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This study introduces a novel approach to achieving nonreciprocal transmission by implementing time modulation to the bianisotropic metasurface. For the first time, we present the analytical solution of the excitation of anti-symmetric surface waves on penetrable metasurfaces depending on the excitation direction. Exploiting this finding, we numerically demonstrate asymmetric control of the transmission coefficient under a fast-time scale by employing solely time modulation. This approach lowers the complexity of the modulation scheme and implementation encountered in the space-time modulation technique. We develop and simulate a 3D unit cell model in the microwave domain, which forms a surface cavity that incorporates time-varying capacitors. The impedance transfer matrix method and harmonic balance numerical solutions are applied to the retrieved equivalent circuit for the numerical simulations. The results reveal optimized phase-coherent and incoherent nonreciprocal transmission at the significant isolation level (≥ 40 dB) for forward and backward transmissions. We discuss the consistency and discrepancies between numerical methods and consider the impact of the losses and nonlinearity on the metastructure performance. [ABSTRACT FROM AUTHOR]

The article "Minkowski Diagrams Aid Understanding of Special Relativity" discusses the use of Minkowski space-time to visually represent special relativity, providing students with an alternative perspective to equations and thought experiments. It explains how Minkowski's work connects space and time, allowing for the visualization of events in both dimensions. The article also delves into the twin paradox and length contraction, showcasing how Minkowski spacetime can help resolve complex physics problems. Overall, the paper aims to encourage educators to incorporate visual approaches like Minkowski diagrams when teaching special relativity. [Extracted from the article]

Recent advancements in optical, terahertz, and microwave systems have unveiled non-transverse optical toroidal pulses characterized by skyrmionic topologies, fractal-like singularities, space-time nonseparability, and anapole-exciting ability. Despite this, the longitudinally polarized fields of canonical toroidal pulses notably lag behind their transverse counterparts in magnitude. Interestingly, although mushroom-cloud-like toroidal vortices with strong longitudinal fields are common in nature, they remain unexplored in the realm of electromagnetics. Here, we present strongly longitudinally polarized toroidal pulses (SLPTPs), which boast a longitudinal component amplitude exceeding that of the transverse component by over tenfold. This unique polarization property endows SLPTPs with robust propagation characteristics, showcasing nondiffracting behavior. The propagation-invariant strongly longitudinally polarized field holds promise for pioneering light–matter interactions, far-field superresolution microscopy, and high-capacity wireless communication utilizing three polarizations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of viscosity solutions on the past set (future set) of the Cauchy surface. Furthermore, when the time orientation of viscosity solution is consistent, the uniqueness and stability of viscosity solutions are also obtained. [ABSTRACT FROM AUTHOR]

For the Stückelberg-modified massive Abelian 3-form theory in any arbitrary D-dimension of spacetime, we show that its classical gauge symmetry transformations are generated by the first-class constraints. We establish that the Noether conserved charge (corresponding to the local gauge symmetry transformations) is same as the standard form of the generator for the underlying local gauge symmetry transformations (expressed in terms of the first-class constraints). We promote these classical local, continuous and infinitesimal gauge symmetry transformations to their quantum counterparts Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations which are respected by the coupled (but equivalent) Lagrangian densities. We derive the conserved (anti-)BRST charges by exploiting the theoretical potential of Noether's theorem. However, these charges turn out to be non-nilpotent. Some of the highlights of our present investigation are (i) the derivation of the off-shell nilpotent versions of the (anti-)BRST charges from the standard non-nilpotent Noether conserved (anti-)BRST charges, (ii) the appearance of the operator forms of the first-class constraints at the quantum level through the physicality criteria with respect to the nilpotent versions of the (anti-)BRST charges, and (iii) the deduction of the Curci–Ferrari-type restrictions from the straightforward equality of the coupled (anti-)BRST invariant Lagrangian densities as well as from the requirement of the absolute anticommutativity of the off-shell nilpotent versions of the conserved (anti-)BRST charges. [ABSTRACT FROM AUTHOR]

This article explores the complexities surrounding the definition of weight in physics education. It argues that current definitions of weight in pedagogical and metrological literature are unsatisfactory. The article examines weight as a local phenomenon that varies with the location and motion of the reference frame, as well as the forces and interactions involved. It discusses different definitions of weight, the concept of weightlessness, and emphasizes that weight is not an inherent characteristic of an object but depends on nongravitational forces. The article proposes a comprehensive definition of weight that considers all gravitational forces and the local centrifugal force due to celestial object rotation. Examples of weightlessness are provided, such as objects in free fall and astronauts in space. The author, Eugene Hecht, is an emeritus professor of physics with a background in optics, quantum mechanics, and college physics. [Extracted from the article]

