Initial Data for General Relativity Containing a Marginally Outer

Home , Apparent horizon

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R p, 2 ˜ u u.Tephysical The out. cut hti ae obe to taken is that g + , g dz and , 2 + R p g ˜ r ∂ 2 stetrace the is M on dϕ ˜ making , 2 M ˜ on plus p/ (4) (3) (1) (2) R M ˜ 2 3 Lδψ δ ψ =0 on M,˜ (5) By using conformal symmetries, the PDE can be de- ≡−△ composed into an infinite set of uncoupled linear ODEs Bδψ =0 on ∂M,˜ (6) that were solved numerically. A convenient cutoff of the ψ> 0, ψ 1 as r2 + z2 . (7) → → ∞ infinite series yields a highly accurate solution for the constraint. The existence of solutions can be decided by We are left with the freedom to fix the boundary sur- solving a single linear ODE, together with the high ac- face, taking it to be a standard torus in flat space, defined curacy obtained for the actual solution, this enables a in cylindrical coordinates as a solution of secure grip on the two interesting limiting cases. z2 + (r A)2 = a2. (8) The key technical input is the utilization of a U(1) − U(1) conformal isometry that leaves the boundary torus× Since a finite overall scaling of the coordinates bears no invariant. In order to benefit from the conformal isome- physical significance, we can fix A = 1 for the boundary try the conformal method is reformulated for a compact tori considered, which leaves us with the one parameter base manifold M, taken to be the one-point compactifi- ˜ 3 family 0 a 1. The extremal case a = 0 corresponds cation of M, that is S with the interior of a torus cut 4 to a circle≤ of≤ unit radius in the z = 0 plane, the other out. The conformal ansatzg ˜ = G g, with g the standard 3 extreme, a = 1, defines a torus which touches its axis of metric on S , results in an elliptic PDE similar to (5), symmetry. L G =4πcδ on M, (9) Using the method described below it is possible to con- g Λ tinue the data smoothly inside the minimal tori to some but with a distributional source term: δΛ is the Dirac finite extent, assuming vacuum and conformal flatness in- delta distribution concentrated at the point Λ on M, side, but in general this assumption will lead to a nodal which corresponds to asymptotic infinity of M˜ , and c surface, and thus to a curvature singularity, if one pro- is a positive constant that can be chosen arbitrarily for −1 ceeds too far. The obvious question whether it is possible overall scaling. G is thus a Green function for c Lg with to get a regular interior and nonnegative energy density boundary condition of the matter is not dealt with here. Regular solutions to the constraint equation (5) with BgG =0 on ∂M. (10) boundary conditions (6, 7) exist for a ]a0 0.14405, 1[ (all numerical results are rounded to the∈ last∼ digit), where The existence of a positive Green function G is deter- the parameter a labels the boundary tori as defined above mined by the sign of λ1, the first eigenvalue of the conformal Laplacian with the present boundary condition, in Eq. (8). In the limit a a0 the mass blows up, and the Sobolev quotient (an extension→ of the definition of the or equivalently by the sign of the Sobolev quotient Q(g), Yamabe number to manifolds with boundary) vanishes. a conformal invariant defined as 2 1 2 1 2 Also the apparent horizon moves outward to infinity and ( ϕ + Rgϕ )d v + pϕ d σ becomes spherical in the limit. This behavior is famil- Q(g) := inf M |∇ | 8 4 ∂M , ϕ ( ϕ6d v)1/3 iar from the case when the (usual) Yamabe number ap- R M R proaches zero [3]. If a is further decreased, the Sobolev R (11) quotient becomes negative, the mass diverges to minus infinity as a a0, and the conformal factor develops a which is a generalization of the Yamabe functional [6] nodal surface.ր In the other extremal case, when a 1, (defined for manifolds without boundary) to manifolds the mass stays finite, but a curvature singularity develops→ with boundary [7] and has the same sign as λ1. at the center, where the toroidal minimal surface touches It can be shown in a manner equivalent to the no- the axis of symmetry. In this last case the solution can boundary case dealt with in [8], that a positive G exists be given explicitly. iff λ1 is positive. Furthermore, when G> 0, the positive 4 The toroidal minimal surfaces are all unstable, and mass theorem holds forg ˜ = G g [9]. part of their spectrum of perturbations was computed. The rest of this paper is organized as follows: In the The ADM mass can be cast into an explicitly positive next section (II) conformal compactification via inverse 3 3 form in a way similar to Brill’s result [4], additionally it stereographic projection from R to S and toroidal coor- can be bounded from below. dinates thereon will be discussed. In Sec. III the method Note that the initial data discussed here are perfectly to solve the constraint numerically using toroidal coor- viable for evolution, since an apparent horizon boundary dinates will be developed, including numerical solution condition [5] can be applied at the topologically spheri- for the first eigenvalue of the conformal Laplacian. Sec- cal apparent horizon that surrounds the boundary torus. tion IV deals with the instability of the minimal tori by The minimal torus could then be viewed as a model of a considering normal variations of the surfaces. Apparent bulk of matter or gravitational radiation that has enough horizons are located and analyzed with respect to the energy to confine all information about it inside of the isoperimetric inequality for black holes [10] and the hoop horizon. This comment also holds in the case a 1, conjecture [11] in Sec. V. A discussion of the results, when a curvature singularity appears at the center.→ with emphasis on the limiting cases, is given in section VI.

