Rindler Space Research Articles

Entanglement in ( 1+1 )-dimensional free scalar field theory: Tiptoeing between continuum and discrete formulations

We review some classic works on ground state entanglement entropy in (1+1)-dimensional free scalar field theory. We point out identifications between the methods for the calculation of entanglement entropy and we show how the formalism developed for the discretized theory can be utilized in order to obtain results in the continuous theory. We specify the entanglement spectrum and we calculate the entanglement entropy for the theory defined on an interval of finite length L. Finally, we derive the modular Hamiltonian directly, without using the modular flow, via the continuum limit of the expressions obtained in the discretized theory. In a specific coordinate system, the modular Hamiltonian assumes the form of a free field Hamiltonian on the Rindler wedge. Published by the American Physical Society 2024

Spacetime symmetry breaking on nongeodesic leaves and a new form of matter

We examine the permanent damage caused by the historical breakdown of full diffeomorphism invariance induced by a foliation. We focus on the case where the foliation is allowed to be nongeodesic after the interactions with the foliation switch off. Gravity and other forms of matter recover full diffeomorphism invariance only at the expense of introducing a new matterlike component, carrying the nonvanishing Hamiltonian (and momentum, as it turns out) left over from the violating past interactions. This matter form must be stress-free in the preferred frame; this is the only way a matter action can mimic the evolution of the leftover Hamiltonian (and momentum) driven by the Dirac hypersurface deformation algebra. Hence, if the preferred frame is nongeodesic, the equivalent matter component must have energy and a momentum current in this frame, but still no spatial stresses: an unusual form of “matter.” It is equivalent to a fluid with anisotropic stress in some regimes, reducing to dust in others, or even displaying completely new features in extreme situations. Its stress energy tensor is conserved. We provide two examples based on accelerated frames: Rindler space-time and the canonical Schwarzchild frame. Published by the American Physical Society 2024

Strings near black holes are Carrollian

We demonstrate that strings near the horizon of a Schwarzschild black hole, when viewed by a stationary observer at infinity, probe a string Carroll geometry, where the effective light speed is given by the distance from the horizon. We expand the Polyakov action in powers of this light speed to find a theory of Carrollian strings. We show that the string shrinks to a point to leading order near the horizon, which follows a null geodesic in a two-dimensional Rindler space. At the next-to-leading order the string oscillates in the embedding fields associated with the near-horizon two-sphere. Published by the American Physical Society 2024

Black-body radiation in an accelerated frame

We derive Planck’s radiation law in a uniformly accelerated frame expressed in Rindler coordinates. The black-body spectrum is time-dependent in its temperature and Planckian at each instantaneous time, but it is scaled by an emissivity factor that depends on the Rindler spatial coordinate and the acceleration magnitude. An observer in an accelerated frame will perceive the black-body as black, hyperblack, or grey, depending on their position relative to the source (moving away or toward it), the acceleration magnitude, and whether they are accelerating or decelerating. For an observer accelerating away from the source, there exists a threshold on the acceleration magnitude beyond which they no longer receive radiation from the black-body. Since the frequency and the number of modes in Planck’s law evolve over time, the spectrum is continuously red- or blue-shifted towards lower or higher frequencies as time progresses, and the radiation modes (photons) may be created or annihilated, depending on the observer’s position and their acceleration or deceleration relative to the source of radiation.

