Static brane--like vacuum solutions in D \ge 5 dimensional spacetime with positive ADM mass but no horizon
aa r X i v : . [ h e p - t h ] M a y IMSc/2011/11/15
Static brane–like vacuum solutionsin D ≥ dimensional spacetimewith positive ADM mass but no horizon S. Kalyana Rama
Institute of Mathematical Sciences, C. I. T. Campus,Tharamani, CHENNAI 600 113, India.email: [email protected] describe static, brane–like, solutions to vacuum Einstein’sequations in D = n + m + 2 dimensional spacetime with m ≥ n ≥ r = 0 where r is the radial coordinate in the m + 1 dimensionalspace. The presence of n ≥ . Introduction We consider a D = n + m + 2 dimensional spacetime with m ≥ n dimensional space may be taken to be compact and toroidal, or to be R n .We assume a static, brane–like ansatz for the line element, hence the metriccomponents are independent of time and the n dimensional coordinates, anddepend only on the radial coordinate r of the m + 1 dimensional space.We study the solutions to vacuum Einstein’s equations R MN = 0 where R MN is the D dimensional Ricci tensor. The solutions are given by ds = − e a F dt + X i e a i F ( dx i ) + dr f + r d Ω m where i = 1 , , · · · , n and ( a , a i ) are constants obeying a + P i a i = . Thesolutions are required to be asymptotically Minkowskian with positive ADMmass. Einstein’s equations can be solved in a closed form for F ( f ) and r ( f ) which, however, is cumbersome to analyse. But it turns out that thequalitative properties of the evolution of f ( r ) and e F ( r ) can be understoodwithout this closed form. We find the following results.For K = a + P i a i − = 0 , we get e F = f = 1 − M ∞ r m − where M ∞ > n − brane solutionthen follows for 2 a − a i = 0 ; for other values of ( a , a i ) satisfying K = 0 , one obtains solutions studied e.g. in [1].For K > r decreases from ∞ to 0 : • f ( r ) decreases from 1 , reaches a minimum f > r ,increases to 1 again at some r = r < r , and then increases to ∞ inthe limit r → f ( r ) always remains positive and nonzero. • e F ( r ) decreases monotonically from 1 to 0 . • Define a ‘mass’ function M ( r ) = r m − (1 − f ) . As r decreases from ∞ to 0 , M ( r ) decreases from M ∞ but remains positive for r < r ,vanishes at r , becomes negative for r < r , and decreases to −∞ inthe limit r → M ( r ) from M ∞ to −∞ is monotonic.2hese features are in contrast to the standard Schwarzschild or black n − brane solution where, as r decreases from ∞ to 0 , M ( r ) = M ∞ remains constant and e F = f = 1 − M ∞ r m − decreases from 1 , vanishesat some r h , and decreases to −∞ in the limit r → f and e F and, hence, all metric components remain non zeroand finite for 0 < r ≤ ∞ . This implies that there is no horizon and thatthe curvature invariants are all finite for 0 < r ≤ ∞ . As r → f → ∞ and e F → n − dimensional space is crucial for theseproperties of the solutions. The absence of the n − dimensional space, orthe trivialty of its metric, means that a i = K = 0 which leads to thestandard Schwarzschild, or black n − brane, solution. We assume that a i =0 generically and, further, that K >
K > , we present the equationsand write them in a convenient form. In section , we analyse the equationsand describe the evolution of f , e F , and M . In section , we discuss thephysical relevance of the K > .
