Holographic thermalization of charged operators

We study a light-like charged collapsing shell in AdS-Reissner-Nordstrom spacetime, investigating whether the corresponding Vaidya metric is supported by matter that satisfies the null energy condition. We find that, if the absolute value of the charge decreases during the collapse, energy conditions are fulfilled everywhere in spacetime. On the other hand, if the absolute value of the charge increases, the metric does not satisfy energy conditions in the IR region. Therefore, from the gauge/gravity perspective, this last case is only useful to study the thermalization of the UV degrees of freedom. For all these geometries, we probe the thermalization process with two point correlators of charged operators, finding that the thermalization time grows with the charge of the operator, as well as with the dimension of space.


I. INTRODUCTION
The study of non-equilibrium processes in quantum field theory is an important and active area of research (see [1] for references). The subject has most diverse applications that range from astrophysics and cosmology, i.e. the study of particle production at the end of inflation and the generation of density fluctuations, to relativistic heavy ion collisions, i.e. the dynamics of the quark-gluon plasma at RHIC. For the case of near-equilibrium states there exist well developed formalisms: linear response, kinetic theory or fluid dynamics. This techniques rely on the existence of a dilute equilibrium ensemble or weak coupling. However in many physical instances the system to be studied can be strongly coupled or in a dense state or both, and therefore in these situations the standard techniques do not apply.
The question of whether a given subset of the degrees of freedom of a certain system reaches equilibrium starting from an arbitrary initial state, and whether such equilibrium can be described by a thermal density matrix, is an open one whose answer seems to depend on the details of the underlying dynamics [2]. A large number of numerical and analytical research on such quenching process has been published (see [3] and references therein), regarding both weakly coupled systems in the perturbative regime [4], and integrable systems [5]. Moreover, strong coupling QFT's have been studied numerically in the lattice [6].
In this paper we will consider the study of far from equilibrium strongly coupled systems using the gauge/gravity correspondence. As it is well known, the holographic correspondence is the natural arena for analyzing strongly coupled QFT problems translating them into classical gravity computations. For near-equilibrium states, the correspondence has been extensively studied with particular emphasis in the linear response and the hydrodynamic regime, in terms of perturbations of the black hole geometry (see the recent reviews [7,8]). More recently, attention has been paid to farfrom-equilibrium states. Reference [9] made a holographic proposal to model the sudden injection of energy into the QFT vacuum state, and its subsequent thermalization, in terms of an AdS Vaidya geometry. The dynamical Vaidya spacetime physically corresponds to the collapse of a homogeneous massless shell in AdS, leading to the formation of a black hole [10]. It interpolates between pure AdS spacetime (vacuum) in the distant past to an AdS black hole (thermal state) in the distant future. Two point functions of operators O of large conformal dimensions, which in the dual picture can be computed in the semiclassical approximation in terms of geodesics, and Wilson loops and entanglement entropy, which in the gravity perspective relate to minimal (hyper)surfaces, were used as probes of thermalization. The resulting picture is that the UV degrees of freedom thermalize first, followed later by the IR ones (top-down thermalization [9]). From the gravity perspective this is an expected result since IR probes explore deeply in the radial direction being therefore sensitive to the shell position for longer times.
The aforementioned results were generalized in [11,12] to situations with a non-trivial chemical potential or, in the dual perspective, to the case in which the system experiences a sudden injection of both energy and particles. The dual geometry was chosen to be that of a charged collapsing null shell interpolating between AdS in the distant past and asymptotically AdS Reissner-Nordstrom black hole (AdSRN) in the distant future. The thermalization was probed by two point functions of chargeless operators, Wilson loops, and entanglement entropy. The emergent picture of [11] was that as the final chemical potential is increased, it takes longer for the system to thermalize (see [13]- [32] for related works).
It is the aim of the present paper to extend the studies mentioned above. In particular, since the Vaidya geometry is known to be supported by an energy-momentum tensor satisfying null energy conditions, in the present work we analyze the energy conditions of the previously explored charged Vaidya geometries [11,12] and its further generalizations. We afterwards study the thermalization process by probing the system with charged operators.
The organization of the paper is the following: In section II we analyze whether the Vaidya metric used in [11,12], that represents a quench on the chemical potential and temperature in the dual field theory, is supported by matter satisfying null energy conditions, in section III we probe the thermalization process with two point functions of charged operators. The results are presented and discussed in section IV. We conclude with a summary in section V. In the appendices we discuss Eddingtong-Finkelstein coordinates, the worldline formalism and the WKB approximation, used in the text to obtain the two-point functions of charged operators.

