Classical Solutions to Quantum Corrected Gravity K.S. Stelle Imperial College London Abdus Salam Memorial Meeting Nanyang Technological University, Singapore January 27, 2016 Work together with Alun Perkins, Hong L¨ u and Chris Pope PRL 114, 171601 (2015), Phys.Rev. D92 (2015) 12, 124019 1 / 38
1978 Discussions with Salam When I arrived at Imperial College as a postdoc in the autumn of 1978, many of the early conversations I had with Professor Salam revolved around the rˆ ole of higher derivative terms in quantum gravity. 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 Poincar´ e 20 , 69 (1974) d 4 x √− g ( α C µνρσ C µνρσ + β R 2 ) terms ab initio in � Inclusion of 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 R 2 )]. 2 / 38
PHYSICAL REVIK% 18, NUMBER 15 DECEMBER 1978 VOLUME 12 D 'I Remarks on high-energy stability and renormalizability of gravity theory Abdus Salam Center for Theoretical International Physics, Trieste, Italy of Theoretical and Department Physics, Imperial College, London, England J. Strathdee Center for Theoretical International Physics, Trieste, Italy (Received 2 March 1978; revised manuscript received 5 June 1978) that high-energy (Froissart) boundedness of gravitational cross sections may make it necessary to Arguing R ' and R""R„„, we suggest Einstein's if supplement Lagrangian with terms containing criteria which, satisfied, could make the tensor ghost in such a theory innocuous. I. PROPOSALS FOR RENORMALIZING GRAVITY H. STABLE HIGH-DERIVATIVE THEORIES At present there are two views about renormal- Since the Lagrangians we wish to consider con- ization prospects of quantum gravity. tain higher than second-order derivatives, we (i) S-matrix elements, as contrasted to Green's first examine these for high-energy stability. may be finite. This result substanti- A theory is stable if, in each order of a perturba™ functions, in ex- ated at the two-loop level for the S matrix in mo- tion expansion, the high-energy behavior tended supergravities, it is hoped, hold also menta k does not increase, may, except to the extent of for Green's once supergravities are re- functions, powers of logarithms (logk'). Conventional formalism. ' within a superfield formulated normalizable theories are stable'; so are higher- (ii) Gravity may be renormalizable, but non- derivative theories, provided the number of perturbatively. Two nonperturbative techniques derivatives in the interaction does not Lagrangian tech- have been suggested: (a} the nonpolynomial exceed the number in the free Lagrangian. nique, ' which relies of "cocoon" on a summation A. theories. Pro- Conventional renomsalizable totype 4 = z(&y)' - Ap . Since (yrp) = 1/x', y -1/x using the formula graphs, for x-0 in the Wilson-product-expansion sense, —, ); n (y"(x)y" (O)) =n l and 4 is no more singular 1/x~. For such than matrix elements F (k) with E external theories, 3 / 38 like k' lines are stable and behave (barring (b) the gauge technique, ' which relies on a solu- A y' theory (y'-1/x') is logarithmic factors). with I' (k)- k' tion of Dyson-Schwinger' equations, ", where n is the by making suPerstable use of a nonperturbative solution of gauge iden- order of perturbation. function &"' tities connecting the inverse Green's B. Hi gher-derivati ve theories. with the vertex operators I'. I z+ Lzz+Lrzz (i) and (ii) (a) but not (ii) (b) suffer Both proposals from one serious defect. The high-energy behavior of matrix elements in each order of approximation like (K'k')". Thus any (Froissart} increases boundedness of cross sections' can become man- ifest only after a further summation of the per- series — turbation a task surely not to be under- taken lightly. In order to improve high-energy behavior, we wish to revive the suggestion' that the Einstein 4 4xzz -" (g4M ) (4) Lagrangian (R) should be supplemented by higher- derivative Lagrangians containing terms of the All g's are dimensionless. type' 8""A~„and 8'. Such Lagrangians The theory contains have been shown to be renormalizable. ' a positive-norm massless and a negative-norm they con- However, massive particle of mass M. Since tain ghosts. Based on a renormalization-group (yq ) „=, M' logx', i. e. , ( — - (logx') ~', (y we suggest criteria which, if investigation, satisfied, could make the ghosts innocuous. 18 4480
