Finite Action Principle and Horava-Lifshitz Gravity: early universe, black holes and wormholes
aa r X i v : . [ g r- q c ] F e b Finite Action Principle and Horava-Lifshitz Gravity: early universe, black holes andwormholes.
Jan Chojnacki , Jan Kwapisz , Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark (Dated: January 2021)The destructive interference of the neighbouring field configurations with infinite classical actionin the gravitational path integral approach serves as a dynamical mechanism resolving the blackhole singularity problem. It also provides an isotropic and homogeneous early universe without theneed of inflation.In this work, we elaborate on the finite action in the framework of Horava-Lifshitz gravity- a ghost-free QFT. Assuming the mixmaster chaotic solutions in the projectable H-L theory, we show thatthe beginning of the universe is homogeneous and isotropic. Furthermore, we show that the H-Lgravity action selects only the regular black-hole spacetimes. We also comment on possibility oftraversable wormholes in theories with higher curvature invariants.
I. INTRODUCTION
The path integral approach yields a powerful frame-work in the quantum theory since it emphasizes Lo-rentz covariance and allows for the description of non-perturbative phenomena. In the path integral, one is sup-posed to sum over all possible configurations of a field(s)Φ weighted by e iS [Φ] , where S [Φ] is the classical action ofthe theory. In the Minkowski path integral, the classicalaction approaching infinity causes fast oscillations in theexponential weight and hence the destructive interferenceof the neighbouring field configurations [1]. Hence suchconfigurations does not contribute to the physical quan-tities. Furthermore, in Wick rotated path integral, thefield configuration is weighted by e − S [Φ] , and the field(s)configurations on which the action is infinite do not con-tribute at all. Hence a Finite Action Principle, sayingthat an action of the Universe should be finite [2], is well-motivated theoretically (see also a newly proposed finiteamplitudes principle [3]). This principle has a significantimpact on the nature of quantum gravity and the evo-lution of the Universe, once the higher-curvature termsare included [4, 5]. Following this principle, unlike for theEinstein action, in Stelle gravity [6] the presence of the R term implies a homogeneous and isotropic conditionsfor the early universe. Furthermore the highly symmetricstate yields a vanishing Weyl tensor [7], explaining thelow entropy of the early universe.Note that the presented reasoning is customary in con-text of QFT. In similar spirit for Yang-Mills theories onerequires that instanton configurations have finite actionand hence A → − dU U − and F →
0, where U is thegauge transformation of the gauge group, A is the gaugefield and F is the field strength.Recently, this principle has been applied to the studyof black holes [1]. Since it is expected that the quantumgravity should resolve the black-hole singularity problem,one may ask which of the microscopic actions remain fi-nite for non-singular black holes and conversely interfe-re destructively for the singular ones. This we shall call the finite action selection principle. Only after the inclu-sion of higher-curvature operators, beyond the Einstein-Hilbert term, such selection principle can be satisfied [1].Furthermore in asymptotic safety the quantum correc-tions to the Newtonian potential eliminate the classical-singularity [8]. One should mention that the metrics donot need to be on-shell, i.e. solutions of equations of mo-tion, to enter the path-integral and make it infinite.These findings suggests that by taking into account thehigher curvatures one can resolve the singularities in theearly universe and the black holes. Yet, an issue with thehigher-curvature theory of quantum gravity is the exi-stence of the particles with the negative mass-squaredspectrum, known as ghosts , which makes the theory non-unitary. It is the consequence of the Ostrogradsky The-orem [9] and the presence of higher than second-ordertime derivatives in the terms beyond R in the action.However this might be resolved by additional symmetry[10], giving up mixing the causality prescriptions [11, 12]or taking into