Causality in Gauss-Bonnet Gravity

Transcription

1 Causality in Gauss-Bonnet Gravity K.I. Phys. Rev. D 90, July Keisuke Izumi ( 泉圭介 ) (National Taiwan University, LeCosPA) -> (University of Barcelona, ICCUB)

2 From Newton to Einstein Newton s gravity theory (1687) Physics in solar system Perihelion precession of Mercury Dark Planet Vulcan?? Einstein s general relativity (1916) Cosmology Cosmic acceleration and galaxy rotation problem Dark energy and dark matter??? New gravity theory?? (20??)

3 High Energy physics of Gravity General relativity Singularity theorem (Hawking, Penrose 1970) Singularity inside BH Initial singularity of universe Quantization of Gravity Problem of renormalization Algebraic structure of generators

4 Modification of Gravity General relativity Singularity problem Quantum gravity Problem of renormalization Cosmology Dark energy, Dark matter Modified Gravity Modification of Lagrangian : f(r), Gauss-Bonnet Modification of vacuum state : ghost condensation Modification of concept of geometry higher dimension : Braneworld other manifold : Teleparallel gravity Introducing mass of graviton : massive gravity

5 Consistency Check of modified gravity 0-th order (of cosmology) : FLRW universe without perturbation Consistency with standard cosmology DM and DE?? 1-th order : perturbation on FLRW background Consistency with standard cosmology Consistency with solar system physics Stability Nonlinear property Causal structure Nonlinear stability Main topic in this talk Quantization

6 Gauss-Bonnet gravity action S = R d n x â ã 1 fr à 2Ë + ë(r 2 à 4R 2ô Dà2 AB R AB + R ABCD R ABCD )g + L m GB terms Matter contribution EoM G AB + Ëg AB à 2 ë H AB = 2ô Dà2 T AB H AB := (R 2 à4r CD R CD +R CDEF R CDEF )g AB à 4(RR AB à 2R AC R C B à 2R ACBDR CD + R ACDE R CDE ) B Higher curvature but no higher order derivative in EoN Defined in higher dimension (In 4-dim., GB term becomes surface term.)

7 Superluminal mode by curvature coupling Curvature coupling in kinetic term (g ö + ër ö )r ö r þ + V 0 (þ) = 0 GB gravity G ö + fr ã Rg ö = 0 linerlization g ö r ö r h ëì + frg ö r ö r h ëì = 0 Causal relation based on null curve is meaningless. We need to use the fastest propagation for analysis of causal structure. Questions When do superluminal modes appear? Are black holes well-defined? Where is the horizon in the sense of causality? C. Aragone (1988) Choquet-Bruhat (1988) Superluminal modes potentially appear. (GB gravity )

8 1,introduction 2, Origin of superluminal modes 3, Brief review of characteristic 3.1, intuitive understanding 3.2, Way of analysis 4,Characteristics in GB gravity 5,Summary Contents

9 1,introduction 2, Origin of superluminal modes 3, Brief review of characteristic 3.1, intuitive understanding 3.2, Way of analysis 4,Characteristics in GB gravity 5,Summary Contents

10 Example: Scalar field With canonical kinetic term g ö r ö r þ + V 0 (þ) = 0 High energy limit g ö k ö k þ k = 0 S. Mukohyama, J.-P. Uzan (2013) k ö is null for metric g ö With non-canonical kinetic term (g ö + ër ö r )r ö r þ + V 0 (þ) = 0 R ö High energy limit gà ö k ö k þ k = 0 gà ö := g ö + ër ö r R ö k ö is null for effective metric gà ö

11 Gravity theory Many complications h nonlinear kinetic terms h n :: : graviton ::@h :: Nonlinear kinetic terms make the discussion non-trivial. constraints The structures of kinetic term depend on constraints.. gauges Due to gauge, some variables are not physical. General Relativity The causal structure has been well studied, and gravitational waves propagate with speed of light.

