions UHsA (); UHsB (); BBM (); CLMV (); SVV (). In particular, it was shown that the universal horizon radiates like a blackbody at a fixed temperature, and obeys the first law of black hole mechanics BBM (). The main idea is as follows: In a given space-time, a timelike foliation parametrized by Constant might exist globally. Since the surfaces are timelike, we must have , where is the normal unit vector of the surface, defined as

 uμ=ϕ,μ√X, (1.2)

with . The signatures of the metric are . Among these surfaces, there may exist a surface at which diverges, while physically nothing singular happens there, including the metric and the space-time. Given that defines an absolute time, any object crossing this surface from the interior would necessarily also move back in absolute time, which is something forbidden by the definition of the causality in the theory. Thus, even particles with superluminal velocities cannot penetrate this surface, once they are trapped inside it. For more details, we refer readers to BS11 ().

In this paper, our purposes are twofold: First, we shall generalize the above definition of the universal horizons to any gravitational theory that may or may not violate the Lorentz symmetry, although such a generalization might be useful only for theories that violate the Lorentz symmetry. In BS11 (), was referred to as the “khronon” field, and considered as describing one degree of freedom of the gravitational field, a spin-0 graviton. To generalize the definition of the universal horizons to other theories, in this paper we shall promote it to the same role as played by a Killing vector of a given space-time, so its existence does not affect the background, but defines the properties of a given space-time. By this way, such a field is no longer part of the underlaid gravitational theory and it may or may not exist in a given space-time, depending on the properties of the space-time considered. Second, we shall find static charged solutions of the healthy extensions of the HL gravity BPS () to show that the universal horizons exist in some of these solutions. Such horizons exist not only in the IR limit of the HL gravity, as has been considered so far in BS11 (); UHsA (); UHsB (); BBM (); CLMV (); SVV (), but also in the full HL gravity, that is, when high-order operators are not negligible.

The rest of the paper is organized as follows: In Sec. II, we generalize the definition of the universal horizons first discovered in BS11 () in the khronometric theory, which is equivalent to the Einstein-aether theory with the hypersurface-orthogonal condition Jacobson10 (); Wang13 (), to any theory by considering the khronon field as a probe field, quite similar to a Killing vector field existing in a given space-time, so that whether a khronon field exists or not depends totally on the properties of a given space-time. In Sec. III, we consider charged static spacetimes in the framework of the non-projectable HL gravity in the IR limit, and present various classes of solutions. In Sec. IV, we study the existence of universal horizons in some of the solutions presented in Sec. III, and find that universal horizons indeed exist. The paper is ended with Sec. V, in which we derive our main conclusions and present some discussing remarks. An appendix is also included, in which we briefly review the non-projectable HL gravity in ()-dimensional spacetimes when coupled with an electromagnetic field.

It should be noted that electromagnetic static spacetimes in other versions of the HL theory were studied in EMa (), while a new mechanism for generation of primordial magnetic seed field in the early universe without the local U(1) symmetry was considered in EMb (). However, so far no studies of the existence of the universal horizons have been carried out in these models.

## Ii Causal Structure of Gravitational Theories with Broken Lorentz Symmetry and Universal Horizons

As shown in the last section, once the Lorentz symmetry is broken, the speed of particles can become superluminal, and even instantaneous propagation exists. Then, the causal structure will be quite different from that in theories with Lorentz symmetry, in which light-cones play a central role. It should be noted that the violation of the Lorentz symmetry does not mean the violation of the causality. In fact, in such theory the causality still exists, but different from that given, for example, in GR. In particular, now the causal structure of a given point is uniquely determined by the time difference, , between the two events and . If , the event is to the past of ; if , it is to the future; and if , the two events are simultaneous [cf. Fig.1].

As a result, all the definitions of black holes in terms of event horizons HE73 (); Tip77 (); Hay94 (); Wang () become invalid, as a particle initially trapped inside such a horizon now can penetrate it and propagate to infinity, as long as its velocity is sufficiently large. To provide a proper definition of black holes, anisotropic conformal boundaries HMT2 () and kinematics of particles KM () have been studied in the framework of the HL gravity. In particular, defining a horizon as the infinitely redshifted 2-dimensional (closed) surface of massless test particles KKb (), it was found that for test particles with sufficiently high energy, the radius of the horizon can be made as small as desired, although the singularities can be seen in principle only by observers with infinitely high energy GLLSW (). This is expected, as such horizons are similar to the event horizon defined in GR HE73 ().

