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’