Massive gravity and Caustics: A definitive covariant constraint analysis

Transcription

1 Massive gravity and Caustics: A definitive covariant constraint analysis Cosmo Chicago George Zahariade UC Davis August 29, / 18

2 Goals Motivation and goals de Rham-Gabadadze-Tolley massive gravity: no Boulware-Deser ghost Presence of superluminalities (cf Galileons in the decoupling limit): intense debate in the community First fully non-linear propagation analysis S. Deser, M. Sandora, A. Waldron, GZ, arxiv: Method of characteristic surfaces Reliance on covariant constraint analysis Potential propagation pathologies... 2 / 18

3 Outline Covariant constraint analysis Spacelike characteristic surfaces Discussion 3 / 18

4 Constraint analysis Covariant constraint analysis Spacelike characteristic surfaces Discussion 4 / 18

5 Constraint analysis General setting First order Cartan formalism in 4D 4 vierbein 1-forms e m := e µ m dx µ (16 fields) 6 connection 1-forms ω mn := ω µ mn dx µ (24 fields) drgt action (4 fiducial vierbein 1-forms f m := f µ m dx µ ) S = 1 4 m 2 ɛ mnrs e m e n [ dω rs + ω r tω ts] ï ɛ mnrs e m β0 4 en e r e s + β 1 3 en e r f s + β ò 2 2 en f r f s + β 3 f n f r f s 5 / 18

6 Constraint analysis Equations of motion Zero torsion condition T m := e m := de m + ω m ne n 0 Einstein equations G m := G m m 2 t m 0 Einstein 3-forms G m := 1 2 ɛ mnrse n [dω rs + ω r tω ts ] Mass term 3-forms t m := ɛ mnrs [β 0 e n e r e s + β 1 e n e r f s + β 2 e n f r f s + β 3 f n f r f s ] 40 eoms for 40 dynamical fields 6 / 18

7 Constraint analysis Primary constraints Space-time decomposition of a p-form (p < 4) θ := θ + θ where θ dt = 0 Primary constraints: purely spatial part of the eoms T m = de m + ω m ne n =16 constraints G m m 2 t m 7 / 18

8 Constraint analysis Secondary constraints Symmetry constraint So t [m e n] 0 and generically Vector constraint G [m e n] = 1 2 ɛ mnrse r T s 0 F := e m f m 0 G m = 1 2 ɛ mnrst n [ dω rs + ω r tω ts] 0 So t m 0 which reduces to V := ɛ mnrs M mn K rs 0 where M mn (e r, f s ) and K mn := ω mn ω mn (f r ) 6+4=10 constraints 8 / 18

9 Constraint analysis Tertiary constraints Curl of symmetry constraint F := K mn e m f n 0 where the purely spatial part is not new! Curl of vector constraint V := ɛ mnrs M mn K rs + 0 K rs : no time derivatives on-shell Scalar constraint: S := V 0 3+1=4 constraints =10 first order dofs OR 5 physical dofs 9 / 18

10 Characteristic surfaces Covariant constraint analysis Spacelike characteristic surfaces Discussion 10 / 18

11 Characteristic surfaces Spacelike characteristic surfaces Spacelike characteristic surfaces: hyspersurfaces which cannot be used to as initial data surfaces for the eoms (cf A. Terana s parallel session talk: Cauchy breakdown) Analogous problem for first order differential equations: a(y, t)ẏ + b(y, t) = 0 with y(t 0 ) = y 0 IF a(y 0, t 0 ) = 0: impossible to evolve the differential equation for the given initial conditions at t 0! For mgr PDEs: non-invertibility of the coefficient of the highest order derivatives on some spacelike hypersurface Σ 11 / 18

12 Characteristic surfaces Other interpretations Σ is the world-sheet of a superluminal shock wave-front Equivalently: propagation of a superluminal wave-front in the infinite frequency limit over some mean-field solution of the eoms Causal structure: Light-cone of the theory larger than the dynamical metric light-cone!potentially dangerous (cf M. Trodden s plenary talk) 12 / 18

13 Characteristic surfaces Analysis Investigation of the eoms+constraints Suppose Σ is spacelike characteristic surface (with timelike normal ξ µ ) Propagation of a shock wave-front along which first order derivatives are discontinuous µ e m ν Σ+ µ e m ν Σ := ξ µ e ν m µ ω mn ν Σ+ µ ων mn mn Σ := ξ µ w ν Compute the discontinuities in the eoms+constraints See whether the e ν m and w ν mn are allowed to be non-zero 13 / 18

14 Characteristic surfaces Results Characteristic equation: Ç χ eν m w ν mn å 0 Characteristic matrix χ depends on the initial conditions (compatible with the constraints) given on Σ OR equivalently on the mean-field solution of the eoms over which shocks propagate Invertibility is not warranted (and even dubious generically) 14 / 18

15 Characteristic surfaces Results l o m K mn f n f oo + e m f n w omn 0 2ɛ mnrs l o m (β 1 e n + β 2 f n )K rs f oo ɛ mnrs M mn w o rs 0 ɛ mnrs Ä β1 e m e t 2β 2 e (m f t) 3β 3 f m f tä (K nr w o s t K s tw o nr ) +2ɛ mnrs Ä β1 l o [m e t] β 2 l o (m f t)ä K nr K s tf oo +2m 2 ɛ mnrs l o m ( 4β 0 β 1 e n e r e s + 3(β β 0 β 2 )e n e r f s +6β 1 β 2 e n f r f s + (β 1 β 3 + 2β 2 2)f n f r f s) f oo 3ɛ mnrs β 3 f m f n Å ρ rs + 2m 2 ξ [r [ τ s] 1 2 τ t te s]] ã f oo 4β 2 l o m Ḡ m f oo 2ɛ mnrs β 1 l o m e n R rs f oo 0 15 / 18

16 Discussion Covariant constraint analysis Spacelike characteristic surfaces Discussion 16 / 18

17 Discussion How bad is this? Σ can be any spacelike hypersurface! Light-cone of the theory completely flat? Maybe non-invertibility occurs only in strong-coupling regimes where the theory cannot be trusted anyway... Maybe for some region of parameter space OR some fiducial background choice, χ is generically invertible... Thorough analysis difficult / 18

18 Conclusion Conclusion First covariant constraint analysis of drgt mgr Study of characteristic surfaces The theory has potentially pathological behavior unless... Strong-coupling and quantum corrections save the day Magic happens (in the characteristic matrix) 18 / 18