Remarkably, studying the behavior of a khronon field in the fixed Schwarzschild black hole background,

 ds2=−(1−rsr)dv2+2dvdr+r2dΩ2, (2.1)

where , Blas and Sibiryakov showed that a universal horizon exists inside the Schwarzschild radius BS11 () 3. This surface, in contrast to the event horizon, now is spacelike, and on which the time-translation Killing vector becomes orthogonal to ,

 uμζμ=0. (2.2)

Since is well-defined in the whole space-time, and remains timelike from the asymptotical infinity () all the way down to the space-time singularity (), Eq.(2.2) is possible only inside the Killing horizon (), as only there becomes spacelike and can be possibly orthogonal to .

The khronon defines globally an absolute time, and the trajectory of a particle must be always along the increasing direction of . Thus, once the particle across the universal horizon, the only destination is to move towards the space-time singularity, and arrive at it within a finite (proper) time, as shown in Fig.2. From this same figure, one can also see that the normal vector is pointing outwards for some at the event horizon . That is, for a particle with a sufficient large velocity (larger than that of light), it can escape from the interior of the event horizon to asymptotically-flat infinities. In particular, near the universal horizon the khronon behaves like BS11 (),

 ϕ≃v+log(ξUH−ξ)ξ2UHu′τ√ξ2UH−1, (2.3)

where , and (or is the location of the universal horizon. is the -component of the khronon field, and . Here is the Schwarzschild timelike coordinate, defined as BS11 ()

 τ=v−[r+rsln(rrs−1)]. (2.4)

The hypersurfaces of Constant are illustrated in Fig.2, from which we can see that these curves are all accumulated to the one , which is the location of the universal horizon and only particles with infinitely large velocities can move around on this surface. A particle inside this surface cannot get out of it, no matter how large its velocity would be.

It should be noted that the singularity of the khronon on the universal horizon is not physical, and can be removed by the gauge transformation,

 ~ϕ=F(ϕ), (2.5)

allowed by the symmetry of the khronon field, where is an arbitrary monotonic function of . In this sense, the khronon is quite different from a usual scalar field.

To generalize the above definition of the universal horizon to other gravitational theories, one can see that two important ingredients are essential: the existence of the khronon field , and the existence of the asymptotically timelike Killing vector . For a given space-time, the latter can be obtained by solving the Killing equation,

 Dνζμ+Dμζν=0, (2.6)

where denotes the covariant derivative with respect to the ()-dimensional metric .

To find the equation for the khronon, we start with its general action EA (),

 Sϕ = ∫dD+1x√|g|[c1(Dμuν)2+c2(Dμuμ)2 (2.7)

where , and ’s are arbitrary constants. It should be noted that the above action is the most general one in the sense that the resulting differential equations in terms of are second-order EA (). However, when is hypersurface-orthogonal, that is, when satisfies the relation,

 u[νDαuβ]=0, (2.8)

only three of them are independent 4, as in this case we have the identity EA (),

 ΔLϕ≡aμaμ−(Dαuβ)(Dαuβ)+(Dαuβ)(Dβuα)=0. (2.9)

Then, one can always add the term,

 ΔSϕ=α∫√|g|dD+1xΔLϕ, (2.10)

into , where is an arbitrary constant. This is effectively to shift the coupling constants to , where

 c′1=c1+α,c′2=c2,c′3=c3−α,c′4=c4−α. (2.11)

Thus, by properly choosing , one can always set one of to zero. However, in the following we shall leave this possibility open.

Hence, the variation of with respect to yields the khronon equation,

 DμAμ=0, (2.12)

where Wang13 () 5,

 Aμ ≡ ≡ DγJγν+c4aγDνuγ, Jαμ ≡ (c1gαβgμν+c2δαμδβν+c3δανδβμ (2.13) −c4uαuβgμν)Dβuν.

