Mass-gap for black hole formation in higher derivative and ghost free gravity
aa r X i v : . [ h e p - t h ] M a y Mass-gap for black hole formation in higher derivative and ghost free gravity
Valeri P. Frolov Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1 ∗ We study a spherical gravitational collapse of a small mass in higher derivative and ghost freetheories of gravity. By boosting a solution of linearized equations for a static point mass in suchtheories we obtain in the Penrose limit the gravitational field of an ultra-relativistic particle. Takinga superposition of such solutions we construct a metric of a collapsing null shell in the linearizedhigher derivative and ghost free gravity. The latter allows one to find the gravitational field of athick null shell. By analysing these solutions we demonstrate that in a wide class of the higherdimensional theories of gravity as well as for the ghost free gravity there exists a mass gap for themini black hole production. We also found conditions when the curvature invariants remain finiteat r = 0 for the collapse of the thick null shell. PACS numbers: 04.70.s, 04.70.Bw, 04.20.Jb Alberta-Thy-7-15
It is generally believed that the theory of general rela-tivity (GR) should be modified to improve its ultraviolet(UV) behavior and remove singularities. One of the op-tions is to allow terms in the gravitational action thatcontain more than two derivations. The UV propertiesof the higher derivative theory of gravity are usually bet-ter than in GR. In particular, the forth order gravitycan be made renormalizable [1]. At the same time, thegravitational potential of a point mass in the Newtonianlimit of such theories is usually finite (see e.g. [2, 3] andreferences therein). However the higher derivative grav-ity possesses new unphysical degrees of freedom (ghosts)[1, 2]. The problem of ghosts can be solved if one allowsan infinite number of derivatives in the gravity action,that makes it non-local. Ghost-free theories of gravityare discussed in [4–7]. Their application to the problemof singularities in cosmology and black holes can be foundin [8]In this paper we study gravitational collapse of a smallmass in higher derivative (HD) and host-free (GF) the-ories of gravity. We obtain solutions of the linearizedequations for such theories for a spherical collapse of nullfluid. We demonstrate that if a static gravitational fieldof a point mass in the HD and GF gravity is regular at r = 0 [3], then the metric for the collapsing object hasthe same property. This means, that the perturbation ofthe metric, which is proportional to the collapsing mass M , is smooth and uniformly bounded, so that the higherin M corrections can be neglected in the leading order.This implies that for the collapse of a small mass an ap-parent horizon is not formed. In other words, for thiswide class of HD and GF theories of gravity there existsa mass gap for mini black hole production. This propertyis a consequence of the existence of the UV length scale,where such theories become different from GR. For theWeyl modified gravity this was shown long time ago in[9].We study the linearized gravity equations on the flatMinkowski background η µν and write the metric in the ∗ [email protected] form g µν = η µν + h µν . The most general action for thehigher derivative theory of gravity which contains nothigher than second power of h µν is [5, 6] S = − Z d x (cid:20) h µν a ( (cid:3) ) (cid:3) h µν + h σµ b ( (cid:3) ) ∂ σ ∂ ν h µν + hc ( (cid:3) ) ∂ µ ∂ ν h µν + 12 hd ( (cid:3) ) (cid:3) h + h λσ f ( (cid:3) ) (cid:3) ∂ σ ∂ λ ∂ µ ∂ ν h µν (cid:21) . (1)Five, in general non-linear functions of the box-operatorobey the following three relations a + b = 0 , c + d = 0 , b + c + f = 0 . (2)Thus the action S contains in fact only two independentarbitrary functions of the box operator. In order to re-cover GR in the infra-red domain these functions mustsatisfy the following conditions a (0) = c (0) = − b (0) = − d (0) = 1. Let us list some special interesting examples[6]:1. General relativity (GR), L = R : a = c = 1;2. Gauss-Bonnet (GB) gravity, L = R + α ( (cid:3) ) G , where G = R − R µν R µν + R µναβ R µναβ is the Gauss-Bonnet invariant: a = c = 1;3. L ( R ) gravity, L ( R ) = L (0)+ L ′ (0) R +1 / L ′′ (0) R + . . . : a = 1, c = 1 − L ′′ ( (cid:3) );4. Weyl gravity, L = R − µ − C µναβ C µναβ : a = 1 − µ − (cid:3) , c = 1 − µ − (cid:3) ;5. Higher derivative (HD) gravity: a = Q ni =1 (1 − µ − i (cid:3) ), c = Q n c k =1 (1 − ν − k (cid:3) ). For simplicity, inwhat follows we assume that masses µ i are differ-ent;6. Ghost free gravity: a = c = exp( − (cid:3) /µ ).It is evident that the linearized GB gravity has the sameproperties as GR. In the same linearized approximationthe Weyl and L ( R ) theories of gravity are nothing butspecial cases of the general HD gravity.Let us consider first static solutions of the linearizedgravity equations. In the Newtonian limit the stress-energy tensor is τ µν = ρ ( ~r ) δ µ δ ν , and the metric is of theform ds = − (1 + 2 ϕ ) dt + (1 − ψ + 2 ϕ ) dℓ . (3)The functions ϕ and ψ obey the equations [5] a ( △ ) △ ψ = 8 πGρ , (4)( a ( △ ) − c ( △ ))( △ ϕ − △ ψ ) = 8 πGρ . (5)Here △ is a usual flat Laplace operator in a flat 3D spacewith metric dℓ , and G is the gravitational coupling con-stant. After solving (4) and finding the potential ψ , onecan find the second potential ϕ by solving (5).For a point mass ρ = mδ ( ~r ) the solution (3) is spher-ically symmetric. We call it finite if ϕ ( r ) and ψ ( r ) near r = 0 have the form ψ ( r ) ∼ ψ + ψ r + 12 ψ r + O ( r ) , (6) ϕ ( r ) ∼ ϕ + ϕ r + 12 ϕ r + O ( r ) . (7)A finite solution is not necessary regular one. Really, theKretschmann tensor invariant R = R αβγδ R αβγδ for themetric (3), (6) and (7) is the form R = A r + A r + O (1) , (8) A = 8(4 ψ − ψ ϕ + 3 ϕ ) ,A = 16[ ψ (5 ψ − ϕ ) − ϕ ( ψ − ϕ )] . The quantity A is a positive definite quadratic form ofvariables ψ and ϕ , and it vanishes only when ψ = ϕ = 0. In such a case the quantity A vanishes as well,so that R is finite at r = 0. We call such a solutionregular. We also call a solution ψ -regular, if ψ = 0. Fora special class of theories, where a = c , one has ψ = 2 ϕ and a solution which is ψ -regular is at the same time aregular one.We denote ˆ O = a ( △ ) △ , Q ( ξ ) = ˆ O − ( △ = − ξ ) = − [ ξa ( − ξ )] − . (9)and assume that Q ( ξ ) can be written as the Laplacetransform of some function f ( s ) Q ( ξ ) = Z ∞ dsf ( s ) e − sξ , (10) f ( s ) = 12 πi Z α + i ∞ α − i ∞ dξQ ( ξ ) e sξ . (11) Notice that the first of the equations (17) of this paper contains amisprint and it should be written as 2( a − c )[ ∇ Φ − ∇ Ψ] = κρ . The second relation is nothing but the inverse Laplacetransform. A parameter α must be chosen so that theintegration path in (11) lies in the domain of the analyt-icity of Q ( ξ ).A formal solution of the operator equation ˆ O ˆ G = − ˆ I can be written by using the Laplace transform (10).It contains the exponent exp( s △ ), which in the x -representation is nothing but the heat kernel h x ′ | e s △ | x i = K ( | x − x ′ | ; s ) = e −| x − x ′ | / (4 s ) (4 πs ) / . (12)Thus the potential ψ ( r ) for a point mass is ψ ( r ) = 8 πGm Z ∞ dsf ( s ) K ( r ; s ) (13)= Gmπir Z α + i ∞ α − i ∞ dξQ ( ξ ) e −√− ξr . (14)We consider at first a case of HD gravity. We assumethat the function Q ( ξ ) has simple poles and write it inthe form Q ( ξ ) = − [ ξ n Y i =1 (1 + ξ/µ i )] − . (15)This covers the above listed gravitational theories 1-5except for some degenerate cases.The Heaviside expansion theorem [10] gives the follow-ing expression for f ( s ) f ( s ) = − (1 − n X i =1 P − i e − µ i t ) , (16)where P i = Q nj =1 ,j = i (1 − µ j /µ i ). Taking the integral(13) one obtains ψ ( r ) = − Gmr − (1 − n X i =1 P − i e − µ i r ) , (17)For GR f ( s ) = 1 and one has ψ ( r ) = 2 ϕ ( r ) = − Gm/r . (18)For a theory with higher derivatives, where n ≥