2. Einstein’s equations in vacuum
Consider a D = n + m + 2 dimensional spacetime with m ≥ ds given by ds = − e ψ dt + X i e λ i ( dx i ) + e λ dr + e σ d Ω m (1)where d Ω m is the standard line element on an m dimensional unit sphere, i = 1 , , · · · , n , and ( ψ, λ i , λ, σ ) are all functions of r only. Such an ansatz is Solutions with negative masses in the interior and with no horizon occur in [2, 3] whichstudy the back reaction of Hawking radiation in four dimensional spacetime. Solutionswith a central singularity and with no horizon occur in [4, 5] which study static solutionswith incoming radiation matching the outgoing one; the solutions in [4] can be matchedonto negative mass Schwarzschild solutions. n dimensional space described by x i coordinates to be compactand toroidal but it may also be taken to be R n .Let Λ = ψ + mσ + P i λ i . The vacuum Einstein’s equations R MN = 0 ,where R MN is the D dimensional Ricci tensor, then giveΛ r − ( ψ r + m σ r + X i ( λ ir ) ) = m ( m − e λ − σ (2) ψ rr + (Λ r − λ r ) ψ r = 0 (3) λ irr + (Λ r − λ r ) λ ir = 0 (4) σ rr + (Λ r − λ r ) σ r = ( m − e λ − σ (5)where r − subscripts denote derivatives with respect to r . Equations (3) and(4) imply that ψ = a F , λ i = a i F (6)where a and a i are constants and the function F ( r ) is defined by F rr + (Λ r − λ r ) F r = 0 = ⇒ e Λ − λ F r = ( m − N , (7)with N an integration constant. Choose e σ = r so that r denotes thephysical size of the m sphere. ThenΛ = m ln r + A F , A = a + X i a i . (8)Further, replace λ ( r ) by an equivalent function f ( r ) and also define a ‘mass’function M ( r ) analogous to that in the study of stars, as follows: e − λ = f ( r ) = 1 − M ( r ) r m − . (9)The D dimensional line element ds is now given by ds = − e a F dt + X i e a i F ( dx i ) + dr f + r d Ω m . (10)Equations (2), (5), and (7) now give, after some rearrangements,2 A f ( rF r ) = ( m −
1) (1 − f ) + Km f ( rF r ) (11)2 A f ( rF r ) = 2( m −
1) (1 − f ) − rf r (12) e A F f ( rF r ) = ( m − N r m − (13)4here K = a + P i a i − A . From equations (11) and (12), we have Km f ( rF r ) = ( m −
1) (1 − f ) − rf r . (14)We set 2 A = 1 with no loss of generality since this just amounts to defining F by F = 2 ( ψ + P i λ i ) . Thus, we have A = a + X i a i = 12 , K = a + X i a i − . (15)Note that if a i do not all vanish then, generically, K = 0 . Now usingequation (12) for ( rF r ) in equations (14) and (13), we have Km (2( m −
1) (1 − f ) − rf r ) = f (( m −
1) (1 − f ) − rf r ) , (16)which is to be solved for f ( r ) , and e F = ( m − N fr m − (2( m −
1) (1 − f ) − rf r ) . (17)Thus, once f ( r ) is known, e F and the line element ds are completely de-termined. In order to obtain asymptotically Minkowskian solutions withpositive ADM mass, we require that, in the limit r → ∞ , e F → , f ( r ) → − M ∞ r m − , M ∞ = const > . (18)Note that the condition on e F implies that N = M ∞ , and that thecondition on f implies that rf r → ( m − − f ) irrespective of whether More generally, one may also consider σ = ln r + c F and e − λ − cF ) = f ( r ) .Then A , K , and the condition 2 A = 1 are replaced by A = a + mc + P i a i , K = a + mc + P i a i − A , and 2( A − c ) = 1 . Indeed, we have − n n +1) ≤ K ≤ ∞ which can be derived as follows. Let ~a =( a , a , · · · , a n ) and ~ = (1 , , · · · ,
