Geometric properties of a certain class of compact dynamical horizons in locally rotationally symmetric class II spacetimes
GGeometric properties of a certain class of compact dynamical horizons in locallyrotationally symmetric class II spacetimes
Abbas Sherif ∗ Cosmology and Gravity Group, Department of Mathematics and Applied Mathematics,University of Cape Town, Rondebosch 7701, South Africa
Peter K. S. Dunsby † Cosmology and Gravity Group, Department of Mathematics and Applied Mathematics,University of Cape Town, Rondebosch 7701, South AfricaSouth African Astronomical Observatory, Observatory 7925, Cape Town, South Africa
In this paper we study the geometry of a certain class of compact dynamical horizons with a time-dependent induced metric in locally rotationally symmetric class II spacetimes. We first obtain acompactness condition for embedded 3-manifolds in these spacetimes, satisfying the weak energycondition, with non-negative isotropic pressure p . General conditions for a 3-manifold to be adynamical horizon are imposed, as well as certain genericity conditions, which in the case of locallyrotationally symmetric class II spacetimes reduces to the statement that ‘the weak energy conditionis strictly satisfied or otherwise violated’. The compactness condition is presented as a spatial firstorder partial differential equation in the sheet expansion φ , in the form ˆ φ + (3 / φ − cK = 0, where K is the Gaussian curvature of 2-surfaces in the spacetime and c is a real number parametrizing thedifferential equation, where c can take on only two values, 0 and 2. Using geometric arguments, it isshown that the case c = 2 can be ruled out, and the S (3-dimensional sphere) geometry of compactdynamical horizons for the case c = 0 is established. Finally, an invariant characterization of thisclass of compact dynamical horizons is also presented. PACS numbers:
I. INTRODUCTION
A new covariant and gauge invariant way of studying black hole horizons [1–3] has emerged over the last few years.Though used in a limited way to date, it provides a computationally inexpensive method for determining variousgeometric and thermodynamic properties of black hole horizons [2, 3]. This method employs the 1 + 1 + 2 semitetradcovariant splitting of spacetimes, which describes the spacetimes using well defined geometric and matter variables(see the references [4–6]). The first use of this approach in the study of black hole horizons, as far as we are aware,was carried out in 2014 by Ellis et al. , [1], where the authors considered a gravitational collapse scenario in a realisticastrophysical setting, considering examples from the relatively small class of locally rotationally symmetric class IIspacetimes.Ellis et al. [1] considered a case in an astrophysical setting where an initial marginally trapped surface, at thebeginning of a gravitational collapse scenario, bifurcates into evolving surfaces, one being a timelike marginallytrapped tube (to later be defined) that evolves inward and the other being a spacelike marginally trapped tubewhich evolves outward, expanding under the infalling of radiation, and approaches asymptotically a null marginallytrapped tube. The causal character of the marginally trapped tubes (see the references [7–14] for more discussionson marginally trapped tubes) was determined by the slope of the tangent to the marginally trapped surfaces foliatingthem. Noting that the covariant derivative of the outgoing null expansion scalar is normal to the marginally trappedtubes and marginally trapped surfaces, the dot product with the tangent to the marginally trapped surfaces vanishes,which allows one to determine the expression for the slope. In more general spacetimes this approach fails, or at bestan explicit expression for the slope is not possible. This was the primary nature of two works by Sherif et al. [2, 3]which were extensions of the work by Ellis et al. [1], where two approaches were established; in one of the approachesthe norm of the covariant derivative of the outgoing null expansion scalar was used, and in the other approach asmooth function on the marginally trapped tube - an approach by Booth and coauthors [13, 14], albeit restricted tospherically symmetric spacetimes - was expressed in terms of the 1 + 1 + 2 covariant variables. These approaches have ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] O c t allowed Sherif and coauthors to obtain well established results including the stability and instability of marginallytrapped surfaces (see the following references [15–18] for a discussion on this subject) of a Schwarzschild black holeand the Oppenheimer-Snyder dust collapse, as well as the bounds on the equation of state parameter determiningthe causal character of horizons in the Robertson-Walker spacetimes (these results were obtained by Ben-Dov [19]).The third law of black hole thermodynamics was shown for locally rotationally symmetric class II spacetimes and aclassification scheme was provided for diffeomorphically equivalent and causally equivalent marginally trapped tubes.Thurston’s geometrization conjecture [20–22] and Hamilton’s proof of the uniformization theorem [23–25] are two ofthe most fundamental and important