我们在 AdS/CFT 背景下研究 AdS 时空中与 Rindler 楔相关的光面视界附近的量子时空涨落。具体而言，我们求解了光面视界附近的真空 Einstein 方程，并引入一个沿某一光锥方向在 Planck 宽度内被抹平的量子源 Ansatz。这种源的物理解释是由真空能量涨落产生的引力激波，它将 Einstein 方程改变为具有 Langevin 方程形式的随机偏微分方程。通过沿光面积分涨落，我们发现光子往返穿过 Rindler 楔视界所需时间中的累积效应同时依赖于 $d$ 维 Newton 常数 $G_N^{(d)}$ 和 AdS 曲率 $L$，这与以往文献采用不同方法所得结果一致。
英文摘要：
We study quantum spacetime fluctuations near light-sheet horizons associated with a Rindler wedge in AdS spacetime, in the context of AdS/CFT. In particular, we solve the vacuum Einstein equation near the light-sheet horizon, augmented with the Ansatz of a quantum source smeared out in a Planckian width along one of the light-cone directions. Such a source, whose physical interpretation is of gravitational shockwaves created by vacuum energy fluctuations, alters the Einstein equation to a stochastic partial differential equation taking the form of a Langevin equation. By integrating fluctuations along the light sheet, we find an accumulated effect in the round-trip time of a photon to traverse the horizon of the Rindler wedge that depends on both the $d$-dimensional Newton constant $G_N^{(d)}$ and the AdS curvature $L$, in agreement with previous literature utilizing different methods.
In this paper, we study the AdS-Rindler reconstruction. The CFT operators naively given by the holographic dictionary for the AdS-Rindler reconstruction contain tachyonic modes, which are inconsistent with the causality and unitarity of the CFT. Therefore, the subregion duality and the entanglement wedge reconstruction do not hold. We also find that the tachyonic modes in the AdS-Rindler patch lead to arbitrary high-energy or trans-Planckian modes in the global AdS. It means that the mode expansion of the Rindler patch is sensitive to the UV limit of the theory, that is, quantum gravity. In ad…
In anti-de Sitter space a highly accelerating observer perceives a Rindler horizon. The two Rindler wedges in AdS_{d+1} are holographically dual to an entangled conformal field theory that lives on two boundaries with geometry R x H_{d-1}. For AdS_3, the holographic duality is especially tractable, allowing quantum-gravitational aspects of Rindler horizons to be probed. We recover the thermodynamics of Rindler-AdS space directly from the boundary conformal field theory. We derive the temperature from the two-point function and obtain the Rindler entropy density precisely, including numerical f…
Field theories in black hole spacetimes undergo dimensional reduction near horizon (in the Rindler limit) to two dimensional conformal field theories. We investigate this enhancement of symmetries in the context of gauge/gravity duality by considering Rindler space as boundary of Anti-de Sitter space in three spacetime dimensions. We show that the loxodromy conjugacy class of the SO(2,2) isometry group is responsible for generating the special conformal transformations on the boundary under RG flow. We use this approach to present an alternative derivation of the two-point function in Rindler …
We study a holographic construction of conformal blocks in the Regge limit of four-point scalar correlation functions by using coordinates of the two-sided Rindler-AdS black hole. As a generalization of geodesic Witten diagrams, we construct diagrams with four external scalar fields in the Rindler-AdS black hole by integrating over two half-geodesics between the centers of Penrose diagrams and points at the AdS boundary. We demonstrate that late-time behaviors of the diagrams coincide with the Regge behaviors of conformal blocks. We also point out their relevance with the pole-skipping phenome…
If the Hamiltonian of a quantum field theory is taken to be a timelike isometry, the vacuum state remains empty for all time. We search for such stationary vacua in anti-de Sitter space. By considering conjugacy classes of the Lorentz group, we find interesting one-parameter families of stationary vacua in three-dimensional anti-de Sitter space. In particular, there exists a family of rotating Rindler vacua, labeled by the rotation parameter beta, which are related to the usual Rindler vacuum by non-trivial Bogolubov transformations. Rotating Rindler-AdS space possesses not only an observer-de…
In the bulk reconstruction program, one constructs boundary representations of bulk fields. We investigate the relation between the global/Poincare and AdS-Rindler representations for AdS$_2$. We obtain the AdS-Rindler smearing function for massive and massless fields and show that the global and AdS-Rindler boundary representations are related by conformal transformations. We also use the boundary representations of creation and annihilation operators to compute the Bogoliubov transformation relating global modes to AdS-Rindler modes for both massive and massless particles.
