We continue our investigation of formation of trapped surfaces in strongly curved geometries which do not contain gravitational waves. The expansion of open, flat universes does not change substantially the results obtained hitherto in the case of asymptotically and conformally flat space-time. The necessary and sufficient conditions for the formation of trapped surfaces are given, which explicitly demonstrate that the quicker universes are expanding, the more matter is required to develop a trapped surface

Koc, P.

Malec, Edward

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. 23(1992) ACTA PHYSICA POLONICA B No 2TRAPPED SURFACES IN NONSPHERICAL OPEN UNIVERSES** This work is supported by the Government Grant No PB 2526/2/91.P. Koc, E. MalecInstitute of Physics, Jagellonian University Reymonta 4, 30-059 Cracow, Poland(Receiυed February 11, 1992)We continue our investigation of formation of trapped suifaces in strongly curved geometiies which do not contain gravitational waves. The expansion of open, fiat universes does not change substantially the results obtained hitherto in the case of asymptotically and conformally flat space- time. The necessaiy and sufficient conditions for the formation of trapped surfaces are given, which explicitly demonstrate that the quicker universes are expanding, the morę matter is required to develop a trapped surface.PACS numbers: 04.20.Cv1. IntroductionThe attempt to prove the Cosmic Censorship Hypothesis [1] may be di- vided into two steps. The first step would be the proof that singularities are surrounded by trapped surfaces. That of course requires a suitable defini- tion of what is meant by the term “singularities”; our results suggest that it is possible to prove the so called “trapped surface conjecture”, according to which massive singularities have to be trapped. The word “massive” means, in our previous work, geometries that comprise a large amount of matter (or, morę generally, a lot of scalar 3-curvature) inside a fixed volume.The next step should be the proof that trapped surfaces are enclosed, on initial Cauchy slices, by the portion of an event horizon lying in the Cauchy slice. Thus, one would wish to prove, that trapped surfaces, once they appear, always imply the existence of event horizons, that is, the existence of a black hole.The second part of this programme has not been done, at least in suf- ficiently generał situations, but the first problem is relatively well shaped (see [2] for a review).(123)124 P. Koc, E. MalecIn this paper we will examine the formation of trapped surfaces in nonspherical cosmologies. We will study an open, expanding universe, which locally is inhomogeneous and anisotropic, but very far firom bounded regions coincides with a fiat, homogeneous and isotropic Friedman-Robertson-Wal­ker universe. We combine techniques of two'previous papers [3,4], to prove in Sections 3 and 4 that in conformally flat universes (initially) time-indepen- dent perturbations give rise to the formation of trapped surfaces if and only if the amount of energy is large comparing to the size of region in which it is compacted. A part of Section 4 pursues further the previously obtained results on the formation of trapped surfaces in asymptotically and conformally flat geometries.The order of the rest of this work is following. Section 2 is introductory one. Section 5 shows that there is an upper limit for the amount of matter inside a volume of a flxed size. That result has been proven earlier (see [2] for the discussion of known results), and it suggests that the old idea of Einstein [5] that singularities do not exist, may be true, at least for some types of singularities, e.g., massive ones. Morę conclusive statements would require the solving of the problem of the existence of maximal slices, or its equivalent used below, slices of constant mean curvature.Section 6 discusses the formation of surfaces that are pointwise trapped. The last Section comprises some comments on the obtained results and on possible generalizations.2. PreliminariesInitial data for the evolution of Einstein equations consist of energy