II. BACKGROUND GEOMETRY
The gauge/gravity duality relates a strongly coupled quantum field theory (QFT) in d flat spacetime dimensions with a weakly coupled gravity theory in a d + 1 dimensional spacetime which asymptotes to anti-de Sitter spacetime. According to the standard gauge/gravity dictionary, a finite temperature state of the field theory is represented by a geometry with a horizon in the bulk side. Moreover, a global symmetry in the field theory induces a gauge symmetry in the gravity side. As a consequence, the presence of a chemical potential for a global U (1) charge at finite temperature on the QFT side is described in the dual gravitational picture by the presence of an electrically charged black hole [33]. In the simplest example, an equilibrium state with finite temperature and chemical potential is represented by an AdSRN black hole with given mass and charge. On the other hand, a process in which the temperature and chemical potential vary can be represented by a metric which interpolates between two such geometries with different values of mass and charge. In this section, we describe those geometries and analyze the energy momentum tensor that is needed to support them. We model the dynamical geometry by the collapse of a thin shell of null dust. It turns out to be convenient to substitute the standard time coordinate t, which is not constant across the shell, by an infalling radial null coordinate v which is.
In what follows, we take d to be the spacetime dimension of the dual field theory, hence our bulk geometry will be d + 1 dimensional. The indices µ, ν = 0 . . . d denote the bulk coordinates x µ = (v, x, z).

Bulk geometry
In Eddington-Finkelstein ingoing null coordinates, the metric and gauge fields corresponding to a planar AdS d+1 Reissner-Nordstrom black hole take the form (see appendix A) where f and F zv are functions of z that read .
The static background (1)-(2) is a vacuum solution of the (d + 1)-dimensional Einstein-Maxwell system with negative Setting κ 2 = 1 the equations of motion following from S EM are where The geometry (1) asymptotes AdS space with radius L as we approach the boundary located at z = 0, and the constants M and Q correspond to the ADM mass and electric charge respectively. The metric (1) has a curvature singularity at z → ∞ and has horizons whenever the function f vanishes. To characterize the horizons, notice that f has two stationary points, one at z = 0 at which f = 1, and a second one (a local minimum) at z min = (d M/2(d − 1)Q 2 ) 1/(d−2) . The curvature singularity will be hidden from the outside as long as f (z min ) ≤ 0, which implies a constraint on the possible values of M, Q Whenever this inequality is satisfied, we generically have two horizons at z = z ± (inner/outer). Moreover, when the bound is saturated the two horizons coincide and the configuration is called an extremal black hole solution (more on this below). We will demand the constraint (9) on all our solutions in order to have a physically sensible gravitational background.
Although charged black holes generically depend on two arbitrary parameters Q and M , this is not so in the planar horizon case. The absence of a scale on the horizon geometry allows us to get rid of one of the parameters. Explicitly, the rescaling (v, x, z) = z − (ṽ,x,z) maps the (outer) horizon position toz = 1. DefiningM = M z d − andQ = Qz d−1 − one finds thatM = 1 +Q 2 resulting into [33] This is the parametrization often used in the literature for the planar Reissner-Nordstrom AdS black hole. It automatically satisfies the constraint (9), and notice thatQ can take any arbitrary value. The geometry (10) has generically two horizons. Depending on the value ofQ one has: (i) an outer one atz − = 1 and an inner one atz + > 1 for |Q| ≤ d/(d − 2), (ii) coincident horizons atz ± = 1 forQ 2 = d/(d − 2) (extremal BH), and (iii) an inner horizon at z + = 1 and an outer one atz − < 1 for |Q| > d/(d − 2). An appropriate rescaling of the holographic coordinate maps the type (iii) solutions to the type (i) ones. Summarizing, the background (10)- (11) with |Q| ≤ d/(d − 2) parametrizes the most general static planar AdSRN solution with an outer horizon located atz − = 1 and an inner horizon located atz + ≥ 1.

