Upload
milek
View
93
Download
7
Tags:
Embed Size (px)
DESCRIPTION
Entropic Gravity. SISSA, Statistical Physics JC Friday 28, 2011. E. Verlinde , arXiv : 1003.4464v2 [ hep-th ]. F. Entropy. Outlook. Background: Holographic Principle (Black Hole Thermodynamics , Entropy Bound ) Verlinde argument for an entropic gravity - PowerPoint PPT Presentation
Citation preview
Entropic Gravity
SISSA, Statistical Physics JCFriday 28, 2011
E. Verlinde, arXiv: 1003.4464v2 [hep-th]
F
𝑥1 𝑥2
𝑥3
𝑥4
𝑥5
Entropy
𝑥5 ′
∆ 𝑥 𝑭 ∆𝑺
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
Hawking (1971)
Bekenstein (1972)
Hawking (1973)
Black body radiation
𝑑𝑆𝑚𝑎𝑡𝑡+𝑑𝑆𝐵𝐻≥0
𝑆𝑚𝑎𝑡𝑡≠0 =
𝑆𝐵𝐻 𝐴
R Bekenstein (1981)
E
A
𝐸<𝑀𝐵𝐻
𝑺𝒊𝒏=𝑺𝒎𝒂𝒕𝒕+𝑺𝒔𝒉𝒆𝒍𝒍
Susskind (1995)
≤𝑺𝒇𝒊𝒏=𝑺𝑩𝑯=𝑨𝟒𝑀𝐵𝐻−𝐸
Toward the holographic principle…
𝒅=𝐥𝐧𝑵=𝒍𝒏𝒅𝒊𝒎 (𝑯 )
Ex 1 𝑑=100 𝑙𝑛2 100 bits of information
Ex 2 !
Number of degrees of freedom
Ex 3 Quantum field theory
𝑪𝒆𝒍𝒍 𝒔𝒊𝒛𝒆 𝑷𝒍𝒂𝒏𝒄𝒌 𝑳𝒆𝒏𝒈𝒉𝒕 𝑟 𝑆=𝐺𝑚𝑐2h ν=𝑚𝑐2
𝑙𝑝=√ℏ𝐺𝑐3 =1.6×10−33𝑐𝑚
𝐸𝑛𝑒𝑟𝑔𝑦 𝑠𝑝𝑒𝑐𝑡𝑟𝑢𝑚𝑏𝑜𝑢𝑛𝑑𝑒𝑑𝑏𝑦 h𝑡 𝑒 𝑃𝑙𝑎𝑛𝑐𝑘𝑀𝑎𝑠𝑠
𝑚𝑝=√ℏ𝑐𝐺 =1.3×1019𝐺𝑒𝑉V oscillators and n states per oscillator
𝑁=𝑛𝑉 𝑑=𝑉 𝑙𝑛𝑛
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»?
𝑆≤ 𝐴4
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒃𝒊𝒕𝒔=𝒅=𝑨
𝟒 𝑨𝑷
𝑁=𝑒𝐴4
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
Holographic screen
… nothing yet…
Emerged space
110011110010001111101001
We further assume the theory has a notion of time and that its dynamics is traslational invariant
EnergyStat. Phys.
Temperature
Holographic screen
∆ 𝒙
∆𝑺=2𝜋 𝑘𝑚𝑐ℏ ∆ 𝒙 𝐹 ∆𝑥=𝑇 ∆𝑆
𝒌𝑻=𝟏𝟐𝝅
ℏ𝒂𝒄
∆𝑺
𝑻
Unruh Effect
𝑭=𝒎𝒂
Force and Inertia
Newton’s law of gravity
𝑁=𝐴𝑐3𝐺ℏ
𝐸=12𝑁𝑘𝑇 𝐸=𝑀𝑐2
∆𝑺=2𝜋 𝑘𝑚𝑐ℏ ∆ 𝒙 𝐹 ∆𝑥=𝑇 ∆𝑆
Holographic principle
T
𝑭=𝑮𝑴𝒎𝑹𝟐
(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
𝑾𝒉𝒂𝒕𝒂𝒃𝒐𝒖𝒕 𝒕𝒉𝒆𝒖𝒏𝒊𝒗𝒆𝒓𝒔𝒂𝒍 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕𝒔 ? 𝒄 ,ℏ ,𝑮
(iii) There is a change of entropy in the emergent direction
𝑚𝑐2=12𝑁𝑘𝑇 Bekenstein + Unruh
∆𝑺𝑵 =𝒌 𝒂∆ 𝒙
𝟐𝒄𝟐 𝒂=−𝜵𝝓 ∆𝑺𝑵 =−𝒌 ∆𝝓
𝟐𝒄𝟐
∆𝑺=2𝜋 𝑘𝑚𝑐ℏ ∆ 𝒙 𝒌𝑻=𝟏𝟐𝝅
ℏ𝒂𝒄
ɸ is a coarse-graining variable∆𝑺𝑵 =−𝒌 ∆𝝓
𝟐𝒄𝟐
𝟎<− 𝝓𝟐𝒄𝟐<𝟏
Coarse- Graining
Space is emerging!
Amount of coarse graining
Dark Energy
radius of the observable universe
holographic principle
𝑀𝑐2=12 𝑁𝑘𝑇=12 𝐴𝑘𝑇
𝑀=1.4 1060𝑚𝑎𝑠𝑠 𝑜𝑓 h𝑡 𝑒𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑏𝑙𝑒𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑒
𝑘𝑇=𝑀𝑐2𝐴 10− 64
h𝑡 𝑒𝑒𝑛𝑡𝑟𝑜𝑝𝑖𝑐 𝑓𝑜𝑟𝑐𝑒𝐹=𝑘𝑇 𝛻𝑁=𝑘𝑇𝑔𝑟𝑎𝑑 (𝜋 𝑅2 )=2𝜋𝑘𝑇𝑅
1𝑅𝑑2𝑅𝑑𝑡2
=𝐹𝑀𝑅=2𝜋 𝑘𝑇𝑀 1.310−123
𝒌𝑻=𝟏𝟐𝝅
ℏ𝒂𝒄Unruh Effect
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’