Trumpet slices of the Schwarzschild-Tangherlini spacetime
Kenneth A. Dennison, John P. Wendell, Thomas W. Baumgarte, J. David Brown
TTrumpet slices of the Schwarzschild-Tangherlini spacetime
Kenneth A. Dennison, John P. Wendell, and Thomas W. Baumgarte ∗ Department of Physics and Astronomy, Bowdoin College, Brunswick, Maine 04011
J. David Brown
Department of Physics, North Carolina State University, Raleigh, North Carolina 27695 (Dated: draft of April 15, 2019)We study families of time-independent maximal and 1 + log foliations of the Schwarzschild-Tangherlini spacetime, the spherically-symmetric vacuum black hole solution in D spacetime dimen-sions, for D ≥
4. We identify special members of these families for which the spatial slices displaya trumpet geometry. Using a generalization of the 1 + log slicing condition that is parametrized bya constant n we recover the results of Nakao, Abe, Yoshino and Shibata in the limit of maximalslicing. We also construct a numerical code that evolves the BSSN equations for D = 5 in sphericalsymmetry using moving-puncture coordinates, and demonstrate that these simulations settle downto the trumpet solutions. PACS numbers: 04.50.Gh, 04.25.dg
I. INTRODUCTION
Numerical relativity simulations of binary black holeshave matured dramatically in recent years. Starting withthe first complete simulations of binary black hole merg-ers [1–3], a large number of papers on mergers of bina-ries with varying mass ratios and black hole spins haveappeared. Some of the results of these simulations, in-cluding the surprisingly large recoil speed of the mergerremnant for certain spin orientations (e.g. [4, 5]), havealso triggered numerous studies of the astrophysical con-sequences of these findings.Many numerical simulations of black holes adopt theBaumgarte-Shapiro-Shibata-Nakamura (BSSN) formula-tion of Einstein’s equations [6, 7] together with moving-puncture coordinates [2, 3] (see also [8]). The latter con-sist of the 1+log slicing condition for the lapse [9] andthe ¯Γ-driver gauge condition for the shift [10]. The roleof these coordinates in stabilizing the numerical simula-tions has been clarified by considering the late-time be-havior of spatial slices of the Schwarzschild spacetimewhen evolved with moving-puncture coordinates [11–13].In particular, these studies showed that, at late times,these slices form a “trumpet” geometry, meaning thatthe slices asymptotically approach a finite areal radiusand never reach the spacetime singularity (see Fig. 2 in[13] for an embedding diagram that motivates the nameof this geometry).Several groups have also started to simulate black holesin higher dimensions (e.g. [14–22]). Some of these stud-ies aim at exploring the rich geometric structure of blackholes in higher dimensions (see, e.g., [23]), while othersare motived by speculations that miniature black holesmight be created in high-energy collisions in particle col- ∗ Also at Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801 liders (e.g. the LHC), a scenario that requires additionalspacelike dimensions (see, e.g., [24] for a review).Motivated by the success of the BSSN formalism withmoving-puncture coordinates in D = 4 spacetime dimen-sions, several of the above simulations adopt this methodfor D > D ≥ D ≥
4. Wedemonstrate that, as for D = 4, these slices display atrumpet geometry. Parameterizing the 1 + log slicingcondition with a parameter n , we show that we recoverthe results of [18] for maximal slicing in the limit n → ∞ .We also recover the results of [13] for D = 4. Finally, weperform numerical simulations for D = 5 and demon-strate that these simulations indeed settle down to thetrumpet slices.This paper is organized as follows. In Section II, weintroduce the Schwarzschild-Tangherlini spacetime [25]and introduce a “height function” that we will use forcoordinate transformations in the following sections. InSection III, we construct a family of maximal slicesof the Schwarzschild-Tangherlini