Matter Conditions for Regular Black Holes in f(T) Gravity
aa r X i v : . [ g r- q c ] N ov MATTER CONDITIONS FOR REGULAR BLACK HOLES IN f ( T ) GRAVITY
Joshua Aftergood ∗ Department of Physics, Simon Fraser University,Burnaby, British Columbia, Canada, V5A 1S6
Andrew DeBenedictis † Department of Physics and
The Pacific Institute for the Mathematical Sciences,Simon Fraser University, Burnaby, British Columbia, Canada, V5A 1S6
ABSTRACT
We study the conditions imposed on matter to produce a regular (non-singular) interior ofa class of spherically symmetric black holes in the f ( T ) extension of teleparallel gravity.The class of black holes studied (T-spheres) is necessarily singular in general relativity. Wederive a tetrad which is compatible with the black hole interior and utilize this tetrad in thegravitational equations of motion to study the black hole interior. It is shown that in the casewhere the gravitational Lagrangian is expandable in a power series f ( T ) = T + P n =1 b n T n that black holes can be non-singular while respecting certain energy conditions in the matterfields. Thus the black hole singularity may be removed and the gravitational equations ofmotion can remain valid throughout the manifold. This is true as long as n is positive, butis not true in the negative sector of the theory. Hence, gravitational f ( T ) Lagrangians whichare Taylor expandable in powers of T may yield regular black holes of this type. Althoughit is found that these black holes can be rendered non-singular in f ( T ) theory, we conjecturethat a mild singularity theorem holds in that the dominant energy condition is violated in anarbitrarily small neighborhood of the general relativity singular point if the corresponding f ( T ) black hole is regular. The analytic techniques here can also be applied to gravitationalLagrangians which are not Laurent or Taylor expandable. PACS numbers:
Key words: black hole singularities, teleparallel gravity, torsion
There has been extensive study in the literature of the peculiar and not so peculiar propertiesof black holes over the years since the advent of general relativity. The Schwarzschild metric ∗ [email protected] † [email protected] EGULAR BLACK HOLES IN f ( T ) GRAVITY f ( T ) = T , [13]) and also retains second-order equations of motion [14]. Inthis theory the fundamental object is the tetrad, h aµ , and the action is given by S = 116 π Z { f ( T ) + L m } det [ h aµ ] d x . (1)Here L m represents the matter Lagrangian density. The indices employed in this paper havethe following convention: Unadorned Greek indices are spacetime indices, whereas Latinindices span the local tangent spacetime of the gravitational degrees of freedom. In whatfollows we also occasionally project quantities into a local orthonormal frame which is not related to the tetrad of the gravitational degrees of freedom . The indices representing thelocal orthonormal frame are hatted Greek indices. The quantity T is the torsion scalar whichis defined via a linear combination of quadratic contractions of the torsion tensor, T ρµν : T = 14 T ρµν T ρµν + 12 T ρµν T νµρ − T ρρµ T νµν . (2)The torsion tensor itself is defined via the commutator of the curvature-less Weitzenb¨ockconnection, Γ λµν : T λµν = Γ λνµ − Γ λµν = h λa (cid:0) ∂ µ h aν − ∂ ν h aµ (cid:1) . (3)The torsion scalar (2) differs from the Ricci scalar by a total divergence, and hence GeneralRelativity is recovered in the limit that f ( T ) → T . For this reason we will always include Although one can choose the local orthonormal frame to coincide with the tetrad of the gravitational field’sdegrees of freedom, it is generally not convenient to do so.