12 Extension of GR Massive gravity, f(t) gravity Additional constraint No gauge Gauss-Bonnet gravity Superluminal modes, Acausality Y.C. Ong, K.I. J. M. Nester, P. Chen (2013) K.I., J.-A. Gu, Y.C. Ong (2013) K.I., Y.C. Ong (2013) S. Deser, K.I., Y.C. Ong, A. Waldron (2013) EOM : G ö + fr ã Rg ö = 0 linerlization g ö r ö r h ëì + frg ö r ö r h ëì = 0 Kinetic terms are modified. Superluminal modes C. Aragone (1988) Choquet-Bruhat (1988)

13 Superluminal mode and Acausality Superluminal mode Propagation the speed of which is higher than that of light time Light cone Acausality time Pathological causal structure Light cone Superluminal propagation Closed information propagation With Lorentz symmetry, superluminal modes space result in acausality. But without Lorentz symmetry, it is not always true. space

14 1,introduction 2, Origin of superluminal modes 3, Brief review of characteristic 3.1, intuitive understanding 3.2, Way of analysis 4,Characteristics in GB gravity 5,Summary Contents

15 CC Causal structure and Propagation Time evolution Time p Cauchy development CC Initial conditions

16 CC Causal structure and Propagation Time evolution Time Path of light q Superluminal mode Cauchy development Characteristic hypersurface CC Cauchy We can not solve development Initial conditions time evolution beyond this hypersurface

17 Characteristics Î is characteristic á propagation Î

18 1,introduction 2, Origin of superluminal modes 3, Brief review of characteristic 3.1, intuitive understanding 3.2, Way of analysis 4,Characteristics in GB gravity 5,Summary Contents

19 Characteristics (1Dim-ODE) Quasi-linear n-th order differential equation t þ; á á á; (@ t ) nà1 þ](@ t ) n þ t þ; á á á; (@ t ) nà1 þ] = 0 Time evolution 1,Initial condition; t þ; á á á; (@ t ) nà1 þ at t = t 0 If t þ; á á á; (@ t ) nà1 þ]6=0 2, EoM (@ t ) n þj t=t0 ( (@ t ) n þj t=t0 is uniquely fixed) 3, (@ t ) kà1 þ[t = t 0 + Ét] = (@ t ) k þ[t = t 0 ]Ét + (@ t ) kà1 þ[t = t 0 ] (1 ô k ô n)

20 Characteristics (1Dim-ODE) Quasi-linear n-th order differential equation t þ; á á á; (@ t ) nà1 þ](@ t ) n þ t þ; á á á; (@ t ) nà1 þ] = 0 Time evolution 1,Initial condition; t þ; á á á; (@ t ) nà1 þ at t = t 0 If t þ; á á á; (@ t ) nà1 þ] = 0 2, EoM (@ t ) n þj t=t0 ( (@ t ) n þj t=t0 can be arbitrary) 3, (@ t ) kà1 þ[t = t 0 + Ét] = (@ t ) k þ[t = t 0 ]Ét + (@ t ) kà1 þ[t = t 0 ] (@ t ) nà1 þ[t = t 0 + Ét] =?? (1 ô k ô n à 1)

21 Characteristics (Single field PDE) Quasi-linear n-th order differential equation f 1á á á n ö þ; á á ö1 á á á@ önà1 þ]@ 1 á á á@ n þ ö þ; á á ö1 á á á@ önà1 þ] = 0 Time evolution If 1,Initial condition on initial hypersurface t þ; á á á; (@ t ) nà1 þ i i j þ; á á á Î : t = t 0 f tá á át ö þ; á á ö1 á á á@ önà1 þ]6=0 þ[x i ] à Î Î þ[x i + Éx i ] 3, EoM (@ t ) n þj t=t0 ( is uniquely fixed) (@ t ) n þj t=t0 Time evolution is i þ = lim Éx i! 0 þ[x i +Éx i ]àþ[x i ] Éx i