1) be two ( n + 1) − component vectors. Then A = ~a · ~ , A = ( n +1) | ~a | cos θ , and K = | ~a | − A . The inequality follows since | ~a | = A ( n +1) cos θ ≥ A ( n +1) and A = . ∞ is positive or negative. Also, in the limit r → ∞ , it follows fromequations that (11) – (13) f = 1 − M ∞ r m − + ( m − Km (cid:18) M ∞ r m − (cid:19) + ( m − K m (cid:18) M ∞ r m − (cid:19) + · · · e F = 1 − M ∞ r m − + ( m − K m (cid:18) M ∞ r m − (cid:19) + · · · (19)where · · · denote terms of O (cid:16) r − m − (cid:17) . ADM mass is given, using theasymptotic form of e F , by M ADM = m ω m πG D M ∞ − m − m X i a i ! (20)where ω m is the volume of the m dimensional unit sphere and G D the D dimensional Newton’s constant. The last term can be ensured to be < P i a i ) to be sufficiently small. The expression for M ADM isobtained by using an effective m + 2 dimensional metric in Einstein frame,and also by using the formula given in [11] with a modification : the formulagiven there applies to the case where λ = · · · = λ n . The term equivalent to nλ there is replaced by P i λ i when λ i s are unequal.Equations (16) and (17) for f and e F may be written in a more convenientform. Define a new variable R and a constant b by R = r m − , b = 4( m − Km . (21)Then r ( ∗ ) r = ( m − R ( ∗ ) R , where R − subscripts denote derivatives withrespect to R , and equations (16) and (17) become b (2(1 − f ) − Rf R ) = 4 f (1 − f − Rf R ) (22)and e F = M ∞ fR (2(1 − f ) − Rf R ) . (23)Equation (18) now means that, in the limit R → ∞ , e F → , f ( R ) → − M ∞ R , M ∞ = const > . (24)6he quadratic equation (22) can be solved for Rf R . The resulting ex-pressions for Rf R and e F are given, after a little algebra, by Rf R = 2 (1 − f ) ( f − f ) f − f ± q α f ( f − f ) (25)and e F = M ∞ (cid:16) √ f − f ± √ α f (cid:17) α R (1 − f ) (26)where square roots are always to be taken with a postive sign and α = 11 + b , f = 1 − α = b b . (27)Among the ± signs in equation (25) for Rf R , and correspondingly in equa-tion (26) for e F , + sign is to be chosen in the limit R → ∞ so that, for any α > one has Rf R → − f in that limit. This is easily checked since f → f − f → α . This branch choice also gives Rf R = 1 − f in thelimit b → α → f → α → h by √ f − f = ǫ h √ α h where ǫ h = Sgn h ,equation (25) becomes dRR = dh h ± ǫ h √ − α + αh − h , (28)which can be integrated, and thus R ( h ) obtained, in a closed form. But thisclosed form involves ln and Sinh − terms; it is difficult to invert it toobtain h ( R ) ; and its analysis is cumbersome even in special limits. Hence,we work with equation (25) itself.
3. Analysis of solutions We have taken the solution to the quadratic equation ˜ ax + ˜ bx + ˜ c = 0 in the form x = − c ˜ b ± √ ˜ b − a ˜ c which is more convenient here. Note that the inequalities on K given in footnote and the definitions b = m − Km and α = b imply that − n ( m − m ( n +1) ≤ b ≤ ∞ and m ( n +1) n + m ≥ α ≥ = 0 case If there are no compact directions, i.e. if n = 0 , or if a i = 0 for all i then we have 2 a = 1 and K = 0 . But K = 0 for other choices of ( a , a i )also, see equation (15). If K = 0 then b = 0 and equation (22), togetherwith the boundary conditions (24), implies that1 − f − Rf R = M R = 0 = ⇒ M ( R ) = M ∞ (29)and, hence, f = 1 − M ∞ R . It then follows from equation (23) that e F = f .Schwarzschild or black n − brane solution follows when 2 a = 1 , and n = 0or a i = 0 , but there are solutions for other values of ( a , a i ) which satisfy2 A − K = 0 . Such solutions, including also the parameter c mentionedin footnote , have been used in [1] to generate, following the methods of[12] – [14], the multi parameter solutions studied in [15] – [19] in the contextof non BPS branes and tachyon condensation. K > case : b > , α < a i do not all vanish and that K = 0 ,hence b = 0 . Equation (22) implies that M R = 1 − f − Rf R = 0 and,hence, the mass function M ( R ) = R (1 − f ) is non trivial. We now studythe solutions f ( R ) to the equation Rf R = 2 (1 − f ) ( f − f ) f − f + q α f ( f − f ) . (30)We have chosen the positive square root branch, for reasons explained belowequation (27). Assuming that f ( R ) → − M ∞ R in the limit R → ∞ , with M ∞ > f ( R ) as R decreases from ∞ .For K < f ( R ) for all R , with M ∞ > M ∞ < K >
