from black hole entropy to the cosmological constant modified dispersion relations: from black hole...
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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