In 1978, Salam and Strathdee argued, on the basis of Froissart boundedness for gravitational cross sections, that quadratic curvature terms ought to be included in the initial gravitational action and proposed ideas on how the resulting tensor ghost could be made innocuous. At the end of the paper, they suggested that the ghost might be avoided should there be a nontrivial ultraviolet fixed point for the quadratic curvature coefficients. A. Salam and J. Strathdee, Phys. Rev. D18 , 4480 (1978) More recently, this perspective has been turned on its head. We now know that the quadratic curvature theory is asymptotically free Fradkin & Tseytlin 1982, Avramidy & Barvinsky 1985 in the higher-derivative couplings. This has been exploited in the asymptotic safety scenario, considering the possibility that there may be a non-Gaussian renormalization-group fixed point for Newton’s constant and the cosmological constant with associated flow trajectories on which the ghost states arising from the (Weyl) 2 term could be absent. S. Weinberg 1976, M. Reuter 1996, M. Niedermaier 2009 4 / 38
Another context in which quadratic curvature has seriously been considered is inflation. At the linearized level, I had showed that the − R + β R 2 theory is equivalent to a theory with ordinary massless spin-two plus a non-ghost massive spin-zero mode. K.S.S., Gen.Rel.Grav. 9 (1978) 353 Brian Whitt later generalized this to the nonlinear level. B. Whitt, Phys. Lett. B145 (1984) 176 d 4 x √− g ( − R + β R 2 ) model � This was the basis for Starobinsky’s for inflation. A.A. Starobinsky 1980; Mukhanov & Chibisov 1981 It has has been quoted (at times) as a good fit to CMB fluctuation data from the Planck satellite. J. Martin, C. Ringeval and V. Vennin, 1303.3787 In order for this to work, the dimensionless R 2 coefficient needs to be large, giving a scale for the spin-zero mode mass around five orders of magnitude below the Planck scale. 5 / 38
Classical gravity with higher derivatives Here, we shall simply adopt the point of view that it may be appropriate to take the higher-derivative terms and their consequences for gravitational solutions seriously in an effective theory of quantum gravity. We consider the gravitational action d 4 x √− g ( γ R − α C µνρσ C µνρσ + β R 2 ) . � I = The field equations following from this higher-derivative action are � � R µν − 1 + 2 H µν = γ 2 g µν R 3 ( α − 3 β ) ∇ µ ∇ ν R − 2 α � R µν � � +1 β + 2 3 ( α + 6 β ) g µν � R − 4 α R ηλ R µηνλ + 2 3 α RR µν +1 � � β + 2 � � = 1 2 α R ηλ R ηλ − R 2 3 α 2 g µν 2 T µν 6 / 38
Full nonlinear field equations for spherical symmetry Use Schwarzschild coordinates ds 2 = − B ( r ) dt 2 + A ( r ) dr 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) The first equation contains the third-order derivative B (3) = B ′′′ 7 / 38
The second equation contains the third-order derivative A (3) = A ′′′ : 8 / 38
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 restricting attention to field equations linearized in h µν , or equivalently by restricting attention to quadratic terms in h µν in the action. The action then becomes � d 4 x {− 1 4 h µν (2 α � − γ ) � P (2) µνρσ h ρσ I Lin = +1 2 h µν [6 β � − γ ] � P (0; s ) µνρσ h ρσ } ; 1 P (2) 2( θ µρ θ νσ + θ µσ θ νρ ) − P (0; s ) = µνρσ µνρσ 1 P (0; s ) = 3 θ µν θ ρσ θ µν = η µν − ω µν ω µν = ∂ µ ∂ ν / � , µνρσ where the indices are lowered and raised with the background metric η µν . 9 / 38
From this linearized action one deduces the dynamical content of the linearized theory: ◮ positive-energy massless spin-two 1 2 (2 α ) − 1 ◮ negative-energy massive spin-two with mass m 2 = γ 2 1 2 (6 β ) − 1 ◮ positive-energy massive spin-zero with mass m 0 = γ 2 K.S.S. 1978 10 / 38
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