The cosmic no-hair conjecture and black-hole formation:

an exact model with gravitational radiation

Jiří Bičák

Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,

V Holešovičkách 2, 180 00 Prague 8, Czech Republic

bicak@aci.cvut.cz and

Department of Physics, University of Utah Salt Lake City, UT 84112

Jiří Podolský

Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,

V Holešovičkách 2, 180 00 Prague 8, Czech Republic podolsky@aci.cvut.cz

Abstract

Robinson-Trautman radiative space-times with a positive cosmological constant are studied by analytical methods. They are shown to approach the Schwarzschild-de Sitter solution at large retarded times. Their global structure is analyzed, and it is demonstrated that they represent explicit models exhibiting the cosmic no-hair conjecture and black-hole formation under the presence of gravitational waves.

PACS number(s): 04.30.-w, 04.20.Jb, 98.80.Hw

1 Introduction

Cosmological no-hair conjectures have been discussed in both relativistic and Newtonian contexts for more than a decade (see, e.g., [1], [2] for reviews and a number of references). The most common version claims that all expanding universes with a positive cosmological constant approach the de Sitter space-time locally. If an inflationary phase (described by the de Sitter universe) is really an “attractor” of cosmological space-times with positive Λ, the whole inflationary scenario becomes a more natural indication of the present isotropy and homogeneity of the universe.

The cosmic no-hair conjecture has been studied under various additional assumptions. For example, it was rigorously demonstrated within the Bianchi cosmologies [3]. Counter examples, however, were also found. A simple counter example is given by the Schwarzschild-de Sitter space-time: the metric does not evolve into the de Sitter universe everywhere. Nevertheless, already Gibbons and Hawking [4] speculated that under the presence of a black hole, a space-time with positive cosmological constant would settle down to one of the Kerr-Newman-de Sitter solutions.

Just recently, the cosmic no-hair conjecture was also investigated under the presence of gravitational waves. Maeda, Nakamura and their collaborators [5] have analyzed the dynamical evolution of axisymmetric gravitational waves in the asymptotically de Sitter space-time by using numerical computations. Their numerical work indicates that either the space-time evolves into the de Sitter space-time with small perturbations, or that a Schwarzschild-de Sitter-like space-time arises.

The purpose of the present work is to study the cosmic no-hair conjecture under the presence of gravitational waves by using analytical methods. Simultaneously, we shall discuss the formation of a Schwarzschild-de Sitter black hole in a vacuum radiative space-time with a positive cosmological constant. The model space-times used in our discussion are represented by the Robinson-Trautman metrics with Λ > 0. They will also be referred to as the “cosmological” Robinson-Trautman space-times. (Very recently, black-hole space-times with Λ > 0 have been discussed extensively in various contexts - see, e.g., [6] and references therein.)

It is well known that the Robinson-Trautman metrics are the general vacuum solutions which admit a geodesic, shearfree, and twistfree null congruence of diverging rays [7] [8]. The best candidates for describing gravitational radiation from isolated sources are the Robinson-Trautman metrics of the Petrov type II, which are asymptotically flat and — in contrast to “traditional” belief — contain no nodal singularities although, in general, a naked singularity is present at the center. These space-times have attracted an increased attention in the last decade, in particular in the works by Lukács et al. [9], Schmidt [10], Rendall [11], Singleton [12], and, most recently, by Chruściel [13],[14] and Chruściel and Singleton [15]. (We refer the reader to the last three papers for further references.) In these studies the Robinson-Trautman space-times were shown to exist globally for all positive “times”, and to converge asymptotically to a Schwarzschild metric. Interestingly, the extension of these spacetimes across the “Schwarzschild-like” event horizon can only be made with a finite degree of smoothness. All these rigorous studies are based on the derivation and analysis of an asymptotic expansion describing the long-time behavior of the solutions of the nonlinear, parabolic Robinson-Trautman equation.

The Robinson-Trautman metrics can easily be generalized to solve the vacuum Einstein equations with a non-vanishing Λ [16]. The results proving the global existence and convergence of the solutions of the Robinson-Trautman equation can be taken over from the previous studies since Λ does not explicitly enter this equation.

In the next section we shall review the recent mathematical studies of the Robinson -Trautman equation. We try to give an intuitive (“physical”) summary of the basic results obtained by using rigorous mathematical concepts and techniques. Section 3 is devoted to the analysis of the Schwarzschild-de Sitter solution. Its global structure is first described in terms of Kruskal-type coordinates. Then synchronous (Lemaîtretype) coordinates based on free particles moving radially outwards towards infinity are introduced, and the properties of the Schwarzschild-de Sitter space-time are discussed in the context of the cosmic no-hair conjecture. The global structure of the Robinson-Trautman space-times of the Petrov type II with Λ > 0 is analyzed in Section 4. It is shown that these space-times approach the Schwarzschild-de Sitter space-time in a certain limit. They admit a smooth future (space-like) infinity but, in general, a smooth past infinity does not exist (or, vice versa). The continuation of the metric across the “Schwarzschild-de Sitter-like” black-hole horizon can be made with a higher degree of smoothness than in the corresponding cases with Λ = 0.

Finally, in Section 5 the asymptotic properties of the Robinson-Trautman spacetimes with Λ > 0 are studied at future infinity and the validity of the cosmic no-hair conjecture is explicitly demonstrated. The results are briefly summarized and open problems indicated in Section 6.

Some of the following results were presented in a preliminary form in [17].