In this paper, we prove the existence of global in time small data solutions of semilinear Klein–Gordon equations in space-time with a static Schwarzschild radius in the expanding universe. [ABSTRACT FROM AUTHOR]

The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the Kerr spacetime. Despite the presence of closed timelike curves below the inner horizon, we prove that the timelike geodesics cannot be closed in the Kerr-star spacetime. Since the existence of closed null geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)], this result shows the absence of closed causal geodesics in the Kerr-star spacetime. [ABSTRACT FROM AUTHOR]

We propose an all-optical approach to generating space–time wave packets in a multimode slab waveguide via the multilevel interband stimulated Brillouin scattering process. Two pump sources and a single-mode signal are fed into the waveguide. The pumps generate a single-mode acoustic wave through the electrostrictive process. The acoustic wave then induces an indirect interband photonic transition from the signal wave, resulting in a light bullet, that is, a space–time wave packet that does not change its spatial and temporal shape as it propagates through the waveguide. [ABSTRACT FROM AUTHOR]

We determine the conjugacy classes of the Poincaré group ISO+(n, 1) and apply this to classify the stationary trajectories of Minkowski spacetimes in terms of timelike Killing vectors. Stationary trajectories are the orbits of timelike Killing vectors and, equivalently, the solutions to Frenet–Serret equations with constant curvature coefficients. We extend the 3 + 1 Minkowski spacetime Frenet–Serret equations due to Letaw to Minkowski spacetimes of arbitrary dimension. We present the explicit families of stationary trajectories in 4 + 1 Minkowski spacetime. [ABSTRACT FROM AUTHOR]

We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes. [ABSTRACT FROM AUTHOR]

Here we prove a global gauge-invariant radiation estimates for the perturbations of the 3 + 1 dimensional Minkowski spacetime in the presence of Yang–Mills sources. In particular, we obtain a novel gauge invariant estimate for the Yang–Mills fields coupled to gravity in a double null framework in the Causal complement of a compact set of a Cauchy slice. A consequence of our result is the global exterior stability of the Minkowski space under coupled Yang–Mills perturbations. A special structure present both in the null Bianchi equations and the null Yang–Mills equations is utilized crucially to obtain the dispersive estimates necessary to conclude the global existence property. Direct use of Bel-Robinson and Yang–Mills stress-energy tensor to obtain the energy estimates is avoided in favor of weighted integration by parts taking advantage of the manifestly symmetric hyperbolic characteristics of null Bianchi and null Yang–Mills equations. Our result holds for any compact semi-simple gauge group. This is the first stability result of Minkowski space including a non-linear source. [ABSTRACT FROM AUTHOR]

We propose a systematic generating procedure to construct free Lagrangians for massive, massless and partially massless, totally-symmetric tensor fields on AdSd+1 starting from the Becchi–Rouet–Stora–Tyutin (BRST) Lagrangian description of massless fields in the flat ambient space R d , 2 . A novelty is that the Lagrangian is described by a d + 1 form on R d , 2 whose pullback to AdSd+1 gives the genuine Lagrangian defined on anti de Sitter spacetime. Our derivation uses the triplet formulation originating from the first-quantized BRST approach, where the action principle is determined by the BRST operator and the inner product of a first-quantised system. In this way we build, in a manifestly so(2, d)-covariant manner, a unifying action principle for the three types of fields mentioned above. In particular, our derivation justifies the form of some actions proposed earlier for massive and massless fields on (anti)-de Sitter. We also give a general setup for ambient Lagrangians in terms of the respective jet-bundles and variational bi-complexes. In particular we introduce a suitable ambient-space Euler–Lagrange differential which allows one to derive the equations of motion ambiently, i.e., without the need to explicitly derive the respective spacetime Lagrangian. [ABSTRACT FROM AUTHOR]

