Black Holes in Higher Derivative Gravity

Transcription

1 Black Holes in Higher Derivative Gravity K.S. Stelle Imperial College London With H. Lü, A. Perkins and C.N. Pope Quantum spacetime and the Renormalization Group 674 WE Hereaus-Seminar, Bad Honnef June 19, / 39

2 Quantum Context One-loop quantum corrections to General relativity in 4-dimensional spacetime produce ultraviolet divergences of curvature-squared structure. G. t Hooft and M. Veltman, Ann. Inst. Henri Poincaré 20, 69 (1974) Inclusion of d 4 x g(αc µνρσ C µνρσ + βr 2 ) terms ab initio in the gravitational action leads to a renormalizable D = 4 theory, but at the price of a loss of unitarity owing to the modes arising from the C µνρσ C µνρσ term, where C µνρσ is the Weyl tensor. K.S.S., Phys. Rev. D16, 953 (1977). [In D = 4 spacetime dimensions, this (Weyl) 2 term is equivalent, up to a topological total derivative via the Gauss-Bonnet theorem, to the combination α(r µν R µν 1 3 R2 )]. 2 / 39

3 Despite the apparent nonphysical behavior, quadratic-curvature gravities continue to be explored in a number of contexts: Cosmology: Starobinsky s original model for inflation was based on a d 4 x g( R + βr 2 ) model. A.A. Starobinsky 1980; V.F. Mukhanov & G.V. Chibisov 1981 This early model has been quoted (at times) as a good fit to CMB fluctuation data from the Planck satellite. J. Martin, C. Ringeval and V. Vennin, The asymptotic safety scenario considers the possibility that there may be a non-gaussian renormalization-group fixed point and associated flow trajectories on which the ghost states arising from the α(weyl) 2 term could be absent. S. Weinberg 1976, M. Reuter 1996, M. Niedermaier / 39

4 Asymptotic Safety The asymptotic-safety proposal extends the family of acceptable quantum theories beyond the strictly renormalizable ones to theories where there is a finite set of relevant couplings lying on an ultraviolet critical surface within the infinite space of all coupling constants. This includes ordinary renormalizable and asymptotically free theories, where there is a Gaussian fixed point at the origin of coupling-constant space, but can also include theories with non-trivial fixed points away from the origin, which would be of an essentially non-perturbative nature. An important related feature of the higher-derivative gravity model is asymptotic freedom, meaning that if one writes α = 1/g 2 2 in a fashion similar to Yang-Mills theory, then at large momenta one finds g 2 0. E.S. Fradkin & A.A. Tseytlin 1981, 1982; I.G Avramidy & A.O. Barvinsky 1985 This suggests that perhaps the unstable ghost states of the theory might decouple in the region of phase space where the ghost pole exists. 4 / 39

5 A claim along these lines is that appropriately chosen renormalization-group trajectories exist such that the Hessian matrix for the residues of the effective-theory propagator can be free of negative eigenvalues, so that instabilities arising from ghost states need not actually occur. Niedermaier g Renormalization-group trajectories in coupling-constant space ending on a non-gaussian fixed point with finite g Newton and cosmological constant Λ but with vanishing g 2. 5 / 39

6 Classical gravity with higher derivatives Consider the gravitational action I = d 4 x g(γr αc µνρσ C µνρσ + βr 2 ). The field equations following from this higher-derivative action are ( H µν = γ R µν 1 ) 2 g µνr (α 3β) µ ν R 2α R µν (α + 6β) g µν R 4αR ηλ R µηνλ + 2 (β + 23 ) α RR µν + 1 ( 2 g µν 2αR ηλ R ηλ (β + 23 ) ) α R 2 = 1 2 T µν 6 / 39

7 Nonlinear field equations for spherical symmetry Use Schwarzschild coordinates ds 2 = B(r)dt 2 + A(r)dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) One might expect that the field equations for A and B would be of fourth order. However, there are some modest gifts arising from the diffeomorphism invariance of the theory, which gives rise to Bianchi-type identities for the field equations. One has 0 µ H µr = r H rr + 0 H 0r + θα H θαr + Γ µ µνh νr + Γ r µνh µν Since the second and subsequent terms here are of maximum fourth order in r derivatives, one learns that H rr can be at most of third order in r derivatives. (Analogously in GR, G rr is only of first order in r derivatives.) 7 / 39