2 The global structure of the Robinson-Trautman spacetimes with Λ = 0

In the standard coordinates, the Robinson-Trautman metrics have the form (see [7], [8], [16])

ds^{2} = −Φdu^{2} − 2dudr + 2r^{2}P^{−2}dζdζ̄ ,  (1)

where P = P (u, ζ, ζ̄), ζ is a complex spatial (stereographic) coordinate, r ∈ [0, ∞) is the affine parameter along the rays u = const, ζ = const, and

Φ = Δ lnP − 2r(lnP)_{,u} − \frac{2m}{r} .  (2)

Here Δ = 2P^{2}∂^{2}/∂ζ∂ζ̄ and m is a constant related to the Bondi mass of the system (m = const can always be achieved by a coordinate transformation [16]). The function P satisfies the fourth-order Robinson-Trautman equation

(lnP)_{,u} = −\frac{1}{12m}ΔΔ(lnP) .  (3)

The Robinson-Trautman equation can be formulated (see, e.g., [14], [13], [15]) by introducing a smooth metric g_{ab}^{0}(x^{c}) on a two-dimensional, compact, orientable Riemannian manifold M^{2} and a u-dependent family of 2-metrics

g_{ab} = [f(u, x^{c})]^{−2} g_{ab}^{0} ,  (4)

which, with respect to the local stereographic coordinate ζ, take the form 2P^{−2}dζdζ̄. In general, the last term in (1) can be written as r^{2}f^{−2}gab^{0}dx^{a}dx^{b}, and the function f, as a consequence of the vacuum equations, satisfies

\frac{∂f}{∂u} = −\frac{f}{24m}Δ_{g}R ,  (5)

where R is the curvature scalar and Δ_{g} the Laplacian of the metric g_{ab}. Using R_{0} and Δ_{0} to denote the curvature scalar and the Laplacian of g_{ab}^{0}, one has

R = f^{2}(R_{0} + 2Δ_{0} lnf), Δ_{g} = f^{2}Δ_{0} .  (6)

In general, M^{2} can have the topology of a 2-torus or be a higher genus surface; however, the corresponding space-times are then not asymptotically flat at null infinity [14]. Equation (5) is known in the mathematical literature as Calabi’s equation [18] in dimension 2. It is defined on any manifold, but here we shall primarily concentrate on the physical case M^{2} = S^{2}. Choosing standard coordinates on the sphere,

ζ = \sqrt{2}e^{iφ} tan \frac{θ}{2} ,  (7)

we obtain

P_{0} = 1 + \frac{1}{2}ζζ̄ = (cos^{2} \frac{θ}{2})^{−1} ,

2P_{0}^{−2}dζdζ̄ = dθ^{2} + sin^{2} θd^{2}φ ,  (8)

Δ_{0} lnP_{0} = 1 , R_{0} = +2 .

The metric (1) with P = P_{0} is just the Schwarzschild metric. Writing then (cf. (4))

P = f P_{0} ,  (9)

we find the ‘original’ Robinson-Trautman equation (3) to go over into Calabi’s equation (5) with R, Δ_{g} given by (6) (and R_{0} = 2).

As mentioned in the Introduction, the most general, rigorous and detailed analysis of the existence and behavior of the solutions of the Robinson-Trautman equations was recently given by Chruściel [13], [14] and by Chruściel and Singleton [15] (cf. also [10], [19], [12]). We now very briefly summarize their main theorems and propositions; these are formulated and proved by employing various rigorous mathematical concepts and techniques, but we shall describe them in a simplified, intuitive way.

The main result of [13] (see Proposition 5.1 therein) shows that when f_{0} ≡ f(u = u_{0}, x^{a}) is an arbitrary, sufficiently smooth initial-value function for f, then f satisfying (5), (6) exists for all times u ≥ u_{0}, and there always exists a constant C such that

|f(u, x^{a}) − f_{Schw}| ≤ C e^{−2u/m} ,  (10)

where f_{Schw} corresponds to a — perhaps boosted — Schwarzschild solution. (The constant C depends on the initial data f_{0}.) Therefore, as u → +∞, Robinson-Trautman metrics approach exponentially fast a boosted Schwarzschild metric. Performing this boost, we can without loss of generality assume that

f_{Schw} = 1 .  (11)

(How the boosts can be handled is described in Appendix B in [13].)

In a subsequent paper [14] Chruściel found an asymptotic expansion of f(u, x^{a}) for large u to have the form

f = 1 + f_{1}e^{−2u/m} + f_{2}e^{−4u/m} + ⋯ + f_{14}e^{−28u/m}
+f_{log}ue^{−30u/m} + f_{15}e^{−30u/m} + ⋯,  (12)

where f_{1}, f_{2}, ⋯, f_{log}, ⋯ are smooth functions on S^{2}. Some of these functions may vanish, but Chruściel and Singleton [15] prove that there exist Robinson-Trautman space-times for which f_{log} is non-vanishing.

This implies a rather surprising fact that, in general, the Robinson-Trautman metrics cannot be smoothly extended through the null hypersurface H^{+} given by u = +∞. There exist an infinite number of non-isometric Robinson-Trautman extensions through H^{+} which are obtained by gluing a given Robinson-Trautman space-time to any other Robinson-Trautman space-time with the same mass, as shown in Fig.1. In particular, we may join the radiative metrics to the Schwarzschild metric so that the Robinson-Trautman space-time “settles down” to the Schwarzschild space-time including the interior of the Schwarzschild black hole. The extension through the horizon is C^{5}, in general. There exist extensions which are C^{117} through H^{+} — an example can be obtained by gluing a copy of the Robinson-Trautman space-time to itself, as one does in the Kruskal diagram for two copies of the Schwarzschild spacetime. (The extensions will even be smoother in special cases in which some of the coefficients of the terms ∼ u^{j}e^{−ku/m} vanish.)

In order to see the smoothness across H^{+} , one introduces an advanced time coordinate v by v = u + 2r + 4m ln(r/2m − 1), and Kruskal-type coordinates û, v̂ by (see, e.g., [19])

û = − exp(−u/4m) ,
v̂ = exp(v/4m) .  (13)

(Such coordinates in the context of the Robinson-Trautman metrics were first used by Tod [19] and taken over by Chruściel [14]. Here we modified v used in [19], [14] by a constant additive factor so as to make the dimensions in v and in the final metric (14) correct.) The hypersurface u = +∞ now becomes a boundary given by û = 0. The original metric (1) becomes

ds^{2} = −\frac{32m^{3}}{r} exp(−r/2m)dûdv̂ − 16m^{2}Φ̂dû^{2} + 2r^{2}P^{−2}dζdζ̄ ,  (14)

where

Φ̂ = e^{u/2m} (\frac{1}{2}R − 1 + \frac{r}{12m}Δ_{g}R) ,  (15)

with R and Δ_{g} being given by (6). In terms of û, the expansion (12) becomes

f = 1 + f_{1}û^{8} + f_{2}û^{16} + ⋯ + f_{14}û^{112}
−4mf_{log}(ln|û|)(û)^{120} + f_{15}û^{120} + · · · .  (16)

Due to the presence of the ln|û| terms, the function f is not smooth at the û = 0 — indeed it is C^{119} if f_{log} ≠ 0. The full metric (14) is C^{117}, since Φ̂ entering g_{00} contains the additional factor e^{u/2m} ≈ 1/û^{2}.