spacetime, reproducingearlier results of [18], and identify a special member ofthis family that displays a trumpet geometry. In Sec-tion IV, we derive the main result of this paper, a sta-tionary 1 + log trumpet solution for the Schwarzschild-Tangherlini spacetime. We recover the corresponding D = 4 results from [13] and the maximal slicing resultsof [18] in the appropriate limits. In Section V we performdynamical simulations for D = 5 in spherical symmetryand demonstrate that, at late times, these simulationsindeed settle down to the trumpet slices derived in theprevious sections. We conclude with a brief summary inSection VI.Throughout this paper we adopt units in which the a r X i v : . [ g r- q c ] O c t speed of light is unity, c = 1. However, we do not setto unity the gravitational constant G of a D -dimensionalspacetime, since keeping G in the analytical expressionsmakes their units more transparent. II. SCHWARZSCHILD-TANGHERLINI
The generalization of the Schwarzschild spacetime,which describes spherically symmetric vacuum solutionsin D = 4 spacetime dimensions, to higher dimensions D is the Schwarzschild-Tangherlini solution [25]. Adoptingthe notation of [23], this solution can be written as ds = − f dt + f − dR + R d Ω D − , (1)where we have defined f ( R ) = 1 − µR D − , (2)and where the mass parameter µ is given by µ = 16 πGM ( D −
2) Ω D − . (3)The area of a unit D − D − = 2 π ( D − / Γ (( D − / , (4)and the line element on this sphere is d Ω D − = dχ + sin χ dχ + · · · + (cid:32) D − (cid:89) l =2 sin χ l (cid:33) dχ D − , (5)where the χ l (with l = 2 , , . . . , D −
1) are the angles onthe D − χ and χ D − are often called θ and φ . We also point out that GM has dimensionsof ( length ) D − . For D = 4 we recover µ = 2 GM andΩ = 4 π , as expected.In the line element (1), R is the generalization of theSchwarzschild (or areal) radius. In numerical applica-tions, it is often convenient to express the metric in termsof an isotropic radius r , so that the spatial part of themetric can be transformed to cartesian coordinates veryeasily. To do so, we set the spatial part of the line element(1) equal to its isotropic counterpart, dR − µ/R D − + R d Ω D − = ψ / ( D − (cid:0) dr + r d Ω D − (cid:1) , (6)where the exponent on the conformal factor ψ has beenchosen for convenience. From the identification (6) wesee that R = ψ / ( D − r , (7)and dR − µ/R D − = ψ / ( D − dr . (8) Eliminating ψ we now find ± (cid:90) drr = (cid:90) R D − dR (cid:112) R D − − µR D − . (9)Integrating both sides of the equation we obtain [26] r = µ / ( D − (cid:32) (cid:112) R D − /µ − (cid:112) R D − /µ (cid:33) / ( D − (10)or, solving for R , R = r (cid:16) µ r D − (cid:17) / ( D − . (11)Inserting this into (7) we find the conformal factor ψ = 1 + µ r D − . (12)As expected, these results reduce to the usualSchwarzschild expressions for D = 4. In Section V wewill adopt the above expressions for D = 5 as initialdata.In order to explore alternative slicings (or foliations) ofthe Schwarzschild-Tangherlini spacetime in Sections IIIand IV we now introduce a new time coordinate¯ t = t + h ( R ) , (13)where h ( R ) is the height function [8, 27]. As we will seein the following Sections, different slicing conditions re-sult in different ordinary differential equations for h ( R ).With the new time coordinate, the line element (1) be-comes ds = − f d ¯ t + 2 f h (cid:48) d ¯ tdR + 1 − f h (cid:48) f dR + R d Ω D − (14)where h (cid:48) ( R ) ≡ dh/dR .We can write a general line element in ( D − ds = − α dt + γ ij (cid:0) dx i + β i dt (cid:1) (cid:0) dx j + β j dt (cid:1) , (15)where α , β i , and γ ij are the lapse, shift vector, and spa-tial metric, respectively, and where the indices i, j, . . . runover all D − β R . Comparing terms in the lineelements (14) and (15), we can identify the lapse as α = f − f h (cid:48) , (16)the shift as β R = f h (cid:48) − f h (cid:48) , (17)and the spatial metric as γ ij = α − . . . . . . R . . . . . .