8 The other field equations are indeed of fourth order in r derivatives. However, by r differentiating H rr and taking an appropriate combination of this together with the other field equations, fourth-order r derivatives can be eliminated, giving a second third-order equation. The remaining field equations are related to these two third-order equations by Bianchi-type identities. So the system that one actually has to solve for static and spherically symmetric spacetimes is one of two third-order ordinary differential equations in the radius r. Accordingly, one expects a maximum of six integration constants to appear in the general solution. 8 / 39

9 The first equation contains the third-order derivative B (3) = B Note: rank 1 irregular singularity. This suggests the presence also of non-frobenius solutions see poster by Bonnano and Silvervalle in this conference. 9 / 39

10 The second equation contains the third-order derivative A (3) = A : 10 / 39

11 Separation of modes in the linearized theory Solving the full nonlinear field equations is clearly a challenge. One can make initial progress by restricting the metric to infinitesimal fluctuations about flat space, defining h µν = κ 1 (g µν η µν ) and then considering field equations linearized in h µν, or equivalently by restricting attention to quadratic terms in h µν in the action. The action then becomes I Lin = d 4 x{ 1 4 hµν (2α γ) P µνρσh (2) ρσ hµν [6β γ] P (0;s) µνρσh ρσ } ; P (2) µνρσ = 1 2 (θ µρθ νσ + θ µσ θ νρ ) P (0;s) µνρσ P (0;s) µνρσ = 1 3 θ µνθ ρσ θ µν = η µν ω µν ω µν = µ ν /, where the indices are lowered and raised with the background metric η µν. 11 / 39

12 The two completely transverse spin-sector projectors in I Lin are P (2) µνρσ = 1 2 (θ µρθ νσ + θ µσ θ νρ ) P (0;s) µνρσ P (0;s) µνρσ = 1 3 θ µνθ ρσ where θ µν = η µν ω µν ; ω µν = µ ν /. Note the linearized gauge invariance of the action I Lin : there is no part of the action involving the spin-one projector ω µν. The interaction between two conserved sources T µν (1) and T µν (2) is then given in momentum space by 1 2 κh µν(t (2) )T µν (1) = κ2 2γ [ (T (1) µν T µν (2) 1 2 T ρ (1) ρ T σ (2) σ ) k 2 (1) (T µν T µν (2) 1 3 T ρ ρ (1) Tσ σ (2) ) (k 2 + m2 2) + T ρ ρ (1) Tσ σ (2) (k 2 + m0 2) ]. 12 / 39

13 From this linearized action one deduces the dynamical content of the linearized theory: positive-energy massless spin-two negative-energy massive spin-two with mass m 2 = γ 1 2 (2α) 1 2 positive-energy massive spin-zero with mass m 0 = γ 1 2 (6β) 1 2 K.S.S A simple model of what has happened can be made with a single scalar field and a higher-derivative action coupled to a source J: I hd = d 4 x( 1 2 µφ µ φ α µφ µ φ + Jφ) Going over to momentum space k µ, one can solve for φ and then separate the propagator into partial fractions: φ = J/α k 2 (k 2 + 1/α) = J k 2 J k 2 + 1/α similar to the structure found in quadratic gravity, but without the spin complications. 13 / 39

14 Static and spherically symmetric solutions Now consider spherically symmetric gravitational solutions in the linearised limit of the higher-curvature theory. In the linearized theory, one finds the following general solution to the source-free field equations H L µν = 0, in which C, C 2,0, C 2,+, C 2,, C 0,+, C 0, are the six integration constants: A(r) = 1 C 20 C 2+ em 2r C 2 e m 2r + C 0+ em 0r + C 0 e m 0r r 2r 2r r r C 2+ m 2 e m2r 1 2 C 2 m 2 e m2r C 0+ m 0 e m0r + C 0 m 0 e m 0r B(r) = C + C 20 r + C 2+ em 2r r + C 2 e m 2r r + C 0+ em 0r r + C 0 e m 0r r 14 / 39

