elastic expansion in university

Generic perturbations of black branes viscous + elastic .More general theories of hydrodynamics confined fluids .Fluid membranes | Cellular membranes .AdS/CFT at finite temperature...

Albert Einstel

„ Perturbative construction of higher-dimensional black holes Effective action for higher-dimensional black holes (blackfolds)

Generic perturbations of black branes (viscous + elastic)

.More general theories of hydrodynamics (confined fluids) .Fluid membranes | Cellular membranes

.AdS/CFT at finite temperature

An observation:

yal? + fry Ne?

Need effective theory of embedded fluids

DBl-type action:

minal surface

red blood cell: erythrocyte

Hetfrich-Canham propose in the 70's

an additional piece:

Vi (ot KK) and many more! (se review by fer (1537)

Polyakov and Kleinert make the same proposal for an action of QCD

In the context of cosmic strings the most general elastic action to ond order and codimension > 1 is written dawn:

Consider a fluid brane which is in stationary motion:

y implies:

Therefore the action for non-extremal branes:

and hence the stress-energy tensor

identify:

Along worldvolume directions the brane behaves like a fluid:

Along orthogonal directions it behaves like an elastic brane:

AxiEE: | AUK)? — NODC Aue? — DAUR REK Kụ | Ask)kPkPKY + AufkD2KPk Kat

AxiEE: | AUK)? — NODC Aue? — DAUR REK Kụ | Ask)kPkPKY + AufkD2KPk Kat

The bending moment can be written as:

where the Young modulus is:

aaah Aaah Ag

this can be measured from gravity!

wed — yy(ab)(ed) — yyedab

For codimension-1 surfaces we need to add a piece:

qx" = Me V7 (800K + Ba(lMMKPK Va Ki)

The hydrodynamics modes are coupled to the elastic modes

‘through the Gauss-Codazzi equation:

Raved = Reabea — Kac' Kodi + Kod Koei

a)

‘Summary of the transport coefficients:

3 hydrodynamic, 3 elastic and 1 spin transport coefficient for codimension > 1 surfaces

3 hydrodynamic and 5 elastic transport coefficients for codimension-1 surfaces

Ld 1 hydrodynamic and 4 elastic for fluid membranes in

3-dimensional flat space (hydrodamic transport coefficient and 2 elastic have not been measured yet)

4 Armas arkivl 39047773

Take the general equations of motion:

V7 = uụ0VạV.D99t + D2 R gi, + 80,08

n!,V„V,Ð*t + DOI B 35 +2nt, Ty (Syi°K%,) + SY Rags

nig VS =0 Impose positivity of the entropy current:

5.61 Bhattachary()a, Minwalla, 2011

S.Bhateacheryya, 2012

make the following assumptions:

We assume a spinless fluid

We assume the existence of a worlvolume entropy

current

We consider a first order dissipative theory for

codimension-1 surfaces and a non-dissipative theory

to second order for codimension higher than one

We assume the first law of thermodynamics and the

Gibbs-Duhem relations

tỷ

Armas aPkiv 312.0597

Under these assumptions the equations of motion are:

VaT® = mp!D°iVaKac? —2Va(D“KM)

TY Kap! = nh VaVeD™ + DM Rian,

DK ay!) =0

Need to classify the following structures to second order:

J.Armas , arXiv:1312.0597

—— ~

\We classify all on-shell independent terms ta second order in

the Landau gauge and in a specific choice of surface:

Decompose the derivative of the fluid velocity as:

@ Vay = Yate + oad + Wah + oan

‘Tensors elastic (4)

Classify all terms: second order data eet onter data | Beso imposing BOM OM Tnepende dan Sexist) | FR pKa OnE R he vé, (RA) = 0 | 6H og! eT frideestie(3) | @ubkul , wUeR (PIR) 0 | OB 5 08K Oak EK, | eK RK WRK: Weerars cate | xi gu, vưếth ata KK «uh OU KS WK! we ok ai smtcn) | 9t 2 ong | Y-E9S«) 0| mất, uae _ ewan! ser eK sacl ed Tea KK, KR na ream etn fy | RR PORE WalOKO KE, POKES : avo eat (6) | 0 Peau aa PCR PK Ri

Classify all terms: third order data

6K, , 0° K„, Our Ke Ris 0 Kos |

OM Ka Ks

ou Kae Kha

ww Ky!K, , au Ke Kc WKAR UK PR

WK VaK™ , wT Ki

cK VK

cK Wal web KOT Kod

WKN (T*K a!)

OK Kis oul Kac Kia

Codimension-1 surfaces to first order:

= TE} +n0% + €0P% + or KP® + a2P°P Koa

De = yy"

J2 = su? + 6,0u% + Boa? + B3Kut + ByubKy*

Positivity of the divergence implies:

For higher codimension and to second arder we have:

JP (ar KK; + oak" Koa + agu'u' Ke! Kay)

PPh (aK RE + aK TRY, bull KEK)

Summarizing:

For codimension-1 surfaces and to 1st order we have

2+1 independent transport coefficients (dissipative)

=P For codimension higher we have 10+3 independent

transport coefficients (non-dissipative)

The constraints match those obtained from

equilibrium partition functions

J Armas , arXiv:1312.0597

The fluid becomes black:

Th conga nortan ObeT SN TTTOTESS

equations of motion are obtained by solving:

the total spin is the integral over the current:

We take a Schwarzschild black brane and bend it:

(ig ta? + Pant oath

ty = (nas Bhan 8-+ gts) dot +

‘The dipole moment takes the form:

‘The Young modulus is:

TA Camps.harnadc One 2211220 4035 Camps, Emparan, 2tXiv1201.3505

Pe) fe et) 3+?) Pe) fle) (0) kế

8n =4 PÚ9rÏ(k)ö0)

a = Ae) _ i

Sa ott a s.ấ :

Aring embedded in flat space:

Corrected phase diagram expressed in physical quantities:

Emnpyan,Harmark Nigtchos Oets,Rudlgues,oXiz0208 2181

Empatan,Hamiai, Niatchos, 08ers, Rodrigues,3Xt:0708 1182

Dia, Santos, Way, arkv:1402.6345 1A 2 Harmar, arXiv 402.6350

— > et

Decompose the dipole correction as:

Split the gauge field as:

= A ADO 3 T4 +12— || vá” —ierenssa, s96) , „

The electric dipole moment is of the form:

for charged dilatonic branes from KK reduction

exlonr§ (25.009 + EIPaP)

TA Gath, Obers, P1208 5197, ark 307 504

‘Asummary of the results:

=> =>

—>

<

Generic effective action of fluid branes to second order

First order dissipative theory of (confined) hydrodynamics and second order non-dissipative theory

Measurement of transport coefficients from gravity

Systematic method for finding corrections to black hole charges, good to compare with numerics Can also study stability

Future directions:

ydy AdS/CFT interpretation of the Young modulus | bending D3-brane Including backreaction corrections in the effective theory

Anomalous couplings, Chern-Simons terms Universality of transport coefficients

Full dissipative theory and non-relativistic theory

Spinning actions and thermodynamics to all orders armas, Tioels Harmar (a anpe30

Jay Armas | Albert Einstein Center for Fundamental Physics