Boundary theory
As mentioned above, the AdS boundary is located at z = 0 and, as it is well known, the static geometry (1)-(2) represents a dual QFT equilibrium state characterized by a chemical potential µ and a temperature T [33]. The bulk variable v is identified with the dual gauge theory time t since both coincide at z = 0 (see (12) below).
The standard procedure to relate the boundary and bulk parameters is to impose regularity of the Wick rotated solution. For the static case (3)-(4), the redefinition turns the geometry (1) into the Euclidean form We note in passing that if the function f were v-dependent, as it will become later when we define Vaidya metrics, the redefinition (12) would not be allowed, the right hand side being not an exact differential. The metric (13) is regular at the outer horizon z = z − if we periodically identify t E ≡ t E + β, with the subindex z on f z denotes derivative with respect to z. The boundary theory temperature is then defined as Regarding the chemical potential of the boundary theory, it relates to the black hole charge as follows: we can choose the bulk gauge potential to be with µ arbitrary. From (12) one has iA t E = −γQLz d−2 + µL. Since the t E circle smoothly collapses at the outer horizon, the gauge field must satisfy A t E (z − ) = 0 to avoid singularities. This condition fixes the asymptotic value of the gauge field to For a fixed mass black hole, increasing the chemical potential (i.e. the black hole charge) decreases the black hole temperature. The extremal bound in (9) corresponds to the T = 0 and µ = 0 case. In the gauge/gravity duality context one identifies the gauge field boundary value µ as the source for the QFT conserved charge operator.

Bulk geometry
To construct a time dependent geometry, we promote the z-dependent functions f and F zv on (2) to (z, v)-dependent functions. In other words, we keep the ansatz for the metric and the field strength (1)- (2), but the functions f, F zv now depend on both variables z, v. Under this conditions, extra matter contributions T Matter µν and j µ Matter need to be added to the right hand side of (6)- (7), in order to satisfy Einstein-Maxwell equations of motion. They physically represent an infalling charged matter shell giving birth to the black hole [10]. The additional contribution to the energy momentum tensor is defined as Working out the components of T Matter µν in terms of the functions f, F zv and their derivatives one finds T Matter In the right hand side of these expressions a subindex in f denote partial derivative, A crucial point to be verified is whether the infalling matter that supports the time dependent solution satisfies appropriate energy conditions. In particular, the weakest energy condition that can be imposed is the so called null energy condition, that states that for any null vector n µ , the energy momentum tensor must satisfy everywhere in spacetime. Writing a generic vector as n µ = (n v , n z , n x , 0 d−2 ), where we have used the rotational invariance in the Cartesian coordinates x d−1 to eliminate redundant components, the condition for it to be null reads This equation can be solved for n z . There is one solution n v = n x = 0 under which (22) vanishes identically imposing no constraint. On the other hand, when n v = 0 the solution reads n z = ((n x ) 2 − f (n v ) 2 )/2n v , and upon replacing it into the energy condition (22) one finds In order for this quadratic form to be positive definite for any null vector, we need This set of equations constraint the (z, v)-dependence of the ansatz functions f, F zv . Another important requirement to be imposed on the solution is that any physical matter current sourcing the gauge fields must be time-like or null. Defining the matter current as we find The matter current must satisfy or using (29) We now consider the particular case where the f, F zv functions keep the same z-dependence as in (3)- (4), and M and Q become v-dependent functionsM (v) andQ(v). As expected, this implies that the locus of zeros of f (i.e. the horizon positionsẑ ± ) will be v-dependent, and equations (28)- (29) imply that j µ Matter = (0, j z Matter , 0 ) with j z Matter given by (28). This matter current satisfies condition (30) as an equality, which means that the charged source for the