22 Characteristics (Single field PDE) Quasi-linear n-th order differential equation f 1á á á n ö þ; á á ö1 á á á@ önà1 þ]@ 1 á á á@ n þ ö þ; á á ö1 á á á@ önà1 þ] = 0 Time evolution If 1,Initial condition on initial hypersurface t þ; á á á; (@ t ) nà1 þ i i j þ; á á á f tá á át ö þ; á á ö1 á á á@ önà1 þ] = 0 3, EoM (@ t ) n þj t=t0 Î : t = t 0 ( can be arbitrary) (@ t ) n þj t=t0 Time evolution is not unique þ[x i i þ = lim à Éx i! 0 Î Î þ[x i + Éx i ] þ[x i +Éx i ]àþ[x i ] Éx i

23 1,introduction 2, Origin of superluminal modes 3, Brief review of characteristic 3.1, intuitive understanding 3.2, Way of analysis 4,Characteristics in GB gravity 5,Summary Contents

24 Gauss-Bonnet gravity action S = R d n x â ã 1 fr à 2Ë + ë(r 2 à 4R 2ô Dà2 AB R AB + R ABCD R ABCD )g + L m GB terms Matter contribution EoM G AB + Ëg AB à 2 ë H AB = 2ô Dà2 T AB H AB := (R 2 à4r CD R CD +R CDEF R CDEF )g AB à 4(RR AB à 2R AC R C B Check the characteristic for EoM à 2R ACBDR CD + R ACDE R CDE ) B Assumption : T AB does not involve the highest order derivatives of metric

25 First-order formalism EoM: Second order derivatives of metric Introducing new variables: Levi-Civita connection È ABC = g AD È D := 1 (@ BC 2 C g AB B g AC A g BC ) + F[È EFG ; g EF ] First order differential equation R ABCD C È ABD D È ABC + á á á # of variables (D+1)D g AB : 2 (D+1)D È CAB : 2 2 Symmetric w.r.t A ; B

26 Decomposition A ö are vectors 0 is independent vector ö ñ ø := ð A dx A = dt Projection operator P A B = îa B à øa ð B Decomposition of @x ö V A : V 0 := ø A V A ; V ö := P A ö V A V A : V 0 := ð A V A ; V ö := P ö A VA

27 Trivial characteristics From definition of Levi-Civita connection 2È g 00 2È ë00 = 2@ 0 g 0ë ë g 00 2È 0ëì =@ ë g 0ì ì g 0ë 0 g ëì Characteristics for g AB For Levi-Civita connection 0 = R 00ë0 0 È 00ë ë È á á á 0 = R ìí0ë à R 0ëìí 0 È ìíë ë È ìí0 ì È 0ëí í È 0ëì + á á á Using definition of Levi-Civita connection È ëì0 = à È 0ëì ì g ë0 Characteristics for È 00ë ; È ëìí Replace È ëì0 by à È 0ëì

28 Gauge modes We have fixed time evolution of g AB ; È 00ë ; È ëìí ; È ëì0 Remaining variables are È 000 ; È ë00 ; È 0ëì DoF: 1 ; (D à 1) ; (D à 1)D=2 Diff. inv. : # of gauge DoF is 0 È È ë00 never appear in EoM. Gauge modes È 0ëì are physical. Check the characteristics for È 0ëì!