2 II. COMPACTIFICATION AND CONFORMAL plane z = 0. In flat space γz¯ = α γz¯ = α + π SYMMETRY is a sphere centered at r ={ 0, z = γ}∪{cot γz¯ with radius} γ2(cot2 γz¯ 1) + 2. − For every choice of boundary torus Σ , that is a torus Note that for γ = 0 the radius of S3 becomes infinite, γ p described by Eq. (8) with A = 1 and parameterized by corresponding to R3, and the coordinatesr ¯,z ¯ become γ := √1 a2 1, we define a conformal rescaling from inverted cylindrical coordinates thereon: the flat metric− ≤δ on R3 to a metric g(γ) defined on S3 by 2r 2z r¯ = 2 2 , z¯ = 2 2 . 2 2 r + z r + z g(γ)=Ω(r, z; γ) δ, Ω(r, z; γ)= 2 2 2 , (12) γ + r + z The torir ¯ = const., corresponding to A = a = 1/r¯, all touch the z-axis in this limiting case. which corresponds to an inverse stereographic projection. 3 The notation f(¯r, z¯; γ) is chosen to denote dependence of We have thus constructed a family of foliations of R a function f on both the spatial coordinates and the pa- by standard tori, with each element defined by selection rameter γ, frequently this will be shortened to writing of a boundary torus Σγ that is contained in the foliation. f(γ). With the chosen normalization g(γ) is the stan- For every foliation labeled by γ we defined in Eq. (12) a conformal rescaling with compactifying factor Ω(r, z; γ). dard metric on the 3-sphere of radius 1/γ, which is con- a veniently written as Since Ω is independent of ϕ, (∂/∂ϕ) is a Killing vector of g, but (∂/∂z¯)a is only a conformal Killing vector of g. 2 The reason for the chosen scaling of S3 as opposed to dr¯ 2 2 2 2 2 g(γ)= + (1 γ r¯ )dz¯ +¯r dϕ (13) working on S3 of fixed radius can bee seen when rescaling 1 γ2r¯2 − − to the standard metricg ˆ on S3 of unit radius (with the in toroidal coordinates (¯r, z,¯ ϕ) taking ranges 0 r¯ notation of [12]): ≤ ≤ 1/γ, 0 γz,¯ ϕ 2π. A discussion of the construction 2 ≤ ≤ 3 dρ of this coordinate system on S of unit radius is given gˆ := γ2g(γ)= + (1 ρ2)dχ2 + ρ2d ϕ2, in [12]. The coordinater ¯ as a function on the (r, z)-half 1 ρ2 − − plane can be read off from Eqs. (12) and (13) yielding using rescaled toroidal coordinates (ρ,χ,ϕ) r¯(r, z; γ)=Ω(r, z; γ) r. ρ(r, z; γ) := γ r¯(r, z; γ), χ(r, z; γ) := γ z.¯ The relation between the coordinatez ¯ and the original taking ranges 0 ρ 1, 0 χ, ϕ 2π. The boundary cylindrical coordinates is ≤ ≤ ≤ ≤ torus Σγ that was defined byr ¯ = 1 before is now specified 1 γ2Ω(r, z; γ) by ρ = γ, so that the region of physical interest, 0 ρ cos(γz¯)= − , sign(¯z) = sign(z). γ shrinks to zero volume in the limit γ 0. The problem≤ ≤ 1 γ2r¯(r, z; γ)2 → − of finding a conformal factor solving the rescaled equation ˆ p 3 LgˆG = 4π√2δΛ then becomes ill-defined, the conformal The point Λ on S , corresponding to asymptotic infinity 4 3 factor Gˆ (and thus the mass of Gˆ gˆ) blows up for γ 0. of R , is given byr ¯ =z ¯ = 0. → The vector fields (∂/∂ϕ)a, (∂/∂z¯)a form a pair of com- In order to formulate the constraint equation explicitly muting, orthogonal and hypersurface-orthogonal Killing we need to write down the operators Lg and Bg in our vector fields of g(γ) spanning the surfaces of constantr ¯. toroidal coordinate system. The extrinsic curvature of a Forr ¯ = 0, 1/γ these are flat tori of constant mean cur- torus of constantr ¯ with respect to the metric gab is (the vature6 changing sign at the so called Clifford torus for semicolon denotes the covariant derivative as usual) whichr ¯ = 1/γ√2. Going back to the flat metric and p =r ¯ 1 γ2r¯2 ϕ ϕ γ2z¯ z¯ . (14) cylindrical coordinates, a torusr ¯ = const. is mapped to ab − ;a ;b − ;a ;b a standard torus as defined by Eq. (8) with parame- At ∂M (¯r = 1)p this yields ters A = 1/r¯, a = 1 γ2r¯2/r¯, in particular the torus − r¯ = 1 is mapped to the boundary torus Σγ. The sets 1 2γ2 p p = pa = − , (15) r¯ = 0 (respectivelyr ¯ = 1/γ) are linked great circles on a 2 3 1 γ S , corresponding to the z-axis (respectively the circle − z =0, r = γ) after stereographic projection. which vanishes at the Cliffordp torus given by γ =1/√2, The totally geodesic surfacesz ¯ = const. (ϕ = const.) c where ∂M is maximal. The extrinsic curvaturep ˜ with are metric hemispheres intersecting atr ¯ = 1/γ (¯r = 0) ab respect to the physical metricg ˜ is with γz¯ = α γz¯ = α+π ( ϕ = β ϕ = β+π ) be- { }∪{ 2} { }∪{ } ing smoothly embedded S ’s. Under stereographic pro- p˜ = G2 (p +2g ncG ) , (16) jection, the surfaces γz¯ = const. do not change topology ab ab ab ;c except for γz¯ =0 γz¯ = π which gets decompactified where nc is the unit normal vector field on ∂M. Note into z ={ 0, r γ}∪{ z = 0,} r γ , i.e. the equatorial { ≥ }∪{ ≤ } that since pab has Lorentz signature, in particular it is

3 m not proportional to g (14), by the conformal rescaling ψ =1+ + O(Ω). formula for the extrinsic curvature (16) it is not possible 2√r2 + z2 to getp ˜ = 0 by a conformal transformation. As a ab The limiting cases γ =0, 1 can be considered as singu- consequence the minimal tori cannot be stable, since in lar situations, and the question arises how this will effect that case they would have to be totally geodesic (˜p = 0) ab the physical metricg ˜(γ), in particular its existence. For – (see [1]). γ<γ we have p positive (see Eq. (15)) whence is the Finally we can write down the operators L := L c γ gγ Sobolev quotient Q(g) defined in Eq. (11) – a positive and B := B – simplifying their labels – defined in γ gγ Green function thus exists and the mass of the physical Eqs. (9, 10) explicitly: metricg ˜(γ) is positive. On the other hand p is negative