account infinitely many derivatives [13],see also the discussion [14] on possible resolution in thecontext of asymptotic safety.In this article we explore yet another possibility, namelywe investigate Horava-Lifshitz (H-L) gravity [15], wherethe Lorentz Invariance (LI) is broken at the fundamen-tal level (see [16] for a comprehensive progress report onthis subject). Kinetic terms are first order in the time de-rivatives, while higher spatial curvature scalars regulatethe UV behavior of the gravity. Furthermore, the lower-dimensional lattice studies of Causal Dynamical Triangu-lations (CDT) give the same Hamiltonian as H-L gravity[17–19].In this article we show that the Finite Action argumentsapplied to the projectable H-L gravity result in an iso-tropic, homogeneous, UV-complete, and ghost-free be-ginning of the universe, supporting the topological phaseconjecture [20]. We also show that the finite action selec-tion principle [1] works for H-L gravity in the context ofblack holes (the action is finite for non-singular BH andconversely for the singular). Furthermore, we have foundthat wormholes possess a Finite Action and hence contri-bute to the path-integral of QG, therefore they are consi-stent with ER=EPR hypothesis [21]. On the other hand,the stable, traversable wormholes solutions are known on-ly in the higher derivative gravities [22] (without exoticmatter), so there seems to be a wormhole/non-singularBH trade-off after taking into account the Finite ActionPrinciple. II. HORAVA-LIFSHITZ GRAVITY
In the UV limit of the Horava-Lifszyc gravity, spa-ce and time are scaled in a non-equivalent way. Diffe-omorphism invariance is broken by the foliation of the4-dimensional spacetime into 3-dimensional hypersurfa-ces of constant time called leaves, making the theorypower-counting renormalisable (see also the renormalisa-tion group studies of the subject [23–25]). The remainingsymmetry respects transformations: t −→ ξ ( t ) , x i −→ ξ i ( t, x k ) , (1)and is often referred to as the foliation-preserving diffe-omorphism, denoted by Diff( M, F ). The diffeomorphisminvariance is still present on the leaves. Four-dimensionalmetric may be expressed in the Arnowitt-Deser-Misner(ADM) [26] variables:( N, N i , (3) g ij ) , (2)where N, N i , (3) g ij denote respectively the lapse func-tion, shift vector, and 3-dimensional induced metric onthe leaves. The theory is constructed from the followingquantities: (3) R ij , K ij , a i , (3) ∇ i , (3)where (3) R ij is the 3-dimensional Ricci curvature ten-sor, (3) ∇ i is the covariant derivative constructed fromthe 3-dimensional metric (3) g ij , and a i := N, i N . Extrinsiccurvature K ij is the only object, invariant under gene-ral spatial diffeomorphisms containing exactly one timederivative of the metric tensor (3) g ij : K ij = 12 N (cid:18) ∂ (3) g ij ∂t − (3) ∇ i N j − (3) ∇ j N i (cid:19) . (4)Quantities (2) are tensor/vectors with respect toDiff( M, F ) possessing the following mass dimensions:[ (3) R ij ] = 2 , [ K ij ] = 3 , [ a i ] = 1 , [ (3) ∇ i ] = 1 . (5)One may use (2) to construct, order by order, scalar termsappearing in the Lagrangian of the theory. It may beexpressed as the difference of the kinetic and potentialpart L = K − V , where: K = 1 ζ (cid:0) K ij K ij − λK (cid:1) , (6) where K = K ij (3) g ij . At the 6th order, the potentialpart of the lagrangian contains over 100 terms [16]. Theimmense number of invariants is limited by imposing fur-ther symmetries. The restriction for the potential comesfrom the projectability condition: N = N ( t ), then termsproportional to a i ≡ S ∼ Z dtdx N p (3) g (cid:0) K ij K ij − λK − V (cid:1) , (7)where (3) g denotes the determinant of the 3-dimensionalmetric. Up to the sixth order, the potential V restrictedby the projectability condition is given by: V = 2Λ − (3) R + α
21 (3) R + α
22 (3) R ij (3) R ji + α
31 (3) R (8)+ α
32 (3) R (3) R ij (3) R ji + α
33 (3) R ij (3) R jk (3) R ki , where Λ is the cosmological constant and α ij are thecoupling constants. For our purposes, we drop termscontaining covariant derivatives (3) ∇ i . If they hadbeen present, the finite action reasoning would havebeen even stronger. One should also mention that this minimal theory [28] suffers from existence of spin 0graviton. Various solutions to this problem have beenproposed [16], such as additional symmetries [29]. Yetfor our analysis, it would be largely irrelevant (since itdoesn’t change the conclusions), hence we shall not gointo details and assume the minimal version of the theory. III. ANISOTRPIES AND INHOMOGENEITIESIN THE EARLY UNIVERSE
