„ 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
Trang 3An observation:
yal? + fry Ne?
Trang 7
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
Trang 9y implies:
Therefore the action for non-extremal branes:
and hence the stress-energy tensor
Trang 10Along worldvolume directions the brane behaves like a fluid:
Trang 13Val = ty?VaV-D° + DR oie A(K)KUK, 5 Aa(k)K% Kaye, Aa( Kuch KG, T° Kap! = nÏ,VuVyD*%? + DS RE, Ma(K)K°K?’ Kay Ki, As (K)K"OKK Kas" Ket De Dah Er — A(k)kEivfv — 404) KỲ 2àkhy9Kt AE; |_ 3aÂ)£y°—Aj(l)kEsueu® — 43s(k)K*s,KỲ2 Daath) Aalkes | Av ar — Xjk)kE°v)— 2À(k)k'k”K*, 2rs(hWKO KD!
Trang 14Val = ty?VaV-D° + DR oie A(K)KUK, 5 Aa(k)K% Kaye, Aa( Kuch KG, T° Kap! = nÏ,VuVyD*%? + DS RE, Ma(K)K°K?’ Kay Ki, As (K)K"OKK Kas" Ket De Dah Er — A(k)kEivfv — 404) KỲ 2àkhy9Kt AE; |_ 3aÂ)£y°—Aj(l)kEsueu® — 43s(k)K*s,KỲ2 Daath) Aalkes | Av ar — Xjk)kE°v)— 2À(k)k'k”K*, 2rs(hWKO KD!
Trang 15
The bending moment can be written as: where the Young modulus is:
aaah Aaah Ag
Trang 17For 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
Trang 18
‘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)
Trang 19
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:
Trang 20
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
ý
tỷ
Trang 21
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
Trang 22
\We classify all on-shell independent terms ta second order in the Landau gauge and in a specific choice of surface:
Trang 23Classify all terms: first order data Ist order data | Before imposing EOM | EOM Tndependent data, Sealars fluid (1) 199,7, 0 0 Vectors tid (1) pey,T at | PyWar™=0 “ at a KR äÊuPNuẾ ‘Tensors fluid (1) ot Sealars elastic (1) K Vectors elastic (2) | uy, ut yom, ut
Kem utube Ko utube
KE ueuleK mK! ueuleK bi
‘Tensors elastic (4)
Trang 25
Classify all terms: third order data Srd order data Before imposing EOM Independent data Sealars uid-elastie (9) 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 UKM TK ek Tam ` Out Ke Kies 5 0% Kas! Ki
Trang 26Codimension-1 surfaces to first order:
= TE} +n0% + €0P% + or KP® + a2P°P Koa
De = yy"
J2 = su? + 6,0u% + Boa? + B3Kut + ByubKy*
Trang 27For higher codimension and to second arder we have: = Ti + no® + oP” H+TP* (n2R 7 rome = oR <> guPut + nạo 52+ mại asta!) + ns Reaueu® — naweaw + 750°)
JP (ar KK; + oak" Koa + agu'u' Ke! Kay)
PPh (aK RE + aK TRY, bull KEK)
PK + Ag KM + Azul),
Trang 28Summarizing:
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
Trang 30To connect with gravity we need an equivalent formulation in terms of space-time tensors: rea) = VasileVojnovi,arXic07075395 where: TIP = ugg”)? + ubabe™ + apts
Th conga nortan ObeT SN TTTOTESS
Trang 31Action formulation and mul ipole expansion are equivalent provided: rot es 42d KY,, de 1a, aXWed506.773 the dipole moment is the bending moment: 5
Trang 32
We take a Schwarzschild black brane and bend it:
(ig ta? + Pant oath
Trang 33
‘The dipole moment takes the form: ‘The Young modulus is: "+? +EP()tín) (4°! ~n (vê +ueu8ye)) 3#* = — P(k)r(k)&ín) (aro "` mu
TA Camps.harnadc One 2211220 4035 Camps, Emparan, 2tXiv1201.3505
Pe) fe et) 3+?) Pe) fle) (0) kế
` AE)
8n =4 PÚ9rÏ(k)ö0)
a = Ae) _ i
Trang 34
Aring embedded in flat space:
Trang 35
Corrected phase diagram expressed in physical quantities:
Trang 36Empatan,Hamiai, Niatchos, 08ers, Rodrigues,3Xt:0708 1182 Dia, Santos, Way, arkv:1402.6345
1A 2 Harmar, arXiv 402.6350
Trang 37The same can be done for charged branes:
TAs Cath, Ober aXN209 2200 5157 RL,arXiv2307504
Decompose the dipole correction as:
Split the gauge field as:
= A ADO 3 T4 +12— || vá” —ierenssa, s96) , „
Trang 38
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
Trang 39‘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:
Wy
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