gauge field is light-like. Moreover, the ensuing T Matter µν takes a null dust form, and the energy condition (26) is again satisfied as an equality. In passing, we notice that although we have a time dependent electric field, the matter current leads to no magnetic field in Ampere's law. In summary, generalizing the ansatz to v-dependent functionŝ M (v) andQ(v) in f and F zv leads to a physically sensible gravity solution as long as they satisfy (25), which can be rewritten asM This energy condition (32) was discussed in [11,12]. We would like to remark that (32) coincides with the condition for the horizon not the recede. Indeed, if we define the (time dependent) horizon positionẑ h as the solution to Fig. 3 below). Nevertheless, condition (32) must be satisfied even in the absence of horizons.
We now turn to analyze its consequences: • When the chargeQ vanishes, (32) Therefore, any interpolation between pure AdS and the planar AdS-Schwarzschild metric with a monotonically growing mass function, asymptotically reaching a constant value M , is a healthy solution of Einstein-Maxwell equations. The required additional matter contribution satisfies the null energy condition. The chargeless Vaidya metric case, used in [9] to study thermalization of a strongly coupled plasma at zero chemical potential after an energy quench, allows for an arbitrary time pattern of energy injection (this is a well known result, and we mention it here only for completeness).
• If the charge remains constant as we perform the quench (Q v = 0), condition (32) reduces to (33). As discussed above, the constraint (32) is identically satisfied as long as the mass function is monotonically increasing. The corresponding Vaidya geometry can be used to study thermalization processes at fixed chemical potential after an injection of energy.
• For non-constant charge situations, outside of the support ofQ v , condition (32) reduces to (33). On the other hand, for analyticQ the support is non-compact, implying (32) is violated at large enough z for any fixed value of v. Notice thatQQ v > 0 corresponds to an increasing absolute value of the charge, what we will call a "charging" background in what follows. The null energy condition therefore tell us that we can use a charging AdS Vaidya ansatz to study the thermalization of the dual field theory, as long as our probes explore the near boundary region. From the dual point of view this means that the geometry can only be trusted to describe the UV degrees of freedom of the field theory. These classes of solutions were used in [11,12] to study the thermalization process after a quench on energy and chemical potential.
-No additional constraint follows from (32) for the case of a "discharging" backgroundQQ v < 0, provided the mass is increasing M v > 0.
In summary, the charged metrics obtained by promoting the AdSRN charge and the mass into v-dependent functions (with growing mass), satisfy energy conditions everywhere in spacetime whenever they are "discharging"QQ v < 0, while violate energy conditions in the bulk for large enough z, i.e. in the deep IR, whenever they are "charging" QQ v > 0. In this last case probing the near boundary geometry is still a safe option. Note that in the above definitions, and throughout this paper, we use the words "charging" and "discharging" as referring to an increasing or decreasing absolute value of the charge.
In what follows, we use the aforementioned charging and discharging geometries in order to get information about the thermalization processes of the boundary theory. To such end, we choose our functionsM andQ as interpolating between initial constant values M in and Q in in the asymptotic past v → −∞ and final constant values M and Q in the asymptotic future v → ∞.
In the time-dependent background, the field strength in (2) can be obtained from the gauge potential withμ an arbitrary v-dependent function. The background v-dependence precludes the global definition of a time coordinate, since (12) is not an exact differential, avoiding an Euclidean continuation. Thus, the only constraint we impose on the functionμ arises from the regularity of the Euclidean continuation of the asymptotic regions v → ±∞.
when v → ∞, where z in and z h are the positions of the horizons before and after the quench.