K > n ≥ K > a i such that a i do not all vanish but P i a i = 0. In this case, it follows that8 = , K = P i a i > M ADM > ds = − e F dt + X i e a i F ( dx i ) + dr f + r d Ω m , (31)see equations (15), (20), and (10). Now b > K > α < f = 1 − α > Rf R > f R > f < f < R decreases from ∞ , the function f ( R ) continuously decreases from 1 . Evolution of f ( R ) near R where f ( R ) = f Let f ( R ) = f > R = R . As R approaches R from above, i.e. as R → R , it follows from equation (30) that Rf R → s α − α q f − f → + . (32)Further, using Rf RR = ( Rf R ) R − f R , and after a little algebra, it followsthat, as R → R , Rf RR → R α − α > . (33)This implies that, as one goes from R > R to R < R , the derivative f R goes from positive to negative values, becoming zero and changing sign at R . Hence, the function f ( R ) decreases for R > R , reaches a minimum f > R , and then starts to increase for R < R .Now note that the expression inside the square root in equation (30) canbe written as αf ( f − f ) = ( f − f ) + (1 − α ) (1 − f ) ( f − f ) > ( f − f ) , (34)the last inequality being valid as long as f < f < R since 1 > f > ∼ f there. Therefore, one has to choose the negative squareroot branch for R < R in order to accomodate the change of sign of f R at R . Hence, for R < R , we have Rf R = 2 (1 − f ) ( f − f ) f − f − q α f ( f − f ) . (35)9ote that, as R → R − and f → f , the above equation implies that Rf R → − s α − α q f − f → − . (36)The evolution of f ( R ) near R is similar to that of a particle trajectory x ( t ) near a turning point. Let the particle velocity be ˙ x = − q E − V ( x )) ,in an obvious notation. V ( x ) = E near a turning point x and, as x → x , E − V ( x ) ∝ ( x − x ) generically. As x → x , the particlevelocity ˙ x approaches zero. But its acceleration ¨ x remains finite, non zero,and positive. Hence ˙ x changes sign at x and becomes ˙ x = + q E − V ( x )) ,and the trajectory x ( t ) reverses its path. Evolution of f ( R ) near R where R < R and f ( R ) = 1As R decreases below R , f increases above f since f R < R < R .Let f ( R ) = 1 and R < R . Consider the limit where R → R and g = 1 − f → f − f = α − g and q α f ( f − f ) = α − (1 + α ) g O ( g ) , it follows from equation (35) that, in the limit g → Rf R = − α − α + O ( g ) < . (37)Note that the above expression is valid for both signs of g in the limit g → f < f > f → f R ( R ) remains negative and non zero which implies that as R approaches R and decreases further, the function f approaches 1 and increases further.In equation (35) for Rf R , the numerator is positive for f < f > f < f > αf ( f − f ) < ( f − f ) for f > Rf R is negative for both f < f > g in the limit g → Rf R < f > R < R . Evolution of f ( R ) in the limit f → ∞ R decreases below R , f increases above 1 . Consider the limit f ≫ Rf R ≃ − f − √ α = ⇒ f ≃ ( const ) (cid:18) M ∞ R (cid:19) −√ α (38)and, hence, that f → ∞ as R → f ( R ) for b > R decreases from ∞ to 0 , f ( R ) decreases from 1 , reaches a minimum f =1 − α > R = R , increases to 1 again at R = R < R , and thenincreases to ∞ as f ∼ R − −√ α in the limit R → Evolution of e F The evolution of e F can be easily read off from equations (23) and (26),which we reproduce below: e F = M ∞ fR (2(1 − f ) − Rf R ) f or < R < ∞ (39)= M ∞ (cid:16) √ f − f + √ α f (cid:17) α R (1 − f ) f or R < R < ∞ (40)= M ∞ (cid:16) √ f − f − √ α f (cid:17) α R (1 − f ) f or < R < R . (41)The behaviour of e F in the limit R → ∞ is given by equation (19). It canbe checked that e F remains non zero and finite for 0 < R < ∞ , in particularat R and R ; and that, in the limit R → f ≫ e F ∼ M ∞ R f ∼ (cid:18) M ∞ R (cid:19) − √ α −√ α → . (42)It can be shown that F R = 0 for R < ∞ . If F R = 0 then it follows fromequation (12), and then from equation (22), that2(1 − f ) − Rf R = 0 = 1 − f − Rf R = ⇒ − f = Rf R = 0 . R = ∞ . From the evolution of f ( R ) , we have f R = 0but 1 − f = 1 − f = α = 0 at R = R , and 1 − f = 0 