A nonvanishing value for the Rindler horizon energy is proposed, by an analogy with the "near horizon" Schwarzschild metric. We show that the Rindler horizon energy is given by the same formula $E = α/2$ obtained by Padmanabhan for the Schwarzschild spacetime, where $α$ is the gravitational radius.
In this note, we explain how asymptotically globally AdS spacetimes can be given an alternate dual description as entangled states of a pair of hyperbolic space CFTs, which are associated with complementary Rindler wedges of the AdS geometry. The reduced density matrix encoding the state of the degrees of freedom in one of these CFTs describes the physics in a single wedge, which we can think of as the region of spacetime accessible to an accelerated observer in AdS. For pure AdS, this density matrix is thermal, and we argue that the microstates in this thermal ensemble correspond to spacetime…
We show in this note that the Padmanabhan expression $E = a/2$ for the energy of the Schwarzschild and de Sitter spacetimes (a - the horizon radius) is also valid for the Rindler horizon, using an analogy with the electrostatic case.
We study the proposal by Bredberg et al. (1006.1902), where the fluid is defined by the Brown-York tensor on a timelike surface at r=r_c in black hole backgrounds. We consider both Rindler space and the Schwarzschild-AdS (SAdS) black hole. The former describes an incompressible fluid, whereas the latter describes the vanishing bulk viscosity at arbitrary r_c, but these two results do not contradict with each other. We also find an interesting "coincidence" with the black hole membrane paradigm which gives a negative bulk viscosity. In order to show these results, we rewrite the hydrodynamic st…
The generalized second law is proven for rapidly-evolving semiclassical Rindler horizons at each instant of time, for arbitrary interacting quantum fields minimally coupled to general relativity. The proof requires the background spacetime to have boost and null translation symmetry. Possible extensions to more general horizons and matter-gravity couplings are discussed.
Taking the flat-space limit (zero cosmological constant limit) of the Rindler-AdS spacetime yields the Rindler metric. According to the proposal of Flat/contracted-CFT correspondence, the flat-space limit on the bulk side of asymptotically AdS spacetimes corresponds to the contraction of the conformal field theory on the boundary. We use this proposal for the Rindler-AdS/CFT correspondence and propose a dual theory for the Rindler spacetime, which is a contracted conformal field theory (CCFT). We show that the two-dimensional CCFT symmetries exactly predict the same two-point functions that on…
We find exact formulas for the Extended Uncertainty Principle (EUP) for the Rindler and Friedmann horizons and show that they can be expanded to obtain asymptotic forms known from the previous literature. We calculate the corrections to Hawking temperature and Bekenstein entropy of a black hole in the universe due to Rindler and Friedmann horizons. The effect of the EUP is similar to the canonical corrections of thermal fluctuations and so it rises the entropy signalling further loss of information.
We investigate the past and future Rindler horizons for radial Rindler trajectories in the Schwarzschild spacetime. We assume the Rindler trajectory to be linearly uniformly accelerated (LUA) throughout its motion, in the sense of the curved spacetime generalisation of the Letaw-Frenet equations. The analytical solution for the radial LUA trajectories along with its past and future intercepts ${\cal C}$ with the past null infinity ${\cal J^-}$ and future null infinity ${\cal J^+}$ are presented. The Rindler horizons, in the presence of the black hole, are found to depend on both the magnitude …
Gravitational perturbations of flat Minkowski space make the Rindler horizon dynamical: the horizon satisfies mechanical laws analogous to the ones followed by black holes. We describe the gravitational perturbation of Minkowski space using perturbative field-theoretical methods. The change in the area of the Rindler horizon is described in terms of the deflection of light rays by the gravitational field. The difference between the area of the perturbed and the unperturbed horizon is related to the energy of matter crossing the horizon. We derive consistency conditions for the validity of our …
We assumed a stress tensor is necessary on the event horizon of a Schwarzschild black hole or on the Rindler horizon for the Israel matching conditions to be satisfied. We found the surface energy density $ρ_{s}$ is vanishing but the surface pressure $p_{s}$ equals $1/16πl$ in both cases, where $l$ is the proper distance from the horizon. The junction relations are applied both for $r =$ const. and $T =$ const. surfaces, with the same results for the surface parameters $ρ_{s}$ and $p_{s}$. We emphasize the nonstatic feature of the spacetimes beyond the corresponding horizons.
