Distorting General Relativity: Gravitys Rainbow and f(R) gravity at - PowerPoint PPT Presentation

FFP14 Marseille 18 -7-2014 Distorting General Relativity: Gravity’s Rainbow and f(R) gravity at work Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano

Introduction • Hamiltonian Formulation of General Relativity and the Wheeler-DeWitt Equation • The Cosmological Constant as a Zero Point Energy Computation • Gravity’s Rainbow as a tool for computing ZPE • Gravity’s Rainbow and f(R) theory at work

Part I: Hamiltonian Formulation of General Relativity and the Wheeler-DeWitt Equation

Relevant Action for Quantum Cosmology ( ) 1 ( ) ∫ ∫ = − − Λ + + κ = π 4 4 3 3 2 2 8 S d x g R d x g K S G κ matter 2 ∂ M M → Newton's Constant G Λ → Cosmological Constant

Relevant Action for Quantum Cosmology ( ) 1 1 ( ) ∫ ∫ = − − Λ + + 3 4 4 3 2 S d x g R d x g K S κ κ ∂ matter 2 M M ADM Decomposition   j 1 N −     − + 2 k N N N N 2 2 N N   µν =  k j  = g g  ( )  µν   3 i i j N g N N N   ( ) − 3 ij   i ij g   2 2 N N ( )( ) µ ν = = − + + + 2 2 2 i i j j ds g dx dx N dt g N dt dx N dt dx µν ij is the lapse function is the shift function N N i 1 = − + ∇ + ∇ =  ij K g N N K K g ij ij i j j i ij 2 N

  j 1 N −     − + 2 k N N N N 2 2 N N   µν µ ν =  k j  = = 2 ds g g g dx dx  ( )  µν µν   3 i i j N g N N N   ( ) − 3 ij   i ij g   2 2 N N ( ) ( ) 1 ∫ = − + − Λ + + 3 3 2 3 ij 2 S dtd xN g K K K R S S ( ) ∂ Σ× κ ij matter I 2 Σ× I ( ) ∫ → = + + 3 i Legendre Transformation H d x N H N H H ∂Σ i Σ ( ) g ( ) = κ π π − − Λ = → 3 ij kl 2 2 0 Classical Constraint Invariance by time H G R κ ijkl 2 reparametrization = π = → i ij 2 0 Classical Constraint Gauss Law H | j

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev. 160 , 1113 (1967).   g ( ) ( )   κ π π − − Λ Ψ =  ij kl  2 2 0 G R g   κ ijkl ij   2   G ijkl is the super-metric, • R is the scalar curvature in 3-dim. • Example:WDW for Tunneling   ∂ ∂ π Λ ( )   2 2 9 q = − + Ω [ ] [ ] 2 2 2 2 2 Ψ = − − + − Ψ = 2 4 ds N dt a t d    0 H a a a a ∂ ∂ 3 2 2    4 3  a a a G Formal Schrödinger Equation with zero eigenvalue whose solution is a linear combination of Airy’s functions (q=-1 Vilenkin Phys. Rev. D 37, 888 (1988).) containing expanding solutions

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev. 160 , 1113 (1967).   ∂ ∂ π Λ     2 1 9 [ ] [ ] Ψ = − + − Ψ = 2 4 q      0 H a a a a a ∂ ∂ q   2    4 3  a a a G Sturm-Liouville Eigenvalue Problem     + d d ( ) ( ) ( ) ( ) + λ =   0  p x q x w x  y x     dx dx ∞ b ( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫ + → Ψ Ψ ↔ 4 * q * Normalization with weight a a a da w x y x y x dx w x 0 a 2 2 π π Λ       3 3 [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + → → − → → Ψ λ → 2 4 q q q       p x a t q x a t w x a t y x a       2 2 3 G G

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev. 160 , 1113 (1967).   ∂ ∂ π Λ     2 1 9 [ ] [ ] Ψ = − + − Ψ = 2 4 q      0 H a a a a a ∂ ∂ q   2    4 3  a a a G Sturm-Liouville Eigenvalue Problem  Variational procedure   b   d d ( ) ( ) ( ) ( ) ( ) ∫ − + *   *  y x p x q x y x y x  dx Rayleigh-Ritz     dx dx λ = → a min Variational Procedure ( ) b y x ( ) ( ) ( ) ∫ * w x y x y x dx a ( ) ( ) = = 0 y a y b

Part II: The Cosmological Constant as a Zero Point Energy Computation

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev. 160 , 1113 (1967). Reconsider the WDW Equation as an Eigenvalue Problem Λ can be seen as an eigenvalue • Ψ [g ij ] can be considered as an eigenfunction • Define  ( ) g g ( ) Λ = ˆ κ π π − Λ = − Λ ij kl 2 G R x Σ κ κ ijkl C 2   ( ) ∫ ∫             ∗ ∗ Ψ Λ Ψ Ψ Ψ = Λ D g g g D g g g           x   Σ ij ij ij ij ij ij

