A Multi Gravity ApproachA Multi Gravity Approachtoto

Space-Time FoamSpace-Time Foam

Remo GarattiniRemo Garattini

Università di BergamoUniversità di Bergamo

I.N.F.N. - Sezione di MilanoI.N.F.N. - Sezione di Milano

Low Energy Quantum Gravity Low Energy Quantum Gravity York, 20-7-2007York, 20-7-2007

22

OutlineOutline

The Cosmological Constant ProblemThe Cosmological Constant Problem The Wheeler-De Witt EquationThe Wheeler-De Witt Equation Aspects of Space Time Foam (STF)Aspects of Space Time Foam (STF) Multi Gravity Approach to STFMulti Gravity Approach to STF ConclusionConclusion Problems and OutlookProblems and Outlook

33

The Cosmological Constant The Cosmological Constant ProblemProblem

At the Planck eraAt the Planck era

For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 6161, 1 (1989)., 1 (1989).For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.D 9D 9, 373 (2000), astro-ph/9904398; N. Straumann, , 373 (2000), astro-ph/9904398; N. Straumann, The history of the cosmologicalThe history of the cosmologicalconstant problemconstant problem gr-qc/0208027; T.Padmanabhan, Phys.Rept. gr-qc/0208027; T.Padmanabhan, Phys.Rept. 380380, 235 (2003),, 235 (2003),hep-th/0212290.hep-th/0212290.

76 410C GeV •Recent measuresRecent measures

44710 GeVC

A factor of 10123

44

Action in 3+1 dimensionsAction in 3+1 dimensions The action in 4D with a cosmological termThe action in 4D with a cosmological term

''4 4 3 3

'2

t

M t Bd x g R d x gK d x

Introducing lapse and shift function (ADM)Introducing lapse and shift function (ADM)

4 2 2 i i j jijd s N dt g N dt dx N dt dx

The scalar curvature separates into 4 2 2ij

ijR R K K K Ku a

Conjugatemomentum 8

2ij ij ijg

Kg K G

55

Action in 3+1 dimensionsAction in 3+1 dimensions The action in 4D with a cosmological termThe action in 4D with a cosmological term

''4 4 3 3

'2

t

M t Bd x g R d x gK d x

Legendre transformationLegendre transformation

4 ijijM

S d x g dtH

GravitationalHamiltonian

Classicalconstraint

+EADM

66

Wheeler-De Witt Equation Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.B. S. DeWitt, Phys. Rev.160160, 1113 (1967)., 1113 (1967).

2 2 02

ij klijkl ij

gG R g

GGijklijkl is the super-metric, is the super-metric, 88G and G and is the cosmological constant is the cosmological constant R is the scalar curvature in 3-dim.R is the scalar curvature in 3-dim.

can be seen as an eigenvaluecan be seen as an eigenvalue [g[gijij] can be considered as an eigenfunction] can be considered as an eigenfunction

77

Re-writing the WDW equationRe-writing the WDW equation

Where Where Rg

G klijijkl

2

2ˆ

C

gx

ij ij ij ij ij ijxD g g g D g g g

88

Eigenvalue problemEigenvalue problem

3

1ij ij ij

ij ij ij

D g g d x g

V D g g g

Quadratic ApproximationQuadratic Approximation

Let us consider the 3-dim. metric Let us consider the 3-dim. metric ggijij and and perturb perturb

around a fixed background, around a fixed background, ggijij= g= gSSijij+ h+ hijij

99

Form of the background

0 1 23

1d x

V

2

2 2 2 2 2 2 2sin

1

drds N r dt r d d

b r

r

N(r) Lapse function

b(r) shape function

for example, the Ricci tensor in 3 dim. is

1010

Canonical DecompositionCanonical Decomposition

h is the traceh is the trace (L(Lijij is the longitudinal operator is the longitudinal operator

hhij ij represents the transverse-traceless represents the transverse-traceless

component of the perturbation component of the perturbation graviton graviton

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).

ijijijij hLhgh 3

1

1111

Integration rules on Gaussian wave functionals

11

22

33

44

55

ij ij ijh x h x h

ij

ij

ij hxh

ix

*1 2 1 2 ij ij klD h h h

0

xhij

,ij kl

ijkl

h x h yK x y

5555555555555 55555555555555 5

1212

Graviton ContributionGraviton Contribution

operator czLichnerowi modified theis 2

r)(Propagato 2

:,

klij

yhxhyxK ijkl

iakl

a

jijkl xxKxxKGgxdV

ijkl ,2

1,2

4

1ˆ2

,13

W.K.B. method and graviton contribution to the cosmological constant

1313

Regularization Regularization

i

rm

ii

ii d

rmi

2

2

122

2

216,

• Zeta function regularization Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

2

12ln2ln

1

256,

2

2

2

4

rm

rm

i

ii

1414

Isolating the divergence

finitediv

finitedivergent

G

21218

rmrmGdiv 4

24132

1515

RenormalizationRenormalization

Bare cosmological constant changed intoBare cosmological constant changed into

div 0

The finite part becomes

rG

TTeff ,

8 210

1616

Renormalization Group EquationRenormalization Group Equation

Eliminate the dependance on Eliminate the dependance on and impose and impose

d

rd

G

TTeff ,

8

1 0

must be treated as running

0

42

41000 ln

16,,

rmrm

Grr

1717

Energy Minimization Energy Minimization (( Maximization) Maximization)

At the scale At the scale

2

1

4ln

16,

20

204

0000 Mm

MmG

r

has

a maximum for

40

0 0 32

G

e

Mm 1

4 20

20

with

2 21 03

0

2 22 03

0

3

effective mass

due to the curvature3

MGm r m M

r

MGm r m M

r

Not satisfying

1818

Why Spacetime Foam?Why Spacetime Foam?

