Rindler Transformation Research Articles

Holographic timelike entanglement entropy from Rindler method* *Supported partially by the National Natural Science Foundation of China (12175008)

For a Lorentzian invariant theory, the entanglement entropy should be a function of the domain of dependence of the subregion under consideration. More precisely, it should be a function of the domain of dependence and the appropriate cut-off. In this study, we refine the concept of cut-off to make it applicable to timelike regions and assume that the usual entanglement entropy formula also applies to timelike intervals. Using the Rindler method, the timelike entanglement entropy can be regarded as the thermal entropy of the CFT after the Rindler transformation plus a constant , where c denotes the central charge. The gravitational dual of the 'covariant' timelike entanglement entropy is presented following this method.

Revisit the entanglement entropy with gravitational anomaly

In this paper we study the entanglement entropy in the CFT2, whose gravity dual is AdS3 spacetime with a Chern-Simons term. Using the generalized Rindler method, we obtain the Rindler transformation in the two-dimensional planar CFT and compute the entanglement entropy of the CFT with gravitational anomalies. The conditions under which the entanglement entropy may have anomalous contributions is also discussed. In addition, we present a relatively general form of the Rindler AdS metric and compute its thermal entropy, which agrees with the entanglement entropy in the field theory. Moreover, we utilize the conformal transformation, which maps a cylinder to a plane, to compute the entanglement entropy of the CFT residing on a cylinder, as well as the entanglement entropy of the CFT at finite temperature on a plane. The corresponding contribution of the Chern-Simons term in gravity to the black hole thermal entropy is also obtained from this approach. These results are important for further understandings of the two-dimensional CFT with gravitational anomalies.

Holographic entanglement entropy in flat limit of the generalized minimal massive gravity model

Previously we have studied the Generalized Minimal Massive Gravity (GMMG) in asymptotically AdS_3 background, and have shown that the theory is free of negative-energy bulk modes. Also we have shown GMMG avoids the aforementioned “bulk-boundary unitarity clash”. Here instead of AdS_3 space we consider asymptotically flat space, and study this model in the flat limit. The dual field theory of GMMG in the flat limit is a BMS_3 invariant field theory, dubbed (BMSFT) and we have BMS algebra asymptotically instead of Virasoro algebra. In fact here we present an evidence for this claim. Entanglement entropy of GMMG is calculated in the background in the flat null infinity. Our evidence for mentioned claim is the result for entanglement entropy in filed theory side and in the bulk (in the gravity side). At first using Cardy formula and Rindler transformation, we calculate entanglement entropy of BMSFT in three different cases. Zero temperature on the plane and on the cylinder, and non-zero temperature case. Then we obtain the entanglement entropy in the bulk. Our results in gravity side are exactly in agreement with field theory calculations.

Radiative process of two entangled uniformly accelerated atoms in a thermal bath: a possible case of anti-Unruh event

We study the radiative process of two entangled two-level atoms uniformly accelerated in a thermal bath, coupled to a massless scalar field. First, by using the positive frequency Wightman function from the Minkowski modes with a Rindler transformation we provide the transition probabilities for the transitions from maximally entangled symmetric and anti-symmetric Bell states to the collective excited or ground state in (1 + 1) and (1 + 3) dimensions. We observe a possible case of anti-Unruh-like event in these transition probabilities, though the (1+1) and (1+3) dimensional results are not completely equivalent. We infer that thermal bath plays a major role in the occurrence of the anti-Unruh-like effect, as it is also present in the transition probabilities corresponding to a single detector in this case. Second, we have considered the Green’s functions in terms of the Rindler modes with the vacuum of Unruh modes for estimating the same. Here the anti-Unruh effect appears only for the transition from the anti-symmetric state to the collective excited or ground state. It is noticed that here the (1 + 1) and (1 + 3) dimensional results are equivalent, and for a single detector, we do not observe any anti-Unruh effect. This suggests that the entanglement between the states of the atoms is the main cause for the observed anti-Unruh effect in this case. In going through the investigation, we find that the transition probability for a single detector case is symmetric under the interchange between the thermal bath’s temperature and the Unruh temperature for Rindler mode analysis; whereas this is not the case for Minkowski mode. We further comment on whether this observation may shed light on the analogy between an accelerated observer and a real thermal bath. An elaborate investigation for the classifications of our observed anti-Unruh effects, i.e., either weak or strong anti-Unruh effect, is also thoroughly demonstrated.