Rotor–stator interaction (RSI) is an inherent phenomenon in multi-row turbomachinery. Unsteady reduced-order methods, such as the harmonic balance (HB) method and the space-time gradient (STG) method, have been proposed to capture RSI with fewer computational resources compared to fully unsteady simulation. In this study, the steady mixing-plane method, the HB method, and the STG method are implemented into the open-source external computational fluid laboratory three-dimensional (CFL3D) flow solver to gain the ability to predict turbomachinery flows based on solving Reynolds-averaged Navier–Stokes equations. Additionally, a rotation interpolation approach for adjacent blades is implemented for the unsteady multi-row turbomachinery simulation. For the HB method, the phase-lag periodic conditions and the temporal interpolation approach between two adjacent blade rows are integrated into CFL3D. Then, the steady mixing-plane method, the HB method, the STG method, and the fully unsteady simulation method are conducted on a quasi-three-dimensional radial slice and a three-dimensional geometry of the National Aeronautics and Space Administration Stage-35 compressor. Both the transient and time-averaged flowfield predicted by the reduced-order methods are compared with the unsteady simulations. Results indicate that the STG method and the HB method can accurately simulate the unsteady flow with better predictions of RSI impact. For the HB method, accurate prediction of transient unsteady flow requires a minimum of seven harmonics, whereas the time-averaged flow requires only five harmonics. Additionally, a quantitative assessment of computational speed is conducted, revealing that the HB method with seven harmonics achieved a speed 28 times faster than the fully unsteady simulation. [ABSTRACT FROM AUTHOR]

This article presents characterizations of warped product manifolds based on the flatness and symmetry of the conharmonic curvature tensor. It is proved that when a warped product manifold is conharmonically flat, both the base and fiber manifolds exhibit constant sectional curvatures. In addition, the specific forms of the conharmonic curvature tensor are derived for both the base and fiber manifolds. It is demonstrated that in a conharmonically symmetric warped product manifold, the fiber manifold has a constant sectional curvature, while the base manifold is both Cartan-symmetric and conharmonically symmetric. In this scenario, the form of the conharmonic curvature tensor on the fiber manifold is determined. We characterize the generalized Robertson–Walker (GRW) space-time through the flatness and the symmetry of the conharmonic curvature tensor. It is shown that a conharmonically flat (symmetric) GRW space-time is a perfect fluid. Finally, a conharmonically flat standard static space-time is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we explore the new exact soliton solutions of the truncated M-fractional nonlinear (1 + 1)-dimensional van der Waals equation by applying the exp a function method, extended (G ′ / G) -expansion method, and modified simplest equation method. The concerned equation is a challenging problem in the modeling of molecules and materials. Noncovalent van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science, and physics. The results obtained are verified and represented by two-dimensional, three-dimensional, and contour graphs. These results are newer than the existing results in the literature due to the use of fractional derivative. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter, and other areas of van der Waals equation applications. Therefore, the obtained solutions are valuable for future studies of this model. [ABSTRACT FROM AUTHOR]

Three kinds of Darboux transformations are constructed by means of the loop group method for the complex reverse space-time (RST) nonlocal modified Korteweg–de Vries equation, which are different from that for the P T symmetric (reverse space) and reverse time nonlocal models. The N-periodic, the N-soliton, and the N-breather-like solutions, which are, respectively, associated with real, pure imaginary, and general complex eigenvalues on a finite background are presented in compact determinant forms. Some typical localized wave patterns such as the doubly periodic lattice-like wave, the asymmetric double-peak breather-like wave, and the solitons on singly or doubly periodic waves are graphically shown. The essential differences and links between the complex RST nonlocal equations and their local or P T symmetric nonlocal counterparts are revealed through these explicit solutions and the solving process. [ABSTRACT FROM AUTHOR]

Among the several methods to compute the perihelion precession for bounded orbits in Schwarzschild spacetime, the simplest is to ignore a term in the equations of motion. This is currently justified under the assumption that the eccentricity of the orbit is small. For cases such as Mercury in our solar system, whose eccentricity is not small, this method seems not to be applicable. Yet it gives the right result, the reason being that the term that has been excluded, although responsible for first order—in the ratio of the Schwarzschild radius over the radial coordinate—corrections of the orbit, only produces completely negligible higher order corrections for the perihelion precession. We show this result by two different procedures. We claim, therefore, that as long as the aim of the computation is the perihelion precession, one can safely drop that term regardless of the magnitude of the eccentricity. [ABSTRACT FROM AUTHOR]

The Yang model is an example of noncommutative geometry on a background spacetime of constant curvature. We discuss the Hermitian realizations of its associated algebra on phase space in a perturbative expansion up to sixth order. We also discuss its realizations on extended phase spaces, that include additional tensorial and/or vectorial degrees or freedom. [ABSTRACT FROM AUTHOR]