density p and matter current J⅛, which are prescribed on a space-like hy- persurface Σ with a three-geometry defined by induced metric tensor gab and with an immersion of Σ into 4-dimensional space-time given by an ex- ternal curvature Kab = jftgab∙ This whole set of functions is not arbitrary, sińce they have to satisfy the initial value equations:<3⅛ - ΛΓo6ΛΓβb + (K*a)2 = 16τrp, (2.1)DaKab - DbKaa = -8πJb. (2.2)In (2.1) <3>JJ[<∕] is the scalar curvature of Σ. Da in (2.2) denotes covariant derivative of the 3-dimensional metric. We put c = 1,(7 = 1.As it was mentioned, we consider a cosmological Friedman-Lemaitre model (with k=0) with conformally flat perturbations. We describe solu- tions in terms of their finite deviation from homogeneous-isotropic back- ground. It is necessary to point out that we do not apply linearization procedurę and our analysis is generał one.Trapped Surfaces in Nonspherical Open Univeτse 125Łet us relate the background quantities (denoted by a hat) and per- turbed ones as follows:<7αb = *‰ 9ab9bc = ⅛, gab = Φ~4gab, (2.3) 9 ab = a(i)9ab, (2.4)Kab = ~gab = β(t)gab, (2.5)Kab = Φ4Kab + δKab = β(t)gab + δKab, (2.6)p = p + δpt Jb = δJb. (2.7)Ali indices of non-hatted quantities are raised and lowered with the metric (2.3). ŚF is a conformal factor. In (2.4) gab denotes a fiat metric and α(t) is a scalar function determined by Friedman equation. In (2.5) β(t) = ⅛lnα∕2∙We employ, following [7], special coordinates adapted to equipotential surfaces S (i.e., surfaces of constant Φ). In these coordinates the three- dimensional linę element readsds2 = Φ4(σ)a(t)(gσσdσ2 + gττdr2 + gψφdφ2 + 2gτφdrdφ). (2.8)Here σ > 0, σ foliates the level surfaces of the Φ∙, that from now one are assumed to be convex, and τ, φ are quasi-angle variables.Remark. One can show, that outside matter the surfaces of constant Φ coincide with surfaces of constant g,oo∙ Therefore, the equipotential sur­faces have a transparent physical meaning — these are surfaces of constant redshift — and might be detected experimentally.The cosmological background satisfies the following relation(gabKab)2 - (9adgbcKdcKab) = 6β2 = 16πp. (2.9)We restrict our attention to the case, whenδKab = 0, (2.10)which guarantees us the absence of gravitational waves. This allows us to use the total proper rest mass as the quasilocal measure of the gravitational energy. The constraints (2.1, 2.2) reduce to a single equation= 16πtfp. (2.11)The metric gab is fiat, hence the scalar curvature of the conformally related metric gab reads126 P. Koc, E. MalecV2≠≠5 ’Inserting this into (2.11) we obtain the Lichnerowicz eąuation:V2<f = -2τrfyΦ5.3. Necessary conditionIn this chapter we will find a necessary condition for a surface S to be an averaged trapped surface. We start with expressions for the orthonormal unit vectors to S, in all the three metrics that were defined hitherto:n*  = (τ3-I==,0,0), fιt = ( ^⅛r-=,0,θ), ń*  = (-7⅛=,0,0).∖φzy∕agσσ J ×y∕agσσ ' yV9σσ 7(3-1) We define the mass of the inhomogeneity as followsδM = j SpdV = y 8pΦβdV.v pLet us multiply eąuation (2.12) by Φ and integrate over the volume V. Using the above definition of SM we get(3.2)(3-3)We need to calculate the following quantities, (i) the divergence of the normal unit vector n‘, i.e., the mean curvature of a surface S in the physical metric gikDini = ±∂i(Jgni) = y/3<⅝7-ln√ξ χ ⅛Φ∂σΦ^2y∕agσσ ^4Va9σσ,(ii) the mean curvature of S with respect to the background cosmological metric(3∙4)(3∙5)(2.12)Trapped Surfaces in Nonspherical Open Univeτse 127Above ξ = 9ττ9ψφ ~ 9τφ∙ Inserting (3.4) to (3.5) we obtain the identityf9°f = (l>ini - ⅛ (3.6)We replace the integrand of the last term of (3.3) by the right hand side of (3.6), in order to getSM = f ∂iΦ∂iΦdV [ DinidS + [pdS. (3.7)2τr J 8τr J o7Γ JV S ŚA surface S is called an averaged trapped surface, if the mean expansion d of outgoing futurę directed nuli geodesics which are orthogonal to S is nonpositive,y ddS < 0,Swhere $ relates to the initial data of Einstein equations <7α⅛, Kab byi? = Dana - Kabnanb + gabKab = Dana - Kabhanb + gabKab= Dana + 2/3.In the last two equations we take into account (2.6) and