Gravitational redshift revisited: Inertia, geometry, and charge

Gravitational redshift effects undoubtedly exist; moreover, the experimental setups which confirm the existence of these effects—the most famous of which being the Pound–Rebka experiment—are extremely well-known. Nonetheless—and perhaps surprisingly—there remains a great deal of confusion in the literature regarding what these experiments really establish. Our goal in the present article is to clarify these issues, in three concrete ways. First, although (i) Brown and Read (2016) are correct to point out that, given their sensitivity, the outcomes of experimental setups such as the original Pound–Rebka configuration can be accounted for using solely the machinery of accelerating frames in special relativity (barring some subtleties due to the Rindler spacetime necessary to model the effects rigorously), nevertheless (ii) an explanation of the results of more sensitive gravitational redshift outcomes does in fact require more. Second, although typically this ‘more’ is understood as the invocation of spacetime curvature within the framework of general relativity, in light of the so-called ‘geometric trinity’ of gravitational theories, in fact curvature is not necessary to explain even these results. Thus (a) one can often explain the results of these experiments using only the resources of special relativity, and (b) even when one cannot, one need not invoke spacetime curvature. And third: while one might think that the absence of gravitational redshift effects would imply that spacetime is flat (indeed, Minkowskian), this can be called into question given the possibility of the cancelling of gravitational redshift effects by charge in the context of the Reissner–Nordström metric. This argument is shown to be valid and both attractive forces as well as redshift effects can be effectively shielded (and even be repulsive or blueshifted, respectively) in the charged setting. Thus, it is not the case that the absence of gravitational effects implies a Minkowskian spacetime setting.

Analyticity and the Unruh effect: a study of local modular flow

The Unruh effect can be formulated as the statement that the Minkowski vacuum in a Rindler wedge has a boost as its modular flow. In recent years, other examples of states with geometrically local modular flow have played important roles in understanding energy and entropy in quantum field theory and quantum gravity. Here I initiate a general study of the settings in which geometric modular flow can arise, showing (i) that any geometric modular flow must be a conformal symmetry of the background spacetime, and (ii) that in a well behaved class of “weakly analytic” states, geometric modular flow must be future-directed. I further argue that if a geometric transformation is conformal but not isometric, then it can only be realized as modular flow in a conformal field theory. Finally, I discuss a few settings in which converse results can be shown — i.e., settings in which a state can be constructed whose modular flow reproduces a given vector field.

Numerical approach to the Bell-Clauser-Horne-Shimony-Holt inequality in quantum field theory

The Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality in the vacuum state of a relativistic scalar quantum field is analyzed. Using Weyl operators built with smeared fields localized in the Rindler wedges, the Bell-CHSH inequality is expressed in terms of the Lorentz invariant inner products of test functions. A numerical framework for these inner products is devised. Causality is also explicitly checked by a numerical evaluation of the Pauli-Jordan function. Violations of the Bell-CHSH inequality are reported for different values of the particle mass parameter. Published by the American Physical Society 2024

Non-gaussian Saha’s ionization in Rindler spacetime and the equivalence principle

We investigate the non-Gaussian effects of the Saha equation in Rindler space via Tsallis statistics. By considering a system with cylindrical geometry, we deduce the non-Gaussian Saha ionization equation for a partially ionized hydrogen plasma that expands with uniform acceleration. We demonstrate conditions for the validity of the equivalence principle within the realms of both Boltzmann–Gibbs and Tsallis statistics. In the non-Gaussian framework, our findings reveal that the effective binding energy exhibits a quadratic dependence on the frame acceleration, in contrast to the linear dependence predicted by Boltzmann–Gibbs statistics. We show that an accelerated observer shall notice a more pronounced effect on the effective binding energy for a>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a>0$$\\end{document} and a more attenuated one when a<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a<0$$\\end{document}. We also ascertain that an accelerated observer will measure values of q smaller than those measured in the rest frame. Besides, assuming the equivalence principle, we examine the effects of the gravitational field on the photoionization of hydrogen atoms and pair production. We show that both photoionization and pair production are more intensely suppressed in regions with a strong gravitational field in a non-Gaussian context than in the Boltzmann–Gibbs framework. Lastly, constraints on the gravitational field and the electron and positron chemical potentials are derived.

On geodesics in spherical Rindler space

The geodesics in various spherical Rindler frames are investigated. A display of some kinematical quantities of the spacetime is given. The constant acceleration from the metric acts as the surface gravity of the horizon [Formula: see text]. The radial geodesics are computed both for the Balasubramanian et al. form of the spherical Rindler space and for the non-diagonal metric of Huang and Sun.