1, thepotential ψ ( r ) near r = 0 has a form (6) with ψ = − GmS , ψ = GmS , S k = n X i =1 µ ki P − i . (19)We used here a relation S = 1. Thus a theory withhigher derivatives is ψ -regular, when the condition S = 0is satisfied.For the GF gravity f ( s ) = − ϑ ( s − µ − ) and one repro-duces the result of [5] ψ ( r ) = 2 ϕ ( r ) = − Gm erf( µr/ /r . (20)This solution is regular at r = 0.We demonstrate now, how using a solution of (4) for astatic point mass one can obtain a solution for an ultra-relativistic particle. Let us write the flat space metric inthe form dℓ = dy + dζ ⊥ , and suppose that the source,generating the gravitational field (3), moves along the y -axis with a constant velocity β . To find the gravitationalfield of the moving source we make the following boosttransformation t = λ − v + λ + u , y = λ − v − λ + u . (21)Here λ ± = (1 ± β ) γ/ γ = (1 − β ) − / . In the limit γ → ∞ one gets y ∼ − γu , t ∼ γu , ℓ ∼ γ u + ζ ⊥ , and ds = − dudv + dζ ⊥ + dh ,dh = Φ du , Φ = − γ →∞ ( γ ψ ) . (22)We assume that the energy of the particle, M = γm remains constant in this (Penrose) limit. We use also thefollowing relationlim γ →∞ γ exp( − γ u / (4 s )) = √ πsδ ( u ) . (23)These relations and (13) giveΦ = − GM F ( ζ ⊥ ) δ ( u ) , (24) F ( z ) = Z ∞ dss f ( s ) e − z/ (4 s ) . (25)For GR (as well as for GB and L ( R ) gravity) one has f ( s ) = 1 and F ( z ) = ln( z/η ), where η is an infrared cut-off parameter. The relations (22) and (24) correctly re-produce the well known Aichelburg-Sexl solution for thegravitational field of an ultra-relativistic particle (”pho-ton”) in GR.Using expression (16) for f ( s ) for the HD gravity andtaking integral in (25) one finds F ( z ) = ln( z/η ) + 2 n X i =1 P − i K ( µ i √ z ) . (26)In the limit µ i → ∞ the second term in the right-handside vanishes and one obtains the correct expression forGR. In the presence of the higher derivatives the leadingterm of the function F ( z ) at small z is F ( z ) ∼ C − S z (ln z − c ) − Sz + O ( z ) , (27)where c = 1 + ln 2 − γ and S = P ni =1 µ i ln( µ i ) P − i . Forthe ghost-free gravity one has [11] F ( z ) = ln z + γ + Ei(1 , z ) ∼ z − z + O ( z ) . (28)The obtained metric (22), (25) can be used to find asolution for the linearized HD and GF gravity equationsfor a collapsing spherical thin null shell. For this pur-pose one considers a set of ”photons”, passing through a fixed point P of the Minkowski spacetime. In the contin-uous limit this set fills the surface of the null cone, withthe vertex at P . We additionally assume that the den-sity of this spherical distribution of the ”photons” is uni-form and the corresponding mass per a unit solid angle is M/ π . Since we are working in the linear approximationthe resulting gravitational field for such a distribution is ds = ds + h dh i , where h dh i is obtained by averagingof a single ”photon” metric over their spherical distribu-tion. The calculations give [11] ds = − dt + dr + r dω + h dh i , z = r − t , h dh i = − GM r − F ( z )[( dt − tdr/r ) + zdω / . (29)Let us denote g = ( ∇ ρ ) , ρ ≡ g θθ = r − GMr zF ( z ) , (30)then the equation g = 0 determines a position of theapparent horizon, if the latter exists. In the linear in M approximation this function is g = 1 − GM r − q ( z ) , q ( z ) = zF ′ ( z ) , (31)where ( . . . ) ′ = d ( . . . ) /dz . For GR (as well as for GB and L ( R ) gravity) q ( z ) = 1.Using (26) one finds that for the HD gravity q ( z ) = 1 − √ z n X i =1 µ i P − i K ( µ i √ z ) . (32)For small z one has q ( z ) = − S z (ln z − c + 1) − Sz + O ( z ) . (33)Let us demonstrate now that the function g is positivefor small enough M , and, hence, the apparent horizondoes not exist. Let us notice that outside the null shell | t | /r <
1. We denote t = ± p − β , 0 ≤ β ≤
1, thenone has ( y i = βµ i r ) q ( z ) /r = β n X i =1 µ i P − i Z ( y i ) , Z ( y ) = 1 y − K ( y ) . (34)The function Z ( y ) is positive and takes maximal value0 .