2 for γ>γc, so that a change of sign becomes possible for 2 ∂ 1 2γ Bγ = 1 γ + − , (17) Q(g) at some γ, γc <γ< 1. This indeed happens at the 2 − ∂r¯ 4 1 γ ! numerically calculated value γ0 = 0.98957, correspond- p − ing to a =0.14405. As one expects, this case is analogous p to the Yamabe number going to zero in the usually con- 2 2 2 2 2 ∂ 1 3γ r¯ ∂ sidered case of manifolds without boundary. The mass Lγ = (1 γ r¯ ) 2 − − − ∂r¯ − r¯ ∂r¯ diverges to plus infinity as γ approaches γ0 from below, 1 ∂2 1 ∂2 3 and to minus infinity when taking the limit from above. + γ2. (18) − 1 γ2r¯2 ∂z¯2 − r¯2 ∂ϕ2 4 For γ 0 the metric g becomes the flat metric on the − infinite cylinder→ 0 r¯ 1, < z¯ < , 0 ϕ 2π: ≤ ≤ −∞ ∞ ≤ ≤ g(0) = dr¯2 + dz¯2 +¯r2dϕ2. III. SOLUTION OF THE CONSTRAINT EQUATION The point Λ, representing physical infinity, is atr ¯ =z ¯ = 0, the unphysical infinity,z ¯ = corresponds to the ±∞ A. Formulation physical center r = z = 0. The constraint Eq. (9) reduces to the cylindrically symmetric Poisson equation for a point source From the choices made above our physical data are determined by the single parameter γ, which labels the ∂2 1 ∂ ∂2 position of the boundary torus. This defines the phys- L0G(0) = + + G(0) = 4π√2δΛ, − ∂r¯2 r¯ ∂r¯ ∂z¯2 ical metricg ˜(γ) = G(γ)4g(γ), where G(γ) satisfies the equations (22)

LγG(γ)=4π√2 δΛ on M, (19) with boundary condition Bγ G(γ)=0 on ∂M (20) ∂ 1 + G(0) =0. (23) ∂r¯ 4 |r¯=1 with operators Lγ and Bγ given by the expressions (17,18), the boundary torus is located at the fixed co- The explicit limiting solution is derived in Appendix A ordinate valuer ¯ = 1 and the scale factor is chosen as as c = √2. The rescaling to flat space defined in Eq. (12) corresponds to a Green function G , 1 1 0 G(¯r, z¯;0) = √ √ 2 2 −1/2 2 r¯ +¯z −1/2 2 2 ∞ G0(γ)=Ω(γ) = γ 1 1 γ r¯ cos γz¯ , 2 4ω (K (ω) K (ω)) − − + cos(ωz¯) 1 − 0 I (ωr¯) dω. π 4ωI (ω)+ I (ω) 0 p Z0 1 0 where G0(γ) satisfies Eq. (19) but with the boundary condition (20) replaced by regularity on S3 Λ . The As is discussed in Appendix A, G(¯r, z¯; 0) is a regular asymptotic behavior of G near Λ is known [8]\{ to} be positive function for 0 r¯ 1, < z¯ < ,r ¯2 +¯z2 > 0. The ADM-mass ofg ˜≤(0)≤ = G(0)−∞4g(0) is m∞=3.877, in m −1 particular it is finite. For z¯ large G decays exponentially, G = G0 + + O(G ), (21) 2√2 0 | | G(¯r, z¯; 0) exp( α z¯ ), α =0.6856, where m is the ADM-mass of the physical metricg ˜. By ∝ − | | the positive mass theorem [13] m is positive provided λ1 it is thus zero at the physical origin r = z = 0. There the abcd is positive. Kretschmann invariant diverges as RãbcdR˜ [˜g(0)] The relation to the original formulation of the con- exp(8α z¯ ), the physical metric thus has a curvature sin-∝ straint problem in Eqs. (5-7) is as follows: gularity| at| the origin, where the minimal torus touches the axis of symmetry. − − g˜ = G4g = ψ4δ = ψ4Ω 2g ψ =Ω 1/2G, ⇒

4 B. Existence of Solutions

0.50 As stated above, the existence of a solution to the constraint, that is a positive Green function for the elliptic operator Lγ with boundary condition given by Eq. (10), 0.40 is equivalent to positivity of the lowest eigenvalue λ of 1 0.9890 0.9895 0.9900 Lγ. Calling the eigenfunctions θ we obtain the eigenvalue 0.010 problem 0.30 0.005 λ1 Lγ θ = λθ, Bγ θ r¯=1 =0, | 0.20 0.000 which is separable and the solutions can be written as -0.005 0.10 θλmn(¯r, ϕ, z¯)= ϑλmn(¯r) sin(nϕ + α) sin(mγz¯ + β), -0.010 where α and β are arbitrary angles and we are left to 0.00 solve an equation for ϑ, 0.0 0.2 0.4 0.6 0.8 1.0 γ m2 n2 Lrγ¯ + + ϑλmn = λϑλmn, FIG. 1. The lowest eigenvalue λ1 of the conformal Lapla- 1 γ2r¯2 r¯2 − cian is plotted as a function of γ, the insert shows that λ1 gets negative for γ>γ0. The value for γ = 0 is marked by where Lrγ¯ is the ’radial’ part of the conformal Laplacian an ×. Lγ,

∂2 1 3γ2r¯2 ∂ 3 L = (1 γ2r¯2) − + γ2. (24) rγ¯ − − ∂r¯2 − r¯ ∂r¯ 4 C. Transforming the Constraint to ODEs Since the first eigenfunction is non-degenerate [14] and Since the singular behavior of G(¯r, z¯; γ) at Λ makes a has therefore to depend trivially on the angular coordi- direct numerical solution for the Green function incon- nates ϕ andz ¯, we may set m = n = 0 when looking for venient, the first step in solving the system (9, 10) is to the lowest eigenvalue λ : 1 regularize equation (9). We split G into the singular part G0 and a regular part φ by Lrγ¯ ϑλ00 = λϑλ00. (25) G(γ)= G (γ)+ φ(γ). (26) The existence question has thus been reduced to an 0 ODE of Sturm-Liouville type which can be solved by The asymptotic behavior of φ near Λ is standard methods. In this case routine d02kef from the m NAG Fortran Library [15] was chosen to calculate eigen- −1 φ = + O(G0 ), values and eigenfunctions. 2√2 The lowest eigenvalue is found to be positive for γ < from the asymptotic behavior of the Green function (21), γ0 = 0.98957, and negative for γ greater, see figure (1) m is the ADM-mass of the physical metricg ˜. Calculation and table (I). For γ = 0 Eq. (25) reduces to Bessel’s of the mass is thus as simple as evaluating the numerical equation: solution φ at the point Λ. Insertion of (26) into (9, 10) yields an elliptic problem for φ: 2 ′′ ′ 2 ′ 1 r¯ ϑ +¯rϑ + λr¯ ϑ =0, ϑ + ϑ r¯=1 =0. 4 | Lγφ =0,