Stability of asymmetric Schwarzschild–Rindler–de Sitter thin shell wormhole

The paper examines the dynamics of asymmetric thin shell wormholes that connect two distinct spacetimes using the cut and paste technique. The focus is on analyzing the linear stability of these wormholes by considering radial perturbations and utilizing the modified generalized Chaplygin gas equation of state. The specific case of an asymmetric wormhole connecting Schwarzschild–Rindler spacetime to Schwarzschild–Rindler–de Sitter space–time is analyzed using this formalism. Our investigation uncovers the existence of both stable and unstable regions, which are contingent upon the appropriate selection of various parameters within the metric spacetime and equation of state. Additionally, we determine that stability regions exist as a consequence of the square speed of sound. By increasing the value of the cosmological constant, the stability region is expanded. Furthermore, the stability regions are augmented by the influence of Rindler parameters, while the stability regions are also affected by adjustments in the equation of state parameters, leading to their enlargement.

On the radiation field of a linearly accelerated charged particle in Born–Infeld theory

The electric potential and the electromagnetic field for a linearly accelerated Born–Infeld charged particle are obtained in an inertial frame by a method that can, in principle, be applied to any electromagnetic theory. The method is based on (i) the fact that the metric near the horizon of a Schwarzschild black hole is equivalent to that of Rindler spacetime, and (ii) a theorem that guarantees that the electrostatic potential for a given nonlinear theory in a static, spherically symmetric spacetime is entirely specified by the Maxwellian electrostatic potential in the same background. Using analytical and numerical methods, the features of the radiation field and the radiation-reaction for such an accelerated particle are discussed in detail.

Pseudo entropy and pseudo-Hermiticity in quantum field theories

In this paper, we explore the concept of pseudo Rényi entropy within the context of quantum field theories (QFTs). The transition matrix is constructed by applying operators situated in different regions to the vacuum state. Specifically, when the operators are positioned in the left and right Rindler wedges respectively, we discover that the logarithmic term of the pseudo Rényi entropy is necessarily real. In other cases, the result might be complex. We provide direct evaluations of specific examples within 2-dimensional conformal field theories (CFTs). Furthermore, we establish a connection between these findings and the pseudo-Hermitian condition. Our analysis reveals that the reality or complexity of the logarithmic term of pseudo Rényi entropy can be explained through this pseudo-Hermitian framework.Additionally, we investigate the divergent term of the pseudo Rényi entropy. Interestingly, we observe a universal divergent term in the second pseudo Rényi entropy within 2-dimensional CFTs. This universal term is solely dependent on the conformal dimension of the operator under consideration. For n-th pseudo Rényi entropy (n ≥ 3), the divergent term is intricately related to the specific details of the underlying theory.

Field theory in Rindler frame and more on the correspondence with thermal field theory formalisms

Considering two accelerated observers with same acceleration in two timelike wedges of Rindler frame we calculate the Feynman-like propagators for a real scalar field in a thermal bath with respect to the Minkowski vacuum. Only the same wedge correlators are symmetric under the exchange of the real thermal bath and Unruh thermal bath, while the cross-wedge ones are not. Interestingly, they contain a cross term which is a collective effects of acceleration and thermal nature of field. Particularly the zero temperature description along with no analytic continuation between coordinates in right and left Rindler wedges, as expected, corresponds to usual thermofield-double formalism. However, unlike in later formulation, the two fields are now parts of the original system. Moreover it bears the features of a spacial case of closed-time formalism (CTP) where the Keldysh contour is along the increasing Rindler time in the respective Rindler wedges. Interestingly, we observe a new feature that the analytic continuation between the wedges provides the two more spacial cases of CTP. Hence Rindler-frame-field theory seems to be a viable candidate to deal thermal theory of fields and may illuminate the search for a bridge between the usual existing formalisms.

Crossed product algebras and generalized entropy for subregions

An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III_{1}1, in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II_∞∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.