EGULAR BLACK HOLES IN f ( T ) GRAVITY
3a term linear in T in the subsequent analysis and specifically will consider functions of theform f ( T ) = T + X n =1 b n T n , (4)with the b n constants.Variation of the action (1) with respect to the tetrad yields the gravitational equations ofmotion, h h − h aµ ∂ ρ (cid:0) h h λa S νρλ (cid:1) + T αλµ S νλα i df ( T ) dT + S νλµ ∂ λ T (cid:16) d f ( T ) dT (cid:17) + δ νµ f ( T ) = 4 π T νµ , (5)where h is the determinant of the tetrad and S νρλ is the modified torsion tensor: S νρλ = 12 (cid:0) K νρλ + δ νλ T σρσ − δ ρλ T σνσ (cid:1) , (6)and K νρλ the contorsion tensor: K νρλ = 12 (cid:0) T ρνλ + T νρλ − T νρλ (cid:1) . (7) T νµ is the usual stress-energy tensor.One issue present in teleparallel gravity which is not present in curvature theories wherethe metric is the fundamental gravitational object, such as f ( R ) theories, is that the action(1) is not locally Lorentz invariant. That is, two different tetrads, related to each other via alocal Lorentz transformation, will yield physically distinct equations of motion. Therefore,a certain metric (which still defines the causal structure of the theory) is compatible withmany tetrads all of which yield inequivalent relations between the matter content and thegravitational field via the field equations. Out of this freedom one must find a tetrad whichyields “good” equations of motion. Reasonable criteria for a good tetrad are: • The tetrad chosen should not restrict the form of f ( T ) [15]. That is, the tetrad needs toretain acceptable equations of motion regardless of the function f ( T ) and not just workwell for certain functions. Our minimal conditions for acceptable equations of motionare the following. • The tetrad must produce equations of motion which are compatible with a symmetricstress-energy tensor T µν = T νµ . • The resulting equations of motion should not produce peculiar physics. For example,in spherical symmetry there should be no energy transport in the angular directions. Ina homogeneous scenario, there should be no energy flux from one location to the other,etc.It turns out it is generally non-trivial to find a tetrad that satisfies these conditions. In therealm of cosmology, for example, an array of frames have been analyzed in [16]. We discussthis situation for the case of spherically symmetric black hole interiors in section 2. In section3 we try to identify what presents a physical singularity in teleparallel theory and derive thecriteria required to eliminate this singularity. We also show that it is possible to eliminate
EGULAR BLACK HOLES IN f ( T ) GRAVITY n ≥ in (4) tested, but for negative n regularization fails and thesituation is the same as in general relativity. Finally, in section 4 we summarize the resultsand provide some concluding statements. The line element appropriate for a spherically symmetric, not necessarily vacuum, black holeexterior can be cast in the form ds = A ( t, r ) dt − B ( t, r ) dr − r dθ − r sin θ dφ , (8)with a horizon present where A ( t, r ) = 0 . The obvious tetrad to use for such a line-elementis the diagonal one: (cid:2) h aµ (cid:3) diag = A ( t, r ) 0 0 00 B ( t, r ) 0 00 0 r
00 0 0 r sin θ . (9)However, it is well-known that this tetrad does not produce acceptable equations of motionwhen utilized in (5) [17], unless f ( T ) = T . For example, for f ( T ) = T this tetrad canproduce off-diagonal components to the stress-energy tensor even when the t dependence isnot present and there should be no momentum flux.Note however that one may perform Lorentz transformations on the Lorentz index of thetetrad, which as briefly discussed in the introduction, will alter the equations of motion if thetransformation is a local one. In the literature a common tetrad that is used, which producesacceptable equations of motion is the following rotated tetrad: (cid:2) h aµ (cid:3) rot = A ( r ) 0 0 00 B ( r ) sin θ cos φ r cos θ cos φ − r sin θ sin φ B ( r ) sin θ sin φ r cos θ sin φ r sin θ cos φ B ( r ) cos θ − r sin θ . (10)This tetrad has been utilized in studies of charged black hole exterior spacetimes [18] andspherically symmetric stars [19], [20], and a more generalized version used in [21]. A noveltetrad was utilized in [22] to study a class of static, spherically symmetric solutions. Thediagonal tetrad (9) was utilized in [23] to study higher dimensional models. (10) has alsobeen used to discern some torsion