results in geometry and geometric analysis. These results provide the standardclassification of 2-dimensional and 3-dimensional smooth manifolds. The Ricci flow developed by Hamilton [25] tostudy the evolution of metrics on smooth manifolds, provides a way of classifying the geometry of smooth closed3-manifolds. In particular, Hamilton showed that a closed smooth manifold admitting a metric of positive curvatureis uniquely spherical. As a consequence, a compact Riemannian 3-manifold admitting a positive metric cannot befoliated by hyperbolic planes. This result will be used in Section III when we restrict the allowable geometry underthe compactness conditions that will be imposed on horizons in locally rotationally symmetric class II spacetimes.The compactness condition can be obtained using the
Bonnet-Myers Theorem [26] which gives a bound on the radiusof a manifold with Ricci curvature strictly positive.From a purely geometric viewpoint, finding specific examples of compact 3-manifolds in spacetime to investigateboth local and global geometric properties makes sense as there is a wealth of literature on n -dimensional compactRiemannian manifolds. The effectiveness of the adapting of horizon analysis to the 1 + 1 + 2 covariant variables -as demonstrated in the works by Sherif et al. [1–3] - coupled with standard results for compact 3-manifolds shouldprove very useful in exposing the intricate balance between geometry and thermodynamics on black hole horizons.This work aims to identify certain classes of compact dynamical horizons in spacetimes of the Locally rotationallysymmetric class (LRS II) using existing geometric analytic tools from Riemannian geometry, and investigate theirgeometric and thermodynamic properties.The paper is organized as follows: in section II we briefly discuss the semi-tetrad covariant approach to be followedthroughout this work, and then proceed to provide definitions needed to clarify the discourse of the paper. Black holehorizons, the required energy conditions, and additional properties are introduced in a covariant way. In section III weprovide a compactness theorem for dynamical horizons in LRS II spacetimes, evoking the well known Bonnet-Myerstheorem. The properties of, and interplay between the geometry and thermodynamics of these obtained classes ofcompact horizons are then investigated using the Ricci flow evolution equation. The existence of solutions to theevolution equation and the geometric restrictions are also investigated. Finally, we conclude with a discussions of ourresults in section IV. II. PRELIMINARIES
In this section we provide a review of some background material on the 1 + 1 + 2 covariant splitting of LRS IIspacetimes, as well as useful definitions so that the reader keeps track of concepts that will be used throughout therest of the paper. A. covariant splitting of LRS II spacetimes Any 4-vector U µ in a spacetime manifold may be split into a component along a unit timelike vector field u µ anda component on the 3-space as U µ = U u µ + U (cid:104) µ (cid:105) . The scalar U is the scalar along u µ and U (cid:104) µ (cid:105) is the projected 3-vector [29, 30] projected via the tensor h νµ ≡ g νµ + u µ u ν .This 1 + 3 splitting irreducibly splits the covariant derivative of u µ as ∇ µ u ν = − A µ u ν + 13 h µν Θ + σ µν , (1)and the energy momentum tensor to be decomposed as T µν = ρu µ u ν + 2 q ( µ u ν ) + ph µν + π µν . (2)The vector A µ = ˙ u µ is the acceleration vector, Θ ≡ D µ u µ - the trace of the fully orthogonally projected covariantderivative of u µ - is the expansion and σ µν = D (cid:104) ν u µ (cid:105) is the shear tensor. (Wherever used in this paper, angle bracketswill denote the projected symmetric trace-free part of the tensor.) The quantity ρ ≡ T µν u µ u ν is the energy density, q µ = − h νµ T νγ u γ is the 3-vector defining the heat flux, p ≡ (1 / h µν T µν is the isotropic pressure and π µν is theanisotropic stress tensor.Whenever there is a preferred unit normal spatial direction e µ one may split the 3-space into a direction along e µ and a 2-surface where the projection tensor defined as N µν = g µν + u µ u ν − e µ e ν . (3)The projection tensor N µν projects any 2-vector orthogonal to u µ and e µ onto the 2-surface defined by the sheet( N µµ = 2). Thus u µ N µν = 0 , e µ N µν = 0. The vectors u µ and e µ are normalized so that u µ u µ = − e µ e µ = 1.This is referred to as the 1 + 1 + 2 splitting.This splitting of the spacetime gives rise to four derivatives: • For an arbitrary tensor S µ..νγ..δ , one defines the covariant time derivative (or simply the dot derivative) alongthe observers’ congruence of S µ..νγ..δ as ˙ S µ..νγ..δ ≡ u σ ∇ σ S µ..νγ..δ . • For an arbitrary tensor S µ..νγ..δ one defines the fully orthogonally projected covariant derivative D with thetensor h µν as D σ S µ..νγ..δ ≡ h µρ h ηγ ...h ντ h ιδ h λσ ∇ λ S ρ..τη..ι . • Given a 3-tensor ψ µ..νγ..δ the spatial derivative along the vector field e µ (simply called the hat derivative ) isgiven by ˆ ψ γ..δµ..ν ≡ e σ D σ ψ γ..δµ..ν . • Given a 3-tensor ψ µ..νγ..δ the projected spatial derivative on the 2-sheet (projection by the tensor N νµ ), calledthe delta derivative , is given by δ σ ψ γ..