Strings 2011, Uppsala July 1st, 2011

Erik Verlinde

University of Amsterdam

• Motivation: BH’s + -/.

• Adiabatic reaction forces.

• Hidden phase space: proposal.

• Inertia/ as adiabatic reaction force.

• “Speculations” about /dark matter. GRAVITATIONAL COLLAPSE:

What happens to the phase space occupied by the fermions?

String theory/ AdS-CFT: Y(x)« tr(X nl)

Bulk fields correspond to (short) traces of boundary fields. FD/BE statistics => “D-” statistics

æ x x ö 2 11 12 x1,x2 Þ ç ÷ Z2 ´U(1) ®U(2) è x21 x22 ø

N SN ´U(1) ®U(N) xij = i xˆ j

Gravity results from integrating out “off diagonal” degrees of freedom.

Coordinates as matrices

æ x11 x12 .. x1N ö ç ÷ x x :: : X = ç 21 22 ÷ ç : :: :: xN -1N ÷ ç ÷ è xN1 .. xNN -1 xNN ø

2 2 * I H=tr(PI +[XI , XJ ] + l XI G l)

Eigenvalues: describe position of matter

Space, matter and forces emerge together GRAVITATIONAL æ x11 x12 .. x1N ö COLLAPSE: ç ÷ x x :: : X = ç 21 22 ÷ ç : :: :: xN -1N ÷ The phase space ç ÷ occupied by the è xN1 .. xNN -1 xNN ø fermions goes into the “off diagonal” modes. æ x x .. x ö ç 1 12 1N ÷ Eigenvalues and off ç x21 x2 :: : ÷ X = ç ÷ : :: :: x diagonal modes ç N-1N ÷ ç x .. x x ÷ equilibrate and è N1 NN-1 N ø together form a thermal state describing the . Degenerate Space-time in its usual Fermions sense ceases to exist. æ x11 x12 .. x1N ö ç ÷ x x :: : X = ç 21 22 ÷ ç : :: :: xN -1N ÷ ç ÷ è xN1 .. xNN -1 xNN ø

The off diagonal modes carry a positive gravitational energy

1 2 E = ÑF g 8pG ò 1 2 E = ÑF g 8pG ò

M1 -F2M1 M2 -F1M2 Black Hole

æ ...... ö ç ÷ ç ...... ÷ ç ...... H. ÷ ç ...... ÷ 2 ç ÷ ç ...... ÷ | X × H | ç ...... ÷ ç * ÷ è . H. . . . . X. ø

Matrices gets “Higgsed” Eigenvalues disappear as flat directions Coulomb branch goes over into Higgs branch M Black Hole Horizon R

DW = MgR = TDS

g T = Þ DS = 2pMR 2p Black Hole

æ ...... ö ç ÷ ç ...... ÷ ç ...... ÷ ç . . Higgs...... ÷ ç ÷ ç . . branch...... ÷ McR ç ...... ÷ N = ç ...... ÷ ç ÷ ç ...... ÷ ç ...... ÷ ç ...... ÷ ç ÷ ç ...... ÷ ç ...... Coulomb. . . . ÷ Number of “eigenvalues” ç ...... ÷ ç branch ÷ associated with a region ç ...... ÷ ç ...... ÷ with size R and containing è ø mass M PHASE SPACE

x1

Higgs x2

branch ,

Coulomb . branch

.

xN

Adiabatic Reaction Force Harmonic oscillator with slowly varying frequency E

1 2 2 2 H(p,q;x) = 2 ( p +w (x)q ) x Bohr-Sommerfeld action integral = adiab 1 J º pdq dE =w dJ - Fdx 2p ò

Adiabatic reaction force:

æ ¶E ö æ ¶Jö 1 æ ¶J ö F = - = w = ç ÷ ç ÷ ç ÷ w è ¶E ø è ¶x ø J è ¶xø E x

Adiabatic Reaction Force E For a (chaotic/ergodic) system with many DOF

x

the phase space volume is an adiabatic invariant. Formally it defines an entropy

Born Oppenheimer force: dE = T dS - Fdx

æ ¶E ö æ ¶Sö 1 æ ¶S ö F = -ç ÷ = Tç ÷ = ç ÷ è ¶x ø è ¶x ø T è ¶E ø S E x

Berry Phase and Crossing Eigenvalues

(0) F = -¶x y H y F = F(0) +eF(1) +.. F(1) = xÙd y d y

Systematic expansion of the reaction force magnetic force in terms of a “slowness” parameter. due to Berry phase.

æ z x + iyö xˆ E H = ç ÷ = x × s B = 2 è x - iy -z ø 4p x x Dirac monopool

At the locus of coinciding eigenvalues exists a non-abelian Berry connection. Aij = yi d y j

F /T =¶ S(E, x) Heat Bath x 1/T =¶ S(E, x) T E

An does not lead to Polymer a change in entropy, despite its name.