only effects with non-minimal coupling to scalar fields inthe teleparallel equivalent of general relativity [24]. Dirac field coupling has been consideredin [25]. A tetrad yielding a vacuum Schwarzschild solution for an array of Lagrangians hasbeen introduced in [26]. Also, the Kerr solution has been studied in the teleparallel equivalentof general relativity utilizing a tetrad appropriate for that spacetime in [27]. EGULAR BLACK HOLES IN f ( T ) GRAVITY f ( T ) = T due to a somewhat subtle reason. Ablack hole interior’s line element can be cast as: ds = α ( τ, χ ) dτ − β ( τ, χ ) dχ − τ dθ − τ sin θ dφ , (11)and a horizon exists where β ( τ, χ ) = 0 . In this coordinate chart (sometimes called the T-domain chart, T indicating the time dependence of the interior spacetime, not the torsionscalar) the Schwarzschild black hole interior would read ds Schw = dτ Mτ − − (cid:18) Mτ − (cid:19) dχ − τ dθ − τ sin θ dφ , (12)with the coordinate ranges < τ < M , χ < χ < χ , < θ < π , ≤ φ < π . Althoughline element (11) can be obtained from (8) rather trivially, an interesting complication arisesin f ( T ) gravity which does not occur in the corresponding curvature theories.Due to the switching of the nature of space and time, a rotation in the interior of the blackhole does not generally directly correspond to a rotation in the exterior region. Similarly aboost in the interior does not correspond to a boost in the exterior. One cannot simply takethe tetrad which works in the exterior, tetrad (10), and utilize it in the interior via a simplechange of coordinate roles t → χ , r → τ along with the switching of the zeroth and firstcomponents of the tetrad matrix. Mathematically this is due to the fact that there is no directanalytical extension of coordinate chart (8) to coordinate chart (11). They are distinct chartsdespite their similarity.At this stage there are two choices which will allow us to study the black hole interiors.One choice is to switch to a chart which penetrates the horizon. The advantage will be thatone will then have a tetrad which can describe both the exterior and the interior of a blackhole. The other choice is to attempt to construct from scratch an acceptable tetrad which candescribe the black hole interior. Regarding the first choice, a tetrad we can construct is onecompatible with a Painlev´e-Gullstrand type coordinate chart. In the case of r only dependencean acceptable transformation is given by t pg = t + Z n [ A ( r ) B ( r )] − − P B − ( r ) o / B ( r ) dr , (13)with P a constant. The transformation matrix can readily be formed and applied to (10)yielding (cid:2) h aµ (cid:3) pg = A ( r ) − n [ A ( r ) B ( r )] − − P B − ( r ) o / B ( r ) A ( r ) 0 00 B ( r ) sin θ cos φ r cos θ cos φ − r sin θ sin φ B ( r ) sin θ sin φ r cos θ sin φ r sin θ cos φ B ( r ) cos θ − r sin θ . (14)This tetrad allows for the study of the interior of the black hole as well as the exterior. Italso still yields a symmetric T µν which is a requirement for an acceptable tetrad. The majordisadvantage is that, since the coordinate system is no longer orthogonal, the equations ofmotion become rather complicated and also the energy conditions become more complicatedto analyze than in an orthogonal system. EGULAR BLACK HOLES IN f ( T ) GRAVITY τ only dependent (“T-spheres”[28]-[30]) as adding χ dependence presents an extremely complicated scenario in the blackhole interior even for relatively simple deviations from f ( T ) = T . This class of black hole isnecessarily singular in general relativity, regardless of energy conditions (see below in section3.1.1). We begin with the diagonal tetrad for the black hole interior (cid:2) h aµ (cid:3) diag = α ( τ ) 0 0 00 β ( τ ) 0 00 0 τ
00 0 0 τ sin θ , (15)and consider rotations of this tetrad about the local Euler angles in the tangent space via therotation matrices: [ R x ] = ψ sin ψ − sin ψ cos ψ , [ R y ] = ϑ ϑ − sin ϑ ϑ , [ R z ] = ϕ sin ϕ − sin ϕ cos ϕ
