A Multi Gravity Approach to Space-Time Foam Remo Garattini Università di Bergamo I.N.F.N. - Sezione...
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Transcript of A Multi Gravity Approach to Space-Time Foam Remo Garattini Università di Bergamo I.N.F.N. - Sezione...
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))