Boundary theory
In the dual field theory, the proposed geometry describes a system interpolating between two different values of the temperature as the result of injecting energy homogeneously into the system, and at the same time quenching the chemical potential. The quench can take the chemical potential up or down depending on whether we have a charging or a discharging geometry, and corresponds to a homogeneous injection of particles/antiparticles into the system. Even if, as mentioned in the previous sections, there is no global time coordinate, with the hindsight of the static background we define a time coordinate in the asymptotic region z → 0 as t ∼ v: the variable v at the boundary then coincides with the gauge theory time.

III. PROBES OF THERMALIZATION
As a probe for analyzing the thermalization process, we study equal time two point correlators of charged scalar operators O ∆ of conformal dimension ∆. Decomposing the bulk coordinates as x µ = (x α , z), the AdS/CFT dictionary relates the QFT two point correlator In the large mass limit 1 mL ≈ ∆, the bulk propagator can be approximated by the classical trajectory (see appendix B) with Here S on−shell is the appropriate action for the time-like trajectory, for a particle of mass m and charge e, evaluated on the classical trajectory x µ classical (τ ) that joins (x α 1 , z 1 ) and (x α 2 , z 2 ). The charge to mass quotient e/m is supposed to remain finite in the large m limit (see Appendix B for details). In the limit of large conformal dimension ∆ 1, equations (35)-(37) instruct us to compute the two-point correlator (35) from the classical trajectory of a charged particle whose endpoints sit at the boundary. It is important to realize that any such trajectory lies completely in the classically forbidden region, a complete discussion of the different ways to access such region with classical trajectories is given in Appendix C. For the time being, let us analytically continue the parameter τ and the electric charge e according to τ = −iτ E and e = ie E , resulting in with where we defined q E = e E /m. In what follows we evaluate the two point correlator according to where z is a cutoff in holographic direction, and the on-shell Euclidean action S E on−shell (x α 1 , z ; x α 2 , z ) is given in (38), and evaluated on classical paths starting at (x α 1 , z ) and ending at (x α 2 , z ). In AdS space any geodesic approaching the boundary has a divergent logarithmic contribution to the length ∼ − log z, the z ∆ factors in (35) are present to precisely cancel this contribution. In Appendix D we show that expression (40) for the correlator coincides with the standard holographic definition, as the quotient of the subleading to the leading components of the bulk scalar field as z approaches the boundary, when the Klein-Gordon equation is solved in the WKB approximation.
Therefore, to evaluate the two-point correlator for the quench we insert in (39) the time dependent background of section II B. Afterwards, we compare it with the corresponding correlator obtained by substituting in formula (40) the static background of section II A. The probed degrees of freedom can be said to have reached thermal equilibrium, whenever the quantity vanishes. Our interest is to find a time profile of δS.
In the present work we look for spacelike U-shaped trajectories for mass m and charge e on the the aforementioned backgrounds starting and ending at the boundary, whose endpoints are separated along one of the Cartesian coordinates by a distance , and in time by a distance ∆t (see Figs. 1 and 2). We choose to parametrize the curve as x µ (x) = (v(x), z(x), x 1 (x) = x, x d−2 ) with x d−2 constant, and without any loss of generality we take x i < x < x f , with x i , x f the extremes of the U-shaped curve whose tip sits at x = 0. The boundary conditions for the trajectories we are interested in are where, as mentioned above, we have regularized the problem by considering geodesics whose endpoints lie at z = z , instead of z = 0 (z → 0 should be taken at the end of the computations). Alternatively, the trajectories can be characterized in terms of the boundary conditions at the tip Viewed from the tip, two branches appear: (i) for x i < x < 0 the trajectory describes z ∈ (0, z * ) and hence z > 0, (ii) while for 0 < x < x f it goes from the tip z = z * > 0 to the boundary z = 0 therefore z < 0. These two branches correspond to the two signs in (48) The space and time separation characterizing the geodesic are given by At this point we can already suspect that will be a growing function of z * , implying that any upper cutoff in z * imposes an upper cutoff in . This behavior is well known and can be explicitly checked in the numerical calculations of the forthcoming sections. Moreover, this implies that if we are interested in restricting the region of the geometry to be probed by our geodesics to the near boundary patch, then we must impose an upper bound on < Max . In other words, the violation of energy conditions in the bulk for the charging case imposes an IR cutoff in the degrees of freedom that can be probed (see fig. 3).

A. Equilibrium state
Inserting the eternal AdSRN solution (13) into the action (39) we get where prime ( ) denotes derivative with respect to the Euclidean parameter τ E , that we have gauge fixed to τ E = x. The first term corresponds to the geodesic length, while the second codifies the coupling to the gauge potential. Notice that only for the case of geodesics starting and ending at the same value of v can one drop the µ contribution on the second term. The two resulting second order equations of motion can be shown to have two first integrals: one coming from reparametrization invariance, which in our coordinates turns into the conserved Hamiltonian generating translation along the x parameter, and the second from v-independence of the metric. The equations to be solved are 1  (32) is violated. We also depict the position of the outer/inner horizon, defined as the z(v) solution to f (z, v) = 0, as dashed lines. The yellow and blue geodesics do not explore the sick region and can be used to analyze thermalization. The red geodesics on the other hand enter into the sick region and we cannot trust them to probe the thermalization process, in other words the background geometry can be used to analyze the thermalization of degrees of freedom above an IR cutoff. Note that in concordance with the discussion below (32), the region where the horizon recedes does not satisfy the energy conditions. The background interpolates between pure AdS in the past to a Q = 2 and M = 1 + Q 2 black hole in the future.
where the value of the constants on the right hand side have been fixed at the tip of the U-shaped curve (x = 0) according to the conditions (43):Ẽ = 1/z * 1 − f * v 2 * where f (z * ) = f * . Solving (46)-(47) for z one obtains As explained above, the double sign in this expression corresponds to the two branches of the U-shaped trajectory: the positive sign corresponds to x i < x < 0 while the negative one happens for 0 < x < x f . Solving for v one obtains with z given by (48). The two parameters at the tip (z * , v * ) are related to ( , ∆t) by noting that and Finally, inserting (46) and (49) into (45) we can express the on-shell action as 2 Since the on-shell action diverges when the endpoints of the trajectory reach the boundary, we have regularized (52) by introducing a cutoff z . The factor of two arises from the two branches of the trajectory giving identical contributions. The formulae above enable us to compute the on shell action (52) as a function of and ∆t. Notice that the v-independence of the background implies that no dependence on v * is expected.

