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Distorting General Relativity: Gravity’s Rainbow and f(R) gravity at work

Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano

FFP14 Marseille 18-7-2014

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

• f General Relativity and the

Wheeler-DeWitt Equation

( )

( )

3 4 4 3

1 2 2 8 2 Newton's Constant Cosmological Constant

matter

S d x g R d x g K S G G κ π κ

∂

= − − Λ + + = → Λ →

∫ ∫

M M

Relevant Action for Quantum Cosmology

( )

( )

3 4 4 3

1 1 2 2

matter

S d x g R d x g K S κ κ ∂ = − − Λ + +

∫ ∫

M M

Relevant Action for Quantum Cosmology

( ) ( )

2 2 2 3 3 2 2

1

j k k j i i j ij i ij

N N N N N N N g g N g N N N g N N

µν µν

  −     − +   = =         −    

ADM Decomposition

( )( )

2 2 2

ds is the lapse function is the shift function

i i j j ij i

g dx dx N dt g N dt dx N dt dx N N

µ ν µν

= = − + + + 1 2

ij ij ij i j j i ij

K g N N K K g N = − + ∇ + ∇ = 

( ) (

)

( )

( )

( )

( )

3 3 2 3 3 3

1 2 2 Legendre Transformation 2 2 0 Classical Constraint Invariance by time 2

ij ij matter I I i i ij kl ijkl

S dtd xN g K K K R S S H d x N N H g G R κ κ π π κ

∂ Σ× Σ× ∂Σ Σ

= − + − Λ + + → = + + = − − Λ = →

∫ ∫

H H H

|

reparametrization 2 0 Classical Constraint Gauss Law

i ij j

π = = → H

( ) ( )

2 2 2 2 3 3 2 2

1 ds

j k k j i i j ij i ij

N N N N N N N g g g dx dx N g N N N g N N

µν µ ν µν µν

  −     − +   = = =         −    

Wheeler-De Witt Equation

• B. S. DeWitt, Phys. Rev.160, 1113 (1967).

( ) ( )

2 2 2

ij kl ijkl ij

g G R g κ π π κ     − − Λ Ψ =        

• Gijkl is the super-metric,
• R is the scalar curvature in 3-dim.

( )

2 2 2 2 2 3

ds N dt a t d = − + Ω

[ ] [ ]

2 2 2 4 2 2

9 4 3 q H a a a a a a a G π   ∂ ∂ Λ   Ψ = − − + − Ψ =    ∂ ∂    

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

Example:WDW for Tunneling

Wheeler-De Witt Equation

• B. S. DeWitt, Phys. Rev.160, 1113 (1967).

Sturm-Liouville Eigenvalue Problem

[ ] [ ]

2 2 4 2

1 9 4 3

q q

H a a a a a a a a G π   ∂ ∂ Λ     Ψ = − + − Ψ =      ∂ ∂      

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

* Normalization with weight

b a

d d p x q x w x y x dx dx w x y x y x dx w x λ     + + =         ↔

∫

( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ]

2 2 2 4

3 3 2 2 3

q q q

p x a t q x a t w x a t y x a G G π π λ

+ +

Λ       → → − → → Ψ →            

( ) ( )

4 * q

a a a da

∞ +

→ Ψ Ψ

∫

Wheeler-De Witt Equation

• B. S. DeWitt, Phys. Rev.160, 1113 (1967).

Sturm-Liouville Eigenvalue Problem  Variational procedure

[ ] [ ]

2 2 4 2

1 9 4 3

q q

H a a a a a a a a G π   ∂ ∂ Λ     Ψ = − + − Ψ =      ∂ ∂      

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

* * min *

b a b y x a

d d y x p x q x y x y x dx dx dx w x y x y x dx λ     − +         = →

∫ ∫ ( ) ( )

y a y b = =

Rayleigh-Ritz Variational Procedure

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

Define

( )

( )

ˆ 2 2

ij kl ijkl C

g g G R x κ π π κ κ

Σ

Λ = − Λ = − Λ 

Wheeler-De Witt Equation

• B. S. DeWitt, Phys. Rev.160, 1113 (1967).
• Λ can be seen as an eigenvalue
• Ψ[gij] can be considered as an eigenfunction



( )

ij ij ij ij ij ij

x

D g g g D g g g

∗ ∗ Σ

= Λ

            Ψ Λ Ψ Ψ Ψ            

∫ ∫



Reconsider the WDW Equation as an Eigenvalue Problem

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

Wheeler-De Witt Equation

• B. S. DeWitt, Phys. Rev.160, 1113 (1967).

