Black holes and running couplings: A comparison of two complementary approaches
Benjamin Koch, Carlos Contreras, Paola Rioseco, Frank Saueressig
aa r X i v : . [ h e p - t h ] N ov Black holes and running couplings:A comparison of two complementary approaches
Benjamin Koch*, Carlos Contreras + , Paola Rioseco*, andFrank Saueressig** *Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile,Av. Vicu˜na Mackenna 4860, Santiago, Chile + Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa;Casilla 110-V, Valpara´ıso, Chile**Radboud University Nijmegen,Institute for Mathematics, Astrophysics and Particle Physics (IMAPP),Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Abstract.
Black holes appear as vacuum solutions of classical general relativitywhich depend on Newton’s constant and possibly the cosmological constant. Atthe level of a quantum field theory, these coupling constants typically acquire ascale-dependence. This proceedings briefly summarizes two complementary waysto incorporate this effect: the renormalization group improvement of the classicalblack hole solution based on the running couplings obtained within the gravitationalAsymptotic Safety program and the exact solution of the improved equations of motionincluding an arbitrary scale dependence of the gravitational couplings. Remarkably thepicture of the “quantum” black holes obtained from these very different improvementstrategies is surprisingly similar.PACS numbers: 04.62.+v, 03.65.Ta lack holes and running couplings: A comparison of two complementary approaches
1. Introduction
The emergence of scale-dependent couplings is one of the central phenomena encounteredin quantum field theory. While the quest for a consistent and predictive quantumformulation for gravity is still ongoing, it is natural to expect that this feature willemerge in this case as well. This expectation is supported by perturbative computationsin the framework of higher-derivative gravity [1, 2, 3] as well as the non-perturbativecomputations carried out within the gravitational Asymptotic Safety program [4, 5, 6, 7].An important testing ground for ideas related to modified theories of gravity orquantum gravity is given by the black hole solutions obtained from classical generalrelativity. Striving for a quantum description of these objects, it is natural to study theeffect of scale-dependent coupling constants on the physics of the black holes. In thisproceedings paper we will focus on two complementary strategies for capturing theseeffects: • The first approach discussed in section 2 was pioneered in [8, 9] and performs arenormalization group (RG) improvement of the classical black hole solution. Herethe classical coupling constants are promoted to scale-dependent couplings whoseflow is governed by beta functions computed within Asymptotic Safety. By now,these techniques have been refined by several groups [10, 11, 12, 13, 14, 15, 16]. • The second approach covered in section 3 follows the spirit of [17] and looks forconsistent solutions of the improved equations of motion. These equations can besolved without making further assumptions on the actual scale dependence of thecouplings, leading to a new, spherically symmetric metric. This metric can be seenas a promising candidate for a physical black hole metric that incorporates generaleffects of scale dependent couplings.In section 4 we will compare those results and conclude.
2. Improved solutions from Asymptotic Safety
This section basically follows Ref. [18]. Thus, we restrict ourselves to a summary of thekey concepts and results and refer to [18] for more details and further references.The key ingredient for investigating Weinberg’s Asymptotic Safety conjecture [19]and its phenomenological implications is the gravitational effective average action Γ k [20], a Wilson-type effective action that provides an effective description of physics atthe momentum scale k . As its main virtue, the scale-dependence of Γ k is governed byan exact functional renormalization group equation [20] ∂ k Γ k = 12 Tr (cid:20)(cid:16) Γ (2) k + R k (cid:17) − ∂ k R k (cid:21) . (1)Here Γ (2) k denotes the second variation of Γ k with respect to the quantum fields and R k is an IR-regulator that renders the trace finite and peaked on fluctuations with momenta p ≈ k . lack holes and running couplings: A comparison of two complementary approaches −0.2 −0.1 0.1 0.2 0.3 0.4 0.5−0.75−0.5−0.250.250.50.751 λ g Type IIIaType Ia Type IIaType Ib Type IIIb