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev. 160 , 1113 (1967). [ ] [ ] [ ]  ∫ ∫ ∗ µ Ψ Λ Ψ 3 Induced D h h d x h Σ Λ Cosmological 1 Σ = − ‘‘Constant’’ [ ] [ ] [ ] ∫ κ ∗ µ Ψ Ψ V D h h h [ ] [ ]  ⊥    µ = ξ T D h D h D D h J     ij j Solve this infinite dimensional PDE with a Variational Approach Ψ is a trial wave functional of the gaussian type Schrödinger Picture Spectrum of Λ depending on the metric Energy (Density) Levels

Canonical Decomposition M. Berger and D. Ebin, J. Diff. Geom .3 , 379 (1969). J. W. York Jr., J. Math. Phys., 14 , 4 (1973); Ann. Inst. Henri Poincaré A 21 , 319 (1974). →  g g +h ij ij ij    1 1 ( ) ⊥ → ⇔ = + ξ + ⇔ ∇ − = =  i ij   0 N N h hg L h h g h g h ij ij ij  ij ij  ij ij 3 3  →  0 N i µ µ ν = = ∇ = 0 Equivalent in 4D to 0 No Ghosts Contribution h h h µ µ µ [ ] [ ] [ ]  ∫ ∫ ∗ µ Ψ Λ Ψ 3 D h h d x h Σ Λ 1 Σ = − h is the trace (spin 0) • [ ] [ ] [ ] ∫ κ µ Ψ ∗ Ψ V D h h h (L ξ) ij is the gauge part • [ ] [ ] [ spin 1 (transverse) + spin 0 (long.)] .     µ = ⊥ ξ T D h D h D D h J     ij j F.P determinant (ghosts) h ⊥ ij transverse-traceless  graviton (spin 2) •

Gravity’s Rainbow Doubly Special Relativity G. Amelino-Camelia, Int.J.Mod.Phys. D 11, 35 (2002); gr-qc/001205. G. Amelino-Camelia, Phys.Lett. B 510, 255 (2001); hep-th/0012238. ( ) ( ) − = 2 2 2 2 2 / / E g E E p g E E m 1 2 P P ( ) ( ) = = lim / lim / 1 g E E g E E 1 2 P P → → / 0 / 0 E E E E P P Curved Space Proposal  Gravity’s Rainbow [J. Magueijo and L. Smolin, Class. Quant. Grav. 21, 1725 (2004) arXiv:gr-qc/0305055]. ( ) 2 2 2 2 N r dt dr r r = − + + θ + θ ϕ 2 2 2 2 sin ds d d ( ) ( ) ( ) ( )   2 2 2 / b r / / g E E g E E g E E ( ) 1 − 2 2 2 P   P P 1 / g E E   2 P r   ( ) ( ) ( ) ( ) = − Φ Φ exp 2 is the redshift function N r r r ( ) ( ) ) → = ∈ +∞  is the shape function Condition , b r b r r r  r 0 0 0

Eliminating Divergences using Gravity’s Rainbow [R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]] ( ) ∆ = ∆ − + a 4 h R h Rh ( ) 2 E 2 ia j ij    ∆   ij ⊥ ⊥ = h h ( ) 2 Modified Lichnerowicz operator ij 2 g E ij ( ) 2 ∆ = ∆ − + + jl a a 2 h h R h R h R h     ij ijkl ia j ja i ij Standard Lichnerowicz operator ( ) ( ) ( )     ( 3 2 g E g E 1 − ⊥ ⊥ ( ) ( ) )    ∫ 1, ⊥ Λ = ˆ κ + ∆ 2 3 1 ijkl   2 , , d x g G K x x K x x ( ) ( ) 2 Σ κ 3 Σ   ijkl 4 2 V g E g E   ijkl 2 2    ( )  ( )  ( ) ( ) τ ⊥ τ ⊥ h x h y     ( ) = ∑ , : ij kl (Propagator) K x y ( ) ( ) λ τ 4 2 g E ijkl τ 2

Eliminating Divergences using Gravity’s Rainbow [R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]] ( ) ( ) ( ) We can define an r-dependent radial wave number    3 ' 3 6 1 b r b r b r ( ) ( ) = − + − + 2    2 1 l l m r E ( ) ( ) ( ) = − − ≡ 2 2 1 , , nl 2 2 3 k r l E m r r r x    2 2 ( ) r r r r nl i 2 2 / g E E r  2 P ( ) ( ) ( )    ' 3 6 1 b r b r b r ( ) = − + + 2   m r  2 2 2 3 2 2    r r r r 3 +∞   Λ 2 2 1 E ∑ ∫ d = − − 2   ( / ) ( / ) i ( ) E g E E g E E m r dE π π 1 2 i P P i i 2 2 8 3  ( / )  G dE g E E = 1 i 2 * i P E Λ ω 2 1 +∞ ∫ = − ω i d π π ( ) Standard Regularization 2 1 i 2 ( ) 8 16 G ε − m r ( ) i ω − 2 2 2 m r i i