The regular solution is (apart from arbitrary scaling) ϑ = Bγφ r¯=1 = Bγ G0 r¯=1 | − | J0(√λr¯) (J0 is the Bessel function of order 0) where λ is 2 2 G0 cos γzG¯ 0 1 2γ chosen to satisfy the boundary condition = − 2 =: b(¯z,γ). 2 1 γ2 − 2(1 γ )! − − r¯=1 ′ √ 1 √ J0( λr¯)+ J0( λr¯) r¯=1 =0, Note that the boundaryp condition becomes singular for 4 | γ 1. → which yields λ1 =0.4700. The above partial diﬀerential equation in two dimen- sions can be turned into a system of a countable set of uncoupled ODEs by decomposing φ and b into a cosine series (the sine terms of the Fourier decomposition are zero due to the symmetry with respect to reﬂection at the equator):

5 ∞ ∞ with cutoff-parameter N one may ask how well the φ = φn(¯r)cos(nγz¯), b = bn(¯r) cos(nγz¯). Hamiltonian constraint is satisfied. Due to numerical n=0 n=0 X X inaccuracies and cutting off the Fourier series at finite (27) N, Eqs. (5) and (6) will not be satisfied identically, but instead there will appear two functions E and E such This gives a boundary value ODE problem for every n 1 2 that (prime denoting differentiation with respect tor ¯): n2 δψ = E1 0, Bδψ = E2 0 L φ = 0 (28) △ 6≡ 6≡ rγ¯ − 1 γ2r¯2 n where E1, E2 should be small in a sense that has yet to be −2 ′ 1 2γ determined. From the Fourier series expansion it can be φn + − φn = bn(γ), (29) 4(1 γ2) seen, that increasing N will increase E1 too, since the r¯=1 | | − ′ conformal Laplacian annihilates every single term in the φn r¯=0 =0. (30) | above Fourier series, Lgφn cos(nγz¯) = 0, for φn known The last condition ensures regularity onr ¯ = 0, the exactly. E1 is thus simply the sum of all numerical in- z axis. By integrating the ODEs beyondr ¯ = 1 it is pos- accuracies of the individual Fourier terms, and the sole sible− to continue the solution smoothly inside the minimal purpose of increasing N is to yield better satisfaction of tori to some finite extent, but this will eventually lead to the boundary conditions, that is to decrease E2. a nodal surface, and thus to a curvature singularity. Let us deal with the smallness of E1 first. In the case Note that, for n = 0, Eq. (28) corresponds to Eq. (25), of a vacuum spacetime, where the scalar curvature has to and correspondingly only φ0 blows up for γ0, the higher vanish, there does not exist an a priori measure for E1. order terms pass through λ1 = 0 smoothly. Following [16] the error can be written as R˜ = 16πρres, The equations are solved by a shooting and match- where ρres can be interpreted as the mass-energy density ing method (NAG routine d02agf [15]) for n =1,...,N, of matter for a non-vacuum spacetime. The criterion of where N is suitably chosen to make the truncation error accuracy derived from this viewpoint is the smallness of for ϕ reasonably small. the total amount of mass represented by ρres as compared to the ADM-mass. The residual of the Hamiltonian constraint is defined as 20.0 0.9890 0.9895 0.9900 8 − ρ = ψ 5L ψ 80000.0 res 16π δ 60000.0 15.0 yielding the residual mass 40000.0 6 3 20000.0 mres = ρres dV = ρres ψ d x. m 10.0 V | | V | | 0.0 Z Z Since the solution has been constructed in toroidal co- -20000.0 ordinates, one gets better results, if derivatives are com- 5.0 puted with respect to these, so mres is rather evaluated in the form

1 − 0.0 1/2 0.0 0.2 0.4 0.6 0.8 1.0 mres = ψΩ LgG rd¯ rd¯ zdϕ.¯ 2π V | | γ Z The volume of integration has to be chosen carefully. FIG. 2. The ADM mass is plotted as a function of γ, the Since the domain of computation is infinite with respect insert shows the pole of m at γ = γ0. The value for γ = 0 is to the physical metric, integration over the whole com- marked by an ×. putational domain would yield an infinite result, the integration was thus restricted to √r2 + z2 50 m. An alternative way to control the numerical≤ error that is especially useful to control fulfillment of the boundary D. Tests of Numerical Accuracy conditions can be constructed by putting the ADM mass into an explicitly positive form analogous to the expres- sion derived for Brill-waves in [4]. From Eq. (5) we get For every approximate solution (∂ is the flat derivative operator) N δψ φN = φn(¯r)cos(nγz¯) 0= △ = ∂2 log ψ + (∂ log ψ)2 n=0 ψ X

