Bimetric theory, partial masslessness and conformal gravity Fawad - PowerPoint PPT Presentation

Bimetric theory, partial masslessness and conformal gravity Fawad Hassan Stockholm University, Sweden GGI Conference on “Higher Spins, Strings and Duality” Florence, May 6-9, 2013

Based on ◮ SFH, Angnis Schmidt-May, Mikael von Strauss arXiv:1203.5283, 1204.5202,1208:1515, 1208:1797, 1212:4525, 1303.6940 ◮ SFH, Rachel A. Rosen, arXiv:1103.6055, 1106.3344, 1109.3515, 1109.3230, 1111.2070

Outline of the talk Review: Linear and Nonlinear massive spin-2 fields Ghost-free bimetric theory Mass spectrum of bimetric theory Partially Massless bimetric theory Higher drivative gravity & Conformal gravity Equivalence between CG and PM bimetric theory Higher drivative gravity from bimetric theory Discussion

Outline of the talk Review: Linear and Nonlinear massive spin-2 fields Ghost-free bimetric theory Mass spectrum of bimetric theory Partially Massless bimetric theory Higher drivative gravity & Conformal gravity Equivalence between CG and PM bimetric theory Higher drivative gravity from bimetric theory Discussion

Linear massive spin-2 fields The Fierz-Pauli equation: Linear massive spin-2 field, h µν , in background ¯ g µν � � � � + m 2 ¯ h µν − 1 2 ¯ h µν − ¯ E ρσ µν h ρσ − Λ g µν h ρ g µν h ρ = 0 FP 2 ρ ρ [Fierz-Pauli, 1939] ◮ 5 propagating modes (massive spin-2) ◮ Massive gravity (?) ◮ What determines ¯ g µν ? (flat, dS, AdS, · · · ) ◮ Nonlinear generalizations? [The Boulware-Deser ghost (1972)]

Nonlinear massive spin-2 fields ◮ “Massive gravity” (fixed f µν ): � � √− g R − m 2 V ( g − 1 f ) L = m 2 p ◮ Interacting spin-2 fields (dynamical g and f ): � � √− g R − m 2 V ( g − 1 f ) L = m 2 + L ( ∇ f )(?) p

Nonlinear massive spin-2 fields ◮ “Massive gravity” (fixed f µν ): � � √− g R − m 2 V ( g − 1 f ) L = m 2 p ◮ Interacting spin-2 fields (dynamical g and f ): � � √− g R − m 2 V ( g − 1 f ) L = m 2 + L ( ∇ f )(?) p √ L ( ∇ f ) = m 2 Bimetric: − f R f (?) f [Isham-Salam-Strathdee, 1971, 1977] Generically, both contain a GHOST at the nonlinear level [Boulware-Deser, 1972]

Counting modes: Generic massive gravity: ◮ Linear modes: 5 (massive spin-2) ◮ Non-linear modes: 5 + 1 (ghost) Generic bimetric theory: ◮ Linear modes: 5 (massive, δ g − δ f ) + 2 (massless, δ g + δ f ) ◮ Non-linear modes: 7 + 1 (ghost) Complication: Since the ghost shows up nonlinearly, its absence needs to be established nonlinearly

Construction of ghost-free nonlinear theories Based on “Decoupling limit” analysis: � g − 1 η ) was obtained and shown to be A specific V dRGT ( ghost-free in a “decoupling limit”, also perturbatively in h = g − η [de Rham, Gabadadze, 2010; de Rham, Gabadadze, Tolley, 2010] Non-linear Hamiltonian methods (non-perturbative):

Construction of ghost-free nonlinear theories Based on “Decoupling limit” analysis: � g − 1 η ) was obtained and shown to be A specific V dRGT ( ghost-free in a “decoupling limit”, also perturbatively in h = g − η [de Rham, Gabadadze, 2010; de Rham, Gabadadze, Tolley, 2010] Non-linear Hamiltonian methods (non-perturbative): Questions not answerable by “decoupling limit”: � ◮ Is massive gravity with V ( g − 1 η ) ghost-free nonlinearly? [SFH, Rosen (1106.3344, 1111.2070)] ◮ Is it ghost-free for generic fixed f µν ? [SFH, Rosen, Schmidt-May (1109.3230)] ◮ Can f µν be given ghost-free dynamics? [SFH, Rosen (1109.3515)]