29 General 0 È 0ëì appears only in R 0ë0ì ( = à R ë00ì = á á á ) G AB = R 0ë0ì A AB;ëì +á á á (@ 0 È ëì0 = 0 È 0ëì + á á á ) A 0 È 0ëì + á á á = 0 Characteristic equation A 00;ëì = A 0ö;ëì = A ö0;ëì = 0 A ö ;ëì = g 00 (h ëö h ì à h ëì h ö ) h ö := g ö à g 0ö g 0 =g 00 ( h ö is induced metric of Σ) Null hypersurface g 00 = 0 h ö = diag (0; 1; 1; á á á) h 11 = O[(g 00 ) à1 ] à g 00 h 11 P 0È 0ii + á á á = 0 g 00 h 0 È 01i + á á á = 0 à g 00 h 0 È 011 î ij + á á á = 0 (1; 1) -component (1; i) -component (i; j) -component 1 :null direction i :normal to 1 All component have the same kinetic term D(D à 3)=2 degeneracies Gravitational wave

30 Gauss-Bonnet correction G AB + Ëg AB à 2 ë H AB = 2ô Dà2 T AB H AB = R 0ë0ì B AB;ëì + á á á B 00;ëì = B 0ö;ëì = B ö0;ëì = 0 GR: A 0 È 0ëì + á á á = 0 Null hypersurface à g 00 h 11 P 0È 0ii + á á á = 0 g 00 h 11 P 0È 01i + á á á = 0 à g 00 h 11 P 0È 011 î ij + á á á = 0 ða AB;ëì à 2 ë B AB;ëì 0 È 0ëì + á á á = 0 (1; 1) -component (1; i) -component (i; j) -component

31 Gauss-Bonnet correction Èö 0ëì 0 È 0ëì +á á á = 0 (1; 1) -component +á á á = 0 (1; i) -component (i; j) -component +á á á = 0

32 Example1: All degeneracies are resolved R ijkl = R 1ijk = 0 R 1i1j = Cî ij à g 00 h 11 P i È ö 0ii + á á á = 0 g 00 h 11 P i È ö 01i + á á á = 0 (1; 1) -component (1; i) -component +á á á = 0 Characteristic hypersurface is not null. (i; j) -component The speed of graviton is not that of light.

33 Expample2: still degenerated R 1i1j = R 1ijk = 0 ] +á á á = 0 (i; j) -component Only Èö 011 appears. Characteristic hypersurface is still null. Killing horizon R 1i1j = R 1ijk = 0 Killing horizon is exactly causal edge. It can be extended to Lovelock theory. (Reall, Tanahashi & Way 2014)

34 Dynamical case R AB U A U B = 0 (spherically symmetric case) U A : Radial null vector Null hypersurface tangent to U A is characteristics. R AB U A U B 6=0 Modification to characteristic equation G AB + Ëg AB à 2 ë H AB = 2ô Dà2 T AB Deviate from null balanced If curvature is enough small, we can neglect the contribution from GB term

35 Dynamical case R AB U A U B > 0 R AB U A U B < 0 Subluminal (spherically symmetric case) Superluminal Einstein Branch T AB U A U B õ 0 R AB U A U B õ 0 H. Maeda, M. Nozawa (2008) GB Branch T AB U A U B õ 0 R AB U A U B ô 0 On Einstein Branch with Null Energy condition (This might be physical condition) No superluminal mode, but subluminal modes appear No acausality, but gravitational Cherenkov happens

36 Dynamical case R AB U A U B > 0 R AB U A U B < 0 Subluminal (spherically symmetric case) Superluminal Einstein Branch T AB U A U B õ 0 R AB U A U B õ 0 H. Maeda, M. Nozawa (2008) GB Branch T AB U A U B õ 0 R AB U A U B ô 0 Evaporating BH Contraposition of Area-increasing low R AB U A U B < 0 Superluminal Graviton can escape from Evaporating BH defined by null curves

37 Summary In GB gravity, gravitational propagation potentially becomes superluminal. Causal structure based on null curve is meaningless. We need to analyze it by using the fastest propagation. Causal structure is analyzed with method of characteristics, which is powerful technique. Killing horizon is the event horizon in the sense of causality. On spherically symmetric background, R AB U A U B = 0 R AB U A U B > 0 R AB U A U B < 0 luminal. subluminal. superluminal. gravitational Cherenkov Acausality? Need to check it.