Eq.(2.12) is a second-order differential equation for , and to uniquely determine it, two boundary conditions are needed. These two conditions can be chosen as follows BS11 (): (i) One of them is to require it to be aligned asymptotically with the timelike Killing vector,

 uμ∝ζμ. (2.14)

(ii) The second condition can be that the khronon has a regular future sound horizon, which is a null surface of the effective metric EJ (),

 g(ϕ)μν=gμν−(c2ϕ−1)uμuν, (2.15)

where denotes the speed of the khronon 6.

With the above definition of the universal horizon, several comments now are in order. First, the above definition does not refer to any particular theory of gravity. Therefore, it is applicable to any theory that violates the Lorentz symmetry, including the Einstein-aether theory EA () and the HL gravity Horava (). But, there is a fundamental difference between the khronon introduced in this paper and the one (a particular aether field with hypersurface-orthognal condition) considered in BS11 (); UHsA (); UHsB (); BBM (). In this paper, the khronon plays the same role as a Killing vector field , both of them describe only some properties of a given space-time and have no effects on the given space-time. But, in BS11 (); UHsA (); UHsB (); BBM () the khronon was considered as a part of the gravitational field, although in some cases their effects on the gravitational fields were assumed to be negligible. Second, in the literature it has been often considered that the Einstein-aether theory with the hypersurface-orthogonal condition (2.8) is equivalent to the non-projectable HL gravity in the low energy limit. This is incorrect, as the two theory have different gauge symmetries, and they are equivalent only with a particular choice of the gauge, the -gauge, in which the aether is aligned with the time coordinate , that is, choosing , as shown explicitly in Jacobson10 (); Wang13 (). In the following, we shall refer to these coordinates as the -coordinates. In particular, the Einstein-aether theory is gauge-invariant under the general coordinate transformations,

 t=ξ0(t′,x′k),xi=ξi(t′,x′k), (2.16)

where are arbitrary functions of their indicated arguments. While the HL gravity is gauge-invariant only under the foliation-preserving diffeomorphism,

 t=f(t′),xi=ξi(t′,x′k). (2.17)

A typical example is the Schwarzschild space-time written in the Painleve-Gullstrand coordinates PG (),

 ds2=−dt2+(dr+√rsrdt)2+r2dΩ2. (2.18)

This solution is also a solution of the HL gravity (not only in the IR but also in the UV as now and all high-order operators of vanish) GLLSW (), but the one given by Eq.(2.1) is not. This is because theses two solutions are connected by the coordinate transformations,

 dt=dv−dr1+√rsr, (2.19)

which are forbidden by the gauge transformations of Eq.(2.17), although they are allowed by the ones of Eq.(2.16). Therefore, in the Einstein-aether theory these two solutions describe the same space-time, but in the HL gravity they do not, as they are not connected by any coordinate transformations allowed by its gauge symmetry (2.17). More examples of this kind can be found in CW ().

Therefore, the Einstein-aether theory with the hypersurface-orthogonal condition is equivalent to the non-projectable HL gravity in the low energy limit only in the -coordinates, in which the timelike foliations of the space-time fixed in the HL gravity coincide with the spacelike hypersurfaces Constant 7.

The equivalence shown in Jacobson10 () is actually the equivalence between the Einstein-aether theory with the hypersurface-orthogonal condition (2.8) and the khronometric theory BPS (), as both of them are gauge-invariant under the general covariant coordinate transformations (2.16) and have the same degree of freedom. For more details, we refer readers to Jacobson10 (); Wang13 (); BS11 ().

As an application of the universal horizons defined above, in the next section we shall find static charged solutions in the framework of the HL gravity without the projectability condition in the low energy limit BPS (). In Sec. V, we study their local and global properties, by paying particular attention to the existence of the universal horizons.

## Iii Static charged Lifshitz-type Solutions in non-projectable (D+1)-dimensional HL Gravity

The non-projectable HL gravity in (D+1)-dimensional space-time is briefly reviewed in Appendix A, in which all the field equations are derived, including the generalized Maxwell equations.