00 0 R sin χ . . . R (cid:81) D − l =2 sin χ l . (18)The determinant γ of the spatial metric is γ = α − R D − D − (cid:89) l =2 (cid:0) sin χ l (cid:1) D − − l . (19)We define the extrinsic curvature K ij so that ∂ t γ ij = − αK ij + D i β j + D j β i , (20)where D i is the covariant derivative operator associatedwith the spatial metric. For time-independent spatialmetrics like (18), the left-hand side of (20) vanishes andwe obtain K ij = 12 α ( D i β j + D j β i ) . (21)Taking the trace of this equation we find that the meancurvature is given by K = 1 αγ / ddR (cid:16) γ / β R (cid:17) . (22)This equation is our starting point for imposing maximalslicing in Section III and 1 + log slicing in Section IV. III. MAXIMAL SLICING
Maximal slicing is defined by requiring that the meancurvature vanish, K = 0 . (23)Results for maximal slices of the Schwarzschild-Tangherlini spacetime have already been presented in[18]; the details included here are for the sake of com-pleteness and reference in later sections. In Section III A,we derive a family of time-independent maximal slicesof the Schwarzschild-Tangherlini spacetime for a generalnumber of spacetime dimensions D ≥
4, and we special-ize to D = 4 and D = 5 in Sections III B and III C,respectively. A. General treatment
For maximal slicing, equation (22) reduces to ddR (cid:18) R D − β R α (cid:19) = 0 . (24) Eliminating α and β R with the help of equations (16)and (17) we obtain a first integral R D − (cid:18) f − f h (cid:48) (cid:19) / f h (cid:48) = C, (25)where C is a constant of integration. It is convenient towrite (25) as f h (cid:48) = C f R D − + C , (26)which we can then substitute back into equations (16)and (17) to find the lapse α = f ( R ; C ) , (27)the shift β R = Cf ( R ; C ) R D − , (28)and the spatial line element dl = f − ( R ; C ) dR + R d Ω D − . (29)Here the function f ( R ; C ) is given by f ( R ; C ) = (cid:18) − µR D − + C R D − (cid:19) / . (30)The one-parameter family of spherically-symmetric,time-independent maximal slices of the Schwarzschild-Tangherlini spacetime is now parameterized by the con-stant C . For sufficiently small C the slices end at a radius R at which the lapse vanishes; from equation (27), thislocation is given by the largest root (see [18]) of the equa-tion R D − − µR D − + C = 0 . (31)Two particular members of this family deserve spe-cial mention. For C = 0 the height function h mustbe constant, the spacetime is therefore sliced by slices ofconstant Schwarschild-Tangherlini time t , and we recoverthe metric (1). The other member of the family that wewill be interested in is that for which the slice ends witha double-root of the squared lapse (27). As we will showbelow, this choice singles out a trumpet slice. In Sec-tion V we will see that these slices act as “attractors” indynamical moving-puncture simulations.Setting both α and the first derivative ∂ ( α ) /∂R tozero, we find that this double root occurs at the radius˜ R = (cid:18) µ ( D − D − (cid:19) / ( D − (32)for the value of C given by˜ C ≡ (cid:18) D − D − (cid:19) (cid:18) µ ( D − D − (cid:19) D − / ( D − , (33)where the tilde denotes the special value for the trumpetslice (see also [18]).Before closing this section we analyze the asymptoticproperties of these slices in a neighborhood of R . Giventhe spatial metric (18), we can compute the proper dis-tance ∆ between a point on the limit surface, ( R , χ i ),and a point at ( R , χ i ) from the integral∆ = (cid:90) R R dRα . (34)By definition, the lapse α vanishes at R . Whether or notthis integral is finite therefore depends on the behaviorof α in the neighborhood of R . From its definition inEqs. (27) and (30), we see that the square of the lapsecan be expanded as α = A ( R − R ) + A ( R − R ) + · · · . (35)For generic values of C , R = R is a single root of α and hence