6 which can be integrated with respect to the flat back- where Σ is now taken to be our minimal toroidal bound- ground metric to give ary surface specified byr ¯ = 1 in the physical geometry, Σ is the Laplacian on Σ, Rãb the Ricci cur- na∂ ψ ∂ψ 2 vature△ ofg ˜ on Σ, andp ˜ the extrinsic curvature of a dA + dV =0. ab ∂M ψ M ψ Σ. Stability holds iff all eigenvalues of the operator Z Z a b ab ( Σ + (Rãbn n +p ãbp˜ )) are nonnegative for a min- The boundary ∂M consists of spatial infinity and the −imal△ surface. If Σ is of toroidal topology it is known [1] boundary torus Σ, inserting the definition of the ADM that it can only be stable if the minimal torus is totally mass m yields geodesic, which it cannot be in our case as was concluded from the form of pab given in Eq. (14). Nevertheless it a 2 1 n ∂aψ ∂ψ is interesting to compute the spectrum (or at least the m = dA + dV . first few eigenvalues, since this has to be done numeri- 2π Σ ψ M ψ ! Z Z cally). Using the Gauss law and the constraint conditions a ˜ 2 ˜ Using the boundary condition n ∂aψ + ψp/4 = 0 allows R =p ˜ = 0, and denoting the scalar curvature of Σ as R to calculate the first integral explicitly as the eigenvalue problem becomes a 1 1 n ∂aψ 1 π 2 ˜ ab m := dA = p dA = Σ φ + ( R pãbp˜ )φ = νφ. Σ 2π ψ −8π 2 −△ 2 − ZΣ ZΣ resulting in Straightforward computation yields 2 2 − 1 ∂ ∂ π 1 ∂ψ 2 = G 4 + , m = + dV =: m + m . (31) △Σ 1 γ2 ∂z¯2 ∂ϕ2 2 2π ψ Σ V M −4 − Z 4G − − 2R˜ = G 2G, 2 G 1G, , The minimal mass of the metrics in the sequence consid- 1 γ2 z¯ − z¯z¯ − ered is thus bounded from below by m > π/2, the nu- 2 ab −4 ab p 1 −4 −2 −1 merically calculated minimal value of the mass is 3.623 pãbp˜ = G pabp = G (1 γ ) . − 2 2 − at γ =0.628. In table (I) the relative errors of the numerically cal- With some rearrangement of terms the eigenvalue equa- culated expressions mΣ and mres/m and the choices of tion becomes N are listed along with other results for some values of γ. ∂2φ ∂2φ (1 γ2) + V φ = νG4(1 γ2)φ, As the coordinate range ofz ¯ approaches the entire real − ∂z¯2 − − ∂ϕ2 − line for γ 0, the cosine series (27) has to be replaced by a cosine integral→ that can be written down explicitly as is discussed in Appendix (A), accordingly the cutoff pa- 1 −2 2 −1 V = +2 G G,z¯ G G,z¯z¯ , rameter N of the Fourier series has to be increased for γ −4 − decreasing to keep the accuracy approximately constant. which has to be solved with periodic boundary conditions For practical reasons only few numerical computations inz ¯ and ϕ. Since the geometry is axially symmetric, so were done for γ < 0.15, so that the region 0 <γ< 0.15 is G, and the equation separates. The eigenfunctions can is left out of the figures displayed. be written as

φ(¯z, ϕ)= unν (¯z)vn(ϕ), IV. INSTABILITY OF THE MINIMAL TORI

where vn(ϕ) = sin n(ϕ + α) are the axial eigenfunctions An interesting question concerning surfaces of minimal degenerate in the axial phase shift α and for every n = area is whether they are stable – that is true minimal 0, 1, 2,... one gets an equation for the unν (¯z): surfaces. Stability of a minimal surface Σ means that for ′′ all possible one-parameter (normal) variations Σ of the u + (1 γ2)n2 + V u = ν G4(1 γ2)u , t − nν − nν n − nν surface Σ = Σ0 the second derivative of the area satisfies ′′ (32) A (0) 0. A normal variation can be written as φna ≥a where n is a smooth unit normal vector field along Σ0 which is a regular self-adjoint Sturm-Liouville problem and φ is a smooth function on Σ. The second variation with periodic boundary conditions for γ = 0,γ0. The of the area is then given by spectrum is thus purely discrete and bounded6 below (see e.g. Ref. [17], Sec. 1.4). ′′ A (0) = φ[ φ + (R˜ nanb +˜p pãb)φ]d A, As a result of using periodic boundary conditions the − △Σ ab ab ZΣ eigenfunctions are grouped in pairs by their number of

7 zeros (see e.g. Theorem 1.4.2 of Ref. [17]): The eigen- Fig. (4) for n = 1, the situation is different for n = 0. functions uk, k=0,1,2. . . , with eigenvalue νk have k zeros The general solution u to Eq. (32) for γ = n = 0 and for k even, and k + 1 zeros for k odd, that is u1 has no arbitrary ν is asymptotically trigonometric (see e.g. Ref. zero, u2 and u3 both have two zeros, u4 and u5 four, etc., [18], section 5.11), that is where the endpoints of the interval are identified. For practical purposes it is convenient to convert the u = α sin(¯z + β)+ O(1/z¯), periodic boundary conditions to unmixed boundary con- which is oscillatory with finite amplitude nearz ¯ = . ditions, which is made possible by noting that for Eq.(32) Periodic boundary conditions thus can not be imposed±∞ on it is sufficient to consider only eigenfunctions that are these solutions, for small γ all eigenvalues decrease un- with respect toz ¯ = 0. If a periodic solution u exists boundedly, marking a growing instability of the minimal that is neither symmetric nor antisymmetric, then the surface with respect to axially symmetric deformations. whole solution space is spanned by the linear combina- tions u(x) u( x) and the general solution is periodic, since with ±u(x)− being a periodic solution, so is u( x). The eigenvalue is of multiplicity two in these cases, which− show up in figures (3,4) as the crossing points of branches 1 associated with symmetric and antisymmetric eigenfunc- 2 tions that have both the same number of zeros. 0.2 3 1' The problem of finding solutions with periodic bound- 2' ary conditions can thus be split into the easier problems 0.1 of finding all periodic symmetric and antisymmetric solutions, equivalent to imposing the unmixed boundary ν conditions 0.0

