aa r X i v : . [ g r- q c ] O c t How is a Black Hole Created from Nothing?
Zhong Chao WuDept. of PhysicsZhejiang University of TechnologyHangzhou 310032, China
Abstract
Using the synchronous coordinates, the creation of a Schwarzschild black hole immersed in ade Sitter spacetime can be viewed as a coherent creation of a collection of timelike geodesics. Thepreviously supposed conical singularities do not exist at the horizons of the constrained instan-ton. Instead, the unavoidable irregularity is presented as a non-vanishing second fundamental formelsewhere at the quantum transition 3-surface. The same arguments can be applied to charged,topological or higher dimensional black hole cases.PACS number(s): 98.80.Hw; 04.50.+h; 04.62.+vKeywords: black hole creation, quantum black hole, quantum cosmology1he primordial black hole problem has been extensively studied in the big bang, inflationary andquantum cosmological models. It is usually thought that a black hole is formed from a massive objectthrough a gravitational collape scenario. However, the creation scenario of a black hole from nothing,i.e, from a seed instanton in quantum cosmology, is even much more dramatic. It is believed thatthere had existed an inflationary period in the very early universe, while spacetime was describedby a de Sitter metric. Therefore, we are particularly interested in studying the quantum creation ofa black hole with a de Sitter background.In quantum cosmology, the wave function of the universe is defined as the path integral [1]Ψ( h ij , φ ) = Z d [ g µν ] d [ φ ] exp( − I [ g µν , φ ]) , (1)where the 3-metric h ij and the matter fields φ on it are the arguments of the wave function, and g µν , φ are all closed Euclidean 4-metrics and the matter fields on them with the variables of thewave function as the only boundaries. I is the Euclidean action I = − π Z M ( R −
2Λ + L m ) − π I ∂M K, (2)where R is the scalar curvature of the 4-metrics M , K is the expansion rate of the boundary ∂M , L m is the Lagrangian of the matter fields φ , and Λ is the cosmological constant.The universe is created via a quantum transition from a seed instanton. It can be realized viaan analytic continuation at the equator of the instanton. The relative creation probability of theuniverse, at the W KB level, is evaluated as P ≈ exp( − I ) . (3)It can be derived from gravitational thermodynamics that the entropy of the system is equal tothe negative of the action I [2][3]. Therefore, the relative creation probability is the exponential ofthe entropy.The seed of a non-rotating black hole in the de Sitter spacetime background is a S x S in-stanton. For a regular instanton, the Lorentzian spacetime created is the Nariai spacetime, whichis interpreted as a pair of black holes with the creation probability of exp( − I ) = exp(2 π Λ − )[4]. In comparison, the creation probability of the de Sitter spacetime without a black hole isexp( − I ) = exp(3 π Λ − ) [1]. 2he Nariai black hole and the de Sitter spacetime are two cases of the de Sitter background withthe greatest black hole and no black hole, respectively. Now we are interested in black hole creationwith mass parameter m interpolating these two extrema [5].It is known that an instanton should be an Euclidean regular solution of the Einstein fieldequation, therefore it should have a stationary action. However, for the generic black hole with themass parameter m , the value of the Euclidean action for its Euclidean manifold, once made compact,depends on m . This means that the action is not stationary at least with respect to one degree offreedom, the parameter m itself. Therefore, the associated instanton must be irregular [6].The metric of the Euclidean solution is ds = △ ( r ) dτ + △ − ( r ) dr + r d Ω , (4)where d Ω is the metric of the unit 2-sphere, and △ ( r ) = 1 − mr − Λ r . (5)The black hole horizon and the cosmological horizon are located at zeroes r l and r k of therational expression △ . The surface gravity κ i of the horizon r i is | d △ ( r ) / dr | evaluated there.One can construct an instanton by identifying the τ coordinate with an arbitrary period β on thetwo dimensional spacetime ( τ, r ), and then obtain the instanton between these two horizons. If onechooses β = 2 πκ − i , then the conical singularity at the horizon i can be avoided. Of course, toregularize both conical singularities at the two horizons is impossible due to the reason mentionedabove. The impossibility is represented by the fact that the surface gravities for two horizons aredistinct. Even though, it can be shown as follows that the constructed manifold is of an actionstationary under the condition that the 3-geometry of the quantum transition equator is given. Theinstanton is of topology S x S , where one S represents d Ω , and the other S is the distorted spherein the spacetime ( τ, r ). Strictly speaking, the constrained instanton