J. A. Wheeler established that:J. A. Wheeler established that:

The fluctuation in a typical gravitational potential isThe fluctuation in a typical gravitational potential is

[[J.A. Wheeler, Phys. Rev., 97, 511 (1955)]J.A. Wheeler, Phys. Rev., 97, 511 (1955)]

1 1

3 3 332 2/ / / 1.6 10g G c L G c cm

It is at this point that Wheeler's qualitative description of the quantum geometryof space time seems to come into play [J.A. Wheeler, Ann. Phys., 2, 604 (1957)]

On the atomic scale the metric appears flat, as does the ocean to an aviator far above.The closer the approach, the greater the degree of irregularity. Finally, at distances of the order of lP, the fluctuations in the typical metric component,g, become of the same order as the g themselves.

This means that we can think that the geometry (and topology)of space might be constantly fluctuating

1919

Motivating MultigravityMotivating Multigravity

1)1) In a foamy spacetime, general relativity can be renormalized when a In a foamy spacetime, general relativity can be renormalized when a density of virtual black holes is taken under consideration coupled to N density of virtual black holes is taken under consideration coupled to N fermion fields in a 1/N expansionfermion fields in a 1/N expansion

[L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.]. [L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.].

2)2) When gravity is coupled to N conformally invariant scalar fields the When gravity is coupled to N conformally invariant scalar fields the evidence that the ground-state expectation value of the metric is flat evidence that the ground-state expectation value of the metric is flat space is falsespace is false

[J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.].[J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.].

Merging of point 1) and 2) with N gravitational fields (instead of scalars and fermions) leads to

multigravity

Hope for a betterCosmological constant

computation

2020

First Steps in MultigravityFirst Steps in Multigravity

Pioneering works in 1970s known under the name

strong gravitystrong gravity or

f-g theory (bigravity)[C.J. Isham, A. Salam, and J. Strathdee, Phys Rev. D 3, 867 (1971), A.

D. Linde, Phys. Lett. B 200, 272 (1988).]

2121

Structure of MultigravityStructure of Multigravity T.Damour and I. L. Kogan, Phys. Rev.T.Damour and I. L. Kogan, Phys. Rev.D 66D 66, ,

104024 (2002).104024 (2002).A.D. Linde, hep-th/0211048A.D. Linde, hep-th/0211048

N masslessN massless

gravitonsgravitons 0

1

signature wN

ii

S S g

41 8

2i i i i i ii

S g d x g R g G

0 int 1 2, , ,wTot i NS g S S g g g

2222

Multigravity gasMultigravity gas

: | 22

k ij klijkl ij ij

gD G R g g g

,

k

iN N 0k

iN For each action, introduce the lapse and shift functions

Choose the gauge

Define the followingdomain

1 wk N

0Let No interaction

Depending on the structure You are looking, You could have a

‘ideal’gas of geometries.Our specific case:

Schwarzschild wormholes

2323

i j1

with when wN

i i j

Wave functionals do not overlapWave functionals do not overlap

Additional assumption

3

1ij ij ij

ij ij ij

D g g d x g

V D g g g

3

1

8k

kk k kij ij ijk k

k

k k kk kij ij ijk k

D g g d x g

V GD g g g

The single eigenvalueThe single eigenvalue problem turns intoproblem turns into

2424

And the total waveAnd the total wave functional becomes functional becomes

23

1ij Foam ij Foam ij

ij Foam ij Foam ij

D h h d x h

V D h h h

1 2 wTot N Foam

1

wN

i

The initial problem changes into

1 1,G

2 2,G

,w wN NG

2525

Further trivial assumptionFurther trivial assumptionR. Garattini - R. Garattini - Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090.Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090.

1 2

1 2

2 2 23 3 3

1 2

1 1 1Nw

wNw

N

d x d x d xV V V

1 2 wNG G G Nw copies of the same gravity

Take the maximum

2626

231 1

w

Max d xN V

Bekenstein-Hawking argumentson the area quantization

From Multi gravity we need a proof that

2727

ConclusionsConclusions Wheeler-De Witt Equation Wheeler-De Witt Equation Sturm-Liouville Sturm-Liouville

Problem.Problem. The cosmological constant is the eigenvalue.The cosmological constant is the eigenvalue. Variational Approach to the eigenvalue equation Variational Approach to the eigenvalue equation

(infinites).(infinites). Eigenvalue Regularization with the Riemann zeta Eigenvalue Regularization with the Riemann zeta

function function Casimir energy graviton contribution Casimir energy graviton contribution to the cosmological constant.to the cosmological constant.

Renormalization and renormalization group Renormalization and renormalization group equation.equation.

Generalization to multigravity.Generalization to multigravity. Specific example: gas of Schwarzschild Specific example: gas of Schwarzschild

wormholes.wormholes.

2828

Problems and OutlookProblems and Outlook Analysis to be completed.Analysis to be completed. Beyond the W.K.B. approximation of the Lichnerowicz Beyond the W.K.B. approximation of the Lichnerowicz

spectrum.spectrum. Discrete Lichnerowicz spectrum.Discrete Lichnerowicz spectrum. Specific examples of interaction like the Linde bi-gravity Specific examples of interaction like the Linde bi-gravity

model or Damour et al.model or Damour et al. Possible generalization con N ‘different gravities’?!?!Possible generalization con N ‘different gravities’?!?! Use a distribution of gravities!! Massive graviton?!?Use a distribution of gravities!! Massive graviton?!? Generalization to f(R) theories (Mono-gravity case Generalization to f(R) theories (Mono-gravity case

S.Capozziello & R.G., C.Q.G. (2007))S.Capozziello & R.G., C.Q.G. (2007))