We shall now show that the finiteness of H-L gravityaction requires inhomogeneities and anisotropies to va-nish when approaching t → a. Anisotropies We begin our investigation of theearly universe anisotropies by considering the Bianchi IXmetric, which may be thought of as a nonlinear comple-tion of a gravitational wave: ds IX = − N dt + h ij ω i ω j , (9)where h ij = diag (cid:0) M , Q , R (cid:1) and M, Q, R are functionsof the time only. The usual isotropic FRLW universe isobtained when R ( t ) = M ( t ) = Q ( t ) = a ( t )2 , where a ( t )is the scale factor. The explicit form of the curvatureinvariants in (8) is given by [27]: (3) R = − M Q R (cid:16) M + Q + R − (cid:0) R − Q (cid:1) − (cid:0) R − M (cid:1) − (cid:0) M − Q (cid:1) (cid:17) , (10) (3) R ij (3) R ji = 14( M QR ) h M − M (cid:0) Q + R (cid:1) − M (cid:0) Q − R (cid:1) (cid:0) Q + R (cid:1) + 2 M (cid:0) Q + R (cid:1) + (cid:0) Q − R (cid:1) (cid:0) Q + 2 Q R + 3 R (cid:1) i , (11) (3) R ij (3) R jk (3) R ki = 18( M QR ) (cid:16) (cid:2) ( M − Q ) − R (cid:3) + (cid:2) ( M − R ) − Q (cid:3) + (cid:2) ( Q − R ) − M (cid:3) (cid:17) . (12)The kinetic and the potential part are respectively: N p (3) g K = M QRN h (1 − λ ) ˙ M M + ˙ Q Q + ˙ RR ! − λ ˙ M ˙ QM Q + ˙ Q ˙ RQR + ˙ M ˙ RM R ! i , (13) N p (3) gV = − N ( M QR ) V. (14)The ansatz in [4] stemming from mixmaster chaotic uni-verse solution takes form: M ( t ) ∼ t k , Q ( t ) ∼ t l , R ( t ) ∼ t − p . (15)Suppose N ( t ) = const. We require, that the integrandof the action is finite as t −→
0. As an example, considercosmological constant term proportional to: Z dtN M QR ∼ Z dt t k + l − p . (16)The above integral is convergent as t −→
0, given k + l − p > (3) R and (3) R contributions allow for finite action. These threeterms were sufficient for the isotropic beginning in 4-dimensional gravity. The lack of higher-order time de-rivatives in the action results in less restrictive relationsbetween exponents k, l, and p , opposingly to the usual R action [4].However, adding higher-order curvature terms streng-thens the restriction. For example, the inclusion of the (3) R term results in the simplest contradictory set ofinequalities: K − k − l + 3 p > ,N p (3) g (3) R resulting in l + k − p − > ,N p (3) g (3) R k − l + p > . (17)For the action to be finite at early times, the metric tensormust approach the isotropic FRLW case. Including therest of the terms in the potential (8) adds independentinequalities to the already contradictory set. This doesnot modify the conclusion. b. Inhomogeneities Similar to the anisotropies, thefiniteness of the action suppresses the inhomogeneities.Here, it is enough to consider the first and second-orderof the spatial Ricci scalar curvature. Investigation of theinhomogeneities concerns following metric tensor: ds = − dt + A ′ F dr + A (cid:0) dθ + sin θdφ (cid:1) , (18)where A = A ( t, r ), F = F ( r ) and A ′ = ∂ r A . The ho-mogeneous FRLW metric is retrieved, when F −→
1. Theresulting Ricci scalar and Ricci scalar squared contribu-tion to the action are: p (3) g (3) R ∼ AF ′ + A ′ ( − F ) F , (19) p (3) g (3) R ∼ AF F ′ + A ′ (cid:0) F − (cid:1) A A ′ F . (20)Again, we suppose that each term should be convergentas t −→
0. By the ansatz A ( t ) ∼ t s , inequalities stemmingfrom (3) R and (3) R are contradictory. This means, that F ( r ) −→
1, hence the metric of the early universe washomogeneous.
IV. BLACK HOLES AND WORMHOLES
In this paragraph, we show that H-L gravity satisfiesthe finite action selection principle for the microscopicaction of quantum gravity [1]. In this section we studyboth the solutions of H-L gravity and the known, otherBH solutions, due to the fact that the metric does notneed to be a solution to the equations of motion to enterpath integral and make infinite. We require the singularblack-hole metrics to be destructively interfered, whilethe regular ones have a finite action. We broaden thisanalysis by studying the wormhole solutions.
