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 of General Relativity and the

Wheeler-DeWitt Equation

( ) ( )34 4 31 2 2 82 Newton's Constant Cosmological Constant

matterS d x g R d x g K S G

G

κ πκ ∂

= − − Λ + + =

→Λ→

∫ ∫M M

Relevant Action for Quantum Cosmology

( ) ( )34 4 31 122 matterS d x g R d x g K Sκ κ ∂

= − − Λ + +∫ ∫M M

Relevant Action for Quantum Cosmology

( )( )

2 2 2

33

2 2

1

j

kk j

i i jiji ij

NN N N N N Ng g

N g N N NgN N

µνµν

− − + = =

−

ADM Decomposition

( )( )2 2 2ds

is the lapse function is the shift function

i i j jij

i

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

N N

µ νµν= = − + + +

1 2

ijij ij i j j i ijK g N N K K g

N= − +∇ +∇ =

( ) ( ) ( )

( )

( ) ( )

33 2 3

3

3

1 22

Legendre Transformation

2 2 0 Classical Constraint Invariance by time2

ijij matterI

I

ii

ij klijkl

S dtd xN g K K K R S S

H d x N N H

gG R

κ

κ π πκ

∂ Σ×Σ×

∂ΣΣ

= − + − Λ + +

→ = + +

= − − Λ = →

∫

∫ H H

H

|

reparametrization2 0 Classical Constraint Gauss Lawi ij

jπ= = →H

( )( )

2 2 22

33

2 2

1

ds

j

kk j

i i jiji ij

NN N N N N Ng g g dx dx

N g N N NgN N

µν µ νµν µν

− − + = = =

−

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967).

( ) ( )2 2 02

ij klijkl ij

gG R gκ π π

κ

− − Λ Ψ = • Gijkl is the super-metric, • R is the scalar curvature in 3-dim.

( )2 2 2 2 23ds N dt a t d= − + Ω [ ] [ ]

2 22 4

2 2

9 04 3

qH a a a aa 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 42

1 9 04 3

qqH a a a a a

a a a Gπ ∂ ∂ Λ Ψ = − + − Ψ = ∂ ∂

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0

* Normalization with weight b

a

d dp x q x w x y xdx dx

w x y x y x dx w x

λ + + =

↔∫

( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ]2 2

2 43 3 2 2 3

q q qp x a t q x a t w x a t y x aG Gπ πλ+ + Λ → → − → →Ψ →

( ) ( )4 *

0

qa 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 42

1 9 04 3

qqH a a a a a

a a a Gπ ∂ ∂ Λ Ψ = − + − Ψ = ∂ ∂

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

* *min

*

b

aby x

a

d dy x p x q x y x y x dxdx dx

w x y x y x dxλ

− + = →∫

∫

( ) ( ) 0y a y b= =

Rayleigh-RitzVariational Procedure

Part II: The Cosmological Constant as a

Zero Point Energy Computation

Define

( ) ( )ˆ 2 2

ij klijkl C

g gG R xκ π π

κ κΣΛ = − 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 ijxD 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

Tij j

D h h d x h

V D h h h

D h D h D D h J

µ

κµ

µ ξ

∗Σ

Σ∗

⊥

Λ= −

Ψ Λ Ψ

Ψ Ψ

=

∫ ∫

∫

Induced Cosmological ‘‘Constant’’

Canonical Decomposition

( )ij ij ijg g +h

1 1 03 3

0

i ijij ij ij ij ij ijij

i

N N h hg L h h g h g hN

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).

0Equivalent in 4D to 0 No Ghosts Contributionh h hµ µ νµ µ µ= = ∇ =

[ ] [ ] [ ]

[ ] [ ] [ ][ ] [ ]

3

1

Tij j

D h h d x h

V D h h h

D h D h D D h J

µ

κµ

µ ξ

∗Σ

Σ∗

⊥

Λ= −

Ψ Λ Ψ

Ψ Ψ

=

∫ ∫

∫

Gravity’s Rainbow

( ) ( )

( ) ( )

2 2 2 2 21 2

1 2/ 0 / 0

/ /

lim / lim / 1P P

P P

P PE 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 22 2 2 2

2 2 21 2 22

2

0 0 0

sin/ / /

1 /

exp 2 is the redshift function

is the shape function Condition ,

P P PP

N r dt dr r rds d db rg E E g E E g E E

g E Er

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 1323

2 2

1ˆ 2 , ,4 2

ijkl

ijkl ijkl

g E g Ed x g G K x x K x x

V g E g Eκ

κ− ⊥ ⊥⊥

Σ Σ

Λ = + ∆

∫

( ) ( )( )

( )( )

( ) ( )42

, : (Propagator)2

ij kl

ijkl

h x h yK x y

g E

τ τ

τ λ τ

⊥ ⊥

=∑

( )

( )

2

Modified Lichnerowicz operator

Standard Lichnerowicz operator

4

2

aia j ijij

jl a aij ijkl ia j ja iij

h R h Rh

h h R h R h R h

∆ = ∆ − +

∆ = ∆ − + +

( ) ( )2

2 22

E= ijijh h

g E⊥ ⊥∆

We can define an r-dependent radial wave number

( ) ( )( ) ( ) ( )

22 2

2 22

1, ,

/nl

nl iP

l lEk r l E m r r r xg E E r

+= − − ≡( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

21 2 2 3

22 2 2 3

3 ' 36 12 2

' 36 12 2

b r b r b rm r

r r r r

b r b r b rm r

r r r r

= − + −

= − + +

3222

1 22 21 2*

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

ii P P i i

i i PE

EdE g E E g E E m r dEdE g EG Eππ

+∞

=

− −

Λ=

∑ ∫

( )( )( )2

2

122 2 2

18 16 i

iim r

i i

dG m r

ε

ω ωπ 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

PP

P

Eg E EE

g E E

η

→

= −

=

Failure of Convergence

( )

( )

2

1 2

2

/ exp 1

/ 1

PP P

P

E Eg E EE E

g E E

α β

= − +

=

( ) ( ) ( ) ( )2 2 2 2 21 2 0 0

- - -

/ 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 , =ix xµ ν µνθ

( ) ( )

3 3 3 3NonCommutative 2

3 3VersionNumber of Nodes exp42 2

id xd k d xd k kθ

π π → −

( ) ( )( )( )

( )

2

32 2 22

2 2 2

exp4i

i i i i im r

i i i

m r k d

k m r

θρ θ ω ω

ω

+∞ ∝ − −

= −

∫

0lim 0

8M Gπ→

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

0lim 0

8M Gπ→

Λ=

( )2

22 20

11 1lim ln2 48 2H

E H eHeHV 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