In this paper, we shall study static spacetimes described by,

in the coordinates (), where , and is a constant. The dimensional Ricci scalar curvature is given by

 R=2drg′(r)g3(r)−d(d+1)g2(r). (3.2)

In the IR limit, all the operators higher than order 2 can be safely ignored, as mentioned above. Then, from the Maxwell equations (A.16) and (V.1), we obtain

 A1α′0=0, (3.3) A′′0A′0−g′g−f′f+d+1−zr=0. (3.4)

Thus, must be a constant. Then, from Eqs.(A.4) and (A.5) we can see that now acts like a cosmological constant, and can be absorbed to . Therefore, without loss of the generality, we shall set in the rest of the paper. On the other hand, from Eq.(3.4) we find that,

 A′0=qrd+1−zf(r)g(r), (3.5)

where is an integration constant.

Substituting the above ADM variables and into the rest of field equations, we find that the momentum constraint vanish directly, but the Hamiltonian constraint and - and -components of the dynamical equations are non-trivial. It can be shown that the -component of the dynamical equations can be derived from the Hamiltonian constraint and the -component. Therefore, in the current case there are only two independent field equations, after integrating out the electromagnetic field equations (3.3) and (3.4), which are sufficient to determine the two unknown functions and . The Hamiltonian constraint and the -component of the dynamical equations are given, respectively, by

 −2r2βf′′f+(2r2βg′g−2r(d+1+z)β)f′f +2r(zβ+dγ1)g′g+(2Λ+q22g2er2d)g2 +r2βf′2f2−2dzβ−z2β−d(d+1)γ1=0, (3.6) r2β2f′2f2+(zβ+dγ1)rf′f+z2β2+dzγ1 −(Λ+q24g2er2d)g2+d2(d−1)γ1=0, (3.7)

where

 Λ≡γ0ζ22. (3.8)

To solve the above equations, we first note that Eq.(III) can be cast in the form,

 2dβγ1W+β2W2−d2γ21(r2∗−1)=0, (3.9)

where

 r∗(r) ≡  ⎷r2s+2βg2d2γ21(Λ+2q28g2er2d), W(r) ≡ z+rf′(r)f(r), r2s ≡ 1−(d−1)βdγ1. (3.10)

Inversely, we find that

 g2=d2γ212β(r2∗−r2s)(Λ+2q28g2er2d)−1. (3.11)

From Eq.(3.9), we obtain

 W=s1+ϵr∗(r)1−s, (3.12)

where , and

 s≡dγ1dγ1−β. (3.13)

Then, from the stability conditions (A.24), we get

 1≤s

which implies that

 0

where the equality holds only when or .

When , inserting Eq.(3.12) into the Hamiltonian constraint, we obtain a master equation for ,

 2dq2r∗2q2+C2r2d+rsr2∗r′∗1−d+(d−2)s+sr2∗ =ϵ(11−s−2+d)+(d+s1−s)r∗+rr′∗, (3.16)

which can be further rewritten as

 (s−1)rr′∗+Δ(r)(r2∗−r2s)(r∗+ϵD(r))=0, (3.17)

where , and

 D ≡ [(d−1)+(2−d)s][2q2+C2r2d]2q2s+C2r2d[d+(1−d)s], C2 ≡ 8g2eΛ. (3.18)

When , we find that

 D = dγ1−(d−1)βd(γ1−β),(q=0), (3.19)

which reduces to the case considered in SLWW (); LSWW (). So, in the rest of the paper we consider only the case . Unlike the case without the electromagnetic field, now it is found difficult to find the general solutions of Eq.(3.17). Therefore, in the following we consider some particular cases.

### iii.1 s=1

When , from Eq.(3.13) we find that this implies . Then, the stability and ghost-free conditions require

 λ=1,(β=0) (3.20)

as one can see from Eq.(A.20). Therefore, whenever we consider the case (or ) we always assume that . To study the solutions further, we consider the two cases and , separately.

#### d≥2

Then, we find Eqs.(III) and (III) reduce to

 2rdγ1g′g+(2Λ+q22g2er2d)g2 −d(d+1)γ1=0, (3.21) dγ1rf′f+dzγ1+d2(d−1)γ1 −(Λ+q24g2er2d)g2=0, (3.22)

from which we get

 g2 = 4g2ed(d2−1)γ1r2d+1[(d−1)rd(C2rd+1 +4g0g2ed(d+1)γ1)−2q2(d+1)r]−1, f = f0g−1r1−z, (3.23)

where and are two integration constants. Rescaling , the metric can be cast in the form,

 ds2=−F(r)dt2+dr2F(r)+r2δijdxidxj, (3.24)

where

 F(r)=−2mrd−1+Q2r2d−2−2Λer2d(d+1), (3.25)

with , and

 Q2≡q22|γ1|g2ed(d−1),Λe≡Λ|γ1|. (3.26)