A is nonzero. Therefore the lapse behaveslike α ∼ ( R − R ) / near R and the integral in Eq. (34)is finite. Thus, for generic C , the proper distance to thelimiting surface at R is finite.For the special slice C = ˜ C , on the other hand, R = ˜ R is a double root of α and therefore A vanishes. Thelapse then behaves like α ∼ ( R − ˜ R ) near ˜ R and theintegral in Eq. (34) diverges. The proper distance to thelimiting surface at ˜ R is infinite, even though the arealradius is non-zero and finite (see [18] for an alternativederivation of this property). An embedding diagram ofthe slice would result in a figure similar to that in Fig. 2of [13]; given the appearance in an embedding diagramthese slices are referred to as “trumpet” slices.As in Section II, it would be useful to transform theseslices to isotropic coordinates. Instead of equation (9),we now have ± (cid:90) drr = (cid:90) f dRR = (cid:90) R D − dR (cid:112) R D − − µR D − + C . (36)For C = ˜ C , the expression under the square root onthe right hand side has a double root at R = ˜ R , whichsimplifies the integral for both D = 4 and D = 5. B. Four-dimensional spacetimes
For four-dimensional spacetimes, D = 4, we re-cover the well-known family of time-independent max-imal slices [8, 13, 27–29]. In particular, with Ω = 4 π and µ = 2 GM , the lapse becomes α = f ( R ; C ) = (cid:18) − GMR + C R (cid:19) / . (37)With the lapse known, the shift, spatial metric, and ex-trinsic curvature can be calculated from (28), (18), and (21), respectively. Also, for the special value of ˜ C , whichnow becomes ˜ C = 3 √ GM ) , (38)the transformation to isotropic coordinates can be carriedout by integrating equation (36) (see [30]). C. Five-dimensional spacetimes
For five-dimensional spacetimes, D = 5, we recoverthe results of [18]. We note, however, a difference innotation: our R is their r and vice versa. With Ω = 2 π and µ = 8 GM/ (3 π ) we now find the lapse α = f ( R ; C ) = (cid:18) − GM πR + C R (cid:19) / . (39)With the lapse known, the shift, spatial metric, and ex-trinsic curvature can again be calculated from (28), (18),and (21), respectively. The trumpet geometry is realizedfor ˜ C = 2 / (cid:18) GMπ (cid:19) / , (40)and its limiting surface is at˜ R = 43 (cid:114) GMπ . (41)For the trumpet slice, equation (36) can again be inte-grated analytically (see [18]) to obtain the solution inisotropic coordinates.
IV. STATIONARY 1 + LOG SLICING
We now turn to stationary 1+log slicing β i ∂ i α = nαK, (42)where n is a constant. The most common choice for n is n = 2, but, following [13], we will pursue a more generaltreatment. In particular, we will recover the maximalslicing results of Section III in the limit n → ∞ . Mirror-ing Section III, we derive a family of time-independent1 + log slices of the Schwarzschild-Tangherlini spacetimefor arbitrary D ≥ D = 4 and D = 5 in Sections IV B and IV C. A. General treatment
Using equations (14), (15), and (22), we can rewrite(42) as ddR α = nγ / β R ddR (cid:16) γ / β R (cid:17) . (43)It is convenient to use (16) and (17) to find β R = α (cid:112) α − f , (44)and to eliminate β R in (43). This substitution results in ddR α = n ddR ln (cid:16) R ( D − (cid:112) α − f (cid:17) , (45)which can be integrated immediately to yield α = const + n ln (cid:16) R ( D − (cid:112) α − f (cid:17) , (46)or equivalently, α = f + C ( n ) e α/n R D − . (47)Here C is again a constant of integration, but we pointout that it now depends on the constant n . Equation(47) is a transcendental equation for the lapse α , and so-lutions, when they exist, can be found numerically. Withthe solution for the lapse in hand, the shift, spatial met-ric, and extrinsic curvature can be found from (44), (18),and (21).As for maximal slicing, some values of C ( n ) arespecial. Setting C ( n ) = 0 corresponds to the usual t = const slices, recovering the metric (1). Another so-lution of interest is obtained by requiring equation (45)to be regular for all α ≥