Figure 1: RG flow originating from the Einstein-Hilbert truncation (2). The arrowspoint in the direction of increasing coarse-graining, i.e. of decreasing k . From [21].The simplest setup for obtaining a non-perturbative approximate solution of (1)truncates the gravitational part of Γ k to the (scale-dependent) Einstein-Hilbert actionΓ grav k [ g ] = 116 πG k Z d x √ g [ − R + 2Λ k ] , (2)which includes two running couplings, Newton’s constant G k and the cosmologicalconstant Λ k . The beta functions resulting from this truncation have first been derivedin [20] and are most conveniently expressed in terms of the dimensionless couplingconstants g k = G k k , λ k = Λ k k − . (3)The phase diagram resulting from the flow has been constructed in [21] and is shownin figure 1. The flow is governed by the interplay of a Gaussian fixed point locatedat the origin, g ∗ = 0 , λ ∗ = 0 and a non-Gaussian fixed point (NGFP) governing theUV-behavior of the flow. For the optimized cutoff this NGFP is located at λ ∗ = 0 . , g ∗ = 0 . , g ∗ λ ∗ = 0 . . (4)One way to investigate the implications of the scaling gravitational couplings on(A)dS black holes is the RG improvement of the classical black hole solution. Thisprocedure starts from the classical (Schwarzschild-de Sitter or anti-de Sitter) line-element ds = − f ( r ) dt + f ( r ) − dr + r d Ω (5)with f ( r ) = 1 − GMr −
13 Λ r , (6)and replaces Newton’s constant and the cosmological constant by their scale dependentcounterparts, G → G k , Λ → Λ k . The crucial step following this improvement is thescale setting procedure, which relates the momentum scale k to the radial scale rk ( P ( r )) = ξd ( P ( r )) , (7) lack holes and running couplings: A comparison of two complementary approaches ξ is an a priory undetermined constant. On general grounds the cutoffidentification d ( P ) should be independent of the choice of coordinates and compatiblewith the symmetries of the classical solution. Following [9], a natural candidate for d ( P )is the radial proper distance between the point P and the origin which should providethe physical cutoff of the geometry.Applying this improvement scheme to the classical (A)dS black holes led to variousnovel conclusions, which are largely independent of the details underlying the scalesetting procedure:a) Including the effect of a scale-dependent cosmological constant in the RG-improvement process drastically affects the structure of the quantum-improvedblack holes at short distances . Thus a consistent RG-improvement procedurerequires working in the class of Schwarzschild-(A)dS solutions of Einstein’sequations.b) The short-distance structure of all quantum-improved black holes is governed by theNGFP. This entails that the structure of light black holes is universal. In particularit is independent of the IR-value of Newton’s constant and the cosmologicalconstant and therefore identical for classical Schwarzschild, Schwarzschild-dS andSchwarzschild-AdS black holes.c) In the presence of the cosmological constant, the curvature singularity at r = 0 isnot resolved.
3. Solving improved equations of motion
An alternative strategy for modeling the quantum properties of a classical black hole,based on “improving the equations of motion”, has been developed in [22]. In this case,the scale-setting procedure is carried out at the level of the (wick-rotated) Einstein-Hilbert action (2) where the k -dependence of the couplings is replaced by a generic r -dependence. The resulting equations of motion are [23, 24] G µν = − g µν Λ( r ) + 8 πG ( r ) T µν − ∆ t µν , (8)with ∆ t µν = G ( r ) ( g µν − ∇ µ ∇ ν ) 1 G ( r ) . (9)With the metric ansatz ds = − F ( r ) dt + 1 /F ( r ) dr + r dθ + r sin( θ ) dφ , (10)the equations of motion can be solved exactly, for the functions F ( r ) , Λ( r ), and G ( r ).This solution is non-trivial, leading to four constants of integration c , c , c , c . Theseconstants can be related to familiar properties of the classical solution such as M , G , Λ together with a possible correction. Alternatively, they can be traded for the lack holes and running couplings: A comparison of two complementary approaches λ U ( r ) and g U ( r ) for g ∗ U =0 . λ ∗ U = 0 . g I = 2 .
5, and G = Σ = 1. The different curves correspond to l I = {− . , − . , , . , . } . From [22].adimensional parameters g I , g U , λ I , and λ U which naturally appear in the inducedcoupling flow g U ( r ) = G ( r )Σ , λ U ( r ) = − Λ( r ) r Σ , (11)where Σ is an arbitrary matching constant which has mass dimension one. The valuesof the UV fixed points of this “flow” are g U ( r →
0) = g ∗ U , λ U ( r →
0) = λ ∗ U . (12)The induced “flow” for the couplings (11) is shown in figure 2 and turns out to besurprisingly similar to the genuine RG flow shown in figure 1. Moreover, the mainproperties of the improved solutions can be summarized as followsa) There exists a non-trivial solution of the improved equations of motion (8) which cannot be obtained without the cosmological term. Thus, including a scale dependentcosmological term is crucial for this approach.b) In the UV limit, the new solution is dominated by the fixed points g ∗ U and λ ∗ U .c) The new solution F ( r ) exhibits a singularity at the origin, which is of the samegrade as the singularity of the Schwarzschild solution. In this limit the solution isdominated by the non-trivial fixed point of the induced “flow”.d) Interpreting the solution for the couplings (11) as flow one finds interestingsimilarities with the RG flow derived from the effective average action Γ k .