′ ′ π u (0) = u = 0 for symmetric solutions, γ −0.1 π u(0) = u = 0 for antisymmetric solutions, γ −0.2 0.0 0.2 0.4 0.6 0.8 so that standard methods for Sturm-Liouville problems γ can be applied. Here routine d02kef from the NAG For- FIG. 3. The first five eigenvalues ν0 (n = 0) are plot- tran Library [15] was chosen to calculate eigenvalues and l ted as functions of γ from top down (l = 1,..., 5). The eigenfunctions, as was done already for Eq. (25). curves labelled 1, 2, 3 are the first three branches of eigen- ′ ′ The results are given in figures (3,4) for n = 0, 1. For values with symmetric eigenfunctions, while 1 , 2 label the all values of γ one can find at least one negative eigen- first two branches of eigenvalues with antisymmetric eigenvalue, demonstrating that the tori are not stable, and are functions. thus only saddle points of the area functional. In particular the lowest eigenvalue for n = 0 is found to be negative for all γ, and all other eigenvalues for n = 0 become negative as well for γ small enough, see Fig. (3). For n =1 only the lowest eigenvalue is negative for γ > 0.71, see V. APPARENT HORIZONS Fig. (4). For n = 2 the situation is similar: again only the lowest eigenvalue is negative, but in the much smaller As was shown before, the minimal tori considered here region γ > 0.97. For n > 2 the situation is not totally are all unstable, so that they locally can be deformed to clear: if negative eigenvalues appear, they have to occur an outer trapped surface [1]. By asymptotic flatness the very close to γ0. By explicit computation of the area region of compact outer trapped surfaces has to be finite, of neighboring tori withr ¯ = const., it was found that the outer boundary is called apparent horizon, which is the minimal tori have minimal area for γ < 0.891, and a stable marginally outer trapped surface (see Ref. [1] maximal area for γ > 0.891 among neighboring tori of and references cited therein). Noting thatg ˜ is analytic constantr ¯. by construction of the conformal factor as the solution of In the limiting case γ = γ0 the Green function G di- Eqs. (28-30) and not flat (by analyticity it would have to verges, and all eigenvalues ν go to zero. In the other be flat everywhere if it were flat in some region, but this limiting case, γ = 0, the coordinate range is the entire can not be, since the mass is positive by Eq. (31)), it can real line, and the Sturm-Liouville problem becomes sin- be concluded from Ref. [1] that the apparent horizon has gular. The potential V falls off exponentially to the value spherical topology. n2 1/4 for large z¯ – note that this value is negative The existence and properties of apparent horizons have for−n = 0 and positive| | for n > 0. While nothing spe- received some interest in the context of the cosmic cen- cial happens to the part of the spectrum with n> 0, see sorship hypothesis [19], where they play a double role as

8 circumference various definitions would be possible. In an axially symmetric geometry which is also symmetric with respect to reflection at the equator, straightforward 0.7 1 definitions of circumference exist along the equator (Ce) 0.6 2 and along a meridian circle (Cp): 3 1' 0.5 2 2' Ce = 2πψ R θ=π/2 , 0.4 π/2 2 ′2 2 ν Cp =4 dθ ψ R + R , 0.3 0 Z p 0.2 where R is the radial coordinate in standard polar coor- C 0.1 dinates (R,θ,ϕ). The results for 4πm are given in Fig. (6) and table (I). 0.0 For the problem of locating apparent horizons in the

−0.1 axially symmetric case a variety of different methods is 0.0 0.2 0.4 0.6 0.8 available by now. If one considers configurations which γ are symmetric (or almost symmetric) with respect to reflection at the equator, one can reasonably hope that the FIG. 4. The first five eigenvalues ν1l (n = 1) are plotted apparent horizon will be expressible as a function R(θ). as functions of γ from top down (l = 1,..., 5), the values for This gives the ODE γ = 0 are marked by ×’s. The curves labelled 1, 2, 3 are the ′2 first three branches of eigenvalues with symmetric eigenfunc- ′′ R ′ ′ R R 2 ψR ′ ψθ tions, while 1 , 2 label the first two branches of eigenvalues − ′2 R 4R + R 4 + cot θ 2R =0, 1+ 2 − ψ ψ − with antisymmetric eigenfunctions. R which has to be solved with the boundary conditions

′ ′ in Penrose’s singularity theorem [10] and also as a local R (0) = R (π/2)= 0. (albeit only weak) analogue of the global concept of an event horizon. The second is practically exploited in the This is done numerically using a shooting and matching apparent horizon boundary condition [5] used in numer- method (NAG [15] routines d02agf and d02ebf were used ical relativity and also was used in a new approach for in two different codes for horizon finding). the definition of a black hole itself [20]. The results are shown in figures (5), (6) and table (I). The apparent horizons were located numerically and For γ γ0 the apparent horizon becomes spherical and analyzed with respect to the Penrose and hoop conjec- pinches→ off to infinity as has been discussed for the anal- tures. Penrose [19] formulated the conjecture that if a ogous case when the Yamabe number goes to zero for a collapsing body satisfies the inequality manifold without boundary in Ref. [3]. For γ<γ0 the apparent horizon is oblate, but not very strongly. A m < , r16π VI. DISCUSSION where m is the ADM mass and A the area of a trapped surface, then cosmic censorship should be violated. As pointed out by Horowitz [21], if A is the minimal area In a previous paper [12] the notion of toroidal confor- that surrounds a trapped surface (in the time symmetric mal symmetry was introduced to label a class of initial data where a U(1) U(1) conformal isometry is present. case identical to the apparent horizon) and satisfies the × above inequality, then cosmic censorship does not hold. Working on a compactified background metric allows to For the apparent horizons found here the inequality is easily exploit this conformal symmetry to decompose the always violated, the Penrose conjecture thus confirmed, Lichnerowicz conformal factor in a double Fourier series see figure (5). on the group orbits and give the solution in terms of a The hoop conjecture, due to Thorne [11], states that countable family of uncoupled ODEs on the orbit space. an apparent horizon will exist if and only if a mass m gets The present paper applies these methods to construct compacted into a region such that the circumference C and analyze initial data containing a marginally outer satisfies C 4πm in every direction. The exact defini- trapped torus. Special emphasis is given to the investiga- tion of what≤ should be meant by m and C is left open. tion of the limiting cases when the sequence of solutions For a precise definition of m one would need a notion of intersects the boundary of conformal superspace. quasilocal mass which is available only for the spherically To get a better understanding of the results it seems symmetric case (no gravitational waves), here the ADM useful to look at the usually studied case of vacuum mass was chosen. Also for a general definition of the asymptotically flat data without boundary, where the