Tearing down spacetime with quantum disentanglement

A longstanding enigma within AdS/CFT concerns the entanglement entropy of holographic quantum fields in Rindler space. The vacuum of a quantum field in Minkowski spacetime can be viewed as an entangled thermofield double of two Rindler wedges at a temperature T = 1/2π. We can gradually disentangle the state by lowering this temperature, and the entanglement entropy should vanish in the limit T → 0 to the Boulware vacuum. However, holography yields a non-zero entanglement entropy at arbitrarily low T, since the bridge in the bulk between the two wedges retains a finite width. We show how this is resolved by bulk quantum effects of the same kind that affect the entropy of near-extremal black holes. Specifically, a Weyl transformation maps the holographic Boulware states to near-extremal hyperbolic black holes. A reduction to an effective two-dimensional theory captures the large quantum fluctuations in the geometry of the bridge, which bring down to zero the density of entangled states in the Boulware vacuum. Using another Weyl transformation, we construct unentangled Boulware states in de Sitter space.

A quantum of information in black hole evaporation

Black holes evolve by evaporation of their event horizon. While this process is believed to be unitary, there is no consensus on the recovery of information in black hole entropy. A missing link is a quantum of information in black hole evaporation. Distinct from Hawking radiation, we identify evaporation in entangled pairs by P2 topology of the event horizon consistent with the Bekenstein–Hawking entropy in a uniformly spaced horizon area. It derives by continuation of P2 in Rindler spacetime prior to gravitational collapse, subject to a tight correlation of the fundamental frequency of quasi-normal-mode ringing in gravitational and electromagnetic radiation. Information extraction from entangled pairs by detecting one over the surface spanned by three faces of a large cube carries a quantum of information 2kBlog3 upon including measurement of spin.

Birth and death of entanglement between two accelerating Unruh-DeWitt detectors coupled with a scalar field

We study two accelerating Unruh-DeWitt detectors coupled linearly or quadratically with a scalar field, by using an approach different from previous ones. We use Rindler frame, and integrate over the spacetime coordinates before over the momenta. In this way, the divergent problem previously encountered is avoided. We show that entanglement between the detectors can be created by coupling with the field, and is divergent only in the limit that the two detectors coincide in spacetime. For linear coupling, entanglement increases monotonically with the decrease of acceleration, as previously found, while for quadratic coupling, entanglement behaves non-monotonically, as noted for the first time.

Quantum flux operators for Carrollian diffeomorphism in general dimensions

We construct Carrollian scalar field theories in general dimensions, mainly focusing on the boundaries of Minkowski and Rindler spacetime, whose quantum flux operators form a faithful representation of Carrollian diffeomorphism up to a central charge, respectively. At future/past null infinity, the fluxes are physically observable and encode rich information of the radiation. The central charge may be regularized to be finite by the spectral zeta function or heat kernel method on the unit sphere. For the theory at the Rindler horizon, the effective central charge is proportional to the area of the bifurcation surface after regularization. Moreover, the zero mode of supertranslation is identified as the modular Hamiltonian, linking Carrollian diffeomorphism to quantum information theory. Our results may hold for general null hypersurfaces and provide new insight in the study of the Carrollian field theory, asymptotic symmetry group and entanglement entropy.

Using nanokelvin quantum thermometry to detect timelike Unruh effect in a Bose–Einstein condensate

It is found that the Unruh effect can not only arise out of the entanglement between two sets of modes spanning the left and right Rindler wedges, but also between modes spanning the future and past light cones. Furthermore, an inertial Unruh–DeWitt detector along a spacetime trajectory in one of these cones may exhibit the same thermal response to the vacuum as that of an accelerated detector confined in the Rindler wedge. This feature thus could be an alternative candidate to verify the “Unruh effect”, termed as the timelike Unruh effect correspondingly. In this paper we propose to detect the timelike Unruh effect by using an impurity immersed in a Bose–Einstein condensate (BEC). The impurity acts as the detector which interacts with the density fluctuations in the condensate, working as an effective quantum field. Following the paradigm of the emerging field of quantum thermometry, we combine quantum parameter estimation theory with the theory of open quantum systems to realize a nondemolition Unruh temperature measurement in the nanokelvin (nK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{nK}$$\\end{document}) regime. Our results demonstrate that the timelike Unruh effect can be probed using a stationary two-level impurity with time-dependent energy gap immersed in a BEC within current technologies.

Statistical properties of the one-dimensional Dirac oscillator in Rindler space–time

Statistical properties of the one-dimensional Dirac oscillator in Rindler space–time