4. Comparison of the two approaches
We have summarized two strategies for modeling quantum corrections to classical blackhole solutions based on implementing scale-dependent couplings. The first approach isbased on improving the classical solutions and uses the beta functions obtained fromAsymptotic Safety to fix the scale-dependence of the gravitational coupling constants lack holes and running couplings: A comparison of two complementary approaches r ) and G ( r ) in agenerally covariant theory (8). These a priori unrelated schemes lead to a qualitativelysimilar picture for the improved black hole solutions:a) Λ matters: In both approaches the cosmological constant has a significant effect. In section 2this term strongly dominated the UV behavior of the improved solution, while insection 3 this term was actually crucial for obtaining a non-trivial solution at all.b)
Fixed points control UV:
In both approaches the short distance behavior is dominated by the non-trivialfixed point of the true flow in section 2 or of the induced “flow” in section 3.c)
Singularity persists:
Rather surprisingly, both approaches exhibit the same type of black hole singularitylocated at the origin. Since it is a general expectation that quantum gravity shouldbe capable of resolving this singularity, it would be very interesting to understandthis result in more detail.These coincidences consolidate the findings of both approaches.
Acknowledgements
B.K. thanks the organizers of the first Karl Schwarzschild meeting for hospitality. Thework of B.K. was supported proj. Fondecyt 1120360 and anillo Atlas Andino 10201; theresearch of F.S. is supported by the Deutsche Forschungsgemeinschaft (DFG) within theEmmy-Noether program (Grant SA/1975 1-1). The work of C.C. was supported proj.Fondecyt 1120360 and DGIP grant 11.11.05. [1] K. S. Stelle, Phys. Rev. D , 953 (1977).[2] J. Julve and M. Tonin, Nuovo Cim. B46 , 137 (1978).E. S. Fradkin and A. A. Tseytlin, Phys. Lett.
B104 , 377 (1981); Nucl. Phys.
B201 , 469 (1982).I. G. Avramidi and A. O. Barvinsky,
Phys. Lett.
B159 , 269 (1985).N. H. Barth and S. M. Christensen, Phys. Rev. D , 1876 (1983).G. de Berredo-Peixoto and I. L. Shapiro, Phys. Rev. D , 064005 (2005); hep-th/0412249 .[3] A. Codello, R. Percacci, Phys. Rev. Lett. , 221301 (2006); hep-th/0607128 .M. Niedermaier, Phys. Rev. Lett. , 101303 (2009).K. Groh, S. Rechenberger, F. Saueressig and O. Zanusso, PoS EPS -HEP2011 , 124 (2011);arXiv:1111.1743.[4] M. Niedermaier and M. Reuter, Living Rev. Rel. , 5 (2006).[5] R. Percacci, in Approaches to Quantum Gravity: Towards a New Understanding of Space, Timeand Matter , D. Oriti (Ed.), Cambridge Univ. Press, Cambridge (2009); arXiv:0709.3851.[6] A. Codello, R. Percacci and C. Rahmede, Int. J. Mod. Phys. A , 143 (2008); arXiv:0705.1769.[7] M. Reuter and F. Saueressig, New J. Phys. , 055022 (2012); arXiv:1202.2274.[8] A. Bonanno and M. Reuter, Phys. Rev. D , 084011 (1999); gr-qc/9811026.[9] A. Bonanno and M. Reuter, Phys. Rev. D , 043008 (2000); hep-th/0002196.[10] H. Emoto, hep-th/0511075.[11] A. Bonanno and M. Reuter, Phys. Rev. D , 083005 (2006); hep-th/0602159. lack holes and running couplings: A comparison of two complementary approaches [12] B. Koch, Phys. Lett. B , 334 (2008); arXiv:0707.4644.[13] T. Burschil and B. Koch, Zh. Eksp. Teor. Fiz. , 219 (2010); arXiv:0912.4517.[14] M. Reuter and E. Tuiran, hep-th/0612037; Phys. Rev. D , 044041 (2011); arXiv:1009.3528.[15] K. Falls, D. F. Litim and A. Raghuraman, Int. J. Mod. Phys. A , 1250019 (2012);arXiv:1002.0260. K. Falls and D. F. Litim, arXiv:1212.1821. D. F. Litim and K. Nikolakopoulos,arXiv:1308.5630.[16] R. Casadio, S. D. H. Hsu and B. Mirza, Phys. Lett. B , 317 (2011); arXiv:1008.2768.[17] M. Reuter and H. Weyer, Phys. Rev. D , 104022 (2004); hep-th/0311196.[18] B. Koch and F. Saueressig, Class. Quant. Grav. to appear; arXiv:1306.1546.[19] S. Weinberg in General Relativity, an Einstein Centenary Survey , S.W. Hawking and W. Israel(Eds.), Cambridge University Press, 1979; S. Weinberg, hep-th/9702027.[20] M. Reuter, Phys. Rev. D , 971 (1998), hep-th/9605030.[21] M. Reuter and F. Saueressig, Phys. Rev. D , 065016 (2002); hep-th/0110054.[22] C. Contreras, B. Koch and P. Rioseco, Class. Quant. Grav. , 175009 (2013); arXiv:1303.3892.[23] M. Reuter and H. Weyer, JCAP , 001 (2004); hep-th/0410119.[24] S. Domazet and H. Stefancic, Class. Quant. Grav.29