is the seed of the Schwarzschild-de Sitter spacetime identified into one periodic cell in its Penrose-Carter diagram [5].Now one can evaluate the action of the manifold by recasting it into the canonical form [3][7] I = I l + I k + Z M ′ (cid:16) π ij ˙ h ij − N H − N i H i (cid:17) d xdτ, (6)where M ′ is M minus M l and M k , here M i ( i = l, k ) denotes the small neighbourhood of horizon r i with a boundary of a constant coordinate r , and N and N i are the lapse function and shift vector,3 ij are the conjugate momenta of h ij , H and H i are the Einstein and momentum constraints,which vanish for all classical solutions of the Einstein equations, and dot denotes the derivativewith respect to the imaginary time τ , these associated terms vanish due to the U (1) Killing timesymmetry. Therefore, the action for M ′ equals zero.On the other hand, the action I i can be expressed explicitly as I i = − π Z M i ( R −
2Λ + L m ) − π I ∂M i K. (7)If there is a conical singularity at the horizon, its contribution to the action is reduced to thedegenerate version of the second term, in addition to that from the boundary of M i .The Gauss-Bonnet theorem can be applied to the 2-dimensional ( τ, r ) section of M i ,14 π Z ˆ M i ˆ R + 12 π I ∂ ˆ M i ˆ K + δ i π = χ ( i ) , (8)where the hat notation represents the projection of those objects or quantities onto the 2-dimensional( τ, r ) section, δ i is the deficit angle of horizon i , and χ ( i ) is the Euler characteristic of ˆ M i . Since theexpansion rate of the subspace r d Ω goes to zero at the horizons. K and ˆ K are equal. Comparingeqs (7) and (8), as M i shrinks to the horizon, the action (7) approaches − χ ( i ) A i /
4. where A i isthe surface area of the horizon. It is noted that both the first terms in these equations vanish aftershrinking.Therefore, the entropy or the negative of the total action of the constrained instanton is [3] S = − I = 14 ( χ ( l ) A l + χ ( k ) A k ) . (9)This is a quite universal formula. It is clear that the entropy of a black hole is originated from thetopology of the instanton.The action is independent of the period parameter β . Indeed, from the derivation using canonicalaction form, it is obvious that the action is even independent of the way of gluing the south part andnorth part of the instanton. In the ( τ, r ) section one can join them along two arbitrary continuouscurves connecting the two horizons with discontinuities of the second fundamental forms, or even ofthe first fundamental form. The discontinuity of the second form is a jump of the three expansion,and the consequence of the discontinuity of the first form is not clear. It turns out that the crucialpoint is the existence of the two horizons. 4rom the above argument, it follows that there exist ambiguities in determining the quantumtransition 3-surface, which are associated not only with arbitrariness of the period β . The questionarises as to what is the true scenario of black hole creation? The goal here is to clarify this.We are going to construct an alternative constrained instanton, which may be identified as thetrue instanton. The Schwarzschild-de Sitter black hole metric can be written in the synchronouscoordinates as the following [8]: ds = − dρ + 1cos µ (cid:18) ∂r ( ρ, µ ) ∂µ (cid:19) dµ + r ( ρ, µ ) d Ω . (10)It shows that the classical evolution of the black hole is equivalent to a coherent motion of acollection of timelike geodesics with proper time ρ , labeled by ( µ, θ, φ ), in a potential hill describedby △ , as shown by the following implicit transformation between the usual coordinates ( t, r ) and( ρ, µ ), ρ = Z rr dr [ E − △ ] / , ( E = cos µ ) , (11) t = Z rr Edr [ E − △ ] / △ , (12)where r is an arbitrary constant, which is associated with the gauge freedom of the synchronouscoordinate form of the metric. However, it will be specified for our discussion later.It is noted that although metric (10) appears to be time dependent with respect to ρ , it is timeindependent with respect to the Killing time t . The time ρ is used for the convenience of discussingthe geodesics only.Our alternative constrained instanton can be obtained by going to the regime where ( E −△ ) ≤ r l ≤ r ≤ r k ), so the coherent motion occurs in the potential well −△ in the imaginarysynchronous time and the spacetime is Euclidean. For future convenience, we shall use two complextime coordinates in the following discussion: t + iτ and ρ + iσ , so that the left hand sides of (11)and (12) should be replaced by ρ + iσ and t + iτ respectively. The quantum transition would occurwhen E = △ . Here the time derivative of r vanishes, no matter which time is referred: t, τ, ρ or σ . When ( E − △ ) ≥ µ of the geodesic is determined by the r b ( µ ) value at the moment it issubject to the quantum transition, which is cos µ = △ ( r b ). For example, considering the geodesicspassing through the horizons, one has cos µ = 0, or µ = π/ r c = 3 / m / Λ − / , (13)and we define cos µ c ≡ (1 − / m / Λ / ) / , (cid:0) π > µ c > (cid:1) and for the S equator of the quantumtransition ( r l → r c → r k → r c → r l ) of r values correspond to ( π/ → µ c → π/ → µ c → π/