00 0 0 1 , (16)with ψ , ϑ , and ϕ functions of the coordinates. In order to find acceptable forms for theseangle functions, we must appeal to the conditions outlined in the introduction which providethe criteria for good vs bad tetrads in f ( T ) gravity. After some work, our calculations revealthat setting ψ = 0 , ϑ = θ + π , ϕ = φ (17)will satisfy the criteria that yield acceptable equations of motion (i.e. produces a good tetrad).The resulting tetrad has the form (cid:2) h aµ (cid:3) interior = α ( τ ) 0 0 00 − β ( τ ) cos φ sin θ τ sin φ τ sin θ cos φ cos θ β ( τ ) sin φ sin θ τ cos φ − τ sin θ sin φ cos θ − β ( τ ) cos θ − τ sin θ . (18)Having found an acceptable tetrad to describe the interior of the black hole we now proceedto study the possibility of making the black hole interior regular everywhere, ideally with nonexotic matter. Even in general relativity the issue of what is a serious singularity is often not straight-forward.For example, by definition the best criteria for the presence of a curvature singularity in cur-vature theories is that one or more components of the Riemann tensor, when projected intothe orthonormal frame, becomes infinite. Although one then has a true curvature singularity,it is not necessarily a physically malignant one. One way to “measure” the components of theRiemann tensor is via tidal forces along geodesic paths. However, the equation of geodesic
EGULAR BLACK HOLES IN f ( T ) GRAVITY f ( T ) theory which seem reasonable to us, and we then proceed to alleviate these singular behaviors.We then examine what properties the matter must possess in order to possibly eliminate thesingularities.In the work here we consider several criteria for the presence of a singularity in f ( T ) gravity. The most straight-forward quantity to calculate is arguably the torsion tensor, whichwe project in a local orthonormal frame. Utilizing the tetrad (18) in (3) and projecting thecomponents into a local orthonormal coordinate system yields: T ˆ r ˆ t ˆ r = − T ˆ r ˆ r ˆ t = ˙ β ( τ ) α ( τ ) β ( τ ) , (19a) T ˆ r ˆ θ ˆ φ = − T ˆ r ˆ φ ˆ θ = − T ˆ θ ˆ r ˆ φ = T ˆ θ ˆ φ ˆ r = T ˆ φ ˆ r ˆ θ = − T ˆ φ ˆ θ ˆ φ = 2 τ , (19b) T ˆ θ ˆ t ˆ θ = − T ˆ θ ˆ θ ˆ t = T ˆ φ ˆ t ˆ φ = − T ˆ φ ˆ φ ˆ t = 1 τ α ( τ ) , (19c)where the overdots indicate derivatives with respect to the τ coordinate. Immediately it can beseen that the components in (19b) cannot be made non-singular regardless of the metric func-tions . Therefore, we have a non removable torsion tensor singularity. As stated previously,this does not necessarily mean that the spacetime has a physical singularity. Another quantity Similar pathologies exist if one chooses to project the tensor into the gravitational tetrad (18).
EGULAR BLACK HOLES IN f ( T ) GRAVITY somewhat analogous to the Kretschmann scalar, orvarious permutations thereof. We calculate one as: T µνρ T µνρ = 2 τ α ( τ ) β ( τ ) h ˙ β ( τ ) τ − α ( τ ) β ( τ ) + 2 β ( τ ) i . (20)A series expansion of this quantity about the potential singular point yields: T µνρ T µνρ = 4 τ α (cid:16) − α (cid:17) − α (0) τ α + O ( τ ) , (21)where the subscript indicates that the quantity is to be evaluated at τ = 0 . Note from either(20) or (21) that it may be possible to make this quantity finite for all values of τ .Yet another quantity which can be calculated to analyze the situation is the torsion scalar(2): T = 2 τ β ( τ ) α ( τ ) h α ( τ ) β ( τ ) − β ( τ ) − τ ˙ β ( τ ) i , (22)which may be expanded about τ = 0 as: T = 2 α τ h α − i − α β (0) τ h α (0) ˙ β (0) − ˙ α (0) β (0) i + O ( τ ) . (23)From these expressions it can be seen that this quantity too can be made finite for all values of τ . However, before continuing we note that one of the conditions required to make (21) finiteat τ = 0 , viz. α = 1 / , is incompatible with one of the conditions which make (23) finiteat τ = 0 , α = 1 .Moving on to more physical criteria we consider the equations of motion of a free testparticle. It is a well known interesting fact that in a Weitzenb¨ock spacetime the force equationon a particle subject only to the gravitational field is given by the geodesic equation of thecorresponding spacetime in curvature theory in terms of the Riemann-Christoffel connection[14], e Γ µαβ : d x µ dλ = − e Γ µαβ dx α dλ dx β dλ . (24)The interpretation in teleparallel gravity is that (24) yields the acceleration of the particlesubject to the force of gravity. For constant mass particles if this becomes infinite then thegravitational force experienced by the particle is infinite. We parametrize the normalizedfour-velocity, u α = dx α dλ , as [ u α ] = cosh( ζ ) α − ( τ )sinh( ζ ) cos( ω ) β − ( τ )sinh( ζ ) sin( ω ) cos( σ ) τ − sinh( ζ ) sin( ω ) sin( σ ) ( τ sin θ ) − , (25)with ζ , ω , σ functions of x µ ( λ ) . Considering “radial” motion ( ω = 0 ) we can calculate the Spacetime coordinate invariant but not locally Lorentz invariant.