B. Time dependent state
The action for a charged particle moving in the time dependent metric takes the form (45), but with the constants Q and M substituted by the v-dependent functionsQ andM .
Notice that we have also introduced a v-dependentμ (see discussion below (34)). Due to the lack of v-translation invariance, the on shell action cannot be taken into a pure z integral as in (52). The equations to be solved now are where the value of E is again fixed at the tip of the curve (43) and given by E = 1/z * 1 − f * v 2 * . These equations are solved numerically by the shooting method. In practice, we shoot from the turning point z * with initial (final) conditions (43) for 0 < x < x f and x i < x < 0. For each choice of values (z * , v * , v * ) at the tip we will find an U-shaped geodesic characterized by ( , ∆t, t f ) (see Figs. 1 and 2). The plots below give the thermalization curves for δS given by (41) with the on-shell action (53) being generically a function S E on−shell t−dependent ( , ∆t, t f ).
Notice that, since the time dependent charging background violates the null energy conditions for large enough z, in such case we need to make sure that our geodesics are only probing the healthy part of the geometry (see Fig. 3). As mentioned before, this implies that we can probe thermalization with correlation functions with bounded < Max . On the other hand in the discharging background, we have no such IR cutoff and we can probe the thermalization with any value of .

IV. RESULTS
The considerations of the previous sections were made for arbitrary functionsM andQ. In this section, in order to proceed with the numerical evaluation of the thermalization time for different probes, we will need explicit expressions for those functions. We chooseM Here v 0 parametrizes the shell thickness and the v 0 → 0 case corresponds to the shock wave discussed in [9]. These functions satisfy Therefore the backgrounds we consider interpolate between an AdSRN black hole with mass M in and charge Q in in the distant past v v 0 and an AdSRN black hole with mass M and charge Q in the distant future v v 0 .

A. Vanishing background charge
We start by analyzing the thermalization process in the case of vanishing background charge, which corresponds to a thermal quench with vanishing chemical potential in the boundary theory. To this end, we choosê For the case of vanishing ∆t we reproduced the results of [9], and for nonvanishing ∆t we find results in agreement to those in [35].
The results are summarized in Fig. 4, where the thermalization curves δS involving the on-shell action are plotted as functions of t f . The thermalization time, defined as the approximated value of t f at which the curve reaches the horizontal axis δS = 0, increases with and with ∆t, implying that UV degrees of freedom thermalize first. This is the phenomenon known as "top down thermalization". Moreover, we see that the thermalization time also increases with the dimension of the system.

B. Vanishing probe charge
We now consider a quench in both the temperature and the chemical potential. We first study the effects of the background on a vanishing probe charge q E = 0. For the sake of illustration, we set our background to be pure AdS (T = µ = 0) in the asymptotic past, and AdSRN (T, µ = 0) in the asymptotic future, namelŷ As could have been expected, since we are probing the system with uncharged operators (q E = 0), the background chargeQ has little effect in the form of the thermalization curves. The phenomenon of top-down thermalization is again present, and the thermalization time grows with the dimension of the field theory as in the previous case. The results are depicted in Fig. 5 and are in agreement with those of Refs. [11].