15 As one might expect from the dynamics of the linearized theory, the general static, spherically symmetric solution is a combination of a massless Newtonian 1/r potential plus rising and falling Yukawa potentials arising in both the spin-two and spin-zero sectors. When coupling to non-gravitational matter fields is made via standard h µν T µν minimal coupling, one gets values for the integration constants from the specific form of the source stress tensor. Requiring asymptotic flatness and coupling to a point-source positive-energy matter delta function T µν = δ 0 µδ 0 νmδ 3 ( x), for example, one finds A(r) = 1 + κ2 M 8πγr κ2 M(1+m 2 r) e m 2 r 12πγ r κ2 M(1+m 0 r) e m 0 r 24πγ r B(r) = 1 κ2 M 8πγr + κ2 M e m 2 r 6πγ r κ2 M e m 0 r 24πγ r with specific combinations of the Newtonian 1/r and falling Yukawa potential corrections arising from the spin-two and spin-zero sectors. 15 / 39

16 Note that in the Einstein-plus-quadratic-curvature theory, there is no Birkhoff theorem. For example, in the linearized theory, coupling to the stress tensor for an extended source like a perfect fluid with pressure P constrained within a radius l by an elastic membrane, T µν = diag[p, [P 1 2 lδ(r l)]r 2, [P 1 2 lδ(r l)]r 2 sin 2 θ, 3M(4πl 3 ) 1 ], one finds for the external B(r) function B(r) = 1 κ2 M 8πγr + κ2 e m2r { [ M l cosh(m2 l) γr 2πl 3 m2 2 sinh(m ] 2l) m2 3 [ sinh(m2 l) P m2 3 l cosh(m ]} 2l) m2 2 + l2 sinh(m 2 l) 3m 2 κ2 e m { [ 0r M l cosh(m0 l) 2γr 4πl 3 m0 2 sinh(m ] 0l) m0 3 [ sinh(m0 l) P m0 3 l cosh(m ]} 0l) m0 2 + l2 sinh(m 0 l) 3m 0 which limits to the point-source result as l / 39

17 Frobenius Asymptotic Analysis Asymptotic analysis of the field equations near the origin leads to study of the indicial equations for behavior as r 0. K.S.S Let A(r) = a s r s + a s+1 r s+1 + a s+2 r s+2 + B(r) = b t r t + b t+1 r t+1 + b t+2 r t+2 + and analyze the conditions necessary for the lowest-order terms in r of the field equations H µν = 0 to be satisfied. This gives the following results, for the general α, β higher derivative theory: (s, t) = (1, 1) with 5 free parameters (s, t) = (0, 0) with 3 free parameters (s, t) = (2, 2) with 6 free parameters Lü, Perkins, Pope & K.S.S., ; but see also Bonnano & Silvervalle in this conference 17 / 39

18 (2,2) solutions without horizons For asymptotically flat solutions with nonzero spin-two Yukawa coefficient C 2 0, one finds numerical solutions that can continue on in to mesh with the (2,2) family found by Frobenius asymptotic analysis around the origin. Such solutions have no horizon; numerical solutions have been found in the m 2 = m 0 theory B. Holdom, Phys. Rev. D66 (2002); hep-th and in the R + C 2 theory Lü, Perkins, Pope & K.S.S., Horizonless solution in R + C 2 theory, behaving as r 2 in both A(r) and B(r) functions as r / 39

19 Wormholes Another solution type found numerically has the character of a wormhole. Such solutions can have either sign of M C 20 and either sign of the falling Yukawa coefficient C 2. As an example, one finds a solution with M < 0 in the R αc 2 theory B(r) f(r) 0.1 In this solution, f (r) = 1/A(r) reaches zero at a point where B(r) = a 2 0 > 0. Making a coordinate change r r 0 = 1 4 ρ2, one then has ds 2 = (a B (r 0 )ρ 2 )dt 2 + dρ2 f (r 0 ) + (r r 0ρ 2 )dω 2 which is Z 2 symmetric in ρ and can be interpreted as a wormhole, with the r < r 0 region excluded from spacetime. 19 / 39