It acts adiabatically. Replace all degrees of freedom by just one

ò dE W(E, x) e-E/T = ò [dpdq]eI[ p,q;x]/

I[p,q;x]= ò ( pdq - H(p,q;x)dt)

Action/Entropy correspondence J w S = 2p T = 2p => Adiabatic reaction forces agree

Black Hole F = mg Horizon

æ x .. x z ö ç 11 1N 1 ÷ ç : :: : : ÷ X = ç x .. x z ÷ ç N1 NN N ÷ ç z* .. z* xI ÷ T è 1 N ø

Heat ¶S g ¶S mc bath F = T T = Þ = 2p ¶x 2p ¶x

F = mÑF A F = 1 v2 S = v 2 4G v = escape velocity æ ö ç ...... ÷ ç ...... ÷ ç ...... ÷ ç ÷ ç ...... ÷ ç ...... ÷ ç . . . . . x . ÷ ç 1 ÷ ç ...... x ÷ è N ø

Matter disassociates into N entangled N = MR “quantum energy bits”.

To disentangle one inserts a complete set A C = of states on the boundary. 8pG The micrsocopic phase space is obtained by partitions the N quanta over these states. S = 2p C× N

For a black hole: N=C For metrics of Schwartschild / de Sitter type

dr2 ds2 = -(1- v2 )dt 2 + + r2dW2 1- v2 we have dv 1 dv F = gmv T = v = H0r dr 2p dr and

dS dS A F = T = 2pgmv S = v dr dr 4G The phase space volume of the dynamical system that is underlying our universe saturates the holographic FSB bound

H0V H S = T = 0 4G(d -1) 2p

Inertia is a reaction force due to this system. The energy of the system is (dominated by) the (observed) dark energy in our universe

16pG E H 2 = 0 n(n +1) V PHASE SPACE

x1

Higgs x2

branch ,

Coulomb . branch

.

xN

Black Hole

æ ...... ö ç ÷ ç ...... ÷ ç ...... ÷ ç ...... ÷ ç ÷ ç ...... ÷ ç ...... g...... ÷ ç . . T. . =...... ÷ ç ÷ ç . . . . . 2. p...... ÷ ç ...... ÷ ç ...... ÷ ç ÷ ç ...... ÷ ç ...... ÷ ç ...... ÷ ç ÷ ç ...... ÷ ç ÷ N = MR è ...... ø

1 2 E = ÑF g 8pG ò æ ö ç ...... ÷ Fluctuations in the inertial field ç ...... ÷ ç ...... ÷ ç ÷ 1 ç ...... ÷ - |ÑF|2 2 ò 2 ç ...... ÷ 8pGkBT ÑF = [dF]e ÑF ç . . . . . x . ÷ ò ç 1 ÷ ç ...... x ÷ è N ø A short distance cut off

GkBT 3 ÑF(x)ÑF(0) = 3 R 3 x = x N or a mode cut off. Both give

1 2 2 GNkBT Eg = ò ÑF = NkBT ÑF = 8pG R3 æ ...... ö çæ ÷ ö ç ...... ÷ çç...... ÷ ÷ ç ...... ÷ çç ÷ ÷ çç...... ÷ ÷ ç ...... ÷ çç...... ÷ ÷ ç ...... ÷ çç...... ÷ ÷ ç ...... ÷ çç...... ÷ ÷ ç ÷ çç...... x . . .. . ÷ ÷ ç ...... 1. . . . ÷ çç...... ÷ ÷ çç ...... x ÷ ÷ ç ...... N. ÷ çè ÷ ø è ...... ø

N = MR

H T = 0 2p æ ...... ö çæ ÷ ö ç ...... ÷ çç...... ÷ ÷ ç ...... ÷ çç ÷ ÷ çç...... ÷ ÷ ç ...... ÷ çç...... ÷ ÷ ç ...... ÷ çç...... ÷ ÷ ç ...... ÷ çç...... ÷ ÷ ç ÷ çç...... x . . .. . ÷ ÷ ç ...... 1. . . . ÷ çç...... ÷ ÷ çç ...... x ÷ ÷ ç ...... N. ÷ çè ÷ ø è ...... ø Galaxy! " #$ %&' ( )rotation) &* &%+, -. /0 1&curves&) 232) ) &&&

v(r) 2 GM(r) = r r2 ! " #$ %&' ( ) ) &* &%+, -. /01&&) 232) ) &&&

GM (r) v (r) = obs obs r

GM (r) v (r) = B B r a0 2p a 0 =1.24 ± 0.14 ×10-10 m/s2 4 a0 2p Vobs = GM B 2p Why? a0 » cH0 Eg = NkBT

H. Zhao

1 R R E(R) = dg2 4pr2 dr N(R) = r(r)r 4pr2 dr 8pG ò ò 0 0 J*$"&+.R&)-$-01,($1)%$*.3($%.--100(1+@$

² x11 . . x1N . . xN,N + n % ¢ ...... ¢ ...... ¢ ¢ x . . x . . . N1 NN ¢

. . . . xN +1,N +1 . . ¢ ¢ ...... ¢ ¢ # xN + n,1 . . . . . xN + n,N + n &

! PHASE SPACE

Baryonic Dark Dark x1

M x2 E M a n a , t t e . t r t e . g e y r r xN

! " #$ %&' ( ) ) &* &%+, -. /01&&) 232) ) &&&

! " #$ %&' ( ) ) &* &%+, -. /01&&) 232) ) &&&