EGULAR BLACK HOLES IN f ( T ) GRAVITY (cid:20) d x ˆ µ dλ (cid:21) = α ( τ ) β ( τ ) h ˙ β ( τ ) α ( τ ) sinh ( ζ ) + ˙ α ( τ ) β ( τ ) cosh ( ζ ) i β ( τ ) sinh( ζ ) cosh( ζ ) α ( τ ) β ( τ ) | path , (26)from which it can be seen that as long as α ( τ ) and β ( τ ) do not vanish for finite value of λ ,and the first derivative of α and β remains finite, no infinite force will be experienced by theparticle (these are sufficient, though perhaps not necessary conditions). The analysis can beextended to massless particles in a straightforward manner but we do not pursue that here.Finally, we analyze the physical components of the stress-energy tensor. For the materialmedium we need to choose the form of the stress-energy tensor to be compatible with theequations of motion that result from (18). That is, it must be representable by a matrix oftype- I b and of Segre characteristic [1 , , (1 , . We may express this in a very familiar formas T µν = [ ρ + p ⊥ ] u µ u ν − p ⊥ δ µν + [ p k − p ⊥ ] w µ w ν , (27)with the restrictions u µ u µ = 1 , w µ w µ = − and u µ w µ = 0 . The quantity (27) has athree-fold importance. One is that these quantities correspond to the proper energy density( ρ ) and pressures ( p k , p ⊥ ) of the material medium, and are therefore in principle physicallymeasurable quantities. Secondly, they directly correspond to the equations of motion (5) via ρ = T ˆ τ ˆ τ = T ττ , p k = T ˆ χ ˆ χ = −T χχ and p ⊥ = T ˆ θ ˆ θ = T ˆ φ ˆ φ = −T θθ = −T φφ . Therefore, bydemanding that the gravitational equations of motion do not develop a pathology anywherein the black hole interior we also satisfy non pathological matter. Finally, the quantities ρ , p k and p ⊥ are also coordinate scalar quantities as can be easily checked using (27) and thementioned restrictions on u µ and w µ , via: T µν u µ u ν = ρ and T µν w µ w ν = p k . Now, since ρ and p k are coordinate scalars, the condition T µµ = ρ − p ⊥ − p k establishes that p ⊥ is also ascalar quantity.Explicitly, using (18) we calculate the equations of motion (5) as: π T ττ = 4 πρ = f ( T )4 + df ( T ) dT β ( τ ) α ( τ ) τ (cid:18) β ( τ ) τ + β ( τ ) (cid:19) , (28a) π T χχ = − πp k = f ( T )4 − df ( T ) dT α ( τ ) β ( τ ) τ (cid:18) β ( τ ) ˙ α ( τ ) τ − α ( τ ) β ( τ ) − α ( τ ) ˙ β ( τ ) τ (cid:19) − d f ( T ) dT α ( τ ) β ( τ ) τ (cid:18) α ( τ ) β ( τ ) ¨ β ( τ ) τ − ˙ β ( τ ) β ( τ ) α ( τ ) τ − ˙ β ( τ ) α ( τ ) τ − ˙ α ( τ ) β ( τ ) τ − α ( τ ) ˙ β ( τ ) β ( τ ) τ + α ( τ ) β ( τ ) − α ( τ ) β ( τ ) (cid:19) , (28b) π T θθ = 4 π T φφ = − πp ⊥ = f ( T )4 + df ( T ) dT α ( τ ) β ( τ ) τ (cid:18) β ( τ ) α ( τ ) − β ( τ ) α ( τ ) − β ( τ ) ˙ α ( τ ) τ − ˙ α ( τ ) ˙ β ( τ ) τ + α ( τ ) ¨ β ( τ ) τ + 3 α ( τ ) ˙ β ( τ ) τ (cid:19) + 2 d f ( T ) dT α ( τ ) β ( τ ) τ (cid:18) α ( τ ) β ( τ ) − α ( τ ) β ( τ ) EGULAR BLACK HOLES IN f ( T ) GRAVITY + 2 β ( τ ) ˙ α ( τ ) ˙ β ( τ ) τ − α ( τ ) β ( τ ) ˙ β ( τ ) τ + α ( τ ) ˙ β ( τ ) τ − α ( τ ) β ( τ ) ˙ β ( τ ) ¨ β ( τ ) τ + 2 α ( τ ) β ( τ ) ˙ β ( τ ) τ + 2 α ( τ ) β ( τ ) ˙ β ( τ ) τ − α ( τ ) β ( τ ) ¨ β ( τ ) τ + β ( τ ) ˙ α ( τ ) τ + 3 β ( τ ) ˙ α ( τ ) ˙ β ( τ ) τ (cid:19) . (28c)These gravitational equations will be used to study the interior region of a time dependentblack hole. Originally the energy conditions were seen as reasonable conditions that all physical (and clas-sical) matter was expected to obey and were independent of theories of gravity [36]. Theseconditions impose restrictions on the matter such as positivity of the energy density as mea-sured by all causal observers, no superluminal energy transport, etc. In the scenarios given bya diagonal stress-energy tensor and of the type (27) these conditions can be summarized as:weak energy condition (WEC): ρ ≥ , ρ + p k ≥ , ρ + p ⊥ ≥ , (29a)dominant energy condition (DEC): ρ − | p k | ≥ , ρ − | p ⊥ | ≥ , (29b)strong energy condition (SEC): ρ + p k ≥ , ρ + p ⊥ ≥ , ρ + p k + 2 p ⊥ ≥ . (29c)These conditions will be employed in the analysis below. f ( T ) = T This case corresponds to the teleparallel equivalent of general relativity and therefore