0. Following the method givenin [13] for D = 4, we can use the regularity conditionto determine the corresponding values ˜ C ( n ) for D ≥ ddR α = − n (cid:0) ( D − R − (cid:0) α − f (cid:1) − f (cid:48) (cid:1) f + nα − α . (48)If we require regularity when α ≥
0, the numerator anddenominator must vanish at the same value of R . Thenumerator is zero when α = (cid:115) f + Rf (cid:48) D − . (49)Substituting this value for the lapse and f from (2) intothe right hand side of (48), we see that its denominatorvanishes when R D − + (1 − D ) µ D − R D − − ( D − µ n ( D − = 0 . (50)This equation is quadratic in R D − , and so it is easilysolved for the critical value R c for which the numera-tor and denominator of (48) vanish simultaneously. Thepositive real root is R c = ( D − µ + µ (cid:113) D − n − + ( D − D − D − . (51) The lapse at R c can be calculated from (49) to be α c = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:113) D − + ( D − n − ( D − n (cid:113) D − + ( D − n + ( D − n . (52)Inserting equations (51) and (52) into (47) we now find˜ C ( n ), defined as the special value of C ( n ) that makes(48) regular,˜ C ( n ) = D − D − D − (cid:18) µD − (cid:19) D − D − (cid:16) e − αcn (cid:17) × D − (cid:113) D − + ( D − n n D − D − . (53)For most of the work that follows we consider only thespecial slices with C = ˜ C . The inner boundary of theseslices is defined by the location at which the lapse goes tozero. We will refer to this location as the throat, and willcall its radius ˜ R . To identify ˜ R , we first observe that,at least for sufficiently large R , equation (47) admits twosolutions for the lapse α . For R → ∞ , one of the branchesof solutions approaches α = 1, while the other approaches α = −
1. This behavior is displayed in Fig. 1, where wegraph α versus R for D = 5 and n = 2, for the specialvalue C = ˜ C . We are interested in the “positive” branchof solutions, which approaches α = 1 asymptotically. Ascan be seen from Fig. 1, this positive branch has a root(with α = 0) at a smaller value of R than the negativebranch. In general, then, we define the throat as thelocation ˜ R at which the positive branch of solutions for α vanishes. For D = 4 and D = 5, this corresponds tothe smallest root of equation (47) with α = 0,˜ R D − − µ ˜ R D − + ˜ C ( n ) = 0 (54)(see also the discussion in [13]). In the limit n → ∞ ,the critical value of the lapse α c → largest root in the case of maximal slicing. We willdiscuss explicit solutions for ˜ R ( n ) for D = 4 and D = 5in Sections IV B and IV C below.We can demonstrate that the special slice with C = ˜ C is a trumpet slice. To do so, we assume that close to ˜ R , α can be expanded as a power series α = (cid:18) δ ˜ R (cid:19) m (cid:18) α + α δ ˜ R · · · (cid:19) (55)where R = ˜ R + δ . Since α = 0 for δ = 0 we must have m >
0. Inserting this into equation (47), with C = ˜ C ,and expanding terms to first leading order, we obtain α (cid:18) δ ˜ R (cid:19) m = ˜ C ˜ R D − n (cid:18) δ ˜ R (cid:19) m α + (cid:32) ( D − µ ˜ R D − + 2(2 − D ) ˜ C ˜ R D − (cid:33) δ ˜ R . (56) FIG. 1: The two branches of the lapse α as a function of theradius R for D = 5, n = 2, and C = ˜ C . We are interested inthe “positive” branch that approaches α = 1 at infinity. Thisbranch has a root (with α = 0) at a smaller value of R thanthe “negative” branch. Matching exponents of δ we identify m = 1, so that α ∼ δ = R − ˜ R near R = ˜ R . Therefore the properdistance from any point outside the throat to the throatat ˜ R , given by the integral (34), diverges. As before, thisdemonstrates that this slice features a trumpet geometry.In the limit n → ∞ , the 1 + log slicing condition (42)approaches the maximal slicing