δµ..ν ≡ N ρµ ..N τν N γη ..N δι N λσ D λ ψ η..ιρ..τ .Note that the projections by the tensors h µν and N µν in the definitions of the D and δ derivatives are carried outover all indices (see the references [4, 31, 32] for more discussions). Definition II.1 A locally rotationally symmetric class II (LRS II) spacetime is an evolving, vorticity free andspatial twist free spacetime with a one dimensional isotropy group of spatial rotations defined at each point of thespacetime. It is given by the general line element ds = − A ( t, χ ) + B ( t, χ ) + F ( t, χ ) (cid:0) dy + G ( y, k ) dz (cid:1) , (4) where t, χ are parameters along integral curves of the timelike vector field u µ = A − δ µ of a timelike congruence andthe preferred spacelike vector e µ = B − δ µν respectively. The constant k fixes the function G ( y, k ) ( k = − correspondsto sinh y , k = 0 corresponds to y , k = 1 corresponds to sin y ) [4, 29, 32]. For LRS II spacetimes, all vector and tensor quantities vanish identically and the Weyl tensor is purely electric (seereference [4] for details). Therefore the complete set of 1 + 1 + 2 covariant scalars fully describing the LRS class ofspacetimes are { A, Θ , φ, Σ , E , ρ, p, Π , Q } . The quantity φ ≡ δ µ e µ is the sheet expansion, Σ ≡ σ µν e µ e ν is the scalar associated with the shear tensor σ µν , E ≡ E µν e µ e ν is the scalar associated with the electric part of the Weyl tensor E µν , Π ≡ π µν e µ e ν is the anisotropicstress scalar, and Q ≡ − e µ T µν u ν = q µ e µ is the scalar associated to the heat flux vector q µ .The full covariant derivatives of the vector fields u µ and e ν are given by [4] ∇ µ u ν = − Au µ e ν + e µ e ν (cid:18)
13 Θ + Σ (cid:19) + 12 N µν (cid:18)
23 Θ − Σ (cid:19) , (5a) ∇ µ e ν = − Au µ u ν + (cid:18)
13 Θ + Σ (cid:19) e µ u ν + 12 φN µν . (5b)We also note the useful expression ˆ u µ = (cid:18)
13 Θ + Σ (cid:19) e µ . (6)We will make use of the following commutation relation between the dot and hat derivatives when acting on anarbitrary scalar ψ in LRS II spacetimes: ˆ˙ ψ − ˆ˙ ψ = − A ˙ ψ + (cid:18)
13 Θ + Σ (cid:19) ˆ ψ. (7)The field equations for LRS spacetimes are given as propagation and evolution of the covariant scalars [1, 4]: • Evolution
23 ˙Θ − ˙Σ = Aφ − (cid:18)
23 Θ − Σ (cid:19) + E −
12 Π −
13 ( ρ + 3 p ) , (8a)˙ φ = (cid:18)
23 Θ − Σ (cid:19) (cid:18) A − φ (cid:19) + Q, (8b)˙ E −
13 ˙ ρ + 12 ˙Π = − (cid:18)
23 Θ − Σ (cid:19) (cid:18) E + 14 Π (cid:19) + 12 φQ + 12 (cid:18)
23 Θ − Σ (cid:19) ( ρ + p ) , (8c) • Propagation
23 ˆΘ − ˆΣ = 32 φ Σ + Q, (9a)ˆ φ = − φ + (cid:18)
13 Θ + Σ (cid:19) (cid:18)
23 Θ − Σ (cid:19) − ρ − E −
12 Π , (9b)ˆ E −
13 ˆ ρ + 12 ˆΠ = − φ (cid:18) E + 12 Π (cid:19) − (cid:18)
23 Θ − Σ (cid:19) Q (9c) • Evolution/Propagation ˆ A − ˙Θ = − ( A + φ ) A −
13 Θ + 32 Σ + 12 ( ρ + 3 p ) , (10a)˙ ρ + ˆ Q = − Θ ( ρ + p ) − (2 A + φ ) Q −
32 ΣΠ , (10b)˙ Q + ˆ p + ˆΠ = − (cid:18) A + 32 φ (cid:19) Π − (cid:18)
43 Θ + Σ (cid:19) Q − ( ρ + p ) A. (10c) B. Some useful definitions
We will now give some definitions used in describing black hole spacetimes and associated horizons. What are tofollow are all very familiar definitions and we will follow mostly standard references [7–9, 12, 13].Given an embedded 2-manifold S ⊂ M , one may define two normal vector fields k µ and l µ , called the outgoing andingoing null normal vector fields associated with outgoing and ingoing null geodesics. The vector fields k µ and l µ arenormalized to satisfy the relations k µ k µ = l µ l µ = 0; k µ l µ = − . Associated with S are functions, defined for each of the null normal directions which we denote Θ k and Θ l respectively.These are called the null normal expansions in the k µ and l µ directions. Definition II.2 (Marginally trapped surface (MTS))
An embedded -surface S in M is said to be marginallytrapped if for all points of S , Θ k = 0 and Θ l ∈ R − (where R − denotes the set of negative real numbers). These 2-surfaces foliate hypersurfaces in spacetime that, under certain conditions, may be associated to the boundaryof a black hole. The notion of these hypersurfaces which generalizes Hayward’s future outer trapping horizon (FOTH)was introduced by Ashtekar and Galloway [9]: S S S e a u a l a k a ΞFIG. 1: The usual depiction of a dynamical horizon Ξ foliated by marginally trapped surfaces S, showing the null normal vectorfields k a and l a , as well as the unit timelike and unit normal vector fields u a and e a respectively. Definition II.3 (Marginally trapped tube (MTT))
A marginally trapped tube is a codimension embedded sub-manifold in a spacetime foliated by marginally trapped surfaces. In general, the sign of the induced metric on a marginally trapped tube may vary. In specific cases where the signof the metric is not changing as one moves along the marginally trapped tube, the marginally trapped tube is calleda timelike membrane (TLM), a non-expanding horizon (NEH), or a dynamical horizon depending on the sign of themetric which depends of the formalism used (we will elaborate on this shortly).Above we provide a depiction of a dynamical horizon in figure 1 (original depictions appearing in [7, 8, 11]), whereit is pictured as a hyperboloid in Minkowski space. In this schematic, motions along the unit normal vector field e a are interpreted as time evolution with respect to observers at infinity. Similar picture of a timelike membraneas a hyperboloid in Minkowski space can be presented, where in this case, there is a decrease in the surface areaof the “discs” depicting the marginally trapped surfaces S i in figure 1, as one moves along e a . The case of an non-expanding horizon can be depicted as a cylinder in Minkowski space. In each case the marginally trapped surfacesare intersections of the hyperboloid (cylinder) with spacelike planes. C. Marginal trapping and marginally trapped tubes in LRS II spacetimes