Angular momentum is traditionally taught as a (pseudo)vector quantity, tied closely to the cross product. This approach is familiar to experts but challenging for students, and full of subtleties. Here, we present an alternative pedagogical approach: angular momentum is described using bivectors which can be visualized as "tiles" with area and orientation, and whose components form an antisymmetric matrix. Although bivectors have historically been studied in specialized contexts like spacetime classification or geometric algebra, they are no more complicated to understand than cross products. The bivector language provides a more fundamental definition for rotational physics and opens the door to understanding rotations in relativity and in extra dimensions. [ABSTRACT FROM AUTHOR]

We consider compressible Navier-Stokes equations with hyperbolic heat conduction in R 3 . A space-time description of the classical solution is given when the initial perturbation is suitable small. The result implies that all of the unknowns obey the generalized Huygens' principle as the classical compressible Navier-Stokes equations in Liu and Wang [Commun. Math. Phys. 196, 145–173 (1998)]. Additionally, we show that the decay of the flux q is faster than the other unknowns. [ABSTRACT FROM AUTHOR]

Metric components of potentials of admissible electromagnetic fields in spaces with simply transitive motion group G4 are found. The components of vector tetrads corresponding to the components of the metric tensors found by Petrov are given. The results obtained complement the coordinate-free classification given in Magazev et al. [Theor. Math. Phys. 156, 1127–1141 (2008)]. Previously, admissible electromagnetic fields were found for the case when three- and four-parameter groups of motions act on hypersurfaces of spacetime. Thus, non-equivalent sets of potentials for all electromagnetic fields that admit three- and four-parameter groups of motions are known now. [ABSTRACT FROM AUTHOR]

In this paper, we derive the explicit formula for higher cup products on hypercubic lattices based on the recently developed geometrical interpretation on the simplicial complexes. We illustrate how this formalism can elucidate lattice constructions on hypercubic lattices for various models and derive them from spacetime actions. In particular, we demonstrate explicitly that the (3 + 1)D SPT S = 1 2 ∫ w 2 2 + w 1 4 (where w1 and w2 are the first and second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model constructed on the cubic lattice. Other examples include the double-semion model and also the "fermionic" toric code in arbitrary dimensions on hypercubic lattices. In addition, we extend previous constructions of exact boson-fermion dualities and the Gu-Wen Grassmann integral to arbitrary dimensions. Another result that may be of independent interest is a derivation of a cochain-level action for the generalized double-semion model, reproducing a recently derived action on the cohomology level. [ABSTRACT FROM AUTHOR]

Computational models of reaction–diffusion systems involving low copy numbers or strongly heterogeneous molecular spatial distributions, such as those frequently found in cellular signaling pathways, require approaches that account for the stochastic dynamics of individual particles, as opposed to approaches representing them through their average concentrations. Efforts to remedy the high computational cost associated with particle-based stochastic approaches by taking advantage of Green's functions are hampered by the need to draw random numbers from complicated, and therefore costly, non-standard probability distributions to update particle positions. Here, we introduce an approach that permits the reconstruction of entire molecular trajectories, including bimolecular encounters, in retrospect, after a simulated time step, while avoiding inefficient draws from non-standard distributions. This means that highly accurate stochastic simulations can be performed for system sizes that would be prohibitively costly to simulate with conventional Green's function based methods. The algorithm applies equally well to one, two, and three dimensional systems and can be readily extended to include deterministic forces specified by an interaction potential, such as the Coulomb potential. [ABSTRACT FROM AUTHOR]

The article "Do Objects Move Through Space-Time at the Speed of Light? No." challenges the common misconception that objects move through space-time at the speed of light. It delves into the concept of relativistic 4-vectors and 4-velocity in special relativity to explain why this claim is inaccurate. The author, William G. Delinger, refutes the idea that objects move through space-time at the speed of light, emphasizing the importance of understanding the nuances of relativistic kinematics. The article provides a detailed analysis of the 4-velocity and its relation to the speed of light, ultimately concluding that objects do not move through space-time at the speed of light. [Extracted from the article]

Perturbations of fluid media can give rise to non-equilibrium dynamics, which may, in turn, cause motion of immersed inclusions or tracer particles. We consider perturbations ("activations") that are local in space and time, of a fluid density which is conserved, and study the resulting diffusiophoretic phenomena that emerge at a large distance. Specifically, we consider cases where the perturbations propagate diffusively, providing examples from passive and active matter for which this is expected to be the case. Activations can, for instance, be realized by sudden and local changes in interaction potentials of the medium or by local changes in its activity. Various analytical results are provided for the case of confinement by two parallel walls. We investigate the possibility of extracting work from inclusions, which are moving through the activated fluid. Furthermore, we show that a time-dependent density profile, created via suitable activation protocols, allows for the conveyance of inclusions along controlled and stable trajectories. In contrast, in states with a steady density, inclusions cannot be held at stable positions, reminiscent of Earnshaw's theorem of electrostatics. We expect these findings to be applicable in a range of experimental systems. The phenomena described here are argued to be distinct from other forms of phoresis such as thermophoresis. [ABSTRACT FROM AUTHOR]