it is notexpected that this scenario will respect energy conditions and at the same time yield a non-singular black hole. We briefly discuss this case here to show how it fails and as a segue tomore complicated Lagrangians. Calculating the stress-energy from the equations of motion(5) yields πρ = 1 β ( τ ) α ( τ ) τ h β ( τ ) τ + β ( τ ) + α ( τ ) β ( τ ) i , (30a) πp k = − α ( τ ) τ (cid:2) α ( τ ) − α ( τ ) τ + α ( τ ) (cid:3) , (30b) πp ⊥ = 12 β ( τ ) α ( τ ) h ˙ α ( τ ) β ( τ ) + ˙ β ( τ ) ˙ α ( τ ) τ − α ( τ ) ˙ β ( τ ) − α ( τ ) ¨ β ( τ ) τ i . (30c)Forming the energy conditions (29a)-(29c) and expanding about τ = 0 gives πρ = 12 α τ h α i + 1 β (0) α τ h α (0) ˙ β (0) − ˙ α (0) β (0) i + O ( τ ) , (31a) π [ ρ + p k ] = 1 α β (0) τ h α (0) ˙ β (0) + ˙ α (0) β (0) i + O ( τ ) (31b) π [ ρ + p ⊥ ] = 12 α τ h α i + 12 α β (0) τ h α (0) ˙ β (0) − ˙ α (0) β (0) i + O ( τ ) , (31c) EGULAR BLACK HOLES IN f ( T ) GRAVITY π [ ρ − | p k | ] = 1 β (0) α (cid:16) α + 1 (cid:17) τ h ˙ α (0) β (0) α + ˙ α (0) β (0) + α ˙ β (0) + α (0) ˙ β (0) i + O ( τ ) , (31d) π [ ρ − | p ⊥ | ] = − α τ " α + O ( τ − ) , (31e) π [ ρ + p k + 2 p ⊥ ] = 2 ˙ α (0) α τ + O ( τ ) . (31f)where the subscript (0) indicates that the quantity is to be calculated at τ = 0 . Note thatsufficiently close to τ = 0 it is not possible to eliminate the singular terms in these energyconditions . Since the energy conditions are comprised of linear combinations of ρ , p k , and p ⊥ , this implies it is not possible to eliminate singularities in at least some of these physicalquantities. In order to do so one needs to abandon an everywhere Lorentzian spacetime, anda Riemannian region (-4 signature) is required instead of a pseudo-Riemannian spacetime inthe vicinity of τ = 0 . Therefore the gravitational equations of motion, being singular, are notvalid at τ = 0 . These results are consistent with general relativity as expected for f ( T ) = T .(Specifically, they are problematic from a singularity elimination point of view as can be seenin the lowest order terms in (31a), (31c) and (31e).) n Substituting the desired function for f ( T ) , f ( T ) = T + b n T n , and series expanding the com-ponents of T νµ around τ = 0 , we find that for negative values of n the qualitative singularitysituation does not change in comparison with the general relativity scenario ( n = 1 ). That is,for negative n various components of the mixed stress-energy tensor have singular terms at τ = 0 with expressions such as α in the numerator. This prohibits singularity resolu-tion again unless a Riemannian region is allowed. The reason there is no change for negativepowers of n in comparison to general relativity is because of the way f ( T ) enters into theequations of motion (5); negative powers of n yield higher-order (in τ ) corrections to ρ , p k ,and p ⊥ . That is, the n = 1 term is the term which contributes to lowest power in τ whenconsidering supplements with negative n .For positive n the leading order singular terms are of order O ( τ − n ) . We further notethat in our calculations we find that the conditions α (0) = 1 and β (0) = ˙ β (0) / ˙ α (0) are alwaysnecessary to negate the leading order singular terms. These are the same conditions requiredto regularize the torsion scalar (22), so non-singular stress-energy components necessitate anon-singular torsion scalar. This, of course, is related to the fact that the torsion scalar appearsdirectly in the equations of motion.At this point, finding additional conditions with the components of the stress-energy ten-sor for general n becomes prohibitively difficult. However the goal is to show that for n = 1 it is possible to have regular black holes, so we concentrate on several powers of n to resolvethe issue. Specifically we report below on n = 2 , although we have also studied all the cases − ≤ n ≤ , and all positive n can be made to eliminate singularities in the physical stress-energy and the torsion scalar (22), and hence the gravitational equations of motion are analytic Inner horizons are not considered for T-spheres.