Note that in writing the above expressions we had used the fact

 γ1<0,(d≥2) (3.27)

as one can see from Eq.(A.25). The corresponding Ricci scalar of the surfaces Constant is given by

 R=−r2dC2+2q24g2eγ1r2d, (3.28)

which is singular at when . On the other hand, the -dimensional Ricci scalar is given by

 (d+2)R = −F′′−2dF′r−d(d−1)Fr2 (3.29) = 2d+2dΛe−(d−2)(d−1)q2r2d,

which is also only singular at , provided that .

The above solutions are nothing but the topologically Reissner-Nordstrom (anti-) de Sitter solutions, and the Penrose diagrams have been given for various possibilities for in Yumei (). It can be shown that these diagrams can be easily generalized to the case with any .

#### d=1

In this case, since , from Eq.(A.29) we find that

 γ1is arbitrary,(d=1) (3.30)

and can take any of real values. Thus, from Eqs.(III) and (III) we find that

 f = f0r−z√∣∣C2r2+8g2eγ1g0+4q2ln(r)∣∣, g2 = 8g2eγ1r2C2r2+8g2eγ1g0+4q2ln(r). (3.31)

Then, after rescaling the coordinate , the metric can be cast in the form,

 ds2=−F(r)dt2+dr2F(r)+r2dx2, (3.32)

but now with

 F(r)=−M+Aln(r)+Br2, (3.33)

where

 M≡−g0,A≡q22γ1g2e,B≡Λγ1. (3.34)

The corresponding 2d Ricci scalar is given by

 R=−A+2Br2r2, (3.35)

which is singular only at for . The nature of this singularity depends on the signs of . In particular, when and , the above solutions are identical to the charged Banados-Teitelboim-Zanelli (BTZ) black hole BTZ () with

 (M,Q,Λe)=(−g0,√−A,−B), (3.36)

where and denotes, respectively, the BTZ black hole mass, charge, and the the effective cosmological constant.

But, the solutions of Eq.s(3.33) are more general than the charged BTZ black hole. In particular, is not necessarily negative, as now is a free parameter. To study these solutions further, we consider the cases , and , separately.

Case i) . In this case, one can see that the space-time is always asymptotically anti-de Sitter. However, depending on the signs of , the solutions can have different properties. To see these clearly, let us consider the cases , and , separately.

Case i.1) : In this case, we find that

 γ1>0,Λ>0, (3.37)

for which we have

 F(r)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−∞,r=0,<0,r0,r>rEH,∞,r=∞, (3.38)

where is the unique real root of , as shown in Fig. 3. Thus, in this case the singularity is covered by the Killing horizon located at . The space-time is asymptotically anti-de Sitter with the effective cosmological constant given by . The surface gravity on the Killing horizon is given by

 κEH=B(r2EH+r2c)rEH, (3.39)

which is always positive, where .

Case i.2) : In this case, we have , and the corresponding solutions are not charged. Then, we find,

 F(r)=−M+Br2=⎧⎨⎩−M,r=0,0,r=rEH,∞,% r=∞, (3.40)

where . The surface gravity on the Killing horizon now is given by

 κEH=√BM. (3.41)

Case i.3) : In this case, we find

 γ1<0,Λ<0, (3.42)

and for both and . Thus, depending on the values of (or ), the equation can have two, one or none real roots. In particular, we have , where . Thus, we find

 F(rm) = |A|2[ln(2B|A|)+1]−M (3.43) = ⎧⎨⎩>0,Λ<Λc,=0,Λ=Λc,<0,0>Λ>Λc,

where

 Λc≡−12|γ1A|e2M−|A||A|. (3.44)

Then, the behavior of is illustrated in Fig.4, from which we can see that the singularity at is naked in Case (a), in which we have . In case (c), where , there exist two Killing horizons, located, respectively, at and . In this case, the Penrose diagram is similar to the Reissner-Nordstrom anti-de Sitter solutions. In particular, the surface gravity is negative at , while positive at , as one can see from Fig. 4. In Case (b), the two horizons become degenerate, and the surface gravity is zero.