A. Singular black holes
Singularities may be categorised [30] in the three ma-in groups: scalar, non-scalar and coordinate singulari-ties . Scalar singularities are the ones for which (some of)the curvature invariants, like Kretschmann scalar, beco-me divergent and hence they are the object of interestin our considerations. Non-scalar singularities appear inphysical quantities such as the tidal forces. Finally, thecoordinate singularities appear in the metric tensor, ho-wever, one may get rid of the divergence with a propercoordinate transformation. Yet, coordinate singularitiesof General Relativity may become scalar singularities inthe Horava-Lifshitz gravity [31]. It is due to the fact thatthe spacetime diffeomorphism of GR is a broader symme-try than the foliation-preserving diffeomorphism of H-Lgravity. As an example, consider the Schwarzschild me-tric: ds = − (cid:18) − mr (cid:19) dt + (cid:18) − mr (cid:19) − dr + r d Ω , (21)where d Ω = r ( dθ + sin θ dφ ). The singular pointsare r = 0 and r = r s = 2 m . In GR the singular point r s = 2 m may be removed by the transformation dt P G = dt + √ mrr − m dr. (22)The resulting Painleve-Gullstrand metric is: ds = − dt P G + dr − r mr dt P G ! + r d Ω . (23)For more details see e.g. [32]. In GR metric tensors (21)and (23) describe the same spacetime with singularityat r = 0. Notice, however, that the coordinate transfor-mation (22) does not preserve the spacetime foliation,breaking the projectability condition. Hence, in the fra-mework of H-L gravity, metric tensors (21) and (23) de-scribe distinct spacetimes. Moreover, as we will show,Schwarzschild’s metric singularity at r = r s becomes aspacetime singularity. Hence due to the unique natureof the foliation-preserving diffeomorphism, investigatingthe singularities in H-L gravity is a delicate matter.We consider three representative solutions [31]: (anti-)de Sitter Schwarzschild, which is the simplest spacetimewith the black hole and the cosmological horizon, Kerrspacetime (see also the rotating H-L solution [33]) andthe H-L solution found by Lu, Mei and Pope (LMP) [34].We discuss here the (anti-) de-Sitter Schwarzschild he-re as the simplest, while since the other metrics possesthe have features, we postpone discuss them in the Ap-pendix. All of the presented curvature scalars are three-dimensional unless stated otherwise. a. (Anti-) de Sitter Schwarzschild solution The ge-neral static ADM metric with projectability conditiontakes the form: ds = − dt + e ν ( dr + e µ − ν dt ) + r d Ω , (24)where µ = µ ( r ), ν = ν ( r ). (Anti-) de Sitter Schwarzschildsolutions are be obtained for: µ = ln (cid:0) Mr + Λ3 r (cid:1) , ν = 0 . The resulting kinetic terms and Ricci scalar are: (3) R = 0 ,K = (cid:18) M + Λ r r (cid:19) (cid:18) r − M − r M + Λ r (cid:19) ,K ij K ij = 3 M + Λ r r " (cid:18) M − r M + Λ r (cid:19) . (25)The kinetic part is divergent at r = 0 and r = (cid:16) M | Λ | (cid:17) for the negative cosmological constant. We investigate the finiteness of the function: S s ( r UV , r IR ) := Z r IR r UV dr N √ g (cid:16) K ij K ij − λK + (3) R (cid:17) , (26)which is a part of the action qualitatively describing thesingularities. The r IR is chosen so that the volume in-tegral is finite, hence we do not consider singularitiesstemming from the IR behaviour (large distances) of thespacetime and time integration. The r UV is the minimalradius, which we take r UV →
0. For the scalars (25) andvalue of λ = 1, the function S s ( r UV , r IR ) is divergent atthe expected points r s = r UV = 0 and r s = (cid:16) M | Λ | (cid:17) .However, for λ = 1, which is the value required for lowenergy Einstein-Hilbert approximation, the terms diver-gent at r s = (cid:16) M | Λ | (cid:17) remain finite as one could expect,since r s corresponds to the cosmological horizon. Expli-citly we have: S s ( r UV , r IR ) = 29 Λ r UV − r UV + (cid:18) M − (cid:19) ln r UV − Mr UV + 24 r UV + IR terms . (27)Here, only the spatial Ricci scalar is necessary for the sin-gular solution to be suppressed in the gravitational pathintegral.As mentioned previously, different gauges of the samespacetime in GR, correspond to distinct spacetimes inH-L gravity. Hence, we consider the (anti-) de SitterSchwarzschild metric in the orthogonal gauge, which isnot a