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Entropic Gravity
SISSA, Statistical Physics JC
Friday 28, 2011
E. Verlinde, arXiv: 1003.4464v2 [hep-th]
F
๐ฅ3
๐ฅ1
๐ฅ2
๐ฅ5
๐ฅ4
E
n
t
r
o
p
y
๐น=
โ๐ฅ
๐๐
๐
๐๐ฅ
๐ญ~โ๐บ
๐ฅ5 โฒ
Outlook
โข Background: Holographic Principle
(Black Hole Thermodynamics, Entropy Bound)
โข Verlinde argument for an entropic gravity
(II principle of dynamics, Newtonโs law of gravity)
โข โฆ editorial discussion
๐
๐จ โฅ ๐ Hawking (1971)
๐บ๐ฉ๐ฏ =
๐จ
๐
๐๐๐๐๐ก๐ก + ๐๐๐ต๐ป โฅ 0
Bekenstein (1972)
Black body radiation
๐ป=
โ๐๐
๐๐ฎ๐ด๐
Hawking (1973)
๐๐๐๐ก๐ก โ 0
๐๐๐๐ก๐ก = 0
๐๐ต๐ป ~๐ด
R
๐บ๐๐๐๐ < ๐๐
๐ฌ๐น Bekenstein (1981)
E
๐ธ < ๐๐ต๐ป
๐๐ต๐ป โ ๐ธ
๐บ๐๐
= ๐บ๐๐๐๐ + ๐บ๐๐๐๐๐ โค
A
๐บ๐๐๐๐ โค
๐จ
๐
๐บ๐๐๐
= ๐บ๐ฉ๐ฏ
๐จ
=
๐
Susskind (1995)
Toward the holographic principleโฆ
๐
= ๐ฅ๐ง ๐ต = ๐๐ ๐
๐๐(๐ฏ)
Ex 1 ๐๐๐ ๐๐๐๐ ๐ = 2100 ๐ ๐ก๐๐ก๐๐
Ex 2
๐ฏ๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐
Ex 3
Quantum field theory
Number of degrees of freedom
๐ = 100 ๐๐2
๐ช๐๐๐ ๐๐๐๐ ~๐ท๐๐๐๐๐ ๐ณ๐๐๐๐๐
100 bits of information
๐ = โ!
โฮฝ = ๐๐ 2
๐๐ =
๐บ๐
๐๐ = 2
๐
โ๐บ
= 1.6 × 10โ33 ๐๐
3
๐
๐ธ๐๐๐๐๐ฆ ๐ ๐๐๐๐ก๐๐ข๐ ๐๐๐ข๐๐๐๐ ๐๐ฆ ๐กโ๐ ๐๐๐๐๐๐ ๐๐๐ ๐
V oscillators and n states per oscillator
๐ = ๐๐
๐ = ๐ ๐๐ ๐
๐๐ =
โ๐
= 1.3 × 1019 ๐บ๐๐
๐บ
How many different states can be in a region to describe all the physics inside of it?
๐๐บ ~๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐๐๐
What is the entropy of the «fundamental system»?
๐ด
๐โค
4
๐=
๐ด
๐4
๐จ
๐๐๐๐๐๐ ๐๐ ๐๐๐๐ = ๐
=
๐๐จ๐ท
A region with boundary of area A is fully described by no more than A/4
degrees of freedom, or about 1 bit of information per Planck area
Outlook
โข Background: Holographic Principle
(Black Hole Thermodynamics, Entropy Bound)
โข Verlinde argument for an entropic gravity
(II principle of dynamics, Newtonโs law of gravity)
โข โฆ editorial discussion
SPACE as a storage of information
โฆ nothing yetโฆ
110
011
110
010
001
111
101
001
Emerged space
Holographic screen
We further assume the theory has a notion of time and that its dynamics is traslational invariant
Energy
Temperature
Stat. Phys.
Force and Inertia
โ๐บ
๐ป
โ๐
Holographic screen
๐๐
โ๐บ = 2๐๐
โ๐
โ
๐๐ป =
๐นโ๐ฅ = ๐โ๐
๐ โ๐
๐๐
๐
Unruh Effect
๐ญ = ๐๐
Newtonโs law of gravity
Holographic principle
๐ด๐ 3
๐=
๐บโ
๐๐
โ๐บ = 2๐๐
โ๐
โ
๐ธ=
1
๐๐๐
2
๐ธ = ๐๐ 2
๐นโ๐ฅ = ๐โ๐
๐ฎ๐ด๐
๐ญ=
๐น๐
T
(i) The number of degrees of freedom is proportional to the area
of the screen (Holographic principle)
(ii) The energy is evenly distributed over these degrees of freedom
๐พ๐๐๐ ๐๐๐๐๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐? ๐, โ, ๐ฎ
(iii) There is a change of entropy in the emergent direction
๐๐ 2
1
= ๐๐๐
2
Bekenstein + Unruh
โ๐บ = 2๐๐
โ๐บ
๐โ๐
=๐ ๐
๐ต
๐๐
๐ = โ๐ต๐
๐๐
โ๐
โ
๐๐ป =
๐ โ๐
๐๐
๐
โ๐บ
โ๐
= โ๐ ๐
๐ต
๐๐
ษธ is a coarse-graining variable
โ๐บ
โ๐
= โ๐ ๐
๐ต
๐๐
๐
๐<โ ๐<๐
๐๐
Amount of coarse graining
Coarse- Graining
Space is emerging!
Dark Energy
๐
~2.7 1061 radius of the observable universe
๐ = ๐ด = ๐๐
2 holographic principle
1
1
๐๐ 2 = ๐๐๐ = ๐ด๐๐
2
2
๐ = 1.4 1060 ๐๐๐ ๐ ๐๐ ๐กโ๐ ๐๐๐ ๐๐๐ฃ๐๐๐๐ ๐ข๐๐๐ฃ๐๐๐ ๐
๐๐ 2
๐๐ =
~10โ64
๐ด
๐กโ๐ ๐๐๐ก๐๐๐๐๐ ๐๐๐๐๐ ๐น = ๐๐๐ป๐ = ๐๐๐๐๐๐ ๐๐
2 = 2๐๐๐๐
1 ๐2 ๐
๐น
๐๐
โ123
=
=
2๐
~1.3
10
๐
๐๐ก 2 ๐๐
๐
Unruh Effect
๐ โ๐
๐๐ป =
๐๐
๐
It works for dimensional consistency!
References
โข E. Verlinde โOn the origin of Gravity and the Newton
lawsโ
โข S.Gao Comment on "On the Origin of Gravity and the
Laws of Newton"
โข A. Chivukula โGravity as an entropic phenomenonโ
โข T. Jacobson, โThermodynamics of Spacetimeโ Phys. Rev.
Lett. (1995)
โข R. Bousso โThe holographic principleโ
โข R. Ruffini and H. Ohanian โGravitation and spacetimeโ