9 1.000 1.30

0.998 1.20

0.996 1.10 Ce/4πm 2 Cp/4πm A πm

16 0.994 1.00

0.992 0.90

0.990 0.80 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 γ γ

FIG. 5. Check of the Penrose isoperimetric inequality, the FIG. 6. Equatorial and polar hoop quotients, the values for value for γ = 0 is marked by ×. γ = 0 are marked by × and ◦.

cases understood best are the time-symmetric, notably the other hand studied families of metrics of fixed mass the axially symmetric subclass known as Brill-waves [22]. with a singular limit for which no apparent horizon ap- In order to simplify this comparison I will first discuss peared even in the limiting case. generalizations of the data considered in this paper. The The limiting case γ γ0 of the present paper is sim- obvious generalization is to drop conformal flatness and ilar to the one where the→ Yamabe number goes to zero. consider deformations of the base metric g of the form As the Sobolev quotient defined in Eq. (11) is just an extension of the Yamabe number including an additional dr¯2 boundary term, the mathematical and physical situation g = e2Aq(¯r) + (1 γ2r¯2)dz¯2 +¯r2dϕ2, (33) 1 γ2r¯2 − is analogous. As the mass diverges, the apparent hori- − zon becomes more and more spherical and pinches off to which is the analogue of the Brill-wave ansatz [4] while infinity. Going beyond that critical point the mass be- keeping the conformal symmetry, and is currently inves- comes infinitely negative. Whereas in the no-boundary tigated. The general axially symmetric case corresponds case the amplitude parameter A of Eq. (33) can be in- to form functions q(¯r, z¯) that depend on bothr ¯ andz ¯. creased without bound, which gives rise to an infinite Within the present ansatz one could consider certain non- number of analogous singularities in the mass function axially symmetric situations by placing Λ off the axes (to be discussed in forthcoming work), here the param- (the sets of fixed points of the isometries), the physical eter γ is bounded from above by γ 1, and the mass ≤ metric then has no isometries whatsoever – since an isom- appears to stay negative for γ>γ0. A proof that the etry of g carries over tog ˜ only if Λ is a fixed point of this Sobolev quotient will become negative was unsuccessful isometry [23]. due to the finite range of γ: In the Brill-wave case an The usually studied case is recovered, finally, by impos- estimate can be obtained with simple test-functions by ing regularity on all of R3 instead of thep ˜ = 0 boundary just increasing the amplitude enough [22] – here the same condition, this case has been discussed in [12], the pub- trick can not be carried out. γ0 comes in fact very close lication of numerical results is in preparation. to the upper boundary γ = 1, and so a very good choice For these traditionally considered data sequences of of test function would have to be made. metrics which approach the boundary of the space of ini- The other case where the sequence of solutions hits tial data have been studied both analytically and numer- the boundary of conformal superspace, γ 0, is fun- ically and tested for apparent horizons. Two comple- damentally different. While in the first case→ the base mentary paths have been taken, an overview is given in geometry stays regular, the situation when the boundary [22]. Sequences of metrics for which the Yamabe num- torus touches the axis can be considered as singular, nev- ber goes to zero have been studied analytically by Beig ertheless the mass and location of the apparent horizon and O´ Murchadha [3], who proved that asymptotically remain finite, but a curvature singularity develops at the the mass diverges in indirect proportion to the Yamabe origin. number and that an apparent horizon would always ap- Table (I) lists some numerical results for different val- pear before the mass diverges. Abrahams et al. [24] on ues of γ.

10 φ =0, mΣ △ γ m λ1 C /C m /m − 1 N ∂φ 1 − 1 − p e res π/2 max + φ =(1+¯z2) 3/2 (1+z ¯2) 1/2 0.0 3.8771 0.4700 0.4758 - - - ∂r¯ 4 r¯=1 − 4 −10 −6 0.2 3.8442 0.4639 0.7447 3.7 × 10 −3.7 × 10 100 −9 −7 0.4 3.7444 0.4443 0.7421 1.6 × 10 6.4 × 10 37 from Eqs. (22) and (23). 0.5 3.6771 0.4283 0.7440 2.1 × 10−9 −7.0 × 10−9 33 We solve by cosine transformation: −9 −8 0.6 3.6272 0.4068 0.7567 4.1 × 10 −3.3 × 10 24 ∞ − − 0.7 3.6571 0.3775 0.7891 2.5 × 10 8 −3.9 × 10 8 19 2 −8 −7 φ(¯r, z¯)= cos(ωz¯)fω(¯r) d ω, 0.8 3.9173 0.3353 0.8476 7.6 × 10 −1.1 × 10 15 π 0 −7 −7 Z 0.9 5.0129 0.2636 0.9287 2.5 × 10 −1.1 × 10 12 − − 0.989 226.13 0.0773 0.9998 4.8 × 10 4 7.8 × 10 7 7 resulting in a family of Bessel equations for regular functions fω(¯r) parameterized by ω: TABLE I. Results from numerical calculations, numbers are rounded to the last digit. ′′ 1 ′ f + f ω2f =0. ω r ω − ω The regular solutions are

