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Generalizing Entropic Force Formalism

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Generalizing Entropic Force Formalism Yongpyung 2012, Greenpia Condo, Feb. 21st 2012 GENERALIZING ENTROPIC FORCE FORMALISM Jin-Ho Cho (Hanyang University) w/ Hosin Gong, Hyosung Kim (POSTECH) Motivation (for the entropic force) Quantum Gravity .....not successful so far the role of gravity in AdS/CFT tree level supergravity (closed string) / quantum SYM (open string) Fundamental or Emergent? Entropic Force rubber band tends to increase the entropy S : larger S : smaller [Halliday et al., Fundamentals of Physics] Verlinde’s Idea [Erik Verlinde, 1001.0785] entropic force : F x = T S 4 4 3 holographic principle : Ac /G~ = N E = NkBT/2 T = ~a/2πkBc entropy change : S =2πkB( x mc/~) 4 4 (for a screen shift x ) 4 Newtonian Physics Newton’s law : (a planar screen) T = ~a/2πckB S =2πkB( x mc/~) 4 4 F x = T S 4 4 ~a 2πkBmc x = 4 2πk c ✓ B ◆✓ ~ ◆ = ma x 4 3 Gravitational force : (a spherical screen) N = Ac /G~ 2 Mc = E = NkBT/2 1 Ac3 ~a r2c2 = = a 2 G 2πc ✓ ~ ◆✓ ◆ G Q1: Cosmological Constant ? g R ab R + ⇤g =8⇡GT ab − 2 ab ab 3 Ac /G~ = N F x = T S S =2πkB( x mc/~) 4 4 E = NkBT/2 4 4 T = ~a/2πkBc ‘Volume Energy’ NkBT E = Mc2 +↵V = 2 N~ a ~a = T = 4⇡c 2⇡ckB 2 Ac a 3 = N = Ac /G~ 4⇡G 4⇡GM 4⇡G↵V a = + A Ac2 Determination of ↵ spherically symmetric case GM 4⇡G↵r a = + = Φ r2 3c2 r GM c2⇤r = Newtonian limit of Einstein eq. r2 − 3 c4 ↵ = ⇤ −4⇡G Einstein equation 4 c NkBT E = Mc2 ⇤V = − 4⇡G 2 c4 k c3 Mc2 ⇤ dV = B TdN = adA − 4⇡G 2 4⇡G Z Z Z 1 a b 1 a b 2 T Tg n ⇠ dV ⇤gabn ⇠ dV ab − 2 ab −4⇡G Z⌃ ✓ ◆ Z⌃ 1 = R na⇠b dV 4⇡G ab Z⌃ Q2: Entropic Coulomb Force ? [JHC & Hyosung Kim, 2012 J. Phys.: Conf. Ser. 343 012024 (talk in QTS7)] Erik Verlinde’s new idea : .....gravity as an entropic force [Erik Verlinde, 1001.0785] .....more specifically as an adiabatic reaction force [Erik Verlinde, STRING 2011 talk] Q : what about the Coulomb force? Some earlier attempts : [Tower Wang, Phys.Rev.D81:104045,2010] [Peter Freund, arXiv:1008.4147] Kaluza-Klein addon Moving Sources a cylinder along Kaluza-Klein direction M0 ⇥0 = 0 2πL0 2 E0 =(⇥0 2πL0) c 1 γ = 1 (v/c)2 − q γM0 2 ⇤ = = γ ⇤0 2⇥L0/γ 2 E =(⇤ 2⇥L0/γ) c = γE0 Some Kinematics (I) from M 0 -rest frame to m 0 -rest frame : γ2 A(5)c3 A(4)2πLc3 A(4)c3 N 0 = = = = N G(5)~ G(4)2πL~ G(4)~ (4) 3 2 1 1 A c ~a0 γ2M0c = E0 = N 0kBT 0= 2 2 G(4) 2πc ✓ ~ ◆✓ ◆ r2c2 = a0 G(4) Some Kinematics (II) from m 0 -rest frame to observer-rest frame : γ1 M = γ1γ2M0 m = γ1m0 γ = γ1γ2 γ 2 in terms of γ 1 and γ : 1 2 2 γ = γ− γ = γ γ γ 1 γ 1 2 1 1 − 1 − − qp1 p = γ1γ p − m0c M0c Coulomb Force entropic force exerted on m 0 : GM0 p1 p GM0m0 F = m a0 = m γ = γ γ 0 0 2 r2 1 − m c M c r2 ✓ 0 0 ◆ GmM Gp1p GmM ke e = = 1 r2 − c2r2 r2 − r2 charge as the KK momentum : n ~ G pn G en = lpqp = kn = Rs c r k c r k Conclusions ✴ The cosmological constant is nothing but the coefficient of the ‘volume energy’ contained in the holographic surface. ✴ If gravity is not fundamental but emergent, Coulomb force can be emergent too !! ✴ For the equal sign of charges, the entropic Coulomb force acts in the opposite direction to that of the gravitational force. ✴ Entropic force idea cannot single out the gravitational force from others, but might be a new paradigm applying for all forces..
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