EGULAR BLACK HOLES IN f ( T ) GRAVITY n > , as well as render forces finite in the force equation (26).The WEC and SEC can be respected in all the positive n scenarios. f ( T ) = T + b T For the n = 2 case we substitute f ( T ) = T + b T in (1) and calculate the subsequentequations of motion (5). Series expanding the field equations around τ = 0 allows us todetermine the structure of the matter in the vicinity of the possible singular point. As notedpreviously, the O ( τ − ) and O ( τ − ) terms in all the expanded components of the stress-energytensor are negated with the conditions α (0) = 1 and β (0) = ˙ β (0) / ˙ α (0) . In order to negatethe O ( τ − ) and O ( τ − ) components we must constrain more expansion coefficients. Notethat in general the expansion coefficients are free, meaning there are infinitely many possiblesolutions. In actuality, the metric functions and coupling constants would be constrainedexperimentally. However, as we are considering a black hole interior an experimental methodis not readily apparent. For simplicity, let ˙ α (0) = ¨ α (0) = ... α (0) = .... α (0) = 1 , β (0) = 1 , ¨ β (0) = 15 / , and ... β (0) = 107 / , which also enforces ˙ β (0) = 1 . We again stress that theseare not required conditions, but serve as a specific example out of infinitely many to make theanalysis perspicuous. Under these conditions the series expansions are: π T ττ = 4 πρ = 1 + b τ + 1175 b − b .... β (0) −
18 + O ( τ ) , (32a) − π T χχ = 4 πp k = − (cid:20) b τ − b τ + 1259 b − b .... β (0) + 3 (cid:21) + O ( τ ) , (32b) − π T θθ = 4 πp ⊥ = − b
72 + 4 b .... β (0) −
78 + O ( τ ) . (32c)It is immediately clear that if b = − all of the components are non-singular.The remaining undefined coefficient present in the above expansions is .... β (0) . This finalconstraint comes from the energy conditions, equations (29a) - (29c). From the weak energycondition, we have: .... β (0) − ≥ and − .... β (0) ≥ . (33)Note that satisfying equation (33) also satisfies the null energy condition. The strong energycondition puts a further constraint on the range of .... β (0) : − .... β (0) ≥ . (34)So the range of values of .... β (0) that will satisfy the weak, null, and strong energy conditions is ≤ .... β (0) ≤ < . EGULAR BLACK HOLES IN f ( T ) GRAVITY τ = 0 selects one value out of this range. Thedominant energy condition is satisfied only if .... β (0) = , which enforces πρ = 4 πp ⊥ =1 / and πp k = − / . So the metric functions that satisfy the energy conditions at τ = 0 in this particular case are: α ( τ ) = 1 + τ + 12 τ + 16 τ + 124 τ + O ( τ ) ,β ( τ ) = 1 + τ + 1516 τ + 107144 τ + 47459216 τ + O ( τ ) . We note that the above solution has non-zero neighborhoods about τ = 0 which respect theWEC and SEC, but although the DEC is satisfied at τ = 0 it is violated in a neighborhood asone moves away from τ = 0 . Attempts at a general analysis proved difficult, but no valuesof parameters which were studied yielded a non-zero DEC respecting neighborhood about τ = 0 , although it was found that the region of DEC violation could be made very small.The above can be treated as a local specific solution (one of many allowed which work).Away from τ = 0 one can patch the solution to energy condition respecting, non-singularsolutions. It is not difficult to do so but there is the issue of appropriate junction conditionsin f ( T ) gravity. In our tests we employed Synge’s junction condition [36] as it is a conditionderived on the matter field and does not require detailed knowledge of the particular gravi-tational equations. This condition, [ T µν b n µ ] |± τ = J ( χ ) ( b n µ being a unit normal covector to thejunction surface τ = J ( χ ) ) for the spherically symmetric T-domain may be summarized as (cid:2) T χχ ∂ χ J ( χ ) − T τχ (cid:3) |± τ = J ( χ ) = 0 , (36a) [ T χτ ∂ χ J ( χ ) − T ττ ] |± τ = J ( χ ) = 0 , (36b)where the subscript ± indicates that we are considering the discontinuity in the quantity insquare brackets on the junction surface τ = J ( χ ) . Given that our scenario is χ independent,yielding a diagonal stress-energy tensor, these conditions boil down to continuity of T ττ at thejunction surface τ = J = const., which we find can be easily satisfied. In fact, we find thatan even stronger condition can be met where derivatives of the tetrad functions up to arbitraryorder may be made continuous across the junction. We therefore find that it is possible torespect energy conditions in regular black holes of this type within f ( T ) theories, save for asmall region near τ = 0 .For cases with ≤ n ≤ it is also possible to construct metric functions that regularizethe black hole and describe matter that satisfy the energy