condition K = 0, so weshould recover the maximal slicing results from SectionIII. From (47) we have α = f + C R D − , (57)consistent with (27). Similarly, taking the limit n → ∞ in equation (53) we recover (33). Finally, letting n → ∞ in (51) results in R c = (cid:18) µ ( D − D − (cid:19) / ( D − . (58)Substitution into (54) shows that this coincides with thelocation of the throat, so that R c = ˜ R in this limit,meaning that we have recovered ˜ R in the maximal slic-ing limit (32).As in Section II, it would useful to transform theseslices to isotropic coordinates. While it is impossible tocarry out the necessary integrations analytically, giventhe transcendental nature of equation (47), the equationscan be integrated numerically using the techniques pre-sented in [13]. FIG. 2: The throat location ˜ R as a function of the coefficient n in the 1 + log slicing condition (42) for D = 4 (dashed)and D = 5 (solid). The points are the corresponding valuesfound by our numerical simulations described in Section V.Specifically, we extrapolated the lapse α linearly to zero tofind the throat location ˜ R for any grid resolution, and thenused Richardson extrapolation using our two highest-orderresolution results to obtain the values shown here. B. Four-dimensional spacetimes
For four-dimensional spacetimes, the above results re-duce to those of [13]. In particular, for D = 4, equation(54) becomes a quartic equation˜ R − µ ˜ R + ˜ C ( n ) = 0 . (59)The real root consistent with the positive branch of thelapse is˜ R ( n ) = µ (cid:32)
12 + (cid:114)
14 + Z − (cid:115) − Z + 14 (cid:112) / Z (cid:33) , (60)where Z = 4 (cid:0) (cid:1) / ˜ C ( n ) Y µ + Y / / , (61)and Y = (cid:18)(cid:18) (cid:113) −
768 ˜ C ( n ) µ − (cid:19) µ − ˜ C ( n ) (cid:19) / . (62)We graph these solutions in Fig. 2 (compare with Fig. 3of [13], where these solutions are given only graphically).For comparison with [13], we note that (48) can bewritten as ddR α = − nR ( R D − ( f + nα − α )) × (cid:16) − ( D − R D − f − R D − f (cid:48) + ( D − R D − α (cid:17) , (63)which, for D = 4, reduces to ddR α = − n (cid:0) GM − R + 2 Rα (cid:1) R ( R − GM + nRα − Rα ) . (64)From this equation, the remaining results of SectionII D 1 of [13] can be derived. C. Five-dimensional spacetimes
For D = 5, equation (54) becomes a cubic equation forthe ˜ R ( n ), ˜ R − µ ˜ R + ˜ C ( n ) = 0 , (65)which can again be solved exactly. The root that is real,positive, and consistent with the positive branch of thelapse is˜ R = µ / / (cid:118)(cid:117)(cid:117)(cid:116) (cid:32)
13 arccos (cid:32) −
27 ˜ C ( n )2 µ (cid:33) + 4 π (cid:33) + 1 . (66)We show a graph of ˜ R in Fig. 2.With Ω = 2 π , µ = 8 GM/ (3 π ), and f = 1 − GM/ (3 πR ), the solution (47) of the stationary 1+logslicing condition becomes α = 1 − GM πR + C ( n ) e α/n R . (67)This determines α as a function of R . When solutions tothis equation exist, they can be found numerically witha root-finding routine by starting with an appropriateguess on the positive branch of the lapse. With the lapseknown, the shift, spatial metric, and extrinsic curvaturecan be calculated from (44), (18), and (21), respectively.For some purposes, it is sufficient to find R ( α ) instead of α ( R ). The real, positive solutions of relevance here areclosely related to the solution (66) for ˜ R ( n ): R ( α ) = (cid:114) µ − α ) × (cid:115) (cid:18)
13 arccos (cid:18) − A (cid:19) + 4 π (cid:19) + 1 , (68) and R ( α ) = (cid:114) µ − α ) × (cid:115) (cid:18)