Outline of the talk Review: Linear and Nonlinear massive spin-2 fields Ghost-free bimetric theory Mass spectrum of bimetric theory Partially Massless bimetric theory Higher drivative gravity & Conformal gravity Equivalence between CG and PM bimetric theory Higher drivative gravity from bimetric theory Discussion

Ghost-free bimetric theory Digression: Elementary symmetric polynomials of X with eigenvalues λ 1 , · · · , λ 4 : e 0 ( X ) = 1 , e 1 ( X ) = λ 1 + λ 2 + λ 3 + λ 4 , e 2 ( X ) = λ 1 λ 2 + λ 1 λ 3 + λ 1 λ 4 + λ 2 λ 3 + λ 2 λ 4 + λ 3 λ 4 , e 3 ( X ) = λ 1 λ 2 λ 3 + λ 1 λ 2 λ 4 + λ 1 λ 3 λ 4 + λ 2 λ 3 λ 4 , e 4 ( X ) = λ 1 λ 2 λ 3 λ 4 = det X .

Ghost-free bimetric theory Digression: Elementary symmetric polynomials of X with eigenvalues λ 1 , · · · , λ 4 : e 0 ( X ) = 1 , e 1 ( X ) = λ 1 + λ 2 + λ 3 + λ 4 , e 2 ( X ) = λ 1 λ 2 + λ 1 λ 3 + λ 1 λ 4 + λ 2 λ 3 + λ 2 λ 4 + λ 3 λ 4 , e 3 ( X ) = λ 1 λ 2 λ 3 + λ 1 λ 2 λ 4 + λ 1 λ 3 λ 4 + λ 2 λ 3 λ 4 , e 4 ( X ) = λ 1 λ 2 λ 3 λ 4 = det X . e 0 ( X ) = 1 , e 1 ( X ) = [ X ] , 2 ([ X ] 2 − [ X 2 ]) , e 2 ( X ) = 1 6 ([ X ] 3 − 3 [ X ][ X 2 ] + 2 [ X 3 ]) , e 3 ( X ) = 1 24 ([ X ] 4 − 6 [ X ] 2 [ X 2 ] + 3 [ X 2 ] 2 + 8 [ X ][ X 3 ] − 6 [ X 4 ]) , 1 e 4 ( X ) = e k ( X ) = 0 for k > 4 , e n ( X ) ∼ ( X ) n [ X ] = Tr ( X ) ,

◮ The e n ( X ) ’s and det ( 1 + X ) : � 4 det ( 1 + X ) = n = 0 e n ( X ) ◮ Introduce “deformed determinant” : � 4 � det ( 1 + X ) = n = 0 β n e n ( X )

◮ The e n ( X ) ’s and det ( 1 + X ) : � 4 det ( 1 + X ) = n = 0 e n ( X ) ◮ Introduce “deformed determinant” : � 4 � det ( 1 + X ) = n = 0 β n e n ( X ) ◮ Observation: � � � 4 g − 1 f ) = g − 1 f ) V ( n = 0 β n e n ( [SFH & R. A. Rosen (1103.6055)]

Ghost-free bi-metric theory Ghost-free combination of kinetic and potential terms for g & f : � 4 � � √− gR g − 2 m 4 √− g L = m 2 g − 1 f ) + m 2 β n e n ( − f R f g f n = 0 [SFH, Rosen (1109.3515,1111.2070)] Note, � � 4 4 � � � √− g g − 1 f ) = f − 1 g ) β n e n ( − f β 4 − n e n ( n = 0 n = 0 Hamiltonian analysis: 7 nolinear propagating modes, no ghost! C 2 ( γ, π ) = d C ( γ, π ) = 0 , dt C ( x ) = { H , C } = 0

