66 Chapter 3. Holographic methods at finite temperature
Also from (3.91) it is seen that the partT0iof the energy-momentum tensor (usually interpreted as the heat current) now amounts to
T0i =pg0i+ωu0ui+τ0i = (ǫ+p)u0ui (3.98) if we can assume the stress tensor to have vanishing components here by its general inter-pretation measuring spatial shear effects only. Plugging in our expression for the velocity yields
T0i = (ǫ+p)u0
1 ρCκ∂i
µ
T (3.99)
This completes the first step being the sought after translation ofνµintoT0i viaui. The second step usesdµ= dp/ρC −sdT /ρC in order to translate the gradient:
∂iµ= 1 ρC
(∂ip−s∂iT). (3.100)
Putting the two steps together gives a well known relation T0i =−
ǫ+p T ρC
2
κ
| {z }
≡κ
∂iT − T ǫ+p∂ip
(3.101)
As described in Landau and Lifshitz [106], this expression gives the relativistic hydrodynam-ics heat current. Compared to the non-relativistic one it gets an extra contribution from the pressure gradient throughout the system. The transport coefficient related to heat flow is the heat conductivityκ.
We now have two interpretations of the newκ:
1. It relates the dissipative current with the temperature and chemical potential gradient by (3.93). This is true for general currentsJµ.
2. It also relates the heat current with the temperature and pressure gradient by (3.101). This interpretation though only holds if the charge current vanishes, soJi = 0.
In the application to R-charged black holes [35] the authors conclude by looking at the limit of vanishing charge current, that the dissipative part of the charge current will contribute to the heat current and thus is the heat conductivity.
3.3. Quasinormal modes 67
Quasinormal modes in gravity This paragraph follows closely the work of [111] and details may be obtained from that original work. Normal modes are the preferred time har-monic states e−iωnt of compact classical linear oscillating systems such as finite strings or cavities filled with electromagnetic radiation. The normal frequencies ωn of these systems are real ωn ∈ Rand the general solution can be written as a linear superposition of all pos-sible eigenmodes n. Quasinormal modes in classical supergravity are the analog of normal modes but for a non-conservative system. The quasinormal frequencies assume complex val-uesωqn ∈Cwhere the imaginary part is associated with the dissipation. In the case of a black hole background excitations dissipate energy into the black hole and are therefore damped when traveling through the bulk. Since we would like to utilize AdS/CFT, we are interested in quasinormal modes in theddimensional AdS Schwarzschild metric
ds2 =−h(r)dt2+h(r)−1dr2+r2dΩ2d−2, (3.102) with
h(r) = r2
R2 + 1−r0
r d−3
. (3.103)
This factor for large black holes with r0 ≫ R in AdS5 becomes h(r) = Rr22 − rr02 9. Quasinormal modes are the (quasi) Eigenmodes of fluctuations of fields in presence of a black hole (or black brane) background, also referred to as the ringing of the black hole. As a simple example let us follow [111] and consider the wave equation of a minimally coupled scalarΦ
∇2Φ = 0. (3.104)
Assuming spherical symmetry we may use the product Ansatz
Φ(t, r, θ) =r2−d2 ψ(r)Y(θ)e−iωt, (3.105) with the spherical harmonics Y onSd−2. Splitting the radial from the spherical equation of motion we obtain
[∂r2∗ +ω2−V˜(r∗)]ψ(r) = 0, (3.106) where the tortoise coordinater∗ is given by
r∗ =
Z dr
h(r) + 1. (3.107)
The potential V˜(r∗) vanishes at the horizon r∗ = −∞ and diverges at r = ∞. In general this equation has solutions for arbitraryω. The solutions which are called quasinormal modes are defined to be purely incoming at the horizon Φ ∼ e−iω(t+r∗) (and purely outgoing at infinity Φ ∼ e−iω(t−r∗) , where the boundary of AdS is located in these coordinates). This condition can only be satisfied at discrete complex values ofωcalled quasinormal frequencies.
In the AdS black hole case the potential V˜ diverges at infinityr = ∞, such that we require the solution to vanish at this location.
9Which is identical to the form used e.g. by Myers et al. in [59] up to a scaling withR2
68 Chapter 3. Holographic methods at finite temperature
In order to have a finite variable range we invert the radial coordinater →1/x. The radial equation of motion for the minimally coupled scalar then reads
s(x) d2
dx2ψ(x) + t(x) x−x+
d
dxψ(x) + u(x)
(x−x+)2ψ(x) = 0. (3.108) In ourAdS5-case the coefficients are given by [111]
s(x) = (x+2+ 1)x5
x+4 + (x+2+ 1)x4 x+3 + x3
x+2 + x2
x+2 , (3.109)
t(x) = 4r02x5−2x3−2x2iω , (3.110) u(x) = (x−x+)V(x), (3.111) V(x) = 15
4 + 3 + 4l(l+ 2)
4 x2+9r02
4 x4 (3.112)
r02 = x+2+ 1
x+4 , (3.113)
wherel(l+ 2)is the Eigenvalue of the Laplacian onS3. Note that we do not rewrite (3.108) such that the factor in front of the second derivative becomes one. That is because the coeffi-cientss, t, uhave finite expansions in(x−x+)and thus are more tractable.