[ ] [ ]



[ ] [ ] [ ] [ ] [ ] [ ]

3

1

T ij j

D h h d x h V D h h h D h D h D D h J µ κ µ µ ξ

∗ Σ Σ ∗ ⊥

Λ = −

Ψ Λ Ψ Ψ Ψ     =    

∫ ∫ ∫

Induced Cosmological ‘‘Constant’’

Canonical Decomposition

( )

ij ij ij

g g +h 1 1 3 3

i ij ij ij ij ij ij ij ij i

N N h hg L h h g h g h N ξ

⊥

→     → ⇔ = + + ⇔ ∇ − = =       → 

• h is the trace (spin 0)
• (Lξ)ij is the gauge part

[ spin 1 (transverse) + spin 0 (long.)] . F.P determinant (ghosts)

• h⊥ij transverse-traceless  graviton (spin 2)
• 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).

Equivalent in 4D to 0 No Ghosts Contribution h h h

µ µ ν µ µ µ

= = ∇ =

[ ] [ ]



[ ] [ ] [ ] [ ] [ ] [ ]

3

1

T ij j

D h h d x h V D h h h D h D h D D h J µ κ µ µ ξ

∗ Σ Σ ∗ ⊥

Λ = −

Ψ Λ Ψ Ψ Ψ     =    

∫ ∫ ∫

Gravity’s Rainbow

( ) ( ) ( ) ( )

2 2 2 2 2 1 2 1 2 / /

/ / lim / lim / 1

P P

P P P P E E E E

E g E E p g E E m g E E g E E

→ →

− = = =

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.

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 2 2 2 2 2 2 2 1 2 2 2 2

sin / / / 1 / exp 2 is the redshift function is the shape function Condition ,

P P P P

N r dt dr r r ds d d b r g E E g E E g E E g E E r N r r r b r b r r r r θ θ ϕ = − + + +   −       = − Φ Φ → = ∈ +∞  

Eliminating Divergences using Gravity’s Rainbow

[R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]]

( )

 

( ) ( ) ( )



( ) ( )

  (

)

( )

3 2 1, 2 1 3 2 3 2 2

1 ˆ 2 , , 4 2

ijkl

ijkl ijkl

g E g E d x g G K x x K x x V g E g E κ κ

− ⊥ ⊥ ⊥ Σ Σ

  Λ = + ∆      

∫

(

)

( )

( ) ( ) ( )

( ) ( )

4 2

, : (Propagator) 2

ij kl

ijkl

h x h y K x y g E

τ τ τ

λ τ

⊥ ⊥

= ∑      

( ) ( )

2 Modified Lichnerowicz operator Standard Lichnerowicz operator

4 2

a ia j ij ij jl a a ij ijkl ia j ja i ij

h R h Rh h h R h R h R h ∆ = ∆ − + ∆ = ∆ − + +       

( )

( )

2 2 2 2

E =

ij ij

h h g E

⊥ ⊥

∆  

We can define an r-dependent radial wave number

( ) ( ) ( ) ( ) ( )

2 2 2 2 2 2

1 , , /

nl nl i P

l l E k r l E m r r r x g E E r + = − − ≡

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 1 2 2 3 2 2 2 2 3

3 ' 3 6 1 2 2 ' 3 6 1 2 2 b r b r b r m r r r r r b r b r b r m r r r r r    = − + −           = − + +      