solution to the projectable H-L theory, in contrastto the previous case: ds = − e r ) dτ + e r ) dr + r d Ω , (28)here Ψ( r ) = − Φ( r ) = 12 ln (cid:18) − Mr + 13 Λ r (cid:19) . (29)In the orthogonal gauge, the components of the metrictensor do not depend on the time coordinate, hence thekinetic part vanishes K ij = 0. One finds, that the Ricciscalar is constant (3) R = − (3) R ij (3) R ij = 4Λ M r , (3) R ij (3) R jk (3) R ki = − − M r − M r , (30)yielding an infinite action and suppressing the singu-larity. The same conclusions can be drawn for Kerrspacetime and singular Lu-Mei-Pope metric, derived inthe context of Horava gravity, see Appendix. b. Regular black holes Due to observation’s of thebinary black holes mergers [35] and the Event HorizonTelescope observations [36, 37] the structure of BH canbe investigated on an unprecedented scale [38]. Further-more, due to the expectation that the quantum gravityshall resolve the BH singularity issue, the regular blackholes have been of interest recently, for discussions in va-rious quantum gravity approaches [39–46] (see for moremodel independent viewpoint [47–50]). Following [1] inhere we shall discuss Hayward metric [48]. The Dymni-kova spacetime [47] is discussed in the Appendix. TheHayward metric is an example of the regular black holesolution in GR: ds = − f ( r ) dt + f ( r ) − dr + r d Ω ,f ( r ) = 1 − M r ( r + 2 g ) , (31)where g is an arbitrary positive parameter. The metric isnon-singular in r −→
0. It is not a solution to H-L theory,however, we consider it as an off-shell metric present inthe path integral.The kinetic tensor vanishes K ij = 0, while the Ricci sca-lar and the second-order curvature scalars are regular: (3) R = 24 g GM (2 g + r ) , (3) R ij (3) R ij = 6 M (cid:0) g + r (cid:1) (2 g + r ) , (32)leading to finite action. A similar conclusion arises in thecase of Dymnikova spacetime, see Appendix. These tworegular solutions to GR are also regular in the off-shellH-L theory. B. Wormholes
Here, we take the first step in the direction of the inve-stigations of the consequences of the Finite Action Prin-ciple in the context of wormholes (WH). The wormholesmay be characterized in two classes: traversable and non-traversable. The traversable WH, colloquially speaking,are such that one can go through it to the other side,see [51] for specific conditions. The pioneering Einstein-Rosen bridge has been found originally as a non-static,non-traversable solution to GR. The traversable solutionsare unstable, however, they might be stabilized by an exo-tic matter or inclusion of the higher curvature scalar gra-vity [22]. This is important in the context of finite actionsince usually the divergences of black holes do appear inthe curvature squared terms. Hence, due to the inclusionof the higher-order terms in the actions, the traversa-ble wormholes are solutions to the equations of motionswithout the exotic matter. The exemplary wormhole spa-cetimes investigated here are the Einstein-Rosen bridgeproposed in [52], the Morris-Thorne (MT) wormhole [51],the traversable exponential metric wormhole [53] and the wormhole solution discussed in the H-L gravity [54]. Allof them have an action that is finite. Here, we shall di-scuss the exponential metric WH. The conclusions forthe other possible wormholes are similar and we discussthem in the Appendix. For the exponential metric WH,the line element is given by: ds = − e − Mr dt + e Mr (cid:0) dr + r d Ω (cid:1) . (33)This spacetime consists two regions: “our universe” with r > M and the “other universe” with r < M . r = M corresponds to the wormhole’s throat. The spacial volu-me of the “other universe” is infinite when r −→
0. Suchvolume divergence is irrelevant to our discussion, hencewe further consider only r M . The resulting Ricci andKretschmann scalars calculated in [53] and the measureare non-singular everywhere: R = − M r e − Mr ,R µνσρ R µνσρ = 4 M (12 r − M r + 7 M ) r e − Mr . (34)Resulting in the finite action for the Stelle gravity. Simi-larly for the H-L gravity: (3) R = R, K = K ij K ij = 0 , (3) R ij (3) R ij = 2 M ( M − M r + 3 r ) r e − Mr . (35) V. CONCLUSIONS AND DISCUSSION