20 Black hole solutions If one assumes the existence of a horizon and assumes also asymptotic flatness at infinity, then a no-hair theorem for the trace of the field equations implies that the Ricci scalar must vanish: R = 0. W. Nelson, ; Lü, Perkins, Pope & K.S.S., This significantly simplifies the analysis of the solutions. The field equations then become identical to those in the β = 0 case, i.e. with just a (Weyl) 2 term and no R 2 term in the action. Counting parameters in an expansion around the horizon, subject to the R = 0 condition, one finds just 3 free parameters. The Schwarzschild solution, satisfying R µν = 0, is just such an asymptotically flat solution with a horizon, and it is characterized by two parameters: the mass M of the black hole, plus a trivial g 00 normalization parameter at infinity. So in the higher-derivative theory, there is just one extra non-schwarzschild parameter in the family of asymptotically flat solutions with a horizon. 20 / 39

21 Moving away from Schwarzschild Considering variation of this non-schwarzschild parameter away from the Schwarzschild value, it is clear that changing it generally has to do something to the solution at infinity. For a solution assumed to have a horizon, and holding R = 0, the only thing that can happen initially is that the rising exponential is turned on, i.e. asymptotic flatness is lost. So, for asymptotically flat solutions with a horizon in the vicinity of the Schwarzschild solution, the only spherically symmetric static solution generally is Schwarzschild itself. One concludes that the Schwarzschild black hole is at least in general isolated as an asymptotically flat solution with a horizon. 21 / 39

22 Non-Schwarzschild Black Holes Lü, Perkins, Pope & K.S.S., ; Now the question arises: what happens when one moves a finite distance away from Schwarzschild in terms of the non-schwarzschild parameter? Does the loss of asymptotic flatness persist, or does something else happen, with solutions arising that cannot be treated by a linearized analysis in deviation from Schwarzchild? This can be answered numerically. In consequence of the trace no-hair theorem, the assumption of a horizon together with asymptotic flatness requires R = 0 for the solution, so the calculations can effectively be done in the R αc 2 theory with β = 0, in which the field equations, thankfully, can be reduced to a system of two second-order equations. 22 / 39

23 The study of non-schwarzschild solutions is more easily carried out with a metric parametrization ds 2 = B(r)dt 2 + dr 2 i.e. by letting A(r) = 1/f (r). f (r) + r 2 (dθ 2 + sin 2 θdφ 2 ), For B(r) vanishing linearly in r r 0 for some r 0, analysis of the field equations shows that one must then also have f (r) similarly linearly vanishing at r 0, and accordingly one has a horizon. One can thus make near-horizon expansions [ ] B(r) = c (r r 0 ) + h 2 (r r 0 ) 2 + h 3 (r r 0 ) 3 + f (r) = f 1 (r r 0 ) + f 2 (r r 0 ) 2 + f 3 (r r 0 ) 3 + and the parameters h i and f i for i 2 can then be solved-for in terms of r 0 and f 1. For the Schwarzschild solution, one has f 1 = 1/r 0, so it is convenient to parametrize the deviation from Schwarzschild using a non-schwarzschild parameter δ with f 1 = 1 + δ r / 39

24 The task then becomes that of finding values of δ 0 for which the generic rising exponential behavior as r is suppressed. What one finds is that there does indeed exist an asymptotically flat family of non-schwarzschild black holes which crosses the Schwarzschild family at a special horizon radius r0 Lich. For α = 1 2, one finds the following phases of black holes: M r 0 Black-hole masses as a function of horizon radius r 0, with a crossing point at r0 Lich The red family denotes Schwarzschild black holes and the blue family denotes non-schwarzschild black holes. 24 / 39

25 The Lichnerowicz Operator Now let us study in some more detail the point where the new black hole family crosses the classic Schwarzschild solution family. We can study solutions in the vicinity of the Schwarzschild family by looking at infinitesimal variations of the higher-derivative equations of motion around a Ricci-flat background. For the δr µν variation of the Ricci tensor away from a background with R µν = 0 one obtains γ(δr µν 1 2 g µν δr) + 2(β 1 3 α)(g µν µ ν )δr 2α (δr µν 1 2 g µν δr) 4α R µρνσ δr ρσ = / 39