Although from a physical point of view a C^{5} function may appear as good as a smooth function, the difference between the corresponding metrics is, in principle, observable. The resulting space-times, despite the fact that they have a “Schwarzschildlike” event horizon at r = 2m, should better be called Robinson-Trautman black holes.

Before we turn to the cosmological Robinson-Trautman metrics we shall discuss some properties of the Schwarzschild-de Sitter solution.

3 The Schwarzschild-de Sitter metric and synchronous coordinates

Although the solution generalizing the Schwarzschild metric to the case of a nonvanishing cosmological constant was found in 1918 [20], its global structure has been analyzed in detail only recently [21].

Gibbons and Hawking [4] and Lake and Roeder [22] give a brief discussion of its conformal diagram; however, only Ba żański and Ferrari [21] performed its “Kruskalization” in detail. They considered the general case of a Schwarzschild black hole in a de Sitter universe, thus omitting “extreme” black holes and naked singularities. In the following, we shall also restrict ourselves to such general black-hole cases.

In the standard Schwarzschild-type coordinates the metric reads

ds^{2} = − (1 − \frac{2m}{r} − \frac{Λ}{3}r^{2})dt^{2} + (1 − \frac{2m}{r} − \frac{Λ}{3}r^{2})^{−1} dr^{2} + r^{2}(dθ^{2} + sin^{2} θdφ^{2}) , (17)

where m > 0 is the mass of the central hole and Λ > 0 is the “repulsive” cosmological constant. If Λ = 0 we get the Schwarzschild metric; for m = 0 we get the de Sitter metric in static coordinates. Neither of these coordinate systems covers the whole manifold. The Killing vector ∂/∂t is timelike in a certain region of r provided that the condition

0 ≤ 9Λm^{2} < 1  (18)

is satisfied. Hereafter we shall assume this, if not stated otherwise. As shown in [21], the metric (17) can be written in the form

ds^{2} = −Φ(r)dt^{2} + Φ^{−1}(r)dr^{2} + r^{2}(dθ^{2} + sin^{2} θdφ^{2}) ,  (19)

where

Φ(r) = \frac{(r − r_{+} )(r_{++} − r)(r + r_{+} + r_{++} )}{r(r_{+}^{2} + r_{+} r_{++} + r_{++}^{2} )} .  (20)

The parameters r_{+} , r_{++} , 0 < r_{+} < r_{++} , are two real positive roots of the equation g_{00} = 0; the third root r_{−} is negative. These roots can be parametrized as follows (see, e.g., [23]):

r_{+} = \frac{2}{\sqrt{Λ}} cos(\frac{α}{3} + \frac{4π}{3} ) ,
r_{++} = \frac{2}{\sqrt{Λ}} cos \frac{α}{3} ,  (21)
r_{−} = \frac{2}{\sqrt{Λ}} cos(\frac{α}{3} + \frac{2π}{3} ) ,

where cosα = −3m\sqrt{Λ}. Here r = r_{+} describes the black-hole horizon, and r = r_{++} is the cosmological horizon. Only r = 0 is a real singularity. For r_{+} < r < r_{++} , the solution (17) is static.

In [21], Kruskal-type coordinate systems are found in which the metric (19) is regular in the neighborhood of either the black-hole horizon or the cosmological horizon. Here we shall first convert the metric (17) into a Robinson-Trautman form.

Define the usual “tortoise-type” coordinate r^{∗} by

r^{∗} = ∫ \frac{dr}{Φ(r)} = δ_{+} ln \frac{|r − r_{+} |}{r + r_{+} + r_{++}} −δ_{++} ln \frac{|r_{++} − r|}{r + r_{+} + r_{++}} +δ_{+} [ln(\frac{r_{++}}{r_{+}} ) − \frac{1}{2}] , (22)

where

δ_{+} = \frac{r_{+}}{1 − Λr_{+}^{2}} , δ_{++} = −\frac{r_{++}}{1 − Λr_{++}^{2}} .  (23)

In contrast to [21], we have added a constant term to r^{∗}; this makes the limit Λ → 0 more transparent: r_{++} → ∞, \frac{Λ}{3}r_{++}^{2} → 1, δ_{++} → \frac{1}{2}r_{++} , δ_{+} → r_{+} . Now, by introducing the retarded time coordinate

u = t − r^{∗} ,  (24)

instead of t, we convert the metric (17), resp. (19), into

ds^{2} = − (1 − \frac{2m}{r} − \frac{Λ}{3}r^{2})du^{2} − 2dudr + r^{2}(dθ^{2} + sin^{2} θdφ^{2})
= −Φdu^{2} − 2dudr + r^{2}(dθ^{2} + sin^{2} θdφ^{2}) .  (25)

This is exactly the Robinson-Trautman form (1). (It is also the generalization of the Schwarzschild metric in the outgoing Eddington-Finkelstein coordinates.) It is not difficult to show that for fixed m but increasing Λ from Λ = 0 to Λ = 1/9m^{2}, the parameters r_{+} and δ_{+} monotonically increase within the intervals

2m ≤ r_{+} ≤ 3m , 2m ≤ δ_{+} ≤ +∞ ,  (26)

whereas r_{++} decreases:

+ ∞ > r_{++} ≥ 3m .  (27)

Further, it is easy to see that δ_{++} > 0; it decreases from +∞ for Λ = 0 to a minimum positive value, and then increases back to +∞ for Λ → 1/9m^{2}. At the black-hole horizon, we have t → +∞, r^{∗} → −∞, and u → +∞; at the cosmological horizon, t → +∞, r^{∗} → +∞, and u remains finite. On the other hand, the advanced time coordinate,

v = t + r^{∗} ,  (28)

remains finite at the black-hole horizon, while v → +∞ at the cosmological horizon.