The Finite Action Principle is a powerful tool to studyquantum gravity theories and also the QFT in general.In particular, we have shown that it can be invoked toexplain the homogeneity and isotropy of the early univer-se and can resolve the singularities of black holes in thecontext of Horava-Lifszyc gravity. Since broken LorentzInvariance changes the profile of spacetime singularities,it is necessary to add the third-order terms, for the H-L universe to be isotropic, in contrast to the LI gravity,where R was sufficient.The conditions stemming from the Finite Action Princi-ple justify the Topological Phase hypothesis without theneed of conversion of the degrees of freedom in the earlyuniverse, which is assumed to take place in [20]. Further-more, the anisotropic scaling of Horava gravity admits aninstanton “no-boundary”-like solution ensuring flatness.Moreover, the amplitude of the cosmological perturba-tions are scaling as: δ Φ = H − z z , hence at z = 3 they arealmost scale-invariant [55]. Finally, the Weyl anomaliesstructure in H-L gravity does not lead to strong non-localeffects during the radiation domination epoch [56, 57].This stems from the fact that these anomalies are of se-cond order in derivatives in the flat spacetimes [58–61],hence they are harmless and allow to avoid the vanishingof conformal anomaly criteria [62, 63]. In particular, itwould be interesting to see whether the anisotropic Weylanomalies can also give departure from scale invariance,as it is discussed in [20]. Yet we leave that for furtherinvestigation to be performed elsewhere. Combined withearlier results, our investigation backs up fully the topo-logical phase conjecture, hence making inflation redun-dant. Furthermore, it seems that this is in line with theswampland conjectures and the newly proposed finite-amplitude principle [3], making the asymptotically safequantum gravity to pick initial conditions s.t. inflationceases to be eternal [64], see also [65, 66].From the point of view of finite action selection princi-ple [1] they are equally good theories, resolving the blackholes singularities, assuming that ghost issue is resolvedin the latter case. Yet none of the regular B-H solutionshave been found in context of H-L gravity [67]. Hence itis a strong suggestion that the wormholes may appear inthe UV regime of H-L gravity and can serve as a “cure”for singularities [68–70].In the case of wormholes, both traversable and non-traversable wormholes are on equal footing in the caseof the Finite Action Principle. However, this principlesuggests that there is a trade-off between the resolutionof black-hole singularities and the appearance of worm-hole spacetimes due to higher curvature invariants. Thewormhole solutions will remain in both the LI and H-Lpath integrals. Furthermore the transversable and non-transversable ones stand on the same footing in the con-text of the Finite Action principle. The higher-order cu-rvature scalars, generically present in the quantum gra-vity, stabilize the wormhole solutions without the needfor an exotic matter.Finally, one should mention that there are many experi-ments to test the Lorentz Invariance Violations (LIV) inthe gravitational sector coming from gravitational wavesobservations [71–75], which could in principle validateHorava’s proposal, yet we know much more about theLIV in the matter sector (see for example [76]). Sincethese two can be related [77], one can speculate that theH-L gravity can be explicitly tested in the nearby future. Acknowledgements
We thank J. N. Borrisova, A. Eich-horn, K. A. Meissner, A. G. A. Pithis and A. Wangfor inspiring discussions and careful reading of the ma-nuscript. J.H.K. was supported by the Polish NationalScience Centre grant 2018/29/N/ST2/01743. J.H.K. wo-uld like to thank the CP3-Origins for the extended hospi-tality during this work. J.H.K acknowledges the NAWAIwanowska scholarship PPN/IWA/2019/1/00048.
APPENDIX
Here we present further examples of interesting black-hole and wormhole spacetimes in the context of the Fini-te Action Principle. We find the restrictions on the LMPsolutions necessary to resolve the singularity at the ori-gin. Similarly, the spatial Ricci scalar of Kerr’s spaceti-me yields infinite action. We further give three examplesof wormholes with finite action: Einstein-Rosen bridge, Morris-Thorne wormhole, and a spatially symmetric andtraversable wormhole solution to H-L gravity.