26 Restricting attention to asymptotically flat solutions with horizons, however, we know from the trace no-hair theorem that R = 0 so δr = 0 and the δr µν equation simplifies, upon recalling that m2 2 = γ 2α, to ( ) L + m2 2 δr µν = 0, where the Lichnerowicz operator is given by L δr µν δr µν 2R µρνσ δr ρσ. Restricting attention to the m2 2 > 0 nontachyonic case, one sees that black hole solutions deviating from Schwarzschild must have a λ = m2 2 negative Lichnerowicz eigenvalue for δr µν. 26 / 39

27 The Gross-Perry-Yaffe eigenvalue In a study of the thermodynamic instability of the Euclideanised Schwarzschild solution in Einstein theory, Gross, Perry and Yaffe Phys. Rev. D25 (1982), 330 found that there is just one normalisable negative-eigenvalue mode of the Lichnerowicz operator for deviations from the Schwarzschild solution. For a Schwarzschild solution of mass M, it is λ 0.192M 2 i.e. m 2 M Comparing with the numerical results for the new black hole solutions of the higher-derivative gravity theory, this corresponds nicely with the point where the new black hole family crosses the Schwarzschild family. 27 / 39

28 Time Dependent Solutions and Stability Now consider time-dependent perturbations δr µν away from a Schwarzschild solution in order to search for possible instabilities within the higher-derivative theory. For this one needs to analyse the Lichnerowicz condition ( L + m 2 2 ) δr µν = 0 for time-dependent solutions. For asymptotically flat solutions with a horizon, we still have the R = 0 consequence of the trace no-hair theorem, so δr = 0. Then from the Bianchi identity µ R µν = 1 2 νr we obtain µ δr µν = 0, so δr µν must be a TT quantity. The TT condition for δr µν already indicates a similarity to the situation that obtains in Pauli-Fierz theory, where the linearised field equations for a massive spin-two field ψ µν imply µ ψ µν = ψ ν ν = / 39

29 Regge-Wheeler-Schrödinger Form Start from an ansatz for spherically-symmetric time-dependent TT modes, in which h(r) is a function that can be chosen freely ψ 00 = h ψ 0 (r) e νt, ψ 11 = h 1 ψ 1 (r) e νt, ψ 01 = χ(r) e νt ψ ij = r 2 ψ(r) γ ij e νt The TT conditions imply three equations which can be solved for ψ 0, ψ and χ. The Lichnerowicz eigenvalue equation implies two two-derivative equations for the (01) and (00) components. Inserting the results for ψ 0, ψ and χ from the TT conditions, one can solve for the undifferentiated χ and then obtain a second-order equation purely for ψ 1. Following Zerilli s treatment in Einstein theory PRL 24 (1970), 737, one next introduces a new variable φ(r) defined by φ(r) = ν 1 u(r) χ(r) + v(r) ψ(r) where u(r) and v(r) do not depend on the imaginary frequency ν. 29 / 39

30 In order to obtain a Schrödinger form for the Lichnerowicz eigenvalue equation, one requires that in terms of the tortoise coordinate (ranging from at the horizon to + at spatial infinity) r dr r = h(r ) the function φ should satisfy an equation of the form d 2 φ dr 2 = W (r) φ in which W (r) should be of the form W (r) = ν 2 + V (r) where V (r) does not depend on the frequency ν. These conditions allow one to solve for u(r) and v(r). 30 / 39

31 Picking a unit Schwarzschild radius r 0 = 1 for h(r) = 1 1 spacetime dimension D, one obtains as a result the Schrödinger-form equation d 2 φ dr 2 [ ] + ν 2 + V (r) φ = 0 r D 3 in in which the potential V (r) is given by where h(r) V (r) = r D 1 [ 1 2 (D 2)(D 3) λ r D 1 ] Y (r) 2 Y (r) = 1 16 (D 2)3 (D 3) 2 [(D 2) + (D 4)r D 3 ] (D 2)(D 3)λ r D 1 [2D 2 5D + 6 3D(D 2)r D 3 ] (D + 2) λ2 r 2(D 1) [3(D 2) Dr D 1 ] + λ 3 r 3(D 1) λ = m / 39