Flat-space limit of extremal curves

According to the Ryu-Takayanagi prescription, the entanglement entropy of subsystems in the boundary conformal field theory (CFT) is proportional to the area of extremal surfaces in bulk asymptotically Anti-de Sitter (AdS) spacetimes. The flat-space limit of these surfaces is not well defined in the generic case. We introduce a new curve in the three-dimensional asymptotically AdS spacetimes with a well-defined flat-space limit. We find this curve by using a new vector, which is vanishing on it and is normal to the bulk modular flow of the original interval in the two-dimensional CFT. The flat-space limit of this new vector is well defined and gives rise to the bulk modular flow of the corresponding asymptotically flat spacetime. Moreover, after Rindler transformation, this new vector is the normal Killing vector of the BTZ inner horizon. We reproduce all known results about the holographic entanglement entropy of Bondi-Metzner-Sachs invariant field theories, which are dual to the asymptotically flat spacetimes.

Entanglement contour and modular flow from subset entanglement entropies

The Entanglement contour function quantifies the contribution from each degree of freedom in a region mathcal{A} to the entanglement entropy {S}_{mathcal{A}} . Recently in [1] the author gave two proposals for the entanglement contour in two-dimensional theories. The first proposal is a fine structure analysis of the entanglement wedge, which applies to holographic theories. The second proposal is a claim that for general two-dimensional theories the partial entanglement entropy is given by a linear combination of entanglement entropies of relevant subsets inside mathcal{A} . In this paper, we further study the partial entanglement entropy proposal by showing that it satisfies all the rational requirements proposed previously. We also extend the fine structure analysis from vacuum AdS space to BTZ black holes. Furthermore, we give a simple prescription to generate the local modular flows for two-dimensional theories from only the entanglement entropies without refer to the explicit Rindler transformations.

Quantum simulation of Rindler transformations

We show how to implement a Rindler transformation of coordinates with an embedded quantum simulator. A suitable mapping allows to realise the unphysical operation in the simulated dynamics by implementing a quantum gate on an enlarged quantum system. This enhances the versatility of embedded quantum simulators by extending the possible in-situ changes of reference frames to the non-inertial realm.

Acceleration in de Sitter spacetimes

We propose a definition of uniform accelerated frames in de Sitter spacetimes applying the Nachtmann method of introducing coordinates using suitable point-dependent isometries. In order to recover the well-known Rindler approach in the flat limit, we require the transformation between the static frame and the accelerated one to depend continuously on acceleration, obtaining thus the natural generalization of the Rindler transformation to the de Sitter spacetimes of any dimensions.

Accelerated quark and holography for confining gauge theory

We show a constantly accelerated quark as a string solution of the Nambu-Goto action, which is embedded in the bulk background dual to the $\cal{N}$ $=2$ supersymmetric confining Yang-Mills theory. The induced metric of the world sheet for this string solution has an event horizon specified by the fifth coordinate. By an extended Rindler transformation proposed by Xiao, we move to the comoving frame of the accelerated quark-string. Then we find that this horizon is transferred to the event horizon of the bulk and the causal part of the accelerated quark is transformed to a static free-quark in the Rindler coordinate. As a result, the confinement of the Minkowski vacuum is lost in the Rindler vacuum. This point is assured also by studying the potential between the quark and anti-quark. However, the remnants of the original confining force are seen in various thermal quantities. We also discuss the consistency of our results and the claim that the Green's functions will not be changed by the Rindler transformation.

Campo eletromagnético de uma carga em queda livre num campo gravitacional uniforme

As transformações de Rindler são usadas para obter o campo eletromagnético de uma carga em queda livre num campo gravitacional uniforme.

Campo eletrostático de uma carga em repouso num campo gravitacional uniforme

As transformações de Rindler são usadas para obter o campo eletrostático de uma carga em repouso na presença de um campo gravitacional uniforme.

Relativistic Invariance of Lyapunov Exponents in Bounded and Unbounded Systems

The study of chaos in relativistic systems has been hampered by the observer dependence of Lyapunov exponents (LEs) and of conditions, such as orbit boundedness, invoked in the interpretation of LEs as indicators of chaos. Here we establish a general framework that overcomes both difficulties and apply the resulting approach to address three fundamental questions: how LEs transform under Lorentz and Rindler transformations and under transformations to uniformly rotating frames. The answers to the first and third questions show that inertial and uniformly rotating observers agree on a characterization of chaos based on LEs. The second question, on the other hand, is an ill-posed problem due to the event horizons inherent to uniformly accelerated observers.

Observer-Dependence of Chaos Under Lorentz and Rindler Transformations

The behavior of Lyapunov exponents λ and dynamical entropies h, whose positivity characterizes chaotic motion, under Lorentz and Rindler transformations is studied. Under Lorentz transformations, λ and h are changed, but their positivity is preserved for chaotic systems. Under Rindler transformations, λ and h are changed in such a way that systems, which are chaotic for an accelerated Rindler observer, can be nonchaotic for an inertial Minkowski observer. Therefore, the concept of chaos is observer-dependent.