In order to cover the black-hole horizon by regular coordinates, let us introduce Kruskal-type null coordinates

û = − exp(−u/2δ_{+} ) ,
v̂ = exp(v/2δ_{+} ) .  (29)

The metric (25) now reads

ds^{2} = −\frac{4Λδ_{+}^{2} r_{+} e^{\frac{1}{2}}}{3r_{++} r} (r_{++} − r)^{1+δ_{++} /δ_{+}} (r + r_{+} + r_{++} )^{2−δ_{++} /δ_{+}} dûdv̂ + r^{2}(dθ^{2} + sin^{2} θdφ^{2}) .

(30)

Letting Λ → 0, we obtain back the metric (14) with Φ̂ = 0. Clearly, at the black-hole horizon with r = r_{+} , û = 0, v̂ finite, the metric (30) is regular. Analogously, we can find Kruskal-type null coordinates covering the cosmological horizon in which the metric is again regular; since, however, we shall not need them in the following, we refer to [21]. The conformal diagram of the Schwarzschild-de Sitter space-time is given in Fig.2. Notice that both future and past infinities are spacelike as a consequence of Λ > 0. (In the following section, this point is discussed for a general cosmological Robinson-Trautman space-time.)

Next, we shall introduce the synchronous (Lemaître-type) coordinates in the Schwarzschild - de Sitter space-time. These coordinates have not been discussed so far in literature. We connect new coordinates with free particles moving radially outward across the past black-hole horizon and across the cosmological horizon towards infinity since such coordinates are suitable for the discussion of the cosmic no-hair conjecture. (The worldline of such a particle is illustrated in Fig.2.)

Introduce the function

h(r) = (\frac{2m}{r} + \frac{Λ}{3}r^{2}) ^{\frac{1}{2}} ,  (31)

and the coordinates τ, R by the relations

dτ =  dt − \frac{h}{1 − h^{2}}dr ,  (32)
dR = −dt + \frac{1}{h(1 − h^{2})}dr ,  (33)

or, inversely,

dr = h(dR + dτ) ,  (34)
dt = \frac{1}{1 − h^{2}}(dτ + h^{2}dR) .  (35)

Hence, we obtain r(τ, R) by integrating (34),

R + τ = ∫ (\frac{2m}{r} + \frac{Λ}{3}r^{2} )^{− \frac{1}{2}} dr .  (36)

The integral can be calculated explicitly to yield

R + τ = \frac{1}{\sqrt{3Λ}} ln[2\sqrt{Y (Y + 1)} + 2Y + 1] + const ,  (37)

where

Y = \frac{Λ}{6m}r^{3} .  (38)

The relation (37) can be inverted (we put the additive constant equal zero),

Y = \frac{(1 − Z^{3})^{2}}{4Z^{3}} ,  (39)

where

Z = e^{−H(R+τ)} ,  (40)

and we introduced the “Hubble parameter”

H = \sqrt{\frac{Λ}{3}} .  (41)

In the coordinates τ, R the metric (17) reads

ds^{2} = −dτ^{2} + (\frac{2m}{r} + H^{2}r^{2} ) dR^{2} + r^{2}(dθ^{2} + sin^{2} θdφ^{2}) ,  (42)

with r being given in term of R and τ by (38)-(40). The metric is regular at the horizons r_{+} , r_{++} . The particles with R = const, θ = const, φ = const, satisfy the equations (see (32), (33))

\frac{dr}{dt} = (\frac{2m}{r} + H^{2}r^{2} )^{\frac{1}{2}} (1 − \frac{2m}{r} − H^{2}r^{2}) ,
\frac{dr}{dτ} = (\frac{2m}{r} + H^{2}r^{2} )^{\frac{1}{2}} ,  (43)
\frac{dt}{dτ} = (1 − \frac{2m}{r} − H^{2}r^{2})^{−1} .

Hence, these particles, with τ being their proper time, move from the singularity (r = 0, R + τ = 0), across the past black-hole horizon (r = r_{+} , t = −∞), through the static region (r_{+} < r < r_{++} , dt/dτ > 0), and then continue across the cosmological horizon (r = r_{++} , t = +∞) towards infinity (r = ∞, τ → +∞) (cf. Fig.2). The equation for dr/dτ also shows immediately that the particles cross both horizons in a finite proper time and reach r = ∞ only with τ → ∞. Their local velocity, as measured by local static observers at r = const, θ = const, φ = const in the static region r ∈ (r_{+} , r_{++} ), is given by v_{loc} = dl_{loc}/dτ_{loc} = (2m/r + H^{2}r^{2})^{\frac{1}{2}}. At the horizons v_{loc} = 1 since the local “static” observers themselves “move” with the velocity of light there.

In order to bring the metric (42) into the form in which the de Sitter metric in “standard” coordinates arises explicitly, we relabel the radial comoving coordinate R as a new coordinate χ given by

χ = (\frac{m}{2H^{2}})^{\frac{1}{3}} e^{HR} .  (44)

The metric (42) then reads as follows:

ds^{2} = −dτ^{2} + e^{2Hτ}(1 − Z^{3})^{\frac{4}{3}} [(\frac{1 + Z^{3}}{1 − Z^{3}})^{2} dχ^{2} + χ^{2}(dθ^{2} + sin^{2} θdφ^{2}) ] ,  (45)

where Z is given by (40), or

Z = (\frac{m}{2H^{2}})^{\frac{1}{3}} χ^{−1}e^{−Hτ} .  (46)

This is the exact form of the Schwarzschild-de Sitter metric in the outgoing comoving coordinates. Hence it describes a white hole in the de Sitter universe (cf. Fig.2). The past central singularity at r = 0 is given by Z = 1. As r increases, Z decreases; at r → ∞, or R + τ → ∞, we find Z → 0, and the metric (45) turns into

ds^{2} = −dτ^{2} + e^{2Hτ} [dχ^{2} + χ^{2}(dθ^{2} + sin^{2} θdφ^{2}) ] .  (47)

This is the de Sitter metric written in the standard synchronous Friedmann-Robertson-Walker form with the exponentially growing (“inflationary”) expansion factor. Keeping the leading order terms in the expansion of the Schwarzschild-de Sitter metric (45) for τ → ∞, we obtain

ds^{2} = −dτ^{2} + e^{2Hτ} [dχ^{2} + χ^{2}(dθ^{2} + sin^{2} θdφ^{2}) ]  (48)

+\frac{2m}{3H^{2}}χ^{−3}e^{−Hτ} [2dχ^{2} − χ^{2}(dθ^{2} + sin^{2} θdφ^{2}) ] + O (e^{−4Hτ}) .