32 Gregory-Laflamme Instability Analysis of the possibility of growing (Re(ν) > 0) perturbations can be approached using WKB methods B.F. Schutz and C.M. Will, Ap.J. 291:L33 (1985) or numerically. But in fact, the answer has been known for some time from the 5D string R. Gregory and R. Laflamme, PRL 70 (1993) Considering perturbations about the 5D black string ds(5) 2 = ds2 (4) + dz2 ( ) h µν (4) h µz h zν h zz where the z dependence is assumed to be of the form e ikz, one finds that h µν (4) satisfies an equation of the same Lichnerowicz form ( L + k 2 ) h µν (4) = 0 as for δr µν Y.S. Myung, Phys.Rev. D88 (2013). This form is also found for perturbations about the Schwarzschild solution in drgt nonlinear massive gravity. (1) 32 / 39

33 The Gregory-Laflamme instability is an S-wave (l = 0) spherically symmetric instability from the 4D perspective. In the higher-derivative theory, it exists for low-mass Schwarzschild black holes, but disappears for black hole masses M M max where m 2 M max M 2 Pl =.438 This is precisely the crossing point between the family of new black holes and the Schwarzschild family. Note that this monopole instability depends on the presence in the theory of the m 2 massive spin-two mode. 33 / 39

34 Thermodynamic Implications for Instability The D = 4 Wald entropy formula S = 1 8 hdσ ɛ ab ɛ cd L R abcd gives results that respect the first law of black-hole thermodynamics, dm = TdS. + For non-schwarzschild black holes in D = 4, one obtains the following numerical relations between mass, temperature and entropy: M NSch S S S 3 + T NSch S S / 39

35 Recall that for Schwarzschild black holes, one has the classic mass-temperature relation M Sch = 1 8πT. Eliminating the entropy for the non-schwarzschild black holes, one obtains the corresponding relations between black-hole mass and temperature for Schwarzschild and non-schwarzschild black holes: 2 M T Mass versus temperature relations for Schwarzschld (dashed red) and non- Schwarzschild (solid blue) black holes. 35 / 39

36 Free Energy Consequently, for the free energy F = M TS, one has the following situation, showing a switchover between the Schwarzschild and non-schwarzschild solutions: F 0.8 Schwarzschild BH non Schwarzschild BH T Free energy for Schwarzschild (dashed red) and non-schwarzschild (blue) black holes. Lower free energy corresponds to greater stability. 36 / 39

37 Thermodynamic versus dynamical instabilities Gubser and Mitra proposed a relationship between thermodynamic and dynamical instabilities: time-dependent dynamical instability cannot occur without a corresponding theormodynamic instability in the related finite-temperature Euclidean theory. JHEP 0108 (2001) 018 This has been proved in the context of axisymmetric black holes in Einstein theory by Hollands and Wald Commun.Math.Phys. 321 (2013) 629. Assuming the same relation holds between dynamical and thermodynamic instabilities in the higher-derivative gravity theory, and taking into account the known Gregory-Laflamme instability for Schwarzschild black holes below the Lichnerowicz crossing point, one obtains a clear suggestion for the respective domains of stability and instability of the Schwarzschild and non-schwarzschild black holes. 37 / 39

38 1 M (hot) Unstable (cold) Stable (cold) Stable Lichnerowicz crossing & stability boundary (hot) Unstable r Schwarzschild BH non Schwarzschild BH - 4 Classical stability regimes. The dashed red line denotes Schwarzschild black holes and the solid blue line denotes non-schwarzschild black holes. 38 / 39

39 Some Related papers K.S.S., Phys.Rev. D16 (1977) 953 K.S.S, Gen.Rel.Grav. 9 (1978) 353 H. Lü, Y. Pang, C.N. Pope & J.F. Vásquez-Poritz, Phys.Rev. D86 (2012) ; H. Lü, A. Perkins, C.N. Pope & K.S.S., PRL 114, (2015); H. Lü, A. Perkins, C.N. Pope & K.S.S., Phys. Rev. D92 (2015) 12, ; H. Lü, A. Perkins, C.N. Pope & K.S.S.,Phys. Rev. D96 (2017) 4, ; K. Kokkotas, R.A. Konoplya & A. Zhidenko, Phys. Rev. D96 (2017) 6, ; / 39