A direct relation between the time dependent Milne geometry and the Rindler spacetime is shown. Milne's metric corresponds to that one beyond Rindler's event horizon (in the region $t \succ |x|)$). It was found that the shear tensor from the dissipative term of the RHIC expanding fireball has the same structure as that corresponding to the anisotropic fluid from the black hole interior, even though the latter geometry is curved.
In this paper, we study the spin chain and string excitation in the rotating Rindler-$AdS_3$ proposed in [12]. We obtain a one-parameter deformed $SL(2)$ spin chain at the fast spin limit. Two-spin GKP-like solutions are studied at short and long string limits. General ansatz for the giant magnons and the spiky strings are analyzed in detail for various $β$. At last, we explore its counterpart in analytic continuation and pp-wave limit.
Finite entropy and energy are obtained for the horizon of a Rindler observer on the grounds of the nonstatic character of the geometry beyond the horizon. Edery - Constantineau prescription is used to find the dynamical phase space of this particular spacetime. The number of microstates rooted from the ignorance of a Rindler observer of the parameter $t$ from the nonstationary region is calculated. We suggest that the gravitational energy density constructed by means of the horizon energy and using the Holographic Principle is proportional to $g^{2}$, similar with a result recently obtained by…
In this work we construct metrics corresponding to radiating black holes whose near horizon regions cannot be approximated by Rindler spacetime. We first construct infinite parameter coordinate transformations from Minkowski coordinates, such that an observer using these coordinates to describe spacetime events measures the Minkowski vacuum to be Planckian. Utilizing these results, we construct family of black holes that radiate at spatial infinity. As an illustration we study a subset of the black hole solutions that satisfy the null energy condition.
The Feynman propagator encodes all the physics contained in a free field and transforms as a covariant bi-scalar. Therefore, we should be able to discover the thermality of the Rindler horizon, just by probing the structure of the propagator, expressed in the Rindler coordinates. I show that the thermal nature of the Rindler horizon is indeed contained --- though hidden --- in the standard, inertial, Feynman propagator. The probability $P(E)$ for a particle to propagate between two events, with energy $E$, can be related to the temporal Fourier transform of the propagator. A strikingly simple …
We realize the weak momentum relaxation in Rindler fluid, which lives on the time-like cutoff surface in an accelerating frame of flat spacetime. The translational invariance is broken by massless scalar fields with weak strength. Both of the Ward identity and the momentum relaxation rate of Rindler fluid are obtained, with higher order correction in terms of the strength of momentum relaxation. The Rindler fluid with momentum relaxation could also be approached through the near horizon limit of cutoff AdS fluid with momentum relaxation, which lives on a finite time-like cutoff surface in Anti…
There has been much discussion on the possibility of firewalls at the horizon-scale in black hole physics, including questions regarding the presence or absence of firewalls at apparent horizons, such as the Rindler horizon and the horizon of the Poincaré patch of Anti-de Sitter space. We argue against the presence of such apparent firewalls by demonstrating that one recent argument for firewalls in black holes does not extend to these cases. We also include some brief remarks on some claims in the recent firewall literature.
In this paper we study one-dimensional conformal field theory at finite temperature dual to the two-dimensional anti-de Sitter spacetime in the Rindler coordinates. We show that conformal symmetry for thermal two-point functions manifests itself in a form of recurrence relations in the complex frequency space. It is discussed that all the real-time two-point functions are given by solutions to the recurrence relations.
