Horizon expansion of G

Appendix C

Gauge invariant fluctuations

C.1 Gauge invariant transport coefficients

C.1.1 Gauge transformations acting on fluctuations

In AdS/CFT the background solution represents the field theory ground state.

Perturbing this ground state gives access to its transport properties. As presented in section 3.3, one adds a spacetime dependent fluctuating contribution to each of the background fields. This also includes components of the background fields that are zero in the ground state. As these perturbations are understood to be small, the resulting correction of the action is only taken into account to second order in fluctuations and hence to first order in the equations of motion. In order to obtain information about the physical properties, the analysis of the fluctuations has to be done in terms of gauge invariant modes by which we mean gauge invariant combinations of the introduced perturbations. The prescription of how to obtain these gauge invariant modes can be outlined in full generality. In this review of the methodology we focus on the types of background fields which appear in this thesis: the metricg, a Maxwell field Aand both a charged χand and non-charged scalar φ. They all experience a non-zero change under a transformation of at least one of the two gauge symmetries, diffeomorphism invariance and the U(1) gauge symmetry.

Let us start by writing down the effect of infinitesimal gauge transformation on those fields. The effect of a diffeomorphism symmetry transformationxµ →xµ−Σµ

143

on the fields can be expressed as Lie derivatives along Σ

δ_Σg_µν =L_Σg_µν =∇_µΣ_ν +∇_νΣ_µ, (C.1) δ_ΣA_µ=L_ΣA_µ = Σ^ν∇_νA_µ+A_µ∇_νΣ^ν, (C.2) δ_Σχ=L_Σχ= Σ^ν∇_νχ , δ_Σφ=L_Σφ= Σ^ν∇_νφ . (C.3) The gauge transformation of theU(1) Maxwell symmetry only affects the Maxwell field itself and the charged scalar fields. The infinitesimal version is given by

δΛAµ=∇µΛ, δχe =−ieΛ, (C.4) whereδχ_erepresents any scalar field with chargeeunder theU(1) gauge symmetry.

The background parts of the fields g, A, χ and φ representing the ground state of the field theory are fixed. The gauge transformations are thus understood to affect the perturbations of the ground state only while leaving the ground state itself unchanged. For each component of the fluctuations, we can now compute the individual effect of the transformations on them. Let us arrange each individual component of all perturbations δg, δA, δχ_e and δφ in a vector ϕ schematically transforming as

ϕ → ϕ+ (δ_Σ+δ_Λ)ϕ , (C.5)

It is important to stress, that since we assume both the perturbations and the gauge transformations to be small, we consider the transformed ϕ only up to first order in perturbations or gauge transformations Σ and Λ, i.e. δ_Σ and δ_Λ are computed from the background fields only.

C.1.2 Construction of gauge invariant combinations

We are then interested in constructing linear combinations of the elements of ϕ that are invariant under the gauge transformations. The number of gauge invariant combinations is the number of the dynamical degrees of freedom of the problem.

We start with a general ansatz

Φ = X

i

c_iϕ_i, (C.6)

C.1. Gauge invariant transport coefficients 145

and apply the gauge transformations on it

Φ → Φ + (δ_Σ+δ_Λ) Φ. (C.7)

The equation we have to solve is thus given by

(δ_Σ+δ_Λ) Φ = 0. (C.8)

Φ is the weighted sum of all fluctuations. The transformations δ_Σ,Λ are different for each of those, depending on what component of which field they are. Even-tually we want to know what linear combination, i.e. what values for the c_i yield the gauge invariant modes, and thus spell out (C.8) and set the prefactors of all appearances of δ_Σ,Λ to zero. Solving these equations allows us to express one part of the constants ci in terms of the other part of constants. We call the latter the independent coefficients ˜c_i. The choice of the set ˜c_i is not unique, their total number N_indep however is equal to the number of propagating degrees of freedom.

The final step in constructing the gauge invariant modes is then to build N_indep linear combinations with the coefficients ˜c_i and plug them into the ansatz (C.6).

Let us work through an explicit example, the Einstein-Maxwell dilaton model 5.6 we discussed in chapter 5. As argued in section 5.4 therein, only individual groups among all fluctuations couple and hence the gauge invariant modes, the propagating degrees of freedom can only be formed within those groups. Being interested in the eletrical conductivity along the x-direction we focus on the group which contains δA_x. The ansatz for the gauge invariant combination in this group is

Φ = c₁δg_tx+c₂δg_xr+c₃δA_x+c₄δξ_x. (C.9) We work in Fourier space and expand the fluctuations and the gauge transforma-tion functransforma-tions Σ and Λ in Fourier modes as

ϕ_i = Z

dωe^−iωtϕ_i,ω, Σ = Z

dωe^−iωtΣ_ω, Λ = Z

dωe^−iωtΛ_ω, (C.10) fork= 0. Applying an infinitesimal gauge transformation on Φ, we get a correction

of the following form

Φ→Φ +δΦ =c₁(δg_tx−iωΣ_x) +c₂

δg_xr−Σ_x2 r +g⁰

g

+ Σ⁰_x

+c₃δA_x (C.11) +c₄

δξ_x+L²mΣ_x r²g

.

We see that δA_x does not transform under a gauge transformation and hence already is a gauge invariant mode.

In order to solve the equation (C.8), we note that the infinitesimal gauge trans-formations are arbitrary and hence we treat their derivatives as independent. Had we included all the fluctuations we would have found that the dependent part of the constants c_i expressed in terms of the independent ones ˜c_i do not mix those groups. We find

c₁ = ˜c₁, c₂ = 0, c₃ = ˜c₃, c₄ = iωr²g

m L²c˜₁. (C.12) It is straightforward to construct two linear independent gauge invariant combi-nations. We choose

I ˜c₁ = L²

r²g and ˜c₃ = 0, (C.13)

II c˜₁ = 0 and ˜c₃ = 1, (C.14)

where we essentially rescaled the first invariant combination with thex-component of the background metric’s inverse. The resulting gauge invariant modes are

Φ₁(ω, r) =δg^x_t(ω, r) + iω

α δξ_x(ω, r), Φ₂(ω, r) = δA_x(ω, r). (C.15)

C.1.3 Radial gauge

In many papers the authors choose the radial gauge. This means that the gauge symmetry is partially fixed in a way, that the radial components of the metric perturbations and the vector field perturbations are and remain zero. In this case the above procedure is modified such that the combinations are invariant under the remaining symmetry transformations. Even though this choice highlights the holographic dictionary, it is not necessary to explore the dynamical degrees of freedom.