C. Constant background charge
To analyze the thermalization process for a thermal and chemical potential quench we consider theM andQ functions to beM This background interpolates between a AdSRN black hole of mass M in and charge Q in in the distant past v v 0 , to a heavier AdSRN black hole (M > M in ) in the distant future v v 0 preserving the charge Q in . In the following we choose to re-scale the z-coordinate so as to have M in = 1 + Q 2 in in the far past. This ensures that the background satisfies the bound (9) in the past, and since the evolution increases the mass while keeping the charge constant, (32) is also satisfied. Notice that although the background charge is constant, the chemical potential changes when injecting energy into the system. The reason for this is that we should demand regularity of the euclidean rotated asymptotic geometries (see discussion after eq. (34)). We choose the profile for the chemical potential to bê with where z h < 1 is the horizon position after the quench. The initial position being z in = 1 due to the condition M in = 1 + Q 2 in . Notice that during the quench the absolute value of the chemical potential reduces, but it cannot reach µ = 0.
Figs. 6 to 9 show the results corresponding to an interpolation between M in = 2 and M = 3 with Q in = 1. In Figs. 6 and 7 we see that thermalization time increases as a function of and ∆t respectively, showing again the top-down thermalization effect, and Fig. 8 shows that the thermalization time increases with the spacetime dimension. Finally in Fig. 9 we see that, for positive chemical potential, the thermalization time probed with charged operators grows with the charge q E of the operator, this is an expected result from the gauge theory perspective.
A peculiar feature appears in all the figures: a peak arises at fixed t f where the derivative of δS with respect to t f has a sudden change. This can be understood with the help of Fig. 7, in which it is evident that such peak happens for t f ∆t. Indeed, each point t f in the curves represents a geodesic that, according to our boundary conditions (42), starts at the boundary at t = t f − ∆t and ends at t = t f > 0. Since the shell enters space at t = 0, early trajectories starting at negative times (t f < ∆t) cross the shell once in order to return to the boundary. On the other hand, late trajectories starting at positive times (t f > ∆t) either cross the shell twice, or do not cross it at all, this last case corresponds to a thermalized situation (see Fig.1). These three classes of trajectories prove the spacetime in different ways. Let us first assumeQq E > 0, then, for early trajectories, the gravitational force of the background competes with the electromagnetic interaction during the first part of the trajectory (close to t = t f − ∆t), after the particle crosses the shell the two forces pull in the same direction (close to t = t f ). For late trajectories, the two forces compete only close to the tip of the trajectory, and cooperate at both extremes, close to t = t f − ∆t and t = t f . Finally in the third case the forces cooperate all along the trajectory. The same reasoning can be repeated forQq E < 0, with the regions in which the forces compete or cooperate being interchanged. This can be visualized in Figs. 1 and 2, in which the three types of geodesics are shown. The absence of peaks for vanishing probe charge is another evidence of their relation to the electromagnetic interaction.
As can be seen in Fig. 9, there is a remarkable additional feature on the plots: there exists a value of t f such that the function δS does not depend on the probe charge q E . We do not have an explanation for this behavior at the moment.

D. Discharging background
To study a discharging background of the kind described in section II B we need to choose |Q| < |Q in |. For illustrative purposes we will take the extreme case Q = 0 with the functionsM andQ readinĝ These functions interpolate between an AdSRN black hole with mass M in and charge Q in in the distant past v v 0 and an AdS black hole with mass M (and vanishing charge) in the distant future v v 0 . We again re-scale the radial coordinate so that M in = 1 + Q 2 in in the initial state. In the final state, the bound (9) is satisfied since the charge vanishes. From the dual point of view, we are modeling the process of a sudden decrease of the absolute value of the chemical potential while energy is being injected into the system. We would like to mention that this instance cannot be modeled with a constant charge background and complements the results of the previous section.
Results are shown in Figs. 10 to 13. Again we see that top-down thermalization arises and that the thermalization time grows with , ∆t and the space dimension. The peak at t f = ∆t is also present, the reasons being the same as explained in the previous section. Thermalization time also increases with the charge of the probe.

E. Charging background
We conclude by analyzing the thermalization process for a quench leading to an increase on both the temperature and the chemical potential (this situation has been considered before in [11] for the case of uncharged operators). The functionsM andQ readM These functions interpolate between a pure AdS solution in the distant past v v 0 , to a AdSRN solution in the distant future v v 0 with mass M and charge Q. We rescale the radial coordinates so as to have M = 1 + Q 2 in the future state. From the dual point of view, as energy flows into the system, the absolute value of the chemical potential suddenly increases. As discussed in Sect II B, in the present situation the null energy condition is violated in the deep IR, so we must check that our geodesics do not reach such region (see Fig. 3).
Results are shown in Figs. 14 to 17, with features similar to the previous cases, namely, top-down thermalization, thermalization time growing with the dimension of space, peaks denoting the transition between the different classes of geodesics, etc. It is worth mentioning that the swallow tale structure found in [11] also appears for non-vanishing probe charges as can be seen in Fig. 18.

V. CONCLUSIONS
We have analyzed the energy conditions for the external matter needed to support a family of charged AdS-Vaidya metrics. The metrics studied interpolate between two AdSRN black holes with different mass and charge, or between pure AdS and an AdSRN black hole with non-vanishing charge. They have been used in the literature, via the AdS/CFT correspondence, to model the thermalization process in a strongly coupled plasma after a quench in energy and chemical potential.
We found that the null energy condition is violated in the infrared region of the geometry for increasing mass whenever the absolute value of the black hole charge increases in time. On the other hand, when the absolute value of the black hole charge is kept constant or decreases, the null energy condition is satisfied everywhere. This implies that charged Vaidya metrics can be used to analyze thermalization processes for all energy scales only when the quench decreases the absolute value of the chemical potential. On the other hand, when the quench increases the absolute value of the chemical potential, then the metric is only useful for probing the thermalization process above an IR cutoff1.
We applied the above results to study the thermalization of a strongly coupled plasma after a quench in the energy and chemical potential, considering the cases where the chemical potential either increases or decreases in absolute value. As probe of thermalization we considered charged operators two point functions. We found that the thermalization time increases with the charge of the operator, as well as with the dimension of the field theory. As expected in these kind of holographic constructions, the thermalization is top-down, in the sense that UV degrees of freedom thermalize earlier, followed by IR ones.