3 2 2 2 1 2 2 2 1 2 *

1 ( / ) ( / ) ( ) 3 ( / ) 8

i i P P i i i i P E

E d E g E E g E E m r dE dE g E G E π π

+∞ =

  − −  Λ  =  

∑ ∫

( )

( )

( )

2

2 1 2 2 2 2

1 8 16

i

i i m r i i

d G m r

ε

ω ω π π ω

+∞ −

Λ = − −

∫

Standard Regularization

Eliminating Divergences using Gravity’s Rainbow

[R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]]

Gravity’s Rainbow and the Cosmological Constant

R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]

( ) ( )

1 2

Popular Choice...... Not Promising / 1 / 1

n P P P

E g E E E g E E η →   = −     =

Failure of Convergence

( ) ( )

2 1 2 2

/ exp 1 / 1

P P P P

E E g E E E E g E E α β    = − +       =

( ) ( ) ( ) ( )

2 2 2 2 2 1 2

• /

P

Minkowski de Sitter Anti de Sitter m r m r m r x m r E = = → =

Comparison with the Noncommutative Approach

[R.G. & P. Nicolini Phys. Rev. D 83, 064021 (2011); 1006.4518 [gr-qc] ] Noncommutative ST , =i x x

µ ν µν

θ    

( ) ( )

3 3 3 3 NonCommutative 2 3 3 Version

Number of Nodes exp 4 2 2

i

d xd k d xd k k θ π π     → −        

( ) ( )

( )

( )

( )

2

3 2 2 2 2 2 2 2

exp 4

i

i i i i i m r i i i

m r k d k m r θ ρ θ ω ω ω

+∞

  ∝ − −     = −

∫

lim 8

M

G π

→

Λ =

Problem

Prescription: a metric which has the property of being asymptotically flat MUST satisfy the Minkowski limit In analogy with the SU(2) Yang Mills Constant ChromoMagnetic Field

lim 8

M

G π

→

Λ =

( )

2 2 2 2

11 1 lim ln 2 48 2

H

E H eH eH V π µ

→

  = + −    

Some background, like the Schwarzschild metric, does not satisfy the Minkowski limit

Gravity’s Rainbow and f(R) theory at work

(R.G., JCAP 1306 (2013) 017 arXiv:1210.7760 ) For a f(R) theory in 4D Transformation rule under Gravity’s Rainbow

Gravity’s Rainbow and f(R) theory at work

For spherically symmetric backgrounds

Gravity’s Rainbow and f(R) theory at work

In this case

( ) is an arbitrary function of the 3D scalar curvature

f R R

Gravity’s Rainbow and f(R) theory at work

For this setting the Schwarzschild background can pass the Minkowski test limit

Conclusions and Outlooks

• Application of Gravity’s Rainbow can be considered to

compute divergent quantum observables.

• Neither Standard Regularization nor Renormalization are
• required. This also happens in NonCommutative

geometries.

• The f(R) function can be related to the potential part of

the Horava-Lifshits theory without detailed balanced condition.

• Application to Traversable Wormholes, Topology

Change, Particle Propagation, Black Hole Entropy,Tunneling and Inflation.

Conclusions and Outlooks

 Application to Traversable Wormholes,

Topology Change

R.G. and F.S.N. Lobo, arXiv:1102.3803 [gr-qc] Phys.Rev. D85 (2012) 024043

R.G. and F.S.N. Lobo, arXiv:1303.5566 [gr-qc] Eur.Phys.J. C74 (2014) 2884

 Application to Particle Propagation

R.G. and G. Mandanici Phys.Rev. D85 (2012) 023507 e-Print: arXiv:1109.6563 [gr-qc]

 Naked Singularity and Charge Creation

R.G. and B. Majumder, Nucl.Phys. B884 (2014) 125 e-Print: arXiv:1311.1747 [gr-qc] R.G. and B. Majumder, Nucl.Phys. B883 (2014) 598 e-Print: arXiv:1305.3390 [gr-qc]

 Application to Inflation

R.G. and M.Sakellariadou, arXiV:1212.4987