13 arccos (cid:18) − A (cid:19)(cid:19) + 1 , (69)where A = (cid:0) − α (cid:1) µ C ( n ) e α/n . (70)With C = ˜ C , the positive branch of the lapse is givenby R ( α ) for 0 ≤ α ≤ α c and R ( α ) for α c ≤ α ≤ D = 4 can be constructed similarly fromsolutions to equation (59).The critical radius (51) is now R c = 23 (cid:18) GMπn (cid:16) n + (cid:112) n (cid:17)(cid:19) / , (71)and the lapse (52) at R = R c is α c = (cid:115) √ n − n √ n + n . (72)Finally, we find ˜ C from equation (53),˜ C ( n ) = 2 ( GM ) π (cid:18) n (cid:112) n (cid:19) e − α c /n . (73)As before we can recover the maximal slicing results ofSection III C by letting n → ∞ . From equations (58) and(66) we then obtain R c = ˜ R = 43 (cid:114) GMπ , (74)which agrees with equation (41).Most numerical relativity simulations adopt n = 2. Inthis case equation (51) yields R c = 23 (cid:115) √ π √ GM ≈ . √ GM , (75)and from (53) we have˜ C (2) = 128 (cid:0) √ (cid:1) e − (cid:113) ( √ − ) / ( √ )729 π ( GM ) ≈ . GM ) . (76)The throat (66) is located at˜ R ≈ . µ ≈ . √ GM . (77)For D = 5, the event horizon in the Schwarzschild-Tangherlini spacetime is located at R EH = (cid:112) GM/ (3 π ) = 0 . √ GM . At the horizon,the lapse is α EH ≈ . . (78) V. NUMERICAL WORK
In this section we describe dynamical numerical sim-ulations of the Schwarzschild-Tangherlini spacetime for D = 5. More specifically, we adopt as initial data the t = const slices of the Schwarzschild-Tangherlini space-time in isotropic coordinates, as described in Section II,and evolve these data with moving-puncture gauge con-ditions. We will see that the trumpet solutions derivedin the previous sections indeed act as “attractors” indynamical simulations, meaning that dynamical simula-tions settle down to these solutions at asymptotically latetimes, even when they start with very different initialdata. A. Numerical Method
Our numerical code is based on the third-order finite-difference code of [31]. This code implements the BSSNequations [6, 7] for D = 4 in spherical symmetry, andimposes spherical symmetry with the help of the “car-toon” method [32]. Details of the implementation of thismethod can be found in Appendix A of [31]. Several mod-ifications had to be implemented to adopt this code tofive spacetime dimensions. Clearly, all indices of spatialtensors had to be extended to four spatial dimensions,and the interpolation for the cartoon method had to beadjusted for the extra dimension. In addition, the finitedifferencing stencil was updated to account for the fourthspatial dimension. Furthermore, even though Einstein’sequations take the same form for all spacetime dimen-sions D , the BSSN equations for D = 5 are different fromthe corresponding equations in D = 4 (see, e.g. [17]). Thereason for this is the use of tracefree tensors in the BSSNformalism. In particular, consider the decomposition ofthe extrinsic curvature K ij into its trace K and its trace-less part A ij , A ij = K ij − D − γ ij K. (79)Since the BSSN equations are formulated in terms of A ij and K instead of K ij , factors that depend on D appearin several places. Following [17], we also use the vari-able χ ≡ ψ − instead of the conformal factor ψ in ournumerical simulations.We tested features of our code that are specific to D =5 with the help of a number of test problems, including aspherically symmetric standing wave. We also monitoredconstraint violations during our simulations and verifiedthat they converged to zero at the expected rate. B. Numerical Results
In all our dynamical simulations we start with dataon a slice of constant Schwarzschild-Tangherlini time, for which the conformal factor is given by equation (12). Ifthese data were evolved with the Killing lapse and shiftthat can be identified from the metric (1), the metricwould remain time-independent. Instead, we set α = 1 , β i = 0 (80)initially (at t = 0), and subsequently evolve the lapseand shift using moving puncture coordinate conditions[2, 3], namely the 1+log slicing condition for the lapse[9] and the Γ-driver condition [33]. We will consider twodifferent flavors of the 1+log slicing condition, namely an“advective” and a “non-advective” version.