Observe that the “traces” of the central black hole, characterized by the terms in which the mass m enters, completely disappear as τ → ∞ in full agreement with the cosmic no-hair conjecture. Of course, a completely different picture is seen by observers falling into the hole — those, however, are not described by our “outgoing” comoving coordinates.

Before concluding this section, let us give the asymptotic relation between the Robinson-Trautman-type coordinates {u, r, θ, φ} and the inflationary-type coordinates {τ, χ, θ, φ} for large r and large τ in the present case of the Schwarzschild-de Sitter space-time. It is easy to see that the exact relation (37) for large r and large τ implies

r ∼ (\frac{m}{2H^{2}})^{\frac{1}{3}} e^{H(R+τ)} ,  (49)

i.e., regarding (44), we obtain asymptotically at r → ∞, τ → ∞:

r = χe^{Hτ} + O(1) + O (e^{−Hτ} ) .  (50)

Next, for large r we can integrate (32),

Hτ = Ht + lnHr + const ;  (51)

and by realizing that r^{∗} → const at r → ∞ (see (22)), we can choose a constant in the last relation so that (cf. (24))

e^{Hu} ∼ e^{Hτ}(Hr)^{−1} + ⋯ = \frac{1}{Hχ} + O (e^{−Hτ} ) .  (52)

Finally, let us note that the de Sitter metric (47) preserves its form at large τ up to terms ∼ e^{−2Hτ} under the transformation (τ, χ) → (τ′, χ′) such that

e^{Hτ} = (Hχ′)^{2}e^{Hτ′} − e^{−Hτ′} + ⋯,  (53)
Hχ = \frac{1}{Hχ′} + \frac{1}{(Hχ′)^{3}}e^{−2Hτ′} + ⋯,

which implies χe^{Hτ} = χ′e^{Hτ′} + O (e^{−3Hτ′}). Therefore, omitting primes, we can write down the asymptotic relations at I^{+} (r → ∞, τ → ∞) between the coordinate systems (u, r) and (τ, χ) in the form in which the leading terms are both linear in χ:

r = χe^{Hτ} + O(1) ,  (54)
e^{Hu} = Hχ + O (e^{−Hτ} ) .

These relations obtained in the special case of the Schwarzschild-de Sitter spacetime will motivate our ansatz for an asymptotic transformation for general Robinson-Trautman space-times in Section 5.

4 The global structure of the Robinson-Trautman spacetimes with Λ ≠ 0

When a Robinson-Trautman space-time with Λ = 0 is known in the standard form (1), it is straightforward to generalize it to the case of a non-vanishing Λ (cf. [16]). The metric still keeps the form (1) with P satisfying the Robinson-Trautman equation (3). The only place where Λ enters is the function Φ. The cosmological Robinson-Trautman metric reads

ds^{2} = −Φ_{Λ}du^{2} − 2dudr + 2r^{2}P^{−2}dζdζ̄ ,  (55)

where

Φ_{Λ} = Δ lnP − 2r(lnP)_{,u} − \frac{2m}{r} − \frac{Λ}{3}r^{2} .  (56)

We may still write P = f P_{0}, as in Eq.(9), where P_{0} satisfies (8) and f satisfies (5), (6). Since Λ does not enter the evolution equation for f, we may take over the results for Λ = 0 described in Section 2. The Robinson-Trautman metric will now approach the Schwarzschild-de Sitter metric in the form (25) as u → ∞ (see Fig.3). The approach of f to its Schwarzschild-de Sitter form is again characterized by the inequality (10) and by the asymptotic expansion (12). We shall also assume an analog of (11), i.e.,

f_{Schw−deSi} = 1 .  (57)

The presence of a positive cosmological constant does not affect the smoothness of future infinity in these space-times; however, the future null infinity becomes spacelike in contrast to the cases with Λ = 0 (cf. Fig.3). The smoothness of I^{+} is easily seen if one introduces an inverse radial coordinate l = r^{−1} and uses Ω = l as a conformal factor. One finds

Ω^{2}ds^{2} = 2dudl − l^{2}Φ_{Λ}du^{2} + 2P^{−2}dζdζ̄ ,  (58)

where

Φ_{Λ} = Δ lnP − 2l^{−1}(lnP)_{,u} − 2ml − \frac{Λ}{3}l^{−2} .  (59)

It is easy to see that l = 0 is a regular hypersurface for smooth P (u, ζ, ζ̄); and it is spacelike due to the presence of the last term in Φ_{Λ}. As in the Robinson-Trautman space-times with Λ = 0, past null infinity will not exist, in general, because the Robinson-Trautman (parabolic) equation does not admit smooth solutions both into the future and into the past (cf. e.g [14]).