The technique of using conformal symmetry together fω(¯r)= c(ω)I0(ωr¯), with compactification had several technical advantages. First of all, the ODEs (28) and (25) resulting from the where In and Kn used below are modified Bessel func- Hamiltonian constraint and existence question can be tions of order n and c(ω) is determined from the cosine handled with relatively small numerical expertise and transformed boundary condition desktop workstations. The use of the NAG numerical ∞ ′ 1 2 Fortran library [15] yields an excellent documentation of f (¯r)+ f (¯r)= cos(ωz¯) all numerical routines that have been used. ω 4 ω π r Z0 Another point in favor of the method is that it works − 1 − (1+z ¯2) 3/2 (1+z ¯2) 1/2 dω well for strong data – that is near γ = γ0 in our sequence, × − 4 only the n = 0 term in the Fourier series blows up when the mass does. 1 2 = (4ωK1(ω) K0(ω)) . If there are additional asymptotic regions (representing 4rπ − black holes), discrete symmetries result from an arrange- as ment compatible to the conformal isometry as discussed in [12]. 2 4ωK (ω) K (ω) c(ω)= 1 − 0 . It is hoped, that the technical approach taken here π 4ωI (ω)+ I (ω) and in [12], namely to use compactification and confor- r 1 0 mal Killing vector fields may also prove useful in other We get an explicit formula for the Green function, situations. √2 √2 G(¯r, z¯)= + φ = √r¯2 +¯z2 √r¯2 +¯z2 ACKNOWLEDGMENTS ∞ 2 4ω(K (ω) K (ω) + cos(ωz¯) 1 − 0 I (ωr¯) dω. (A1) π 4ωI (ω)+ I (ω) 0 The author thanks R. Beig for his support, encourage- Z0 1 0 ment and for many stimulating discussions. This work φ is a regular function ofr ¯,z ¯, the integrand diverges was financially supported by Fonds zur Förderung der only logarithmically for small ω and exhibits exponential wissenschaftlichen Forschung, Project No. P09376-PHY. falloff for large z¯ . Its value at the point at infinity gives | | the ADM mass m =3.877 by m =2√2φ(Λ). Alternatively the Green function can be written as APPENDIX A: THE EXPLICIT LIMITING SOLUTION 2 ∞ p(ω,r¯) G(¯r, z¯) = π 0 cos(ωz¯) q(ω) dω, p(ω, r¯) = 4ω(I (ω)K (ωr¯)+ K (ω)I (ωr¯)) R 1 0 1 0 (A2) In the limiting situation γ = 0 the constraint equation +K (ωr¯)I (ω) I (ωr¯)K (ω), (22) can be solved explicitly. Making the ansatz 0 0 − 0 0 q(ω) = 4ωI1(ω)+ I0(ω). √2 G = + φ using the identity [25] √ 2 2 r¯ +¯z ∞ 1 2 (analogous to (26)), we get = cos(ωz¯)K0(ωr¯) d ω. √r¯2 +¯z2 π Z0

11 Positivity of G(¯r, z¯) was checked by numerical evalua- Some information is also available over WWW, at tion, the behavior of G for large z¯ can be seen analyti- http://www.nag.co.uk/. cally by integration of (A2) in the| | complex plane. Both [16] D. Bernstein, D. Hobill, E. Seidel and L. Smarr, Phys. p and q are positive and symmetric on the real line, but Rev. D 15, 50, 3760 (1994). q has infinitely many zeros for purely imaginary argu- [17] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral ments, leading to corresponding poles of p/q. Observ- Theory, Translations of Mathematical Monographs, Vol. ing, that the factor exp ( Im(ω)¯z) makes contributions 38 (American Mathematical Society, Providence, 1975). to the integral small for large− z ¯ in the upper complex half [18] R. Courant and D. Hilbert, Methods of Mathematical plane (forz ¯ the lower half respectively) yields an Physics, Vol. 1 (John Wiley-Interscience, New York, asymptotic formula→ −∞ for the integral by choosing a path 1953). that encircles only the first pole (an asymptotic series [19] R. Penrose, Rivista del Nuovo Cimento 1, 251 (1969), Ann. NY Acad SCi 224, 125 (1973) results from summing over all poles), [20] S. A. Hayward, Phys. Rev. D 49, 6467 (1994). G(¯r, z¯)= c(¯r)exp( α z¯ ), [21] G. Horowitz, in Lecture notes in Physics 202, edited by − | | F. J. Flaherty (Berlin, Springer, 1984). where α = 0.6856 is the first zero of q(ω) on the imagi- [22] N. O´ Murchadha, in Directions in General Relativity, nary axis and Vol. 2, edited by B. L. Hu and T. A. Jacobson (C.U.P., Cambridge, 1993). p(ω, r¯) [23] R. Beig, Class. Quant. Grav. 8, L205 (1991). c(¯r)=4 i Residual ,ω =0.6856 . q(ω) [24] A. M. Abrahams, K. R. Heiderich, S. L. Shapiro and S. A. Teukolsky, Phys. Rev D 46, 24452 (1992). [25] I. S. Gradshteyn, Table of Integrals, Series and Products (Academic Press, New York, 1980).

[1] G. J. Galloway, in Differential Geometry and Mathemat- ical Physics, Contemporary Mathematics, Vol. 170, eds. J. Beem and K. Duggal (American Mathematical Society, Providence, 1994), p. 113. [2] See e.g. Y. Choquet-Bruhat and J. W. York, Jr., in Gen- eral Relativity and Gravitation, ed. A. Held (Plenum, New York, 1980) Vol. 1. [3] See e.g. R. Beig and N. O´ Murchadha, Phys. Rev. Lett. 66, 2421 (1991) and references cited therein. [4] D. R. Brill, Ann. Phys. (N.Y.) 7, 466 (1959). [5] E. Seidel and W.-M. Suen, Phys. Rev. Lett. 69, 1845 (1992). [6] See Ref. [8] and the account of the Yamabe problem and its connection to General Relativity given in N. O´ Mur- chadha, in Proceedings of the Centre for Mathematical Analysis, edited by R. Bartnik (Australian National Uni- versity, Canberra, 1989), Vol. 19, p. 137. [7] J. F. Escobar, J. Diff. Geom. 35, 21 – 84 (1992). [8] Lee and Parker, Bull. Am. Math. Soc. 17, 37 (1987). [9] R. Schoen and S. T. Yau, Comm. Math. Phys. 65, 45 – 76 (1979). [10] R. Penrose, Phys. Rev. Lett. 14, 57 (1965). [11] K. S. Thorne, in Magic Without Magic: John Archibald Wheeler, edited by J. Klauder (Freeman, San Francisco, 1972), p 231. [12] R. Beig and S. Husa, Phys. Rev. D 50, 12 (1994). [13] For an overview on the positive mass theorem of Schoen and Yau [9] and its relation to the Yamabe problem see Ref. [8]. [14] See e.g. J. L. Kazdan, in College on Differential Geome- try, Summer School Report, (ICTP, Trieste, 1989). [15] NAG FORTRAN Library Manual – Mark 16, The Numerical Algorithms Group Ltd., Oxford (1994).