1. 1+log slicing
The “advective” 1+log slicing condition is given by( ∂ t − β i ∂ i ) α = − nαK, (81)where n is again a constant. At late times, when the so-lution settles down and becomes time-independent, thiscondition reduces to the stationary 1+log condition (42).Our numerical results demonstrate that simulationswith the advective 1+log slicing condition (81) indeedsettle down to the stationary 1+log results of SectionIV C. In the following we show results that were ob-tained with N = 5001 uniform gridpoints (not count-ing the buffer points), with the outer boundary imposedat r max = 50 √ GM . We also chose n = 2 for all re-sults shown in this section. (We performed simulationsfor other values of n and N to obtain the results for theradius of the throat ˜ R shown in Fig. 2.)In the following figures we graph several quantities interms of the areal radius R , which simplifies comparisonwith the analytical results for the late-time asymptoticsolutions. In terms of the variables evolved in our code,we compute R from R = (cid:112) ¯ γ θθ /χ , where ¯ γ θθ is the θθ component of the conformally related metric and χ = ψ − .The initial data for our dynamical simulations aregiven in terms of an isotropic spatial metric. However,this spatial metric is evolved in time, and does not re-main isotropic (see Figs. 35 and 36 in [31]), so that wecannot compare the conformal factor in the code directlywith the conformal factor that we would obtain by trans-forming the analytical results of the previous sectionsinto isotropic coordinates. Instead, we compare analyti-cal and numerical values for the metric component γ RR in Fig. 3. We compute numerical values for γ RR from γ RR = ¯ γ rr χ ( dR/dr ) , (82)where ¯ γ rr is the rr component of the conformally relatedmetric and the derivative dR/dr is calculated numericallyfrom the areal radius R given above.We show the evolution of the numerical γ RR at sev-eral different times, starting with the initial data (12) FIG. 3: The metric component γ RR as a function of theareal radius R for a dynamical evolution of the D =5 Schwarzschild-Tangherlini spacetime with the advective1+log slicing condition (81) for n = 2. We show γ RR at anumber of different times, starting with the initial data (12) t = 0, and ending at time t = 42 √ GM . At late times, ournumerical results for γ RR agree very well with the analyticalresults of Section IV C. The latter are included as the solidline, which overlaps with the line representing the numericalresults at t = 42 √ GM .FIG. 4: Same as Fig. 3, but for the lapse α . FIG. 5: Same as Fig. 3, but for the shift β R . at t = 0 and ending at a time t = 42 √ GM before theresults shown in the figure are affected by the presenceof the outer boundary. (In order to avoid double-valuedfunctions, we show the numerical initial data only forisotropic radii outside the event horizon.) Also includedare the analytical results obtained in Section IV C. Wesee that γ RR evolves for a certain period of time, and thensettles down to the expected time-independent solution.In Figs. 4 and 5 we show similar results for the lapseand the shift. The numerical values of the areal shift β R are calculated using β R = ( dR/dr ) β r . Both the lapse andshift start with their initial values (80), go through a dy-namical phase, and then settle down to their asymptoticvalues. At these late times we find excellent agreementwith the stationary solutions that we computed analyti-cally in Section IV C.
2. Maximal slicing
We also performed numerical simulations for the “non-advective” 1+log slicing condition, for which the advec-tive term in (81) is dropped, ∂ t α = − nαK. (83)Time-independent solutions must satisfy the maximalslicing condition, K = 0. If dynamical simulations withthis slicing condition settle down to a time-independentsolution, then the late-time asymptotic solution must begiven by a member of the family of maximal slices that wediscussed in Section III. As it turns out, dynamical simu-lations settle down to the “special” member with C = ˜ C ,0 FIG. 6: The metric component γ RR as a function of theareal radius R for a dynamical evolution of the D = 5Schwarzschild-Tangherlini spacetime with the non-advective1+log slicing condition (83) for n = 2. We show γ RR at anumber of different times, starting with the initial data (12) t = 0, and ending at time t = 42 √ GM . At late times, γ RR approaches the trumpet member of the family of maximalslices discussed in Section III C. The latter is included as thesolid line, which overlaps the line representing the numericalresults at t = 42 √ GM . which displays a trumpet geometry (see also the discus-sion in [34]). We demonstrate this behavior in Fig. 6,where we test the time evolution of the conformal factor ψ by plotting γ RR as a function of areal radius R . As be-fore, the evolution starts with the initial data (12), butnow the evolution settles down to the trumpet memberof the family of maximal slices discussed in Section III C(see also [18]). VI. SUMMARY
We study maximal and stationary 1+log slices of theSchwarzschild-Tangherlini spacetime for D ≥ n in our 1+log slicing con-dition, where the limit n → ∞ corresponds to maximalslicing. We demonstrate that our results for 1+log slicingreduce to the maximal slicing results of [18] in this limit,and we also show that we recover the results of [13] for D = 4.Finally, we perform numerical simulations for spher-ically symmetric black holes in D = 5 spacetime di-mensions. We start with data on a slice of constantSchwarzschild-Tangherlini time, and evolve these datawith moving-puncture gauge conditions. Our resultsdemonstrate that, as for D = 4 spacetime dimensions,the dynamical simulations settle down to the trumpetslices, which can be regarded as “attractors” for moving-puncture simulations. Acknowledgments
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