In a Rindler-type coordinate system spanned in a region outside of a black hole horizon, we have nonvanishing classical holographic charges as soft hairs on the horizon for stationary black holes. Taking a large black hole mass limit, the spacetimes with the charges are described by asymptotic Rindler metrics. We construct a general theory of gravitational holographic charges for a (1+3)-dimensional linearized gravity field in the Minkowski background with Rindler horizons. Although matter crossing a Rindler horizon causes horizon deformation and a time-dependent coordinate shift, that is, gra…
We review the quantization of scalar and gauge fields using Rindler coordinates with an emphasis on the physics of the Rindler horizon. In the thermal state at the Unruh temperature, correlators match their Minkowski vacuum values and the renormalized stress tensor vanishes, while at any other temperature the renormalized stress-energy diverges on the horizon. After giving a new derivation of some of these results using canonical quantization in the thermofield double state, we comment on the relevance of fluxes and boundary conditions at the horizon, which have arisen in calculations of entan…
Orbifold techniques are used to study bosonic, type II and heterotic strings in Rindler space at integer multiples N of the Rindler temperature, and near a black hole horizon at integer multiples of the Hawking temperature, extending earlier results of Dabholkar. It is argued that a Hagedorn transition occurs nears the horizon for all N>1.
Persistent puzzles to do with information loss for black holes have stimulated critical reassessment of the domain of validity of semiclassical EFT reasoning in curved spacetimes, particularly in the presence of horizons. We argue here that perturbative predictions about evolution for very long times near a horizon are subject to problems of secular growth - i.e. powers of small couplings come systematically together with growing functions of time. Such growth signals a breakdown of naive perturbative calculations of late-time behaviour, regardless of how small ambient curvatures might be. Sim…
The time dependent conformally-flat spherical Rindler spacetime is investigated. The geometry has an apparent horizon that coincides with the causal horizon. The scalar acceleration of a static observer is constant and equals to the acceleration $g$ from the static expression of the Rindler-like metric. The timelike radial geodesics are computed and proves to be hyperbolae when a specific choice of the constants of integration is operated.
We study the Thermo-field realization of the duality between the Rindler-AdS higher spin theory and $O(N)$ vector theory. The CFT represents a decoupled pair of free $O(N)$ vector field theories. It is shown how this decoupled domain CFT is capable of generating the connected Rindler-AdS background with the full set of Higher Spin fields.
We investigate the Unruh effect for a massless scalar field in the two dimensional Minkowski space in the presence of a uniformly accelerated perfect mirror, with the trajectory of the mirror chosen in such a way that the mirror completely masks the Rindler horizon from the space-time region of interest. We find that the characteristic thermodynamical properties of the effect remain unchanged, i.e. the response of a uniformly co-accelerated Unruh detector and the distribution of the Rindler particles retain their thermal form. However, since in this setup there are no unobserved degrees of fre…
We show that Rindler horizons are entropic screens and gravity is an entropic force in Rindler space by deriving the Verlinde entropy formula from the focusing of light due to a mass close to the horizon. Consequently, gravity is also entropic in the near horizon regions of Schwarzschild and de Sitter space-times. In different limits, the entropic nature of gravity in Rindler space leads to the Bekenstein entropy bound and the uncertainty principle.