Case ii) . In this case, we find that

 F(r)=−M+Aln(r). (3.45)

Thus, depending on the values of , the solutions can have different properties. In particular, when () the function is monotonically increasing (deceasing) as shown in Fig. 5. Then, there always exists a point at which . The surface gravity on this Killing horizon is positive for and negative for . When , the space-time is flat.

Case iii) : In this case, we have

 F′(r) = 2|B|r2m−r2r, F(rm) = −M+A2[ln(A2|B|)−1] (3.46) = ⎧⎨⎩>0,|B|Bm,

where

 rm≡√A2|B|,Bm≡12Ae−(2M+A)A. (3.47)

Fig. 6 shows the curve of vs . In the case , we can see that is always negative, and the coordinate is timelike, and the corresponding space-time is dynamical, and has a spacelike naked singularity at .

In the case , a coordinate singularity appears at , as it can be seen from Eq.(3.35), which now takes the form,

 R=−2|B|r2m−r2r2. (3.48)

Therefore, in order to obtain a geodesically maximal space-time, extension across this surface is needed. Such an extension is quite similar to the one of the extremal case, , of the Schwarzschild-de Sitter solution, . The corresponding surface gravity is zero, as it can be seem from Eq.(III.1.2).

In the case , is positive only for , where are the two real roots of , with . From Eq.(3.48) we can see that the singularities at represent horizons, and the extensions beyond these horizons are similar to the Schwarzschild–de Sitter solution with . The corresponding surface gravity is given by

 κ±=|B|r2m−r2±r±, (3.49)

which is positive only when .

It should be noted that the above analysis holds only for .

When , we have , and the solutions represent dynamical space-time, in which () is always timelike (spacelike), and the singularity at is naked.

When , we find that

 F(r) = −M−|A|ln(r)−|B|r2 (3.50) = {∞,r=0,−∞,r=∞,

which is monotonically decreasing function of [cf. Curve (c) in Fig.5], and asymptotically approaches to the de Sitter space-time. The surface gravity at now clearly is negative.

### iii.2 Λ=0

In this case, from Eq.(III) we find

 D=r2s=d−1+s(2−d)s. (3.51)

Since , from Eqs.(3.14) and(3.15) we find that the stability condition (A.24) leads to

 0

Without loss of the generality, we set in Eq.(3.12). Then, Eq.(3.17) can be cast in the form,

 drr = [rs−12(d−1)(r∗−rs)−rs+12(d−1)(r∗+rs) (3.53) +1(d−1)(r∗+r2s)]dr∗,

from which we find that the general solution,

 r(r∗)=rEH∣∣r∗+r2s∣∣1d−1|r∗−rs|1−rs2(d−1)|r∗+rs|rs+12(d−1), (3.54)

where is a constant. Then, we obtain

 r(r∗)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩rEH,r∗→−∞,∞,r∗=−rs,0,r∗=−r2s,∞,r∗=rs,rEH,r∗→+∞, (3.55)

Fig. 7 shows the curve of vs .

On the other hand, from Eqs.(III) and (3.12) we find

 dff = [d−1+z−zrs2(d−1)(r∗−rs)+d−1+z+zrs2(d−1)(r∗+rs) (3.56) +1−d−z(d−1)(r∗+r2s)]dr∗,

which has the general solution,

 f=f0|r∗−rs|δ1|r∗+rs|δ2∣∣r∗+r2s∣∣−d−1+zd−1, (3.57)

where is another integration constant, and

 δ1≡d−1+z−zrs2(d−1),δ2≡d−1+z+zrs2(d−1). (3.58)

Finally, from Eq.(3.11) we find

 g2(r∗)=C1r2d(r2s−r2∗), (3.59)

where . Since , we find that is always positive, . Then, to have positive, we must restrict ourselves to the region . Hence, the corresponding metric takes the form,

 ds2=−N2dt2+G2dr2+δijr2dxidxj, (3.60)

where

 N2 = N20r2s−r2∗(r∗+r2s)2, G2 = C1r2(d−1)(r2s−r2∗), (3.61)

where . The corresponding Ricci scalar is given by

 R = R0(r∗−rs)1−drsd−1(r∗+r