A. Black-holes a. LPM black hole
The popular LMP [34] metric isnot a solution to the vacuum H-L equations. However, thesecond class of the LMP solutions written in the ADMframe with projectability condition satisfy the field equ-ations of H-L gravity coupled to anisotropic fluid withheat flow, see [31]. The LMP solutions were found in theorthogonal gauge (28), without the projectability. Thereare two types of solutions. Class A solutions are:Φ = −
12 ln(1 + x ) , Ψ = Ψ( r ) . (A-1)Class B solutions consist of:Φ = −
12 ln (cid:0) x − αx α ± (cid:1) , Ψ = − β ± ln x + 12 ln (cid:0) x − αx α ± (cid:1) , (A-2)where x = p | Λ W | r , Λ = Λ W , α is an arbitrary realconstant, and α ± and β ± = 2 α ± − λ . Their explicit form may be found in [31].The LPM solutions (A-1) and (A-2), have vanishing kine-tic tensor K ij = 0, while the Ricci scalar and the integralmeasure are given respectively by: (3) R = 2 r (cid:0) α (1 + α ± ) x α ± − x (cid:1) , N √ g = r x − β ± . (A-3)The S s function (26) stands: S s ( x UV , x IR ) = − √ Λ W Z x IR / √ Λ W x UV / √ Λ W dx ( α (1 + α ± ) x − α ± − x − α ± ) , (A-4)Where x UV = √ Λ W r UV and x IR = √ Λ W r IR . The ne-cessary condition for the spatial Ricci scalar to be finiteis 2 > α ± . We proceed in the ADM gauge, which descri-bes an independent theory in the H-L gravity. Then, theClass A solution is given by: µ = −∞ , ν = −
12 ln (cid:0) − Λ W r (cid:1) (A-5)applied to (24), we get (3) R = 6Λ W . The S s function isgiven by: S s ( r UV , r IR ) = − Z r IR r UV W r p | − Λ W r | . (A-6)The exact form of S s ( r UV , r IR ) depends on the sign ofthe scaled cosmological constant Λ W , nevertheless, it isalways finite, when r UV −→
0. Indeed, for the negativeΛ W < S s ( r UV , r IR ) = − √− Λ W arcsinh (cid:16)p − Λ W r UV (cid:17) − r UV q − Λ W r UV + 1 . (A-7)Positive cosmological constant splits the space in two re-gions. When r > √ Λ W we get: S s ( r UV , r IR ) = 3 √ Λ W arctanh √ Λ W r UV p Λ W r UV − ! + 3 r UV q Λ W r UV − , (A-8)for a small, positive cosmological constant, above resultis irrelevant for our discussion, since it would describelarge scales. When r < √ Λ W : S s ( r UV , r IR ) = − √ Λ W arcsin (cid:16)p Λ W r UV (cid:17) + 3 r UV q Λ W r UV − . (A-9)The class B solution singularity at the origin, appearingwhen 2 ¬ α + is suppressed by the Finite Action Prin-ciple. Class A solutions are finite and contribute to thepath integral, if the cosmological constant is negative orsmall and positive when r UV −→ √ Λ W . b. Kerr spacetime Kerr spacetime corresponds to anaxially symmetric, rotating black hole with mass M andangular momentum J . It is a solution to the EinsteinEquations in GR, however, it has been shown order byorder in the parameter a = J/M , that it is not a solutionto the H-L field equations [78]. Yet it can still enter thepath integral as an off-shell metric. The line element in the Boyer-Lindquist coordinates is given by: ds = − ρ ∆ r Σ dt + ρ ∆ r dr + ρ dθ + Σ sin θρ ( dφ − ξdt ) , (A-10)where ρ = r + a cos θ, ∆ r = r + a − M r, Σ = ( r + a ) − M r,ξ = 2
M ar Σ . (A-11)We are interested in the singularity on equator planecos θ = 0, r = 0, described in detail in [5]. For the explicitform of the extrinsic curvature scalars and Ricci scalarrefer to [78]. Here, we only show the form of the Ricciscalar on the cos θ = 0 plane: (3) R = − a m ( a + 3 r ) r ( r + a (2 M + r )) . (A-12)It is singular at r = 0. Integrating (3) R with the me-asure N √ g = r , results in the infinite action in theUV limit and the Kerr spacetime does not contribute thepath integral. The vanishing, 4-dimensional Ricci scalaris restored in the LI limit λ = 1. It is then necessary toinclude the Kretschmann scalar to resolve the singularityas discussed in [1]. c. Dymnikova spacetime The Dymnikova spacetimeis a regular solution in GR. It constructed with the lineelement (31) with: f ( r ) = 1 − M ( r ) r , M ( r ) = M (cid:18) − e − r g (cid:19) . (A-13)The corresponding curvature scalars are non-singular: (3) R = 6 Mg e − r g , (3) R ij (3) R ij = 3 M g r e − r g g (cid:18) e r g − (cid:19) − g r (cid:18) e r g − (cid:19) + 9 r ! (A-14)and the action is finite in the limit r UV −→