For LRS II spacetimes the null expansion scalars associated with the outgoing and ingoing null normal vector fieldsto 2-surfaces in a spacetime are given by linear combinations of the shear, expansion and sheet expansion covariantscalars. Explicitly these scalars are given by [1–3]Θ k = 1 √ (cid:18)
23 Θ − Σ + φ (cid:19) , (11a)Θ l = 1 √ (cid:18)
23 Θ − Σ − φ (cid:19) . (11b)The requirement that a marginally trapped surface satisfies Θ k = 0 and Θ l < φ > /
3) Θ − Σ <
0. This then restricts the subclass of LRS II spacetimes potentially admittingmarginally trapped surfaces. From (11a) and (11b), it is clear that whenever φ = 0 we must have (2 /
3) Θ − Σ = 0in which case the 2-surface is minimal (see references [2, 3] for further discussion). Thus we have ruled out minimalsurfaces in LRS II spacetimes for the rest of this paper.As was discussed in the previous subsection, marginally trapped tubes are foliated by marginally trapped surfaces.In the case the metric signature on a marginally trapped tube is fixed, then it may be classed as timelike, non-expandingor spacelike.Determining the metric signature on an marginally trapped tube is not unique. However, one can always relate thedifferent formulations that compute the metric signature. Two explicit formulations have been utilized by applyingthe 1 + 1 + 2 covariant formalism. The first is a formalism developed in reference [2]. Briefly put, a choice of vectorfield, dependent on some smooth function - this function being denoted C - is made, which is tangent to the marginallytrapped tube and everywhere orthogonal to the foliation (this approach was developed by Booth and coauthors (seereference [14] and associated references) and utilized for the well known spherically symmetric spacetimes, but wasgeneralized to all of a more diversed class of spacetimes and interpreted in terms of the 1 + 1 + 2 covariant variables): V µ = k µ − Cl µ . (12)The definition of V µ implies the outgoing null expansion scalar remains fixed as it is Lie dragged along V µ , whichimmediately gives C = L k Θ k L l Θ k , (13)where L n denotes the Lie derivative along the vector field n µ . If C < C = 0 or C >
0, and C is such that it is fixedall over the marginally trapped tube, then the marginally trapped tube is a timelike membrane, a non-expandinghorizon or a dynamical horizon. A second approach (which we will not take into account in this paper but which mayhowever be related to C ) notes that the gradient of Θ k , ∇ µ Θ k , is normal to the marginally trapped tube and as suchthe sign of the norm of ∇ µ Θ k can be used to determine the causal character of the marginally trapped tube. In factit can easily be seen that we can write (the references [2, 3] have the description of the procedures) C ∗ = ∇ µ Θ k ∇ µ Θ k = −L k Θ k L l Θ k , (14)which allows us to write C = − L l Θ k ) C ∗ . (15)Of course then the signs of C and C ∗ are reversed. In this case if C ∗ > C ∗ < L k Θ k ≤ C is explicitly calculated as [2] C = − ( ρ + p + Π) + 2 Q ( ρ − p ) + 2 E . (16)Clearly if L k Θ k = 0 then C = 0, a n0n-expanding horizon. Therefore, if we are assuming the null energy condition issatisfied then this amounts to the energy condition ρ + p + Π > Q . As C must be greater than zero on the dynamicalhorizon we must therefore have ρ < p − E . Thus we obtain the required energy condition on a dynamical horizon inLRS II spacetimes: 2 Q − Π < ( ρ + p ) < p − E ) (see the reference [2] for further details).There are certain cases that may be immediately ruled out, i.e. certain subclass of LRS II spacetimes can bedetermined to not admit a dynamical horizon. One of them is the shear-free case. As has been shown by Sherif et al. [3], for a dynamical horizon the expansion Θ is strictly positive. It is clear then that the shear-free case can admitno dynamical horizon. This is because in the shear-free case one has from the vanishing of (11a) (2 /
3) Θ = − φ , andsince φ > <
0. This in fact clearly shows that any marginally trapped tube in a shear-free LRS IIspacetime will necessarily be a timelike membrane [3].In this work we shall specialize to dynamical horizons. Throughout this paper we shall simply write horizon whenever we are referring to a dynamical horizon. We shall also assume the genericity condition of Ashtekar &Galloway [9], i.e. σ µν σ µν + T µν k µ k ν (cid:54) = 0 , (17)holds true on the horizon, which, in the case of LRS II spacetimes, translates to the condition that the weak energycondition (WEC) is either strictly satisfied or otherwise violated on the horizon ( ρ + p > ρ + p < III. COMPACT DYNAMICAL HORIZONS IN LRS II SPACETIMES