VI. ACKNOWLEDGMENTS
We thank Martin Schvellinger, Walter Baron and Damian Galante for helpful discussions on thermalization of strongly coupled plasmas via AdS/CFT. We also thank Anibal Iucci and Nicolás Nessi for helpful exchange on the general knowledge about thermalization after quenches and Jerónimo Peralta Ramos and Pablo Rodriguez Ponte for discussions. This work was partially supported by Conicet grant PIP2009-0396, PIP 0595/13, ANPCyT grant PICT2008-1426 and UNLP grant 11/X648.

Appendix A: Eddington-Finkelstein null coordinates
For completeness we quote here some well known facts of the Eddington-Finkelstein coordinate system X µ = (v, x, z) chosen in (1). Parametrizing the geodesics as X µ = (v(z), x(z), z) it is immediate to see that the curve X µ = (v 0 , x 0 , z) moving along the holographic direction is null. In what follows we show that the sign of the dvdz term in (1) determines whether the curve is either ingoing or outgoing. This in turn determines whether the mass shell in a Vaydia metric is ingoing or outgoing.
We start by considering the timelike vector ∂ v to be future directed. The null geodesics on the (v, z) plane for the AdS d+1 black hole geometry are obtained from here C is an integration constant and 2 F 1 is Gauss hyper-geometric function, blowing up at M z d = 1 and having an expansion 2 F 1 ≈ z + O(z d+1 ) near the boundary of AdS. The upper sign choice in (A1) therefore implies that the geodesic displayed in (A1) is escaping from the horizon as v increases. Taking into account that ∂ v is timelike outside the horizon, we conclude that the v = v 0 curve corresponds to a radially ingoing null geodesic. The exponential inside the bracket can be understood as the evolution operator for the Hamiltonian H = −g µν D µ D ν + m 2 , which allows us to write G(x α 2 , z 2 | x α 1 , z 1 ) = Here S 1particle [x(τ )] is a one-particle action written in terms of a world line proper time parameter τ ∈ [0 . . . T ], and a dot (˙) means derivative with respect to τ . We can rescale τ → τ T in order to get τ ∈ [0 . . . 1] and we end up with Notice that this one-particle action is not invariant under reparametrization of the world line. Introducing the einbein e τ (τ ), we can interpret (B4) as a gauge fixed expression for an originally reparametrization invariant action [37,38] 3 The gauge fixing amounts to setting e τ = T , and the T integral in (B4) corresponds to the leftover (Teichmüller) parameter after gauge fixing [38]. In the limit of large mass m, in which we are interested, we can take the semiclassical approximation of the integral (B5), to have G(x α 2 , z 2 | x α 1 , z 1 ) = e iS on−shell (x α 1 ,z1;x α 2 ,z2) , with S on−shell (x α 1 , z 1 | x α 2 , z 2 ) = dτ −m −g µνẋ µẋν + eA µẋ µ x µ (τ )=x µ classical (τ ) , where the quotient e/m is assumed finite in the large m limit, and x µ classical (τ ) are the classical trajectories starting at (x α 1 , z 1 ) and ending at (x α 2 , z 2 ).
solution. Nevertheless, with this complex momenta, the imaginary unit i could not be removed from the equations of motion by Wick rotation of the worldline parameter τ = −iτ E , and the there would be no real-valued Euclidean variables to access the forbidden region. As expexcted the square root R in (C9) is real only for spacelike trajectories. In other words, our Euclidean action (C10) allows us to find trajectories joining spacelike separated boundary points.
The conclusion is that, in the absence of charge, spacelike separated points in the forbidden region can be joined by real classical trajectories of the Euclidean worldline time action (C10), obtained from the original one (C4) via the Wick rotation τ = −iτ E . This is a standard prescription and the one used in [9]. and we can thus identify In other words Now, by noticing that the functions S ± correspond to the classical on shell action integrated from the turning point at the tip of the trajectory z = z * into the boundary z = z with the ± identifying each of the geodesic branches, we can establish that S + − S − = S on−shell . With this, we recover formula (35), where we replaced ∆ = mL in the large m limit (see [39] for a related discussion).