By studying global AdS using different foliations, global and Rindler-AdS, we show that there are two different asymptotic Fefferman-Graham expansions possible and thus two different definitions of "boundaries". We demonstrate that imposing boundary conditions on the two boundaries is not mutually compatible even when these boundaries are pushed to infinity. Thus, these two procedures define two genuinely distinct theories that we call global-CFT and Rindler-CFT. We show that the Rindler-CFT is not the same as the theory one gets by "Rindlerizing the global-CFT" as described in hep-th/9804085.…
The Rindler horizon in Minkowski spacetime can be implanted with supertranslation hair by a matter shock wave without planar symmetry, and the hair is observable as a supertranslation memory on the Rindler family of uniformly linearly accelerated observers. We show that this classical memory is accompanied by a supertranslation quantum memory that modulates the entanglement between the opposing Rindler wedges in quantum field theory. A corresponding phenomenon across a black hole horizon may play a role in Hawking, Perry and Strominger's proposal for supertranslations to provide a solution to …
QED in two-dimensional Minkowski space contains a single physical state as seen by an inertial observer or by a constantly accelerating Rindler observer. However in Feynman gauge if one takes a generic representative of the physical Minkowski state and traces over all left Rindler states, one does not arrive at a physical right Rindler state, but rather at a "density matrix" with negative eigenvalues for negative norm states corresponding intuitively to the radiation of uncorrelated temporal photons and ghosts. This reflects the fact that states that are exact under the Minkowski BRST operator…
We show that the dimensionless Rindler energy of a black hole, $E_R$, is exactly the surface Hamiltonian obtained from the Einstein--Hilbert action evaluated on the horizon. Therefore, $E_R$ is given by a surface integral over the horizon and manifestly holographic. In the context of the AdS/CFT duality, Rindler energy corresponds, on the boundary, to a dimensionless energy given by the product of the AdS radius and the extensive part of the CFT energy. We find that, beyond General Relativity, $E_R$ is still holographic but not necessarily given by the surface Hamiltonian of the theory.
The "pole-skipping" phenomenon reflects that the retarded Green's function is not unique at a pole-skipping point in momentum space $(ω,k)$. We explore the universality of the pole-skipping in different geometries. In holography, near horizon analysis of the bulk equation of motion is a simpler way to derive a pole-skipping point and we use this method in Lifshitz, AdS$_2$ and Rindler geometries. We also study the complex hydrodynamic analyses and find that the dispersion relations in terms of dimensionless variables $\fracω{2πT}$ and $\frac{\vert k\vert}{2πT}$ pass through pole-skipping point…
In the AdS/CFT correspondence, CFTs are identified by asymptotic boundary surfaces and the boundary conditions imposed on those surfaces. However, AdS can be foliated in various ways to give different boundaries. We show that the CFTs obtained using certain distinct foliations are different. This difference arises because the asymptotic region of a foliation overlaps with the deep interior region of another. In particular we focus on the CFTs defined on surfaces of large constant radius in global coordinates, Rindler-AdS coordinates, and Poincaré coordinates for AdS$_3$. We refer to these as g…
Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler spacetime admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra, with central charge c=1, defined in the Fock representation. The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated QFT on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens p…
The vacuum expectation value of the current density for a charged scalar field is investigated in Rindler spacetime with a part of spatial dimensions compactified to a torus. It is assumed that the field is prepared in the Fulling-Rindler vacuum state. For general values of the phases in the periodicity conditions and the lengths of compact dimensions, the expressions are provided for the Hadamard function and vacuum currents. The current density along compact dimensions is a periodic function of the magnetic flux enclosed by those dimensions and vanishes on the Rindler horizon. The obtained r…
We compute fluctuations in the modular energy of the vacuum associated with a Rindler-wedge in AdS spacetime in the context of AdS/CFT. We discuss the possible effect of these energy fluctuations on the spacetime geometry, and on the traversal time of a light beam propagating from the boundary to the bulk and back.
The recently suggested interpretation of the universal hadronic freeze-out temperature T_f ~ 170 Mev - found for all high energy scattering processes that produce hadrons: e+ e-, p p, p anti-p, pi p, etc. and N N' (heavy-ion collisions) - as a Unruh temperature triggers here the search for the gravitational black-hole that in its near-horizon approximation better simulates this hadronic phenomenon. To identify such a black-hole we begin our gravity-gauge theory phenomenologies matching by asking the question: which black-hole behind that Rindler horizon could reproduce the experimental behavio…
We study open string configurations in Rindler space suspended from D-branes at finite distance from the Rindler horizon. This is a toy model for strings in the near horizon region of a black hole and has applications for the study of strings configurations of heavy quarks in the AdS/CFT duals of hot field theories, as well as other applications to the study of open strings ending on giant gravitons. We find that this setup produces very similar results to those that have been found in the AdS black hole setup, but it is much more tractable analytically. We also comment on some quantum applica…
An anisotropic fluid with positive energy density and negative pressures is proposed in the black hole interior. The gravitational field is constant everywhere inside and is given by the horizon surface gravity. Even though the geometry is of Rindler type it is curved because of the spherical symmetry used and is singular at the origin. The Israel junction conditions are studied on the stretched horizon instead of using a matching process on the null surface $r = 2M$. The entropy of any inner sphere is mass independent, maximally packed and unaffected by the outer layers.