conditions as above. Note thatfor n outside this range analysis becomes prohibitively difficult, due to the length of theexpressions in the various expansions. We can say that it is necessary in general that b n < .The expansion coefficients of α ( τ ) and β ( τ ) are generally unrestricted, with the exception of α (0) and β (0) . However, artificially restricting the derivatives of α ( τ ) will always generatea working (non-singular) solution in the same manner as shown for n = 2 . The conclusionis that for any f ( T ) = T + b n T n , ≤ n ≤ , singularities which are necessarily presentin general relativity for this class of black hole, can be alleviated by matter that respects theWEC and SEC, although there is still a price to pay in that the DEC is violated in a smallregion about τ = 0 (though it need not be violated right at τ = 0 ). We conjecture that thisis true for all positive integer values of n , and hence there may exist a singularity theorem EGULAR BLACK HOLES IN f ( T ) GRAVITY n sector for T-sphere black holes in the case n > . As discussed above, thesame is not true for the negative sector of n . For the negative sector the spacetime remainsnecessarily singular as in general relativity or its teleparallel equivalent f ( T ) = T . A tetrad has been derived which is suitable for describing the interior of a class of sphericallysymmetric black holes, which are necessarily singular in general relativity, in the extendedteleparallel theory of gravity. This tetrad allows for the study of potentially singular quanti-ties inside the black hole. Specifically, several criteria for singularities were considered andit was found that although the orthonormal torsion tensor cannot be made finite everywhere,finite torsion gravitational forces can be ensured by demanding finite non-zero metric func-tions near the potentially singular points. As well, for Lagrangians which consist of variouspowers of the torsion scalar, it is shown that the matter field quantities remain non-singular,unlike the case in general relativity, and hence the gravitational equations of motion remainvalid throughout the black hole interior manifold. (Extensions to negative values of τ are inprinciple allowed.) At the same time, all energy conditions considered can be satisfied insidethe black hole where the general relativity singularity occurs, although we find that the dom-inant energy condition is violated in an arbitrarily small neighborhood about this point. Thisleads us to speculate that a singularity theorem holds for these black holes in the extendedteleparallel gravity.The above conditions on the matter also ensure that the torsion scalar is finite. Althoughscalars created out of the torsion are scalars under general coordinate transformations, in thetorsion theory statements regarding these quantities are not locally Lorentz invariant. Theyare however globally Lorentz invariant, and for local Lorentz transformations these quantities,although they will change, will not become singular as long as the local Lorentz transforma-tion is not singular. Hence, regular black holes of this type are permitted while preservingthe weak energy and strong energy conditions everywhere, and the dominant energy condi-tion almost everywhere. It was found these results hold for all extensions to the teleparallelequivalent of general relativity studied as long as the powers of n are positive. This is not truefor negative powers. It seems likely therefore that torsion gravitational Lagrangians whichare Taylor expandable allow for regular black holes in cases where general relativity does notand that the matter can obey the WEC and SEC with a minor violation of the DEC. Singular-ities are therefore easier to remedy within f ( T ) theory while still retaining second-order fieldequations, which is not afforded by most curvature extensions of gravity.The analysis presented here can be easily extended to non-Laurent or Taylor expandableLagrangians. There are a number of interesting studies in the f ( T ) literature regarding theability of extended teleparallel gravity to successfully produce the observed acceleration ofthe universe [37]-[39], including recent extensions to anisotropic models [40]. It would beinteresting to see if the same Lagrangians which are capable of yielding the observed cosmo-logical acceleration are also capable of eliminating the big-bang and black hole singularitieswithout the need to resort to exotic matter. Acknowledgments
We would like to thank D. Horvat and S. Iliji´c of the University of Zagreb and K.S.Viswanathan of Simon Fraser University for a number of very helpful discussions.
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