but Rf R = 0 at R = R . Thus, besides at R = ∞ , we see from the evolution of f ( R ) that1 − f and Rf R do not both vanish and, hence, that F R cannot vanish. Theasymptotic behaviour of e F given in equations (19) and (42) in the limits R → ∞ and R → e F decreases monotonically from 1to 0 as R decreases from ∞ to 0 .It can further be shown that e F always remains < f . Note from equations(19) that e F < f in the limit R → ∞ . It also follows, using equations (12)and (30), that f ( RF R − Rf R ) > F R > f R , for R < R < ∞ where f < f < e F < f for R < R < ∞ . Since e F continues to decrease and f increases above f for R < R , it follows that e F < f for all R . Evolution of the mass function M ( R )The mass function is defined by M ( R ) = R (1 − f ) . Since M R =1 − f − Rf R , it follows from equation (22) that M R = 0 and M ( R ) isconstant if and only if b = 0 . For b > f ( R ) that M ( R ) is a positive constant = M ∞ at R = ∞ , remains positivefor R < R < ∞ , vanishes at R = R , becomes negative for R < R , and,in the limit R → f ≫ M ( R ) ∼ − Rf ∼ − R − √ α −√ α → − ∞ . If b > F R , that M R = 1 − f − Rf R cannot vanish for R < ∞ . Its asymptotic behaviourdescribed above then implies that M ( R ) decreases monotonically from apositive constant M ∞ to − ∞ as R decreases from ∞ to 0 . We pointout here that solutions with negative masses in the interior also occur in[2, 3] which study the back reaction of Hawking radiation in four dimensionalspacetime; and in [4] which study static solutions with incoming radiationmatching the outgoing one if such solutions are matched onto negative massSchwarzschild ones. Summary of the solutions
12n summary, we have the D = n + m + 2 dimensional metric components,with m ≥ − g tt = e a F , g ii = e a i F , g rr = 1 f which are all functions of R = r m − , with r denoting the physical size of the m sphere. The solutions are all required to have positive ADM mass and theasymptotic behaviour given in equation (18) in the limit r → ∞ . We alsohave A = a + X i a i = 12 , K = a + X i a i − b = m − Km and α = b . The standard Schwarzschildsolution follows for 2 a − a i = 0 . For other values of ( a , a i ) but with b = 0 , there exist more general solutions. In all these solutions, the metriccomponents vanish or diverge at a non zero, finite value of R = R h , which iseither a regular horizon or, possibly, a curvature singularity depending on thevalues of ( a , a i ) . In all these solutions, the mass function M ( R ) = R (1 − f )remains constant.We assume that b = 0 generically. Then M ( R ) is non trivial and cannotremain constant. Further assuming that b > f , e F , and M . Note that f and e F and, hence, all metriccomponents remain non zero and finite for 0 < R ≤ ∞ . This impliesthat there is no horizon, and that the curvature invariants are all finite, for0 < R ≤ ∞ .Consider the limit R → f ≫ e F → R abcd = e Ma e Nb e Pc e Qd R MNP Q in localtangent frame coordinates. For the D dimensional metric given by equation(1), the non vanishing components of R abcd are given by R ri ′ rj ′ = − δ i ′ j ′ e − λ (cid:16) λ i ′ rr + ( λ i ′ r − λ r ) λ i ′ r (cid:17) (43) R rarb = − h ab e − λ ( σ rr + ( σ r − λ r ) σ r ) (44) R i ′ j ′ k ′ l ′ = ( δ i ′ l ′ δ j ′ k ′ − δ i ′ k ′ δ j ′ l ′ ) e − λ (cid:16) λ i ′ r λ j ′ r (cid:17) (45) R i ′ aj ′ b = − δ i ′ j ′ h ab e − λ λ i ′ r σ r (46) R abcd = e − σ ρ abcd ( h ) + ( h ad h bc − h ac h bd ) e − λ σ r (47)13here i ′ = (0 , i ) , λ i ′ = ( ψ, λ i ) , h ab is the metric on the m dimensional unitsphere given by d Ω m = h ab dθ a θ b , and ρ abcd ( h ) is the corresponding Riemanntensor. It can now be seen that, in the limit R → R abcd ∼ fr → ∞ . (48)This implies that the tidal forces diverge and there is a curvature singularityin the limit R → n − dimensional space is crucialfor these properties of the solutions. The absence of the n − dimensionalspace, or the trivialty of its metric, means that a i = b = 0 , thus leadingto the standard Schwarzschild or black n − brane solution. We have assumedthat a i = 0 generically and, further, that b >