The presence of the cosmological constant, however, may have a considerable effect on the smoothness of the extensions through the null hypersurface H^{+} given by u = +∞. The transformation to the Kruskal-type null coordinates is now given by (29). Hence, instead of (16), we find the expansion of f in terms of û to read

f = 1 + f_{1}(û)^{4δ_{+} /m} + f_{2}(û)^{8δ_{+} /m} + ⋯ + f_{14}(û)^{56δ_{+} /m}
−2δ_{+} f_{log}(ln|û|)(û)^{60δ_{+} /m} + f_{15}(û)^{60δ_{+} /m} + · · · .  (60)

The full metric now takes the form

ds^{2} = −\frac{4Λδ_{+}^{2} r_{+} e^{\frac{1}{2}}}{3r_{++} r} (r_{++} − r)^{1+δ_{++} /δ_{+}} (r + r_{+} + r_{++} )^{2−δ_{++} /δ_{+}} dûdv̂
−4δ_{+}^{2} Φ̂_{Λ} dû^{2} + 2r^{2}P^{−2}dζdζ̄ ,  (61)

where

Φ̂_{Λ} = e^{u/δ_{+}} (\frac{1}{2}R − 1 + \frac{r}{12m}Δ_{g}R) ,  (62)

and R, R_{0}, P, P_{0}, Δ_{g} are given by Eqs. (6)-(9), with f being of the form (60) above. The constant parameters r_{+} , r_{++} , δ_{+} , δ_{++} are the same as in Section 3.

It is interesting to see how the presence of Λ influences the smoothness of the extension of the Robinson-Trautman metrics across the null hypersurface H^{+} (“horizon”) given by u = ∞, or û = 0. In particular, we may join the radiative Robinson-Trautman metrics to the Schwarzschild-de Sitter metric so that a Robinson-Trautman space-time with Λ ≠ 0 “settles down” to the Schwarzschild-de Sitter space-time including the interior of the Schwarzschild-de Sitter black hole (see Fig.3). As summarized in Section 2, such an extension will, in general, be C^{117} in the case of vanishing Λ. With Λ ≠ 0, much higher smoothness can be obtained. Indeed, since δ_{+} = r_{+} /(1 − Λr_{+}^{2} ) (see (23)), the horizon H^{+} can be made “arbitrarily smooth” by letting Λ approach arbitrarily close its extremal value, Λ → 1/9m^{2} (r_{+} → 3m), which corresponds to an almost extreme Schwarzschild-de Sitter black hole. Then δ_{+} becomes arbitrarily large and the terms ∼ û^{k(δ_{+} /m)}, k = 4, 8, ⋯, appearing in (60), will guarantee arbitrarily high smoothness of the function f at û = 0; this in turn determines the smoothness of the functions R, P and thus of the full metric (61) at û = 0. (Notice that Φ̂_{Λ} entering g_{00} contains, in addition, the factor e^{u/δ_{+}} ≈ 1/û^{2} which, however, decreases the smoothness of the metric by 2 only.)

On the other hand, there are cases when the presence of Λ can decrease the smoothness of the metric (61) across û = 0. If 4δ_{+} /m is not an integer, the second term in the expansion (60), f_{1}(û)^{4δ_{+} /m}, is already not smooth. Since 4δ_{+} /m ≥ 8 (see (26)), the function f is then at least C^{8} and the full metric is at least C^{6}. For those values of Λ which imply 4δ_{+} /m equals an integer, the smoothness is always better than for Λ = 0. In general, denoting by n the largest integer smaller than 4δ_{+} /m, the metric (61) is at least C^{n} at û = 0. As mentioned above, n → ∞ with 4δ_{+} /m → ∞.

We also expect to find analogs of the past black-hole horizon and the cosmological horizon of the Schwarzschild-de Sitter metric in a Robinson-Trautman solution with Λ > 0 (see Fig.3). Possible analogs can be considered in a way similar to Tod’s treatment [19] for the case Λ = 0.

5 The cosmic no-hair conjecture

After having examined the behavior of the cosmological Robinson-Trautman metrics at the null hypersurface u → +∞, we shall now turn to their properties close to I^{+} , i.e., near r → ∞, u finite (cf. Fig.2). If the cosmic no-hair conjecture is to be verified in these space-times, we should be able to demonstrate that they all approach the de Sitter space-time locally. For this purpose, it would be sufficient to convert the metrics (55), with Φ_{Λ} given by (56), into the Starobinsky asymptotic form [24]. As will be indicated below, such an asymptotic form automatically guarantees that the cosmic no-hair conjecture is satisfied.

Starobinsky’s form represents the asymptotic behaviour of an inhomogeneous cosmological model with Λ > 0 in synchronous coordinates at large τ (τ being the proper time measured by the observers at rest in these synchronous coordinates). In Section 3, the Starobinsky form of the pure Schwarzschild-de Sitter metric was obtained in Eq.(48). We now wish to find a similar form under the presence of gravitational waves described by a general Robinson-Trautman metric (55).

In order to convert the metric (55) into the desired form, we first make an ansatz which is inspired by the asymptotic form of the transformation (54) between the Robinson-Trautman-type coordinates and the synchronous (“inflationary-type”) coordinates for the Schwarzschild-de Sitter metric.

Assume that the required transformation is of the asymptotic form

r = χe^{Hτ} − H^{−2}(f_{∞,u}/f_{∞}) + ∑_{n=1}^{∞} ψ_{n}e^{−nHτ} ,
e^{Hu} = Hχ − e^{−Hτ} ,  (63)

where H = \sqrt{Λ/3}, ψ_{n} are functions of χ, ζ, ζ̄ to be chosen appropriately, and f_{∞} = f|_{τ→∞} = f(u → H^{−1} ln|Hχ|, ζ, ζ̄) is determined by the function f(u, ζ, ζ̄) entering the metric (55). (Whenever the gravitational waves are present, f ≠ 1; only f_{Schw−deSi} = 1 (cf. (57)).) Under transformation (63), the metric (55) becomes

ds^{2} = −dτ^{2} + e^{2Hτ} [dχ^{2} + (χ/f_{∞}P_{0})^{2} 2dζdζ̄] + h_{ij}^{(0)} dx^{i}dx^{j}
+e^{−Hτ}h_{μj}^{(1)}dx^{μ}dx^{j} + ∑_{m=2}^{∞} e^{−mHτ}h_{μν}^{(m)} dx^{μ}dx^{ν} .  (64)

Here f_{∞} depends only on the spatial coordinates {x^{i}, i = 1, 2, 3} = {χ, ζ, ζ̄}; all h_{μν}’s appearing in (64) also depend only on x^{i}, but {x^{μ}} = {τ, x^{i}} so that the last two terms in (64) involve dτ as well. Nevertheless, the non-diagonal components of the metric h_{0j}^{(1)}, h_{0j}^{(2)}, … can be transformed away by transformations of the following form:

χ^{(m−1)} = χ^{(m)} + a_{m} e^{−(m+2)Hτ} ,
ζ^{(m−1)} = ζ^{(m)} + b_{m} e^{−(m+2)Hτ} ,  (65)

where m = 1, 2, 3, …, {χ^{(0)}, ζ^{(0)}, ζ̄^{(0)}} = {χ, ζ, ζ̄}, and the coefficients a_{m}, b_{m} depend on {χ^{(m)}, ζ^{(m)}, ζ̄^{(m)}}. After choosing a’s and b’s so that h_{0j}^{(m)} = 0, the metric (64) reads

ds^{2} = −dτ^{2} + e^{2Hτ} [dχ̃^{2} + (χ̃/f_{∞})^{2}(1 + \frac{1}{2} ζ̃¯ζ̃)^{−2} 2dζ̃d¯ζ̃]
+ ∑_{m=0}^{∞} e^{−mHτ} h̃_{ij}^{(m)} dx̃^{i}dx̃^{j} + ∑_{m=2}^{∞} e^{−mHτ} h̃_{00}^{(m)} dτ^{2} .  (66)

Furthermore, the metric coefficients h̃_{00}^{(m)}(x̃^{(i)}) can be transformed away by performing a transformation of the form (63) in which ψ_{n} are chosen appropriately.

To summarize, a transformation of the form

r = χe^{Hτ} − H^{−2}(f_{∞,u}/f_{∞}) + ∑_{n=1}^{∞} A_{n} e^{−nHτ} ,
e^{Hu} = Hχ − e^{−Hτ} + ∑_{n=3}^{∞} B_{n} e^{−nHτ} ,  (67)
ζ = η +  ∑_{n=3}^{∞} C_{n} e^{−nHτ} ,

in which A_{n}, B_{n}, C_{n} are suitable functions of χ, η, η̄ brings the metric (55) into the following asymptotic form:

ds^{2} = −dτ^{2} + e^{2Hτ} [dχ^{2} + f_{∞}^{−2}χ^{2} (dθ^{2} + sin^{2} θdφ^{2})]
+ ∑_{m=0}^{∞} e^{−mHτ}h_{ij}^{(m)} dx^{i}dx^{j} ,  (68)

where we have reintroduced the standard angular coordinates θ, ϕ (η = \sqrt{2}e^{iφ} tan(θ/2)

— cf. Eq.(7)), f_{∞} = f|_{τ→∞} = f(u = H^{−1} ln|Hχ|, θ, ϕ), and h_{ij}^{(m)} depend on {x^{i}} = {χ, θ, ϕ} only. This asymptotic form of the metric shows explicitly that for τ → ∞ the space-time metric does not approach the de Sitter metric globally. In contrast to the case of the pure Schwarzschild-de Sitter black hole where outside of the hole all its “traces” disappear as τ → ∞ (cf. Eq.(47)), the gravitational waves now leave “an imprint” on the I^{+} which is demonstrated by the presence of the function f_{∞} in the “angular” part of the metric (68). Only as u → ∞, χ → ∞, r → ∞, so that f_{∞} → 1 (see the point P in Fig.3), do the Robinson-Trautman metrics approach the de Sitter metric since the terms in (68) with the factor e^{2Hτ} dominate those proportional to e^{−mHτ}.

In fact, the cosmic no-hair conjecture claims that a universe with a positive cosmological constant approaches exponentially fast the de Sitter space-time only locally in general. That this is so in our case can easily be demonstrated as follows (cf. [25]).

Write the metric (68) in the form

ds^{2} = −dτ^{2} + e^{2Hτ} a_{ij}(x^{s}) dx^{i}dx^{j} + O(1) .  (69)

Consider a worldline x^{i} = x_{(0)}^{i} = const. It is a geodesic (in Fig.3 indicated by O_{1}). Make a linear transformation x^{i} → x̄^{i}, x^{i} − x_{(0)}^{i} = A_{k}^{i}x̄^{k} such that ā_{ij}(0) = δ_{ij}. Now introduce new coordinates y^{i}, T by

x̄^{i} = y^{i} e^{−HT} /(1 − H^{2}R^{2})^{\frac{1}{2}} ,
e^{Hτ} = e^{HT} (1 − H^{2}R^{2})^{\frac{1}{2}} ,  (70)

where R^{2} = (y^{1})^{2} + (y^{2})^{2} + (y^{3})^{2}. Putting y^{1} = R sin θ̃cos φ̃, y^{2} = R sin θ̃sin φ̃, y^{3} = R cos θ̃, we find that (69) takes the form

ds^{2} = − (1 − \frac{Λ}{3}R^{2}) dT^{2} + (1 − \frac{Λ}{3}R^{2})^{−1} dR^{2}
+R^{2} (dθ̃^{2} + sin^{2} θ̃dφ̃^{2}) + ∑_{m=1}^{∞} e^{−mHT}ã_{μν}^{(m)} dy^{μ}dy^{ν} ,  (71)

where ã_{μν}^{(m)} are functions of of {ỹ^{i}} = {R, θ̃, φ̃}, and ỹ^{0} = T. The transformation (70) cannot be applied globally — it is meaningful only for 1 − H^{2}R^{2} > 0. A geodesic observer (say O_{1} in Fig.3) moving with y^{i} = 0 will see, inside his past light cone, the space-time approach the de Sitter space-time exponentially fast as T → ∞. Thus for a freely falling observer the observable universe becomes quite bald. This is what the cosmic no-hair conjecture claims.

6 Concluding remarks

Starting from the exact model radiative space-times with a positive cosmological constant, we demonstrated the validity of the cosmic no-hair conjecture by purely analytic means. We have also shown that these cosmological Robinson-Trautman solutions settle down to the Schwarzschild-de Sitter solution at large retarded times. The interior of a Schwarzschild-de Sitter black hole can be joined to an “external” cosmological Robinson-Trautman space-time across the horizon with a higher degree of smoothness than in the corresponding case with Λ = 0. As far as we are aware, these models represent the only exact analytic demonstration of the cosmic no-hair conjecture under the presence of gravitational waves. They also appear to be the only exact examples of black-hole formation in non-spherical space-times which are not asymptotically flat. Hopefully, these models may serve as tests of various approximation methods, and as test beds in numerical studies of more realistic situations.