Outline of the talk Review: Linear and Nonlinear massive spin-2 fields Ghost-free bimetric theory Mass spectrum of bimetric theory Partially Massless bimetric theory Higher drivative gravity & Conformal gravity Equivalence between CG and PM bimetric theory Higher drivative gravity from bimetric theory Discussion

Mass spectrum of bimetric theory [SFH, A. Schmidt-May, M. von Strauss 1208:1515, 1212:4525] � � � � d √ √ gR g − 2 m d √ g d d x m d − 2 β n e n ( S )+ m d − 2 S gf = − fR f g f n = 0 Three Questions: ◮ Q1: When are the 7 fluctuations in δ g µν , δ f µν good mass eigenstates? (FP mass) ◮ Q2: In what sense is this Massive spin-2 field + gravity ? ◮ Q3: How to characterize deviations from General Relativity? 2 g µν R ( g ) + V g µν = T g R µν ( g ) − 1 µν R µν ( f ) − 1 2 f µν R ( f ) + V f µν = T f µν

Proportional backgrounds A1: FP masses exist only around, ¯ f µν = c 2 ¯ g µν g and f equations: � Λ g � � T g � g ) − 1 R µν (¯ 2 ¯ g µν R (¯ ¯ µν g ) + g µν = 0 or T f Λ f µν � d − 1 � � d − 1 � d − 1 d � � Λ g = m d Λ f = m d c k β k , c k + 2 − d β k m d − 2 m d − 2 k k − 1 g f k = 0 k = 1 Implication: Λ g = Λ f ⇒ c = c ( β n , α ≡ m f / m g ) (Exception: Partially massless (PM) theory)

Mass spectrum around proportional backgrounds Linear modes: � � � � 1 δ f µν − c 2 δ g µν δ g µν + α d − 2 c d − 4 δ f µν δ M µν = , δ G µν = 2 c � � ¯ δ G µν − 1 2 ¯ g µν ¯ E ρσ µν δ G ρσ − Λ g g ρσ δ G ρσ = 0 , � � ¯ δ M µν − 1 2 ¯ g µν ¯ E ρσ µν δ M ρσ − Λ g g ρσ δ M ρσ � � + 1 2 m 2 δ M µν − ¯ g µν ¯ g ρσ δ M ρσ = 0 FP The FP mass of δ M : � d − 2 � � 1 + ( α c ) 2 − d � d − 1 � m d m 2 c k β k FP = m d − 2 k − 1 g k = 1

Bimetric as massive spin-2 field + gravity A2: The massless mode is not gravity! �� � ρ G µν = g µν + c d − 4 α d − 2 f µν , M G g − 1 f µν = G µρ ν − cG µν G µν has no ghost-free matter coupling!

Bimetric as massive spin-2 field + gravity A2: The massless mode is not gravity! �� � ρ G µν = g µν + c d − 4 α d − 2 f µν , M G g − 1 f µν = G µρ ν − cG µν G µν has no ghost-free matter coupling! Hence: ◮ Gravity: g µν �� � ρ ◮ Massive spin-2 field: g − 1 f M µν = g µρ ν − cg µν ◮ m g >> m f : g µν mostly massless (opposite to massive gr.)

Bimetric as massive spin-2 field + gravity A2: The massless mode is not gravity! �� � ρ G µν = g µν + c d − 4 α d − 2 f µν , M G g − 1 f µν = G µρ ν − cG µν G µν has no ghost-free matter coupling! Hence: ◮ Gravity: g µν �� � ρ ◮ Massive spin-2 field: g − 1 f M µν = g µρ ν − cg µν ◮ m g >> m f : g µν mostly massless (opposite to massive gr.) A3: M µν = 0 ⇒ GR. M µν � = 0 ⇒ deviations from GR, driven by matter couplings

Outline of the talk Review: Linear and Nonlinear massive spin-2 fields Ghost-free bimetric theory Mass spectrum of bimetric theory Partially Massless bimetric theory Higher drivative gravity & Conformal gravity Equivalence between CG and PM bimetric theory Higher drivative gravity from bimetric theory Discussion