4. Physical relevance of the solutions
Physical relevance of the present solutions can be naturally motivatedand, indeed, such solutions may be naturally anticipated if one assumes thatMathur’s fuzzball proposal for black holes is correct. See [6] – [10] for a reviewof this proposal. Broadly speaking, according to this proposal, the blackhole entropy arises due to the microstates of M theory objects, equivalentlystring theory objects, which are typically bound states of intersecting braneconfigurations with a large number of low energy excitations living on them.For example, an effective four dimensional black hole may be described by a22 ′ ′ configuration which consists of two sets of M M O (1) times the Schwarzschild radius. At shorter dis-tances, the spacetime is different from that of black holes and, in particular,has no horizon.If this picture is correct then it should be possible to construct a star,modelling its M theory brane constituents by appropriate matter sources. Atlarge distances, it should appear as a spherically symmetric four dimensional(more generally, ( m + 2) dimensional) star; should have a finite radius, be14table, and have no horizon irrespective of how high its mass M ∗ is; and thethermodynamics of its constituents should give an entropy ∝ M ∗ .Technically, one constructs the interior of the star and, at its surface,matches the interior solution onto vacuum solutions. If the matching vac-uum solution is the standard Schwarzschild one then, for any choice of mattersources that the author can think of, it seems impossible to obtain a star solu-tion with the above properties. Also, such a matching seems to miss a crucialingredient : that, at a fundamental level, both the spacetime and the consti-tutents of the star are higher dimensional and this higher dimensionality islikely to play an important role.In [20, 21, 22], we had studied early universe using 22 ′ ′ intersectingbrane configuration. Starting with a eleven dimensional universe, we foundthat, at later times, the seven toroidal brane directions cease to expandor contract and stabilse to constant sizes; and, in the limit t → ∞ , thecorresponding metric components e λ i → e v i (1 + c ( t ) t δ ) where v i and δ > | c ( t ) | is finite. This results in an effectively four dimensionalexpanding universe. The tailing–off behaviour of e λ i suggests that, in thecontext of stars also, the internal directions are likely to have non trivial r dependence in the limit r → ∞ .This line of reasoning is what led us to study the higher dimensionalvacuum solutions, in particular to study the general solutions with non trivialdependence of e λ i . It turned out that such solutions exist indeed, withthe properties described in this paper. The early universe study mentionedabove also suggests that stars whose exterior solutions are similar to the onespresented here may form in a physical collapse, and that one has to carefullytake into account the higher dimensional nature of the constituents.
5. Conclusions
Finding the more general vacuum solutions is only a beginning. It is im-portant to actually construct both equilibrium and collapsing star solutions,study their stability, thermodynamic entropy, and other properties.Also, it will be interesting to generalise the present solutions to includerotation and charges. One may also start from the present solutions and,using the techniques of e.g. [12, 13, 14, 1], generate string and M theorybrane solutions. 15 cknowledgement:
We thank Steven G. Avery for discussions and forpointing out references [2, 3].
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