It has been proposed that a certain Z_N orbifold, analytically continued in N, can be used to describe the thermodynamics of Rindler space in string theory. In this paper, we attempt to implement this idea for the open-string sector. The most interesting result is that, although the orbifold is tachyonic for positive integer N, the tachyon seems to disappear after analytic continuation to the region that is appropriate for computing ${\mathrm {Tr}} ρ^{\mathcal N}$, where $ρ$ is the density matrix of Rindler space and Re $\mathcal N$>1. Analytic continuation of the full orbifold conformal field…
The notions of temperature, entropy and `evaporation', usually associated with spacetimes with horizons, are analyzed using general approach and the following results, applicable to different spacetimes, are obtained at one go. (i) The concept of temperature associated with the horizon is derived in a unified manner and is shown to arise from purely kinematic considerations. (ii) QFT near any horizon is mapped to a conformal field theory without introducing concepts from string theory. (iii) For spherically symmetric spacetimes (in D=1+3) with a horizon at r=l, the partition function has the g…
The entropy S of the horizon $theta = pi/2$ of the Hawking wormhole written in spherical Rindler coordinates is computed in this letter. Using Padmanabhan's prescription,we found that the surface gravity of the horizon equals the proper acceleration of the Rindler observer. S is a monotonic function of the radial coordinate $xi$ and vanishes when $xi$ equals the Planck length. In addition, its expression is similar with the Kaul - Majumdar one for the black hole entropy, including logarithmic corrections in quantum gravity scenarios.
The tensionless limit of string theory has recently been formulated in terms of worldsheet Rindler physics. In this paper, by considering closed strings moving in background Rindler spacetimes, we provide a concrete exemplification of this phenomenon. We first show that strings probing the near-horizon region of a generic non-extremal blackhole become tensionless thereby linking a spacetime Carroll limit to a worldsheet Carroll limit. Then, considering strings in $d$-dimensional Rindler spacetime we find a Rindler structure induced on the worldsheet. Novelties, including folds, appear on the c…
We substantiate the Hawking radiation as quantum tunneling of fields or particles crossing the horizon by using the Rindler coordinate. The thermal spectrum detected by an accelerated particle is interpreted as quantum tunneling in the Rindler spacetime. Representing the spacetime near the horizon locally as a Rindler spacetime, we find the emission rate by tunneling, which is expressed as a contour integral and gives the correct Boltzmann factor. We apply the method to non-extremal black holes such as a Schwarzschild black hole, a non-extremal Reissner-Nordström black hole, a charged Kerr bla…
In this paper, we discuss Rindler space string thermodynamics from a thermal scalar point of view as an explicit example of the results obtained in JHEP 1402 (2014) 127. We discuss the critical behavior of the string gas and interpret this as a random walk near the black hole horizon. Combining field theory arguments with the random walk path integral picture, we realize (at genus one) the picture put forward by Susskind of a long string surrounding black hole horizons. We find that thermodynamics is dominated by a long string living at string-scale distance from the horizon whose redshifted t…
In the hydrodynamic regime of field theories the entropy is upgraded to a local entropy current. The entropy current is constructed phenomenologically order by order in the derivative expansion by requiring that its divergence is non-negative. In the framework of the fluid/gravity correspondence, the entropy current of the fluid is mapped to a vector density associated with the event horizon of the dual geometry. In this work we consider the local horizon entropy current for higher-curvature gravitational theories proposed in arXiv:1202.2469, whose flux for stationary solutions is the Wald ent…