fiKab = 0. The condition that S is an averaged trapped surface reads nowy(Dana + 2β)dS < 0 (3.8)sWe rewrite (3.7) in the following form, by adding and subtracting a term containing /3:(3-9)Now let us assume that S is an averaged trapped surface. In such a case the flrst two terms of the above equation are positive. A classical 128 P. Koc, E. Malecisoperimetric inequality (see, e.g., [7]) gives for a convex surface S in the three-dimensional fiat space the estimation/ pdS ≥ ⅛y∕~"κSŚ(5 denotes an area of S and p is just the mean curvature of S with respect to the Euclidean metric gik). Thus we may conclude thatLet us remark also that the third term on the right hand side of (3.9) is equal to (∕3∕4τr)S,. Taking into account this fact, the last inequality and Eq. (2.9), we may infer the followingTheorem 1. (a necessary condition). Under the assumption of:δKaa = 0,if an equipotential convex surface S is trapped thenwhere S denotes the area of a surface S.4.Sufficient conditionLet us start from the assumption that an equipotential surface S is not tτappe<L Then its averaged expansion is positive,The equation (3.9) and the above inequality imply(4-1)We define quantitiesTrapped Surfaces in Nonspherical Open Uniυeτse 129(4.2)(4.3)Rad(S(σ)) equals to the proper radius in the case of spherical symmetry W- It is reasonable to conjecture that Rad(S,(σ)) is bounded from above by the largest proper radius of the volume enclosed by the surface S in all geometries when the equipotential surfaces are convex [4,7]. We prove now an important technical estimation:Lemma. Assume that 6p > 0. ThenRad(S(σ)) ≥ D(S(σ)). (4-4)Proof.For σ — 0 the equation (4.4) becomes an identity. Thence it is sufficient to show that δσRad(S(σ)) ≥ 5σZ>(S,(σ)), t.e.,I⅛om the maxiιnum principle and δp > 0 : ∂σΦ < 0, hence the above equation holds true assuming that(4-5)It is possible to prove even a stronger inequality(4-6)Eq. (4.6) was proven in [4], under certain (conjectured) property of convex foliations in fiat geometry. Here we shall present a different proof, communicated to one of us (EM) by Eanna Flanagan.130 P. Koc, E. MalecAt the beginning, we prove (4.6) under a morę stringent condition than stated in Łemma, namely we assume that δp = 0 outside S.Let Φ be a function satisfying the Łaplace equation outside S and having the values —1 on S and 0 at infinity. It can be seen that Φ = 1 + (1 — A)!?, where A is a value of Φ on S. The inequality (4.6) is now equivalent toLet us define quantitiesThe first quantity is related to the electrostatic capacity and the relation between the two was discussed by Szegδ [9], who proved that for any convex surface embedded in Euclidean 3-space M > 4τrC,. This ends the proof of Łemma in the case of 8p = 0.(4.6) holds also in the case of non-vanishing (outside S) matter density. Let us assume that Φ and χ have the same boundary values, A at S and 1 at infinity and V2χ = 0, V2Φ = — 2πδpχ5. Notice that V2(Φ — χ) < 0; thus, from the maximum principle, u = Φ — χ achieves its maximal value somewhere between S and infinity. Since u = 0 at S, this implies that at S we have ∂σ{Φ — χ) > 0, i.e.,χ satisfies the inequality (4.6) at S, according to the preceding proof. Be- cause of the last inequality and the boundary condition u = 0 at S, we conclude that also Φ fulfills the property (4.6). This ends the proof of Łemma.Now we may replace the last two terms in (4.1) by Rad(5(σ)), as to get8M < 5√∕- + Rad(S,(σ)). (4.7)V 6 τrIf S is not trapped, then the amount of perturbative mass inside it can not exceed expression on the right hand side of (4.7). Thus we have proven the following:Trapped Surfaces in Nonspherical Open Universe 131Theorem 2. (a sufficient condition). Assume that:Sp > 0,SKab = 0.If the amount of perturbative mass satisfies the inequalitySM > + Rad(S(σ)),then a convex surface S is trapped.5.Mass of perturbations is boundedErom Eq. (3.6) we haveThis equation can be rewritten in the formSM = Z>(S(σ)) + [ pdS - f (Φp + 2fισ∂σΦ)dS.o7T J 4τr JŚ ŚErom (4.2) and the inequality (4.5) we getSM < 2P(5(σ)).By the preceding Łemma, the following holds true. Theorem 3. Under the assumptions:Sp > 0, SKab = 0.the amount of perturbing mass is bounded from above,SM < 2Rad(5,(σ)).Inserting the above equation in (3.7) and replacing the first and the last terms of (3.7) by Z>(S(σ)) (see Eq. (4.2)) we obtain132 P. Koc, E. Malec6. The existence of pointwise trapped surfacesUsing (3.4), one gets the following from the definition of the mean expansion d6.1)If S is not pointwise trapped, then certainly the largest value of the expan- sion i? is positive, i.e.,4Φ(-∂σΦ) ≤ Φ2 max(np). (6-2)Here and below max(∕) means the maximal value of f(x), for x ∈ S. Combining (6.1) and (6.2) one obtains thenThe last equality follows from Eq. (3.9), as expressed in terms of the mean expansion and D(S(σ)) (see Eq. (4.2)).The inequality (6.3) has always to be true, if S is not pointwise trapped, as indicated above. Thus, if (6.3) is broken, then S has to be trapped. In particular, from the inequality (4.4) of our Lemma, we may conclude the followingTheorem 4. Assume that inside an equipotential surface S the follow- ing conditions are fulfilled:(i) δx*  = o,sp>o,(ii) SM > Rad(S,(σ)) + ⅛βS + ⅛ ∕ nσ[max(nσp) - nσp].Then S is pointwise trapped.S7. Concluding remarksWe have discussed the formation of trapped surfaces in the class of conformally fiat deformations ( which initially do not change the expansion of the geometry) of Friedman fiat expanding universes. Trapped surfaces have a well defined physical sense, namely a bundle of light rays which emanates from a trapped compact surface S (orthogonally to it) increases Trapped Surfaces in Nonspherical Open Universe 133its intensity, at least initially, when moving outwards. That is unlike our everyday,s experiencej we know surfaces (say, concave mirrors) that have the property of focusing light rays, but they are not closed. A compact trapped surface S plays a role of a compact reflector; from this intuitive definition it is obvious that trapped surfaces can exist only in strongly curved geometries.We have proven that, assuming the absence of gravitational waves, trapped surfaces can appear only in the presence of large masses (Section 3) and that they must appear if concentration of matter in a fixed volume is large. Surprisingly, the amount of perturbed mass in a Unitę volume has to be bounded from above (Section 5). The formation of trapped sur­faces requires the morę matter the larger is their nonsphericality (Section 6). Moreover, the expansion of the Universe (as represented in the above statements by the Hubble constant β or the averaged matter density p ) makes it morę difficult to form a trapped surface, in the perfect agreement with intuition.The all results are strict, sińce all inequalities are saturated in some spherically symmetric situations [3].These results might be proved also in another foliations of the space- time. A generalization of the radial gauge (in which Tr A = K° — no summation over σ) allows one to consider spacetimes with initial data off time symmetry, sińce then the notions of minimal surfaces and apparent horizons are identical1.1 I thank Niall O’ Murchadha and Jemal Guven for discussing that point.The existence problem of these foliations is not solved as yet, similarly as for the constant mean curvature foliations that are used in this paper.REFERENCES[1] R. Penrose, Techniques of Differeniial Topology in Relativity, Society for In- dustrial and Applied Mathematics, Philadelphia, 1972.[2] E. Malec, Acta Phys. Pol. B22, 829 (1991).[3] U. Brauer, E. Malec, Trapped surfaces due to spherical inhomogeneities in expanding open univeτses, to appear in Class. Quantum Grav..[4] E. Malec, Phys. Rev. Lett. 67, 949 (1991).[5] A. Einstein, Ann. Math. 40, 922 (1939).[6] R. Bartnik, P. Chruściel, N.O. Murchadha, Commun. Math. Phys. 130, 95 (1990).[7] E. Malec, Acta Phys. Pol. B22, 347 (1991).[8] Yu.D. Burago, V.A. Zalgaller, Geometrie Inequalities, Springer Verlag, 1988.[9] G. Szego, Buli. Am. Math. Soc. 51, 325 (1945).

Trapped surfaces in nonspherical open universes

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Trapped surfaces in nonspherical open universes

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