For LRS II spacetimes, with both the unit vectors u µ and e µ being hypersurface orthogonal, the Ricci tensor forany spacelike 3-surface is given by [4] R µν = − (cid:18) ˆ φ + 12 φ (cid:19) e µ e ν − (cid:20) (cid:16) ˆ φ + φ (cid:17) − K (cid:21) N µν , (18)where K is the Gaussian curvature of the 2-sheet which is given by [4] K = 13 ρ − E −
12 Π + 14 φ − (cid:18)
23 Θ − Σ (cid:19) , (19)whose dot and hat derivatives are respectively given by˙ K = − (cid:18)
23 Θ − Σ (cid:19) K (20a)ˆ K = φK. (20b)The Ricci scalar on the 3-manifold is given by R = − (cid:18) ˆ φ + 34 φ − K (cid:19) . (21)In all that is to follow we will usually set α = − (cid:18) ˆ φ + 12 φ (cid:19) and β = − (cid:20) (cid:16) ˆ φ + φ (cid:17) − K (cid:21) . We will emphasize that R without an index specifying the space we are working in, we will always be referring toembedded 3-submanifolds. Whenever we are referring to the ambient spacetime or the marginally trapped surfacesfoliating the 3-manifolds R will be specifically indexed for that purpose. In the case that the spacelike 3-manifold isfoliated by 2-surfaces that are marginally trapped, the Ricci tensor can be written as R µν = (cid:20) ρ + (cid:18) E + 12 Π (cid:19) − φ (Θ + φ ) (cid:21) e µ e ν + (cid:20) ρ − (cid:18) E + 12 Π (cid:19) − φ (Θ + φ ) (cid:21) N µν , (22)where we have used (9b) and the vanishing of Θ k . Thus the Ricci scalar becomes simply R = 2 ρ − φ (Θ + φ ) . (23)Throughout we will assume that the Ricci tensor on the 3-manifolds do not vanish. Consequently we have therestriction α, β (cid:54) = 0; R (cid:54) = 0on the horizon, which combines to give 25 ρ (cid:54) = − (cid:18) E + 12 Π (cid:19) . (24)The relationship between the geometry and the Ricci curvature of a Riemannian manifold has been extensivelystudied (see the references [25, 27]). In particular, bounds on the Ricci curvature have shed many insights ontopological properties of Riemannian manifolds [26]. If these manifolds are foliated by marginally trapped surfaces,what general properties, both geometric and topological, can be obtained?We state and prove a compactness result for embedded 3-manifolds in LRS II spacetimes, which depends on a firstorder spatial differential equation in the sheet expansion. Theorem III.1
Let M be an LRS II spacetime and Ξ an embedded spacelike -manifold with R µν (cid:54) = 0 . Furthermore,suppose ˆ φ + 34 φ − cK = 0 (25) is satisfied at all points of Ξ . If on the marginally trapped surfaces foliating Ξ we have K > with c = 0 or K < with c = 2 for finite K , then Ξ is compact. Proof
The mode of the proof will be to invoke the well known
Bonnet-Myers theorem [26] which, simply put, impliescompactness of an n -dimensional Riemannian manifold M if its Ricci curvature R is bounded below by( n − m > , (26)for some constant m .First it is clear that for K > K, ˆ K >
0, so K is positive all over Ξ. We may rewrite(21) as R = (cid:20) − K (cid:18) ˆ φ + 34 φ (cid:19) + 2 (cid:21) K, (27)so that R assumes the form of the left hand side of (26), with m = K . The bracketized term of (27) can now beequated to ( n −
1) as (for n = 3): ( n −
1) = 2 = − K (cid:18) ˆ φ + 34 φ (cid:19) + 2 , (28)so that ˆ φ + 34 φ = 0 . (29)Then if K > n − K = 2 K >
0. Similarly, we can set − ( n −
1) = − − K (cid:18) ˆ φ + 34 φ (cid:19) + 2 , (30)so that ˆ φ + 34 φ − K = 0 . (31)Then if K < n −
1) ( − K ) = − K >
0. In either case we have
R > R = ± K (‘+’ is for thecase of K > − ’ for the case of K < given an LRS II spacetime M and a compact -surface Ξ ⊂ M , if ˆ φ + (3 / φ − cK = 0 is satisfied on Ξ for c ∈ { , } - where K is the Gaussian curvature of -surfaces S in M , then the smooth embedding ϕ : S −→ Ξ of S into Ξ preserves the Ricci curvature .Notice here that we do not specify the geometry of the marginally trapped surfaces. While we are stating that weare considering the cases for K >
K < K may vary over the marginally trapped surface but the sign is required to be fixed.Our interest throughout this work will be to study various properties of the compact horizon types of TheoremIII.1. Notice that as long as the kernel of the left hand side of (25) is nonempty (here we are viewing the left handside of (25) as a function Φ : Ξ −→ R ), then there is always a subset of Ξ that is compact, and all results that are tofollow will hold as well on such subsets. A. Geometry of compact dynamical horizons in LRS II spacetimes
As a dynamical horizon evolves its geometry and topology may change. A consequence of this is that the ther-modynamic quantities also evolve. The relationship between the geometry and thermodynamics may be investigatedthrough analysis of the time evolution of the 3-metric on the horizon. The condition in (25) provides a constraint onthe subclass of spacetimes admitting compact horizons foliated by marginally trapped surfaces on which
K >
K <
0. There is an obstruction to the existence of the compact case with
K <
0, and this will be shown. As thecompactness we have determined is governed by differential equations of geometric quantities on Ξ, we will now studyproperties of these horizons and in effect how the geometric evolution constrains the geometry and thermodynamicsof Ξ.
