Modified Dispersion Relations: from Black Hole Entropy to the from Black Hole Entropy to the

Cosmological Constant Cosmological Constant

Remo GarattiniRemo Garattini

Università di BergamoUniversità di Bergamo

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

Benasque 19 September 2011

IntroductionIntroduction

•Part I Black Hole EntropyPart I Black Hole Entropy•Part II The Cosmological ConstantPart II The Cosmological Constant

Part I: Black Hole EntropyPart I: Black Hole Entropy

33

Black Hole EntropyBlack Hole Entropy[J. D. Bekenstein, Phys. Rev. D 7, 949 (1973). S. W. Hawking, Comm. Math. Phys. 43, 199 (1975).]

The prescription for assigning a Bekenstein-Hawking entropy

to a black hole of surface area A was first inferred in the mid '70s from the formal similarities between black hole dynamics and thermodynamics, combined with Hawking's discovery that black hole radiates thermally with a characteristic (Hawking) temperature

0

0

2 surface gravity

HT

24BHP

AS

l

44

55

2

2 2 2 2 2 2 2exp 2 sin

1

drds r dt r d r d

b r

r

b(r) is the shape function (r) is the redshift function

0 0

0 ,

b r r

r r

Gerard 't Hooft,1 in 1985 considered the statistical thermodynamics of quantum fields in the Hartle-Hawking state (i.e. having the Hawking temperature TH at large radii) propagating on a fixed Schwarzschild background of mass M. To control divergences, 't Hooft introduced a brick wall" with radius slightly larger than thegravitational radius 2MG. He found, in addition to the expected volume-dependent thermodynamical quantities describing hot fields in a nearly at space, additional contributions proportional to the area. These contributions are, however, also proportional to , where is the proper distance from the horizon, and thus diverge in the limit 0. For a specic choice of , he recovered the Bekenstein-Hawking formula [G. 't Hooft, Nucl. Phys. B256, 727 (1985).]

How it works?? Consider a massless scalar field in the following background

66

0

max

2

0

0

1, , ,

1 Free energy F= ln 1 exp

2 1 ,

R

r h

l

l k r l dr

dgd

d

g dl l l

From the equation of motion, we can define an r-dependent radial wave number

2

22

11, , exp 2

1 1

nlnl

l lk r l r

b r b r r

r r

0

32 2

3 2

exp 216The entropy is simply S= S=

901

R

r h

rFr dr

b r

r

77

In proximity of r0

3230 00

3 3 2 200

0

2 2 2

exp exp16 A S=

90 / 290 4 4

exp 1

90 4P P

r rr T

r AIdentification S

l l

88

Eliminating the brick wall from QFT Eliminating the brick wall from QFT ProceduresProcedures

•Renormalization of Newton's constant [L. Susskind and J. Uglum, Phys. Rev. D50, 2700 (1994). J. L. F. Barbon and R. Emparan, Phys. Rev. D52, 4527 (1995) 1995), hep-th/9502155. E. Winstanley, Phys. Rev. D63, 084013 (2001) 2001), hep-th/0011176.]•Pauli-Villars regularization [J.-G. Demers, R. Lafrance and R. C. Myers, Phys. Rev. D52, 2245 (1995), gr-qc/9503003. D. V. Fursaev and S. N. Solodukhin, Phys. Lett. B365, 51 (1996), hep-th/9412020. S. P. Kim, S. K. Kim, K.-S. Soh and J. H. Yee, Int. J. Mod. Phys. A12, 5223 (1997) gr-qc/9607019.]

99

Eliminating the brick wall using Eliminating the brick wall using Generalized Uncertainty PrincipleGeneralized Uncertainty Principle

2

2

Planck Length

Planck Constant

P

P

x p p

Counting of Quantum States Modified by GUP

3 3

33 2 22 1 P

d xd p

p

• X. Li, Phys. Lett. B 540, 9 (2002), gr-qc/0204029.• Z. Ren, W. Yue-Qin and Z. Li-Chun, Class. Quant. Grav. 20 (2003), 4885.• G. Amelino-Camelia, Class.Quant.Grav. 23, 2585 (2006), gr-qc/0506110.• G. Amelino-Camelia, Gen.Rel.Grav. 33, 2101 (2001), gr-qc/0106080.

1010

Eliminating the brick wall using Eliminating the brick wall using Modified Dispersion RelationsModified Dispersion Relations

[[R.Garattini P.L.B.R.Garattini P.L.B. B685 (2010) 329 B685 (2010) 329 e-Print: e-Print: arXiv:0902.3927 [gr-qc]]arXiv:0902.3927 [gr-qc]]

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

exp 2 sin/ / /

1 /P P PP

dt dr r rds r d d

b rg E E g E E g E Eg E E

r

1111

0

3 3 22

0

exp 32 1Free energy F= ln 1 exp /

31

R

P

r h

rdE E h E E dE r dr

dE b r

r

From the equation of motion, we can define an r-dependent radial wave number

2 212

22

/ 1 /1, , exp 2 /

/1 1

P PP

P

E h E E l l g E Ek r l E r h E E

b r b r r g E E

r r

0 0 0Assumption = / 1 /

Modification of the brick wall due to the rainbow's functions

P Pr h r h E E r E E

1212

0

40 3 30

r 2000

exp 3 r ln 1 exp2r 1Free energy F = /

r / 31 ' rP

P

E dE h E E dE

E E dEb

In proximity of the throat, we consider the approximate free energy

Property / 0 when / 0P PE E E E

0

40 3 30

r 2000

exp 3 r ln 1 exp2r 1Int. by parts F =- /

r / 31 ' r

This is possible if / is rapidly convergent Good choice / exp /

PP

P P P

EdE h E E dE

dE E Eb

h E E h E E E E

Assumption on (E/EP)

/ /

Two interesting cases

Case a) =0 >0

Case b) 0 >0

P PP

EE E h E E

E

1313

0

20

0

exp 2 2

4 1 ' 9r P rA E

Sb r

Case a)

In the limit EP>>1

To recover the area law, we have to set

This corresponds to a changing of the time variablewith respect to the Schwarzschild time.

