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```Entropic Gravity
SISSA, Statistical Physics JC
Friday 28, 2011
E. Verlinde, arXiv: 1003.4464v2 [hep-th]
F
𝑥3
𝑥1
𝑥2
𝑥5
𝑥4
E
n
t
r
o
p
y
𝐹=
∆𝑥
𝜕𝑆
𝑇
𝜕𝑥
𝑭~∆𝑺
𝑥5 ′
Outlook
• Background: Holographic Principle
(Black Hole Thermodynamics, Entropy Bound)
• Verlinde argument for an entropic gravity
(II principle of dynamics, Newton’s law of gravity)
• … editorial discussion
𝒅𝑨 ≥ 𝟎 Hawking (1971)
𝑺𝑩𝑯 =
𝑨
𝟒
𝑑𝑆𝑚𝑎𝑡𝑡 + 𝑑𝑆𝐵𝐻 ≥ 0
Bekenstein (1972)
𝑻=
ℏ𝒄𝟑
𝟖𝑮𝑴𝒌
Hawking (1973)
𝑆𝑚𝑎𝑡𝑡 ≠ 0
𝑆𝑚𝑎𝑡𝑡 = 0
𝑆𝐵𝐻 ~𝐴
R
𝑺𝒎𝒂𝒕𝒕 < 𝟐𝝅𝑬𝑹 Bekenstein (1981)
E
𝐸 < 𝑀𝐵𝐻
𝑀𝐵𝐻 − 𝐸
𝑺𝒊𝒏
= 𝑺𝒎𝒂𝒕𝒕 + 𝑺𝒔𝒉𝒆𝒍𝒍 ≤
A
𝑺𝒎𝒂𝒕𝒕 ≤
𝑨
𝟒
𝑺𝒇𝒊𝒏
= 𝑺𝑩𝑯
𝑨
=
𝟒
Susskind (1995)
Toward the holographic principle…
𝒅 = 𝐥𝐧 𝑵 = 𝒍𝒏 𝒅𝒊𝒎(𝑯)
Ex 1 𝟏𝟎𝟎 𝒔𝒑𝒊𝒏 𝑁 = 2100 𝑠𝑡𝑎𝑡𝑒𝑠
Ex 2
𝑯𝒂𝒓𝒎𝒐𝒏𝒊𝒄 𝒐𝒔𝒄𝒊𝒍𝒍𝒂𝒕𝒐𝒓
Ex 3
Quantum field theory
Number of degrees of freedom
𝑑 = 100 𝑙𝑛2
𝑪𝒆𝒍𝒍 𝒔𝒊𝒛𝒆 ~𝑷𝒍𝒂𝒏𝒄𝒌 𝑳𝒆𝒏𝒈𝒉𝒕
100 bits of information
𝑁 = ∞!
ℎν = 𝑚𝑐 2
𝑙𝑝 =
𝐺𝑚
𝑟𝑆 = 2
𝑐
ℏ𝐺
= 1.6 × 10−33 𝑐𝑚
3
𝑐
𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑝𝑒𝑐𝑡𝑟𝑢𝑚 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑃𝑙𝑎𝑛𝑐𝑘 𝑀𝑎𝑠𝑠
V oscillators and n states per oscillator
𝑁 = 𝑛𝑉
𝑑 = 𝑉 𝑙𝑛 𝑛
𝑚𝑝 =
ℏ𝑐
= 1.3 × 1019 𝐺𝑒𝑉
𝐺
How many different states can be in a region to describe all the physics inside of it?
𝒆𝑺 ~𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒎𝒊𝒄𝒓𝒐𝒔𝒕𝒂𝒕𝒆𝒔
What is the entropy of the «fundamental system»?
𝐴
𝑆≤
4
𝑁=
𝐴
𝑒4
𝑨
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒃𝒊𝒕𝒔 = 𝒅 =
𝟒𝑨𝑷
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
• Background: Holographic Principle
(Black Hole Thermodynamics, Entropy Bound)
• Verlinde argument for an entropic gravity
(II principle of dynamics, Newton’s law of gravity)
• … editorial discussion
SPACE as a storage of information
… nothing yet…
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
∆𝑺
𝑻
∆𝒙
Holographic screen
𝑚𝑐
∆𝑺 = 2𝜋𝑘
∆𝒙
ℏ
𝒌𝑻 =
𝐹∆𝑥 = 𝑇∆𝑆
𝟏 ℏ𝒂
𝟐𝝅 𝒄
Unruh Effect
𝑭 = 𝒎𝒂
Newton’s law of gravity
Holographic principle
𝐴𝑐 3
𝑁=
𝐺ℏ
𝑚𝑐
∆𝑺 = 2𝜋𝑘
∆𝒙
ℏ
𝐸=
1
𝑁𝑘𝑇
2
𝐸 = 𝑀𝑐 2
𝐹∆𝑥 = 𝑇∆𝑆
𝑮𝑴𝒎
𝑭=
𝑹𝟐
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
𝑾𝒉𝒂𝒕 𝒂𝒃𝒐𝒖𝒕 𝒕𝒉𝒆 𝒖𝒏𝒊𝒗𝒆𝒓𝒔𝒂𝒍 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕𝒔? 𝒄, ℏ, 𝑮
(iii) There is a change of entropy in the emergent direction
𝑚𝑐 2
1
= 𝑁𝑘𝑇
2
Bekenstein + Unruh
∆𝑺 = 2𝜋𝑘
∆𝑺
𝒂∆𝒙
=𝒌 𝟐
𝑵
𝟐𝒄
𝒂 = −𝜵𝝓
𝑚𝑐
∆𝒙
ℏ
𝒌𝑻 =
𝟏 ℏ𝒂
𝟐𝝅 𝒄
∆𝑺
∆𝝓
= −𝒌 𝟐
𝑵
𝟐𝒄
ɸ is a coarse-graining variable
∆𝑺
∆𝝓
= −𝒌 𝟐
𝑵
𝟐𝒄
𝝓
𝟎<− 𝟐<𝟏
𝟐𝒄
Amount of coarse graining
Coarse- Graining
Space is emerging!
Dark Energy
𝑅~2.7 1061 radius of the observable universe
𝑁 = 𝐴 = 𝜋𝑅2 holographic principle
1
1
𝑀𝑐 2 = 𝑁𝑘𝑇 = 𝐴𝑘𝑇
2
2
𝑀 = 1.4 1060 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑒
𝑀𝑐 2
𝑘𝑇 =
~10−64
𝐴
𝑡ℎ𝑒 𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑓𝑜𝑟𝑐𝑒 𝐹 = 𝑘𝑇𝛻𝑁 = 𝑘𝑇𝑔𝑟𝑎𝑑 𝜋𝑅2 = 2𝜋𝑘𝑇𝑅
1 𝑑2 𝑅
𝐹
𝑘𝑇
−123
=
=
2𝜋
~1.3
10
𝑅 𝑑𝑡 2 𝑀𝑅
𝑀
Unruh Effect
𝟏 ℏ𝒂
𝒌𝑻 =
𝟐𝝅 𝒄
It works for dimensional consistency!
References
• E. Verlinde ‘On the origin of Gravity and the Newton
laws’
• S.Gao Comment on "On the Origin of Gravity and the
Laws of Newton"
• A. Chivukula ‘Gravity as an entropic phenomenon’
• T. Jacobson, ‘Thermodynamics of Spacetime’ Phys. Rev.
Lett. (1995)
• R. Bousso ‘The holographic principle’
• R. Ruffini and H. Ohanian ‘Gravitation and spacetime’
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