1. Compact dynamical horizons in LRS II spacetimes foliated by marginally trapped surfaces with
K > Let us start with the case of c = 0, where K > c = 0 and (9b) we obtain − ρ − E −
12 Π + (cid:18)
13 Θ + Σ (cid:19) (cid:18)
23 Θ − Σ (cid:19) = 0 , (32)which, on the horizon, simplifies K as K = ρ − (cid:18)
23 Θ − Σ (cid:19) (cid:18)
43 Θ + 54 φ (cid:19) = ρ + φ (cid:18)
43 Θ + 54 φ (cid:19) . (33)Therefore since we must have φ, Θ > > K is always positive. Proposition III.2
Let M be an LRS II spacetime and let Ξ be a compact horizon in M foliated by marginally trapped -surfaces with K > , satisfying (25) for c = 0 . If the induced metric h µν on Ξ is time dependent, then at some time T ∈ ( a, t end ] for a > t (where t is the initial time of the metric evolution and t end < ∞ ) , there exists a metric on Ξ for which Ξ neither absorbs nor emits radiation. Proof
We shall utilize the Ricci flow geometric evolution equation [25] and show that the constraints generated bythe flow implies Q must vanish. We will assume that the time coordinate parametrizes the family of metric on Ξ.The covariant time derivative of the metric on Ξ is given by u δ ∇ δ h µν = u δ ∇ δ ( g µν + u µ u ν )= u δ ∇ δ ( u µ u ν )= u ( µ ˙ u ν ) = 2 Ae ( µ u ν ) . (34)0For a compact 3-manifold (Riemannian) the Ricci flow equation is normalized as u δ ∇ δ h µν = − R µν + 23 Rh µν , (35)which for LRS II spacetimes can explicitly be written as Ae ( µ u ν ) = 13 ( β − α ) (2 e µ e ν − N µν ) , (36)where the round brackets on the indices denote symmetrization. Contracting (36) by u µ u ν , e µ e ν , u ( µ e ν ) and N µν weobtain the two independent equations A = 0 , (37a)( β − α ) = 0 , (37b)in which case (36) is satisfied. Thus, as the metric evolves, A tends to zero and α tends to β , and this occurs at time t = T ( A =0 ,α = β ) . (38)The condition α = β implies that K = − (1 /
2) ˆ φ , which using (25) for c = 0 gives K = 38 φ . (39)Taking the dot derivative of (39) and comparing to (20a) we obtain34 φ ˙ φ = − (cid:18)
23 Θ − Σ (cid:19) K, (40)which, upon inserting (8b) and noting that A = 0, yields a constraint on Q φQ = − (cid:18)
23 Θ − Σ (cid:19) (cid:18) K − φ (cid:19) . (41)On the horizon (41) simplifies as (using (39) to substitute for K ) φQ = 0 , (42)in which case we must have either φ = 0 or Q = 0 on Ξ. If φ = 0, then, from (39), one has K = 0 and consequently R = 0. It is well known [23] that, for a compact 3-manifold, if R > R = 2 K > t , and hence we can rule out the case φ = 0.Therefore we have that Q = 0 and the result follows.We see in this case that K is always positive from (39). In fact it can also be shown that for times T (cid:54) = T ( A =0 ,α = β ) ,we have the following estimate for K : K ≥ φ . (43)To see this, we recall [23] that, for a 3-manifold with positive scalar curvature (or Ricci tensor), one has the estimate13 R ≤ | R µν | ≤ R . (44)1Using R µν = αe µ e ν + βN µν , we write (44) explicitly as13 ( α + 2 β ) ≤ α + 2 β ≤ ( α + 2 β ) , (45)which can be split as 13 ( α + 2 β ) ≤ α + 2 β = ⇒ −
23 ( α − β ) ≤ , (46a) α + 2 β ≤ ( α + 2 β ) = ⇒ − βR ≤ . (46b)Notice that (46a) always holds. For (46b) to hold, since R >
0, we must have β ≥
0. Explicitly, noting thatˆ φ + (3 / φ = 0, we may write β as β = K − φ , (47)and hence the result follows. As a consequence we have that R ≥ φ . (48)(Notice that (39) satisfies the estimate (43)). Indeed, it makes sense intuitively that the sheet expansion controls the‘growth’ of the curvature R , since the Gaussian curvature of the marginally trapped surfaces determines R .Hypothetically, consider this case: Let us consider a scenario where the horizon Ξ evolves along u a , so that at eachtime t of the horizon evolution we have an associated metric, a solution to (35). Then Proposition III.3 presents asituation where it is possible that i ) . the horizon may radiate for some time, after which it stops radiating, ii ) . thehorizon has been non-radiating since its formation (we are assuming here a formation from an astrophysical collapse)or, iii ) . the horizon is initially absorbing radiation and after time T ( A =0 ,α = β ) it stops absorbing radiation. From (39) K stays positive throughout the evolution of the metric on Ξ, and so the geometry of the foliation is fixed. Insightinto the general properties of these horizon types would require a thorough analysis to check consistency of the fieldequations on these horizons. For example, without a detailed and careful analysis of the field equations on thesehorizon types, one might wrongly assert that during the evolution of the metric, one goes from a positive definitemetric to a negative definite one. To see this, we recall from [14] that if ˜ (cid:15) denotes the area form on the 2-surfaces,then Lie dragging ˜ (cid:15) along V gives L V ˜ (cid:15) = − C Θ l ˜ (cid:15), (49)so that expansion and contraction of an marginally trapped tube is in essence determined by the metric signature onthe marginally trapped tube (noting that Θ l < C <
0) if and only if itcontracts (Θ <
0) and spacelike (
C >
0) if and only if it expands (Θ > t ≥ T ( A =0 ,α = β ) when Q vanishes. Take the dot derivative of (32) and use (8a), (8c) and (10b). After some simplification, the resultingequation on the horizon simplifies to Θ φ (cid:18)
2Θ + 12 φ (cid:19) = 0 . (50)Since Θ φ cannot be zero on the horizon we must have Θ = − (1 / φ , and noting that φ > < non-expanding phase if Θ is to become negative, i.e. to occur as Θ → → R − . We shall see that in fact the metricdoes become degenerate at time T ( A =0 ,α = β ) , i.e. Θ = 0.2 Proposition III.3
Let M be an LRS II spacetime and let Ξ be a compact horizon in M foliated by marginally trapped -surfaces with K > , satisfying (25) , and let the induced metric h µν on Ξ is time dependent. Then (35) admits nosolutions for time t = T ( A =0 ,α = β ) . Proof