2) of µ values.We consider the bottom of the potential well r = r c as the south pole of the ( τ, r ) space, andspecify r = r c for equations (11) and (12). The total expansion rate around the south pole is I ∂g / µµ ∂σ dµ = 4 Z π/ µ c µ ∂ r∂σ∂µ dµ = 4 Z π/ µ c µ ∂ r∂µ∂σ dµ = 4 Z π/ µ c µ ∂ [ △ − E ] / ∂µ dµ = 4 Z π/ µ c [cos µ c − cos µ ] − / sin µdµ = 2 π. (14)This result implies that the south pole is regular, therefore by setting r = r c , one obtains anseed constrained instanton. If one chooses r = r c as the south pole, then some irregularity mustoccur there, so (14) is not valid, this means that one does not get a seed instanton. Once we set r = r c , it follows from (11) and (12) that t = τ = ρ = σ = 0 at the south pole.A geodesic with given parameter µ can be thought of as a particle moving in the potential well,with σ referred to as the Newtonian time. The time it takes from the bottom r c to the highest point r b ( µ ) in the imaginary time is σ b ( µ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z r b ( µ ) r c dr [ △ − cos µ ] / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (15)and τ b ( µ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z r b ( µ ) r c cos µdr [ △ − cos µ ] / △ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (16)Since the well does not take the form of that of a linear oscillator, the time spent to reach thehighest point depends on the parameter µ . This fact implies that the quantum transitions do notoccur simultaneously in the imaginary synchronous time σ . The same argument is valid in theimaginary time τ . One can consider the role of △ in (16) as a redshift effect of the gravitationalfield. In general, the second fundamental form at the boundary of the south part of the instantondoes not vanish, except for those which happen to be located at the two horizons. In contrast, inthe earlier literature, the irregularities are supposed to be concentrated as conical singularities atthe horizons [5]. 6n the Lorentzian regime, on the other hand, the creation of all geodesics does occur simulta-neously at t = 0 in the real time. At the horizons τ b is identified as one quarter of the inversetemperature there. Its significance elsewhere at the quantum transition 3-surface is not clear. Howthe fact that the quantum transitions for all geodesics do not occur simultaneously in the imaginarytime τ is related to any observable effect is not known, at least for the time being. The above dis-cussion suggests that describing a spacetime by complex coordinates may reveal richer phenomenain gravitational physics.To recast the de Sitter spacetime creation into the synchronous coordinates is quite instructive.The metric is ds = − dρ + 3Λ cosh r Λ3 ρ ! dµ + 3Λ cosh r Λ3 ρ ! sin µd Ω . (17)And the values of both (15) and (16) are √ π/ √ Λ, which are independent of parameter µ , sincethe potential −△ = − Λ r is of a linear oscillator form, whose period is independent of theamplitude. The instanton is formed by joining two standard hemispheres of S , as has been knownfor a long time [1]. The fact that integration value of (16) is independent of µ is well expected inphysics, while quite surprising in mathematics, given the fact that the integrand in (16) is a rathercomplex function of E or µ .Our discussion can be applied to any d -dimensional solution of the Einstein equation with metricform (4), in which △ ( r ) becomes a more general function with two zeroes as black hole and cosmo-logical horizons, and Ω is replaced by a d − d − .For 4-dimensional topological black hole cases, the metrics take form (4), with △ and Ω replacedby △ = p − mr − Λ r p = 1 , d Ω = dθ + sin θdφ ,p = 0 , d Ω = dθ + θ dφ ,p = − , d Ω = dθ + sinh θdφ , (19)where the 2-spaces Ω are made compact topologically for flat ( p = 0) and hyperbolic ( p = − △ of metrics (4) is replaced by △ = p − mr + Q r − Λ r , (20)where Q is the electromagnetic charge. If the charge is electric, a Legendre transform for the actionof the instanton is required to recover the duality between the magnetic and electric cases. It isnoted that in the canonical form of the action, there is a Gauss constraint term, in addition to themomentum and Einstein constraints [3][9].To generalize our argument to higher dimensional black holes with metric form (4) is straight-forward, where Ω is replaced by Ω d − and △ = p − ηr d − − r l , (21)where d is the dimensionality, η is proportional to the mass and l is the radius of curvature of thede Sitter background [10].In fact, our arguments can also be applied to the BT Z or Lovelock black hole cases.
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