399 at y = 1 . | q ( z ) | /r < . n X i =1 µ i | P i | − . (35)This implies that for small enough value of the mass M the invariant g is positive everywhere outside the shell.In other words, for such mass M the collapse of the nullshell does not produce a mini black hole. This meansthat for the class of the higher derivative theory of gravity(15) with n ≥ µ − of the theory. Theapparent horizon does not exist if M µ .
1. The sameconclusion is valid for the GF theory of gravity [11].It is possible to calculate the curvature invariants forthe metric (29). In particular, the Kretschmann curva-ture invariant R in the lowest order in M is R = 48 G M r F , F = 2 z q ′ − zqq ′ + q . (36)Using expression (33) for small z one finds F ∼ z [( w + 4 w + 5) S + 2( w + 2) SS + S ] , (37)where w = ln z − c . This means that Kretschmann cur-vature vanishes on the null shells. However, in a generalcase it is divergent at r = 0.We demonstrate now that this divergence is a resultof the (unphysical) assumption, that the thickness of thenull shell is zero, and for the collapse of the shell withthe finite thickness this divergence becomes softer or isabsent. To obtain a solution of the HD gravity equationsfor such a thick shell we proceed as follows [11]. Considera set of spherical null shells collapsing to the same spatialpoint r = 0, but passing it at different moments of time t .In the continuous limit, one obtains a distribution of thematter, that describes a spherical thick null shell whichinitially collapses and has a mass profile M ( t + r ), andafter passing through the center it re-collapses with themass profile M ( t − r ). In the linear in M approximationthe gravitational field of such a shell can be obtained byaveraging the solutions (29). We denote by hh dh ii theresult of the averaging of the perturbation h dh i . Forsimplicity we present here the expression for hh dh ii forthe case when ˙ M is constant, and the time duration ofthe thick shell is b , so that the total mass M of the shellis ˙ M b . In the domain of the intersection of the incomingand outgoing null fluid fluxes the metric is static. Thecalculations give (see [11] for more details) hh dh ii = − GMbr [ c dt + c dr r + 12 ( c r − c ) dω ] , (38) where c k = R r − r dxx k F ( r − x ). It is easy to check thatconstant C , which enters (27) does not contribute to thecurvature. For this reason we put C = 0. Using expan-sion (27) of F ( z ) one obtains c = − r u − S + 3 S ] ,c = − r
225 [(30 u − S + 15 S ] , (39)where u = ln r − c − ln 2. For small M the function g remains positive, while the Kretschmann invariant is R ∼ G ˙ M [(36 u + 5) S + 36 uS S + 9 S ] . (40)Hence, a collapse of a thick null shell in the theory withhigher derivatives results in the logarithmic singularityof the curvature. However, if such a theory is ψ -regular,the curvature is finite. In particular, this property is validfor any regular theory with higher derivatives. For theghost-free theory of gravity the Kretschmann invariant R ∼ G ˙ M µ is always finite at r = 0 [11].Let us summarize the results. Our main conclusionis that in the higher dimensional and ghost free theo-ries of gravity there exist a mass gap for a mini blackhole production. Such theories possess a characteristiclength scale µ − and the apparent horizon is not formedwhen the gravitational radius of the mass M , 2 GM , issmaller that µ − . For a regular (and ψ -regular) theorythe Kretschmann curvature invariant for the thick nullshell collapse is regular. Whether in the strong gravitylimit this property remains valid, and the black hole inte-rior does not contain singularities, is an interesting openquestion.The author thanks the Killam Trust and the NaturalSciences and Engineering Research Council of Canada fortheir financial support. [1] K. S. Stelle, Phys.Rev. D
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