We will proceed with the proof by showing that as the induced metric is evolved Ξ becomes null for time t = T ( A =0 ,α = β ) , i.e. Θ = 0. In this case we shall show that either φ = 0 (this was ruled out on grounds thatthe case φ = 0 = ⇒ K = 0 = ⇒ R = 0 which is not possible), or the shear scalar Σ is complex valued, or thestrong energy condition has to be violated in which case it can be shown that the weak energy condition has to beviolated or otherwise the isotropic pressure is negative (here we are assuming the generecity condition in which case ρ + p (cid:54) = 0). We apply the commutation relation in (7), on the pairs of evolution and propagation equations (8a) and(9a), (8b) and (9b) and (8c) and (9c). Taking the hat and dot derivatives of (8a) and (9a) we obtain respectively(after simplifications) 23 ˆ˙Θ − ˆ˙Σ = − φ (cid:20) Σ (cid:18)
23 Θ − Σ (cid:19) + E + 12 Π (cid:21) − (cid:16) ˆ p + ˆΠ (cid:17) = 32 φ (cid:20) − Σ (cid:18)
23 Θ − Σ (cid:19) − E + 12 Π (cid:21) , (51)and 23 ˙ˆΘ − ˙ˆΣ = 32 φ (cid:18)
49 Θ − ΘΣ − E + 12 Π (cid:19) . (52)Subtracting (52) from (51) we obtain (cid:18)
23 ˆ˙Θ − ˆ˙Σ (cid:19) − (cid:18)
23 ˙ˆΘ − ˙ˆΣ (cid:19) = − φ (cid:18)
49 Θ −
13 ΘΣ − Σ (cid:19) . (53)Now, using the commutation relation (7) we have (cid:18)
23 ˆ˙Θ − ˆ˙Σ (cid:19) − (cid:18)
23 ˙ˆΘ − ˙ˆΣ (cid:19) = 32 φ (cid:18)
13 ΘΣ + Σ (cid:19) . (54)Comparing (53) and (54) we have the constraint 0 = φ Θ . (55)From (55) we must have Θ = 0 or φ = 0 (the case φ = 0 has already been ruled out).Next, substituting Θ = 0 and taking the hat and dot derivatives of (8b) and (9b) we obtain (after simplifications)ˆ˙ φ = 12 Σ (cid:20) − φ − Σ − ρ − (cid:18) E + 12 Π (cid:19)(cid:21) , (56)and ˙ˆ φ = 12 Σ (cid:20) − φ − − ρ − p + (cid:18) E + 12 Π (cid:19)(cid:21) . (57)Subtracting (57) from (56) we obtainˆ˙ φ − ˙ˆ φ = Σ (cid:20)
12 Σ − φ − ρ + 32 p − (cid:18) E + 12 Π (cid:19)(cid:21) . (58)Now, using the commutation relation (7) we have3ˆ˙ φ − ˙ˆ φ = − Σ (cid:20) φ + Σ + 23 ρ + (cid:18) E + 12 Π (cid:19)(cid:21) . (59)Comparing (58) to (59) we have the following constraint:0 = Σ (cid:20)
32 Σ + 12 ( ρ + 3 p ) (cid:21) . (60)Hence from (60) we have that either Σ = 0 or Σ + ( ρ + 3 p ) = 0 in which case either the strong energy conditionis violated, i.e. ρ + 3 p < ρ + p >
0, then it isnot very difficult to see that p <
0, and we are not interested in this case), or that the shear scalar Σ ∈ C , the set ofcomplex numbers, which we can rule out. If Σ = 0, then from the vanishing of Θ − Σ + φ we also have that φ = 0 onΞ (since Θ is also zero), and we have already ruled out the case φ = 0. Consequently, we rule out solutions at time t = T ( A =0 ,α = β ) .
2. Compact dynamical horizons in LRS II spacetimes foliated by marginally trapped surfaces with
K < Next, we consider the case of c = 2, where the marginally trapped surfaces are 2-surfaces on which K < c = 2 and (9b) we obtain0 = (cid:18)
23 Θ + 12 Σ (cid:19) (cid:18)
23 Θ − Σ (cid:19) − ρ − φ + E + 12 Π , (61)which, on the horizon, simplifies K as K = − φ − ρ + (cid:18)
23 Θ − Σ (cid:19) (cid:18) Θ + 12 φ (cid:19) = − φ − ρ − Θ φ. (62)Again, it is sufficient to specify the energy density as positive on Ξ, in which case K is always negative. Let us nowstate and prove the following Theorem III.4
Let M be an LRS II spacetime and let Ξ be a compact horizon in M foliated by marginally trapped -surfaces with K < , satisfying (25) for c = 2 . If the induced metric h µν on Ξ is time dependent, then at some time T ∈ ( a, t end ] for a > t (where t is the initial time of the metric evolution and t end < ∞ ), K is strictly positive. Proof
This is easy to show as we note that (34) to (38) holds here as well, and so the implication α = β = ⇒ K = − (1 /
2) ˆ φ consequently holds. For c = 2 in (25) this gives K = 316 φ , (63)which is always positive.We see that the situation here gets a little more complicated. The horizon Ξ is compact if it satisfies (25) for c = 2,where K < K changes sign.For a smooth evolution we have K → → R − , in which case the Ricci curvature scalar R on Ξ is negative. Thechange of sign of the Gaussian curvature occurs only if the energy density satisfies: ρ = − φ − Θ φ, (64)which is always negative.A consequence of Theorem III.4 is the following4 Corollary III.5
There cannot exist a compact dynamical horizon Ξ in LRS II spacetimes foliated by marginallytrapped -surfaces with K < , satisfying (25) for c = 2 where the induced metric h µν on Ξ is time dependent. Corollary III.5 is a consequence of a combination of results: As have been discussed in the introduction, Hamiltonproved that any smooth closed 3-manifold admitting a metric with
R > R = − K > c = 2. Notice that on the marginally trapped surfaces R = 2 K <
R >
R > t = T | ( A =0 ,α = β ) implies that R is nownegative ( R = − K <
K >
0) at time t = T | ( A =0 ,α = β ) , which is not possible.Note that in Corollary III.5 we specified that we are ruling out the existence of compact horizon types in LRS IIspacetimes satisfying (25) for c = 2 where the induced metric h µν on Ξ is time dependent. While one may speculatethat it is therefore possible that there are these horizon types if there is no time dependence of the metric, the firstpart of the discussion in the previous paragraph has already ruled out this possibility.The statement of Proposition III.3 combined with that of Corollary III.5 may allow us to then state the following: every compact dynamical horizon of class satisfying (25) in LRS II spacetime, if it exists, is of spherical geometry,i.e. the geometry S , and their cross sections are topological -spheres (the Gaussian curvature K remains positive).The fact that the Gaussian curvature K not only determines the geometry of the marginally trapped surfaces, butalso characterizes the geometry of the foliated horizons is a very interesting property of the horizon types consideredin this work, and it is definitely not trivial.In the next subsection we look to obtain a characterization of compact dynamical horizons satisfying the compactnesscondition (25) in invariant way. B. An invariant set characterizing compact dynamical horizons in LRS II spacetimes