0. In particu-lar, in this limit we have (3) R ij (3) R ij −→ M /g d. Higher-order curvature scalars Here we give a ge-neral expression for the higher-order scalars present inthe H-L potential (8) for the projectable ADM and or-thogonal gauge metric tensors. The metric tensor in pro- jectable ADM gauge (24) yields: (3) R ij (3) R ij = 2 e − ν ( r ) (cid:16) r ν ′ ( r ) + (cid:0) rν ′ ( r ) + e ν ( r ) − (cid:1) (cid:17) r , (3) R ij (3) R jk (3) R ki = 2 e − ν ( r ) (cid:16) r ν ′ ( r ) + (cid:0) rν ′ ( r ) + e ν ( r ) − (cid:1) (cid:17) r . (A-15)In the orthonormal gauge they are given by (3) R = 2 e − r ) (cid:0) r Φ ′ ( r ) + e r ) − (cid:1) r , (3) R ij (3) R ij = 2 e − r ) (cid:16) r Φ ′ ( r ) + (cid:0) r Φ ′ ( r ) + e r ) − (cid:1) (cid:17) r , (3) R ij (3) R jk (3) R ki = 2 e − r ) (cid:16) r Φ ′ ( r ) + (cid:0) r Φ ′ ( r ) + e r ) − (cid:1) (cid:17) r . (A-16) B. Wormholes a. Einstein-Rosen bridge
The Einstein-Rosen (E-R)bridge smoothly glues together two copies of the Schwarz-schild spacetime: black-hole and the white-hole solutionscorresponding to the positive and negative coordinate u .Metric tensor of the Einstein-Rosen wormhole proposedin [52] and discussed in e.g. [79] is given by: ds = − u u + 4 M dt + ( u + 4 M ) du + 14 ( u + 4 M ) d Ω . (B-1)The E-R bridge is non-traversable and geodesically in-complete in u = 0. This fact, however, does not impactthe regularity of the curvature scalars. The 4-dimensionalRicci scalar: R = 2 (cid:0) M + 32 M u + 4 u + u (cid:1) (4 M + u ) . (B-2)second order curvature scalar R µν R µν :4 (cid:16) M + 8 (cid:0) M + u (cid:1) + (32 M − (cid:0) M + u (cid:1) (cid:17) (4 M + u ) . (B-3)Both of which integrated with the measure are non-singular: √ g = 14 u (4 M + u ) . (B-4)The wormhole solutions analyzed in this paper generallyyield the finite action in both GR and H-L. The finiteAction Principle suggests, that in the quantum UV regi-me, singular black-hole spacetimes may be replaced withthe regular wormhole solutions. b. Morris-Thorne wormhole The MT wormhole isdefined in the spherically symmetric, Lorentzian spaceti-me by the line element: ds = − e r ) dt + dr − b ( r ) r + r d Ω (B-5)where Φ( r ) is known as the redshift and there are no ho-rizons if it is finite. Function b ( r ) determines the worm-hole’s shape. We choose Φ( r ) , b ( r ) to be:Φ( r ) = 0 , b ( r ) = 2 M (cid:0) − e r − r (cid:1) + r e r − r , (B-6)where r is the radius of the throat of the wormhole,such that b ( r ) = r . 4-dimensional Curvature scalars forthis spacetime have been calculated in [80]. The Riccicurvature scalar is singular at r = 0, however, the radialcoordinate r varies between r > R = − M − r ) e r − r r . (B-7)The resulting S s = R r IR r UV = r √ gR function is divergent as r UV −→ r and cannot be expressed in terms of simplefunctions:2(2 M − r ) Z r IR r UV = r r rr − M (1 − e r − r ) + r e r − r e r − r dr. (B-8)However, this is only a coordinate singularity and onemay get rid of it with a proper transformation.Higher-order curvature scalars for Morris-Thorne worm-hole are: (3) R = 2 b ′ ( r ) r R ij (3) R ij = 3 r b ′ ( r ) − rb ( r ) b ′ ( r ) + 3 b ( r ) r , (3) R ij (3) R jk (3) R ki = − r b ( r ) b ′ ( r ) + 5 r b ′ ( r ) + 15 rb ( r ) b ′ ( r ) − b ( r ) r , (B-9)and integrated give action that is finite. c. H-L wormhole Static spherically traversablesymmetric wormholes have been constructed in [54] inthe H-L theory through the modification of the Rosen-Einstein spacetime: ds = − N ( ρ ) dt + 1 f ( ρ ) dρ + ( r + ρ ) d Ω , (B-10)with additional Z symmetry with respect to the worm-hole’s throat. There are solutions with λ = 1 asymptoti- cally corresponding to the Minkowski vacuum. Explicitlywe have: f = N = 1 + ω ( r + ρ ) − p ( r + ρ ) ( ω ( r + ρ ) + 4 ωM ) . (B-11)Radius of the wormhole’s throat is given by r . The para-meters ω, and M are connected to the coupling constantsin H-L action. See [54] for their explicit form. 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