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IRGAC 2006 Barcelona, 15-7-2006. Multigravity and Spacetime Foam. Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano. The Cosmological Constant Problem. For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 61 , 1 (1989).

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multigravity and spacetime foam

IRGAC 2006Barcelona, 15-7-2006

MultigravityandSpacetime Foam

Remo Garattini

Università di Bergamo

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

the cosmological constant problem
The Cosmological Constant Problem

For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).

For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.

D 9, 373 (2000), astro-ph/9904398; N. Straumann, The history of the cosmological

constant problem gr-qc/0208027; T.Padmanabhan, Phys.Rept. 380, 235 (2003),


  • Recent measures
  • At the Planck era

A factor of 10118

wheeler de witt equation b s dewitt phys rev 160 1113 1967
Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967).
  • Gijkl is the super-metric, k =8pG and L is the cosmological constant
  • R is the scalar curvature in 3-dim.

L can be seen as an eigenvalue

eigenvalue problem
Eigenvalue problem

Quadratic Approximation

Let us consider the 3-dim. metric gij and perturb around a fixed background, (e.g. Schwarzschild)

gij= gSij+ hij

canonical decomposition
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).

  • h is the trace
  • (Lx)ij is the longitudinal operator
  • h^ij represents the transverse-traceless component of the perturbation  graviton
graviton contribution
Graviton Contribution

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

  • Zeta function regularization  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect
  • Bare cosmological constant changed into

The finite part becomes

renormalization group equation
Renormalization Group Equation
  • Eliminate the dependance on m and impose

L0 must be treated as running

energy minimization l maximization
Energy Minimization (L Maximization)
  • At the scale m0

L0 has

a maximum for


Not satisfying

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

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

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

[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 better

Cosmological constant


first steps in multigravity
First Steps in Multigravity

Pioneering works in 1970s known under the name

strong gravity


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).]

structure of multigravity t damour and i l kogan phys rev d 66 104024 2002 a d linde hep th 0211048
Structure of MultigravityT.Damour and I. L. Kogan, Phys. Rev.D 66, 104024 (2002).A.D. Linde, hep-th/0211048

N massless


multigravity gas
Multigravity gas

For each action, introduce the lapse

and shift functions

Choose the gauge

Define the following


No interaction

Depending on the structure You are looking, You could have a ‘ideal’gas of geometries.

Our specific case: Schwarzschild wormholes


Additional assumption

The single eigenvalue

problem turns into

  • Wave functionals do not overlap
and the total wave functional becomes
And the total wave functional becomes

The initial problem

changes into


further trivial assumption r garattini int j mod phys d 4 2002 635 gr qc 0003090
Further trivial assumptionR. Garattini - Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090.

Nw copies of

the same gravity

Take the maximum


There are arguments

leading to

Nevertheless, there is no

Proof of this

  • Wheeler-De Witt Equation  Sturm-Liouville Problem.
  • The cosmological constant is the eigenvalue.
  • Variational Approach to the eigenvalue equation (infinites).
  • Eigenvalue Regularization with the Riemann zeta function  Casimir energy graviton contribution to the cosmological constant.
  • Renormalization and renormalization group equation.
  • Generalization to multigravity.
  • Specific example: gas of Schwarzschild wormholes.
  • Analysis to be completed.
  • Beyond the W.K.B. approximation of the Lichnerowicz spectrum.
  • Discrete Lichnerowicz spectrum.
  • Specific examples of interaction like the Linde bi-gravity model or Damour et al.
  • Possible generalization con N ‘different gravities’?!?!
  • Use a distribution of gravities!!