Finally, let us note that the ideas discussed here can also be applied to the Robinson-Trautman space-times containing an incoherent homogeneous radiation field (null fluid). For Λ = 0, such space-times were studied by Bičák and Perjés [26] who showed that they approach the Vaidya metric asymptotically. With Λ ≠ 0 the radiating Vaidya metric in the de Sitter universe should arise.

7 Acknowledgments

We want to thank Piotr Chruściel for several very enlightening discussions of his work on the Robinson-Trautman space-times, and Joe Romano for reading the manuscript. J.B. is grateful to Karel Kuchař for the warm hospitality in Salt Lake City where this work was finished.

We also acknowledge the support of the U.S.-Czech Science and Technology Grant No 92067, and grants GACR-0503 and GAUK-318 of the Czech republic and Charles University.

References

[1] K.Maeda, in Fifth M.Grossman Meeting on General Relativity, Proceedings, Perth, Australia, 1988, edited by D.G.Blair, M.J.Buckingham and R.Ruffini (World Scientific, Singapore, 1989), p.145.

[2] J.D.Barrow and G.Götz, Class.Quantum Grav. 6, 1253 (1989).

[3] R.W.Wald, Phys.Rev. D28, 2118 (1983).

[4] G.W.Gibbons and S.W.Hawking, Phys.Rev. D15, 2738 (1977).

[5] K.Nakao, T.Nakamura, K.Oohara, and K.Maeda, Phys.Rev. D43, 1788 (1991); K.Nakao, K.Maeda, T.Nakamura, and K.Oohara, Phys.Rev. D47, 3194 (1993); M.Shibata, K.Nakao, T.Nakamura, and K.Maeda, Phys.Rev. D50, 708 (1994).

[6] D.R.Brill and S.A.Hayward, Class.Quantum Grav. 11, 359 (1994).

[7] I.Robinson and A.Trautman, Phys. Rev.Lett. 4, 431 (1960).

[8] I.Robinson and A.Trautman, Proc.Roy.Soc.Lond. A265, 463 (1962).

[9] B.Lukács, Z.Perjés, J.Porter, and Á.Sebestyén, Gen.Rel.Grav. 16, 691 (1984).

[10] B.G.Schmidt, Gen.Rel.Grav. 20, 65 (1988).

[11] A.D.Rendall, Class.Quantum Grav. 5, 1339 (1988).

[12] D.B.Singleton, Class.Quantum Grav. 7, 1333 (1990).

[13] P.T.Chruściel, Commun.Math.Phys. 137, 289 (1991).

[14] P.T.Chruściel, Proc.Roy.Soc.Lond. A436, 299 (1992).

[15] P.T.Chruściel and D.B.Singleton, Commun.Math.Phys. 47, 133 (1992).

[16] D.Kramer, H.Stephani, M.A.H.MacCallum, and H.Herlt, Exact Solutions of the Einstein’s Field Equations (VEB Deutscher Verlag der Wissenschaften, Berlin / Cambridge University Press, Cambridge, 1980).

[17] J.Bičák and J.Podolský, in Relativistic Astrophysics and Cosmology, edited by S.Gottlöber, J.P.Mücket and V.Müller (World Scientific, Singapore, 1992), p.167.

[18] J.E.Calabi, in Seminar on differential geometry, ed. S.T.Yau (Princeton University Press, Princeton, 1982).

[19] K.P.Tod, Class.Quantum Grav. 6, 1159 (1989).

[20] F.Kottler, Ann.Phys. (Leipzig) 56, 410 (1918).

[21] S.L.Ba żański and V.Ferrari, Nuov.Cim. 91B, 126 (1986).

[22] K.Lake and R.C.Roeder, Phys.Rev. D15, 3513 (1977).

[23] M.J.Jaklitsch, C.Hellaby, and D.R.Matravers, Gen.Rel.Grav. 21, 941 (1989).

[24] A.A.Starobinsky, Pisma ZETF 37, 55, in Russian (1983).

[25] W.Boucher and G.W.Gibbons, in The Very Early Universe, edited by G.W.Gibbons, S.W.Hawking and S.T.C.Siklos (Cambridge University Press, Cambridge, 1983), p.273.

[26] J.Bičák and Z.Perjés, Class.Quantum Grav. 4, 595 (1987).

Figure Captions

Figure 1. Starting with arbitrary, smooth initial data for the Robinson-Trautman equation at u = u_{0}, the Robinson-Trautman metrics converge exponentially fast to a Schwarzschild metric as u → ∞. However, the vacuum extension beyond the null hypersurface H^{+} (u = +∞) that includes, for example, the interior of the vacuum Schwarzschild black hole, can only be done with a finite degree of smoothness. (See the text for more details.)

Figure 2. The conformal diagram for the Schwarzschild-de Sitter space-time. The worldline R = const (θ = const, φ = const) represents the history of a free particle moving radially outward across the past black-hole horizon and across the cosmological horizon towards future (spacelike) infinity. For an observer moving with such a particle, all “traces” of the hole disappear as he approaches I^{+} ; only the de Sitter geometry persists.

Figure 3. Starting with arbitrary, smooth initial data at u = u_{0}, the Robinson-Trautman metrics with positive cosmological constant converge exponentially fast to a Schwarzschild-de Sitter metric as u → ∞. In the neighborhood of P at future infinity I^{+} , the metric is approaching the de Sitter metric exponentially fast. Although traces of gravitational waves will persist in other regions of future infinity, all geodesic observers (such as O_{1}) will observe the metric to approach the de Sitter metric exponentially fast within their past light cone. Observer O_{2} falls into the black hole. The metric at the horizon H^{+} has only a finite degree of smoothness, however, although this can be higher than in the case with Λ = 0.