Discrepancy of a factor of 3/2 with the ’t Hooft result

0

0

exp 2 9

1 ' 2

r

b r

0

The internal energy where we have used4

r 2 and 8

MU

MG MG

In the limit EP>>1

To recover the area law, we have to set

Discrepancy of a factor of 3/2 with the ’t Hooft result

0

20

0

exp 2 1

4 1 ' 6r P rA E

Sb r

0

0

exp 26

1 '

r

b r

In the limit EP>>1

0

20

0

exp 2 1 2

4 1 ' 3 6 3 3r P rA E

Sb r

Case b) with =3

Case b) with ≠3

4

MU

1

0

0

exp 29 2

1 ' 2 3

r

b r

+6

2 12-

MU

1414

Part II: The Cosmological Part II: The Cosmological ConstantConstant

1515

1616

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 is the

cosmological constantcosmological 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

1717

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

1818

Solve this infinite dimensional PDE with a Variational Solve this infinite dimensional PDE with a Variational ApproachApproach

is a trial wave functional of the gaussian typeis a trial wave functional of the gaussian typeSchrSchröödinger Picturedinger Picture

Spectrum of Spectrum of depending on the metric depending on the metricEnergy (Density) LevelsEnergy (Density) Levels

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

1919

Canonical DecompositionCanonical Decomposition

ij ij ijg g +h

1 1 0

3 30

i ijij ij ij ij ij ijij

i

N N h hg L h h g h g h

N

h is the trace (spin 0)h is the trace (spin 0) (L(Lijij is the gauge part is the gauge part [spin 1 (transverse) + spin 0 (long.)].[spin 1 (transverse) + spin 0 (long.)]. F.P determinant (ghosts)F.P determinant (ghosts)

hhij ij transverse-traceless transverse-traceless graviton (spin 2) 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 h

3

1

Tij j

D h h d x h

V D h h h

D h D h D D h J

2020

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 x

g E E r

21 2 2 3

22 2 2 3

3 ' 361

2 2

' 361

2 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 ( / )8i

i P P i ii i PE

EdE g E E g E E m r dE

dE g EG E

2

2

122 2 2

1

8 16 i

iim r

i i

dG

m r

Standard Regularization

2121

Gravity’s Rainbow and the Cosmological ConstantGravity’s Rainbow and the Cosmological ConstantR.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]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 E

E

g E E

, parameters

Failure of Convergence

1

2

Promising Choice

/ exp 1

/ 1

PP P

P

E Eg E E

E E

g E E

2222

=1,=1/34

8 PEG

0

P

m r

E

=1,=-1/4

0

P

m r

E

4

8 PEG

A double limit on

0 0 0 0

0 0

, ,0

8 8limlim limlimt t t trr r r r r

r r

G G

2323

Gravity’s Rainbow and the Cosmological ConstantGravity’s Rainbow and the Cosmological Constant

2

1 2

2

Another Promising Choice

/ exp 1

/ 1

PP P

P

E Eg E E

E E

g E E

1

1

2

1 4

4 3

1

4 2.7744164292 1.818231873x

21 2 2 3

22 2 2 3

3 ' 361

2 2

' 361

2 2

b r b r b rm r

r r r r

b r b r b rm r

r r r r

2 21 03

2 22 03

3,

effective masswith

due to the curvature3

,

MGm r m M r

r

MGm r m M r

r

2424

Gravity’s Rainbow and the Cosmological ConstantGravity’s Rainbow and the Cosmological Constant

21

1 1 2

2

Another Promising Choice

4/ exp 1

3

/ 1

PP P

P

E Eg E E

E E

g E E

2

21 2 2

1 2

/ exp 12

1 .7744164292

4

1.818231873

PP P

E Eg E E

E E

x

2525

Gravity’s Rainbow and the Cosmological ConstantGravity’s Rainbow and the Cosmological Constant

2

1 2

2

Another Promising Choice

/ exp 1

/ 1

PP P

P

E Eg E E

E E

g E E

2 21 2

- - - Minkowski de Sitter Anti de Sitter

m r m r

2626

Connection with the Noncommutative Approach Connection with the Noncommutative Approach [R.G. & P. Nicolini 1006.4518 [gr-qc] ]

Noncommutative ST , =ix x

3 3 3 3NonCommutative 2

3 3VersionNumber of Nodes exp

42 2i

d xd k d xd kk

2

32 2 22

2 2 2

exp4i

i i i i im r

i i i

m r k d

k m r

2727

Conclusions, Problems and OutlookConclusions, Problems and Outlook Black Hole Entropy can be computed without a Black Hole Entropy can be computed without a

brick wall with the help of Rainbow’s Gravitybrick wall with the help of Rainbow’s Gravity The same application of Rainbow’s functions can The same application of Rainbow’s functions can

be considered on the Cosmological Constant with be considered on the Cosmological Constant with the help of the Wheeler-De Witt Equation the help of the Wheeler-De Witt Equation Sturm-Liouville Problem.Sturm-Liouville Problem.

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

Regularization substituted by a Modified Regularization substituted by a Modified Geometry Finite Results!!! Geometry Finite Results!!!

Effect on particles motion Effect on particles motion In preparation In preparation