We compute the quasinormal modes numerically by expanding the solution in a power series about the horizon atx = x+. In order to find the near-horizon behavior we determine the indices (as explained in section 3.1.2) α = 0 and α = iω/(2πT). Only the first index describes ingoing modes at the horizon and we discard the second one. This fixes the leading order(x−x+)0 for our solution and we expand the remaining analytic part of it in a Taylor series about the horizon [105]
ψ(x) = (x−x+)α X∞ n=0
an(x−x+)n, (3.114)
Then we demand this series to vanish at infinity r = ∞ equivalent tox = 0. The expan-sion (3.114) is substituted in the equation of motion (3.108) in order to compare coefficients of(x−x+)nin each ordern. From this we find the recursion relations
an =− 1 Pn
n−1
X
k=0
[k(k−1)sn−k+ktn−k+un−k]ak, (3.115) with the expressionPn =n(n−1)s0+nt0 = 2x2+n(nκ−iω). Only the coefficienta0remains undetermined as expected for a linear equation.
Together with the condition that the solutionψshould be normalizable and therefore has to vanish at spatial infinityψ(x= 0) = 0, we have mapped the problem of finding quasinormal frequencies to the problem of finding the zeroes of
X∞ n=0
an(ω)(−x+)n= 0, (3.116)
3.3. Quasinormal modes 69
in the complex ω plane. Equation (3.116) can only be satisfied for discrete values of com-plexω. We approach the exact result by truncating the series atnand finding the zeroes of the partial sumψN(ω, x= 0) = PN
n=0
an(−x+)n. To be more specific, we really find the minima of the absolute value squared|ψN(ω, x = 0)|2 of the partial sum and then check if the value at that minimum is (numerically) zero. The accuracy can be increased by going to larger nand the error is estimated from the change ofw(n)asnis increased.
Alternative QNM computations In more complicated backgrounds (such as the D3/D7-setup) it is hard or even impossible to write down analytical expressions as those used in the previous paragraph, especially if some factors like the embedding function in the metric components are only given numerically. In this case one has to reside to numerical methods.
Numerically we can compute|φ|2directly starting with two boundary conditions at the hori-zon and search its minimum. In some cases (especially if the solution is oscillating heavily on one boundary) the numerical method of matching in the bulk [25, section 7.2] has proven more adequate to find solutionsφ. Numerics may also be improved by a coordinate transfor-mation to more tractable (non-singular) coordinates. An application of this latter method is given in [60].
Quasinormal modes in AdS/CFT In the context of AdS/CFT it has been shown [27, 104]
that the lowest lying (i.e. those with the smallest absolute value) quasinormal frequency of the perturbation of a distinct gravity field φcoincides with the pole of the two-point function for the operator O dual to this distinct field. We can see this by approaching the problem with the question: what is the two-point correlator of two gauge-invariant operators? As described above, the correlator is given by
hOOi= lim
r→rbdy
B(r)φ(r)∂rφ(r), (3.117) where φ(r) is the solution to the gravity equation of motion (ordinary differential equation ODE) for the fieldφdual to the operatorO. Here we use the same radial coordinaterdefined above in equation (3.102). At the boundary the solution can be written as linear combination of two local solutions
φ(r) = Aφ1(r) +Bφ2(r), (3.118) with A and B being determined by the coefficients in the differential equation for φ. The coefficientsAandBgive that particular linear combination which satisfies the incoming wave boundary condition at the horizon. Near the boundary the solution (3.118) splits into the normalizable and non-normalizable parts
φ(r) = Ar−∆−(1 +. . .) +Br−∆+(1 +. . .), (3.119) The action quadratic in field fluctuationsφreduces to the boundary term
S(2) ∝ lim
r→rbdy
Z
dωdpq B(r, ω,q)φ(r)∂rφ(r) +contact terms. (3.120)
70 Chapter 3. Holographic methods at finite temperature
Applying (3.117) and assuming∆+ > ∆−, ∆+ >0we obtain the two-point function of op-eratorsOby an expansion in the radial coordinaterand taking the boundary limit afterwards
hOOi ∝ B
A +contact terms. (3.121)
The poles of the retarded correlator thus correspond to the zeroes of the connection coeffi-cientA. On the other handAis determined by the coefficients of the equation of motion for the field fluctuationφ and therefore A = 0 is a particular choice of boundary condition for that field fluctuationφ. As an example consider∆− = 0, ∆+ = 2and B(r, ω,q) ∝ r3 and rbdy=∞. Then
hOOi ∝ lim
r→rbdy
r3 −2Br−3
A+Br−2 +contact terms. (3.122) Now we are ready to connect our holographic considerations back to the gravity definition of quasinormal modes given above (3.116). Comparing the two approaches we conclude that the condition for having quasinormal modes coming from gravity (3.116) and the boundary condition for the field fluctuation in AdS/CFTA = 0are identical. For this reason the quasi-normal frequencies of black hole excitations are identical to the poles of the retarded two-point correlator of their AdS/CFT-dual operators.