The new covariant way of studying black hole horizons, initiated by [1] and extensively exploited by [2, 3], has provedvery useful in unveiling properties of horizons in a relatively straightforward way compared to other approaches. Thevanishing of the outgoing null expansion scalar Θ k characterizes marginally trapped surfaces, and by extension thehorizons foliated by marginally trapped surfaces. We present an analogue of such characterization of the compacthorizon types considered in this work. The primary motive of this construction is to unify, in a formal manner, therelationship between geometry and thermodynamics of the compact dynamical horizon types considered here.First notice that the condition (37b) implies (37a) since e ( µ u ν ) (cid:54) = 0. Interestingly, since (39) implies Q = 0, thereis a nice geometric condition that gives the condition Q = 0: One defines an “Einstein-like” symmetric (0 , G µν = R µν − Rh µν , (65)which has played a crucial role in the study of conformal geometry of Riemannian manifolds [27, 28] (this tensor givesthe deviation from Einstein space). In particular, the sign of the norm of this tensor and its contraction with certaincomplete vector fields have been used to test when a Riemannian manifold is conformorphic or isometric to spheres ofvarying dimensions. For the cases considered throughout this work, if we calculate the norm of this tensor we obtain G = G µν G µν = R µν R µν − R = 23 ( α − β ) . (66)Clearly G = 0 gives (37b), which consequently gives the vanishing of the heat flux, i.e. Q = 0. We also have thescalar ˜ G = k ν h µδ ∇ δ G µν , whose integral plays a crucial role in the study of conformal transformations on Riemannianmanifolds of arbitrary dimensions (see the references [27, 28]). Explicitly we calculate˜ G = 1 √ (cid:18)
23 Θ − Σ + φ (cid:19) K = Θ k K. K (cid:54) = 0 (which has been assumed throughout this work), then we must have ˜ G = 0 = ⇒ Θ k = 0. Now,define the invariant set I = { ( G , ˜ G ) } i with ( G , ˜ G ) : ¯ M × ¯ M → R , and indexed by the triple (Θ , Σ , φ ) where ¯ M is a3-manifold (Riemannian) in the LRS II class. Then the subset J ⊆ I defined by the constant map ( G , ˜ G ) (cid:55)→ (0 , J = { ( G , ˜ G ) ∈ I | ( G , ˜ G ) (cid:55)→ (0 , ∈ R } provides a collection of marginally trapped tubes which includes theclass of compact dynamical horizons in LRS II spacetimes considered throughout this work. If we assume that thecompactness condition (25) is satisfied, then, indeed, the set J characterizes the class of compact horizons consideredhere.Notice how the scalar ˜ G not only identifies the horizon for non-zero K , but also that it is sufficient to determinethe topology and geometry of the black hole itself: if there is indeed a trapped surface, then Θ k < G < K >
0, in which case the trapped 2-surfaces are spherical, which is as one wouldexpect.
IV. DISCUSSION
Initiated purely out of mathematical curiousity, this work set out to investigate the geometry of a certain class ofcompact dynamical horizons with a time-dependent induced metric in LRS II spacetimes. The geometry of Riemannianmanifolds and transformations to metrics on them (conformal, homothetic or isometric) is a well grounded area ofstudy in differential and Riemannian geometry. As mentioned in the introduction, the study of marginally trappedtubes and their evolution, using the 1 + 1 + 2 semitetrad covariant formalism, has been successfully carried out inrecent works [1–3], yielding established results as well as providing clear insights into the nature of the matter andthermodynamic variables on the marginally trapped tubes. Here, we have derived a class of compact horizons andhave established geometrical results on these horizons, employing a range of well established results for n -dimensionalcompact Riemannian manifolds.The compactness condition is established - using the Bonnet-Myers theorem - as the requirement that the sheetexpansion, φ , satisfies the spatial first order differential equation (25), parametrized by a real constant c which takeson the value ‘0’ and ‘2’ in which case the Ricci curvature on the horizon takes the simple form R = ± K , with K being the Gaussian curvature of 2-surfaces in the spacetime (the ‘+’ is for ‘ c = 0’ and the ‘ − ’ for ‘ c = 2’). For the c = 0 case it is seen that R = 2 K = R .Let Ξ be a compact DH of type considered here. Using the Ricci flow evolution equation for compact 3-manifoldsit is shown, for the case c = 0 ( K > t = T α = β for which Ξ does not radiate,i.e. Q = 0. These solutions to the Ricci evolution equations on grounds that, either Ξ is minimal (this was ruled outsince φ = 0 would imply that K = 0 = ⇒ R = 0; this is not possible on a smooth compact Riemannian manifoldwith positive Ricci curvature) or the shear scalar Σ is complex valued (which can be ruled out), or the strong energycondition (SEC) has to be violated in which case it is not difficult to show that the weak energy condition is eitherviolated or otherwise the isotropic pressure is negative. Since K > R = 2 K > S geometry of the compact dynamical horizon.LRS II spacetimes admit no compact dynamical horizons of the type considered here for the case c = 2 ( K <
K >
Acknowledgements
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