Hypersurfaces embedded in conformal manifolds appear frequently as boundary data in boundary-value problems in cosmology and string theory. Viewed as the non-null conformal infinity of a spacetime, we consider hypersurfaces embedded in a Riemannian (or Lorentzian) conformal manifold. We construct a finite and minimal family of hypersurface tensors—the curvatures intrinsic to the hypersurface and the so-called "conformal fundamental forms"—that can be used to construct natural conformal invariants of the hypersurface embedding up to a fixed order in hypersurface-orthogonal derivatives of the bulk metric. We thus show that these conformal fundamental forms capture the extrinsic embedding data of a conformal infinity in a spacetime. [ABSTRACT FROM AUTHOR]

Light propagation through diffusive media can be described by the diffusion equation in a space–time domain. Furthermore, fluorescence can be described by a system of coupled diffusion equations. This paper analyzes time-domain measurements. In particular, the temporal point-spread function is measured at the boundary of a diffusive medium. Moreover, the temporal profile of fluorescence is considered. In both cases, we refer to the maximum temporal position of measured light as the peak time. In this paper, we provide proofs of the existence and uniqueness of the peak time and give explicit expressions of the peak time. The relationship between the peak time and the object position in a medium is clarified. [ABSTRACT FROM AUTHOR]

Newman–Penrose (NP) constants of massless spin-0 fields propagating in Minkowski spacetime are computed close to spatial and null infinity by means of Friedrich's i0-cylinder. Assuming a certain regularity condition on the initial data ensuring that the field extends analytically to critical sets, it is shown that the NP constants at future I + and past null infinity I − are independent of each other. In other words, the classical NP constants at I ± stem from different parts of the initial data given on a Cauchy hypersurface. In contrast, it is shown that, using a slight generalization of the classical NP constants, the associated quantities (i0-cylinder NP constants) do not require the regularity condition being satisfied and give rise to conserved quantities at I ± that are determined by the same piece of initial data, which, in turn, correspond to the terms controlling the regularity of the field. Additionally, it is shown how the conservation laws associated with the NP constants can be exploited to construct, in flat space, heuristic asymptotic-system expansions, which are sensitive to the logarithmic terms at the critical sets. [ABSTRACT FROM AUTHOR]

In Schwarzschild spacetime, time-like geodesic equations, which define particle orbits, have a well-known formulation as a dynamical system in coordinates adapted to the time-like hypersurface containing the geodesic. For equatorial geodesics, the resulting dynamical system is shown to possess a conserved angular quantity and two conserved temporal quantities, whose properties and physical meaning are analogs of the conserved Laplace–Runge–Lenz vector, and its variant known as Hamilton's vector, in Newtonian gravity. When a particle orbit is projected into the spatial equatorial plane, the angular quantity yields the coordinate angle at which the orbit has either a turning point (where the radial velocity is zero) or a centripetal point (where the radial acceleration is zero). This is the same property as the angle of the respective Laplace–Runge–Lenz and Hamilton vectors in the plane of motion in Newtonian gravity. The temporal quantities yield the coordinate time and the proper time at which those points are reached on the orbit. In general, for orbits that have a single turning point, the three quantities are globally constant, and for orbits that possess more than one turning point, the temporal quantities are just locally constant as they jump at every successive turning point, while the angular quantity similarly jumps only if an orbit is precessing. This is analogous to the properties of a generalized Laplace–Runge–Lenz vector and generalized Hamilton vector which are known to exist for precessing orbits in post-Newtonian gravity. The angular conserved quantity is used to define a direct analog of these vectors at spatial infinity. [ABSTRACT FROM AUTHOR]

In this work, we describe the partial collapse of a compact object and the emission of spacetime waves as a result of back-reaction effects. As a source mass term, we propose a non-smooth continuous function that describes a mass-loss, and we then obtain the solution of such setting. We present three distinct examples of the evolution of the norm |Rnl(t, r * )| in terms of t, and four different results are shown for the parameter l = 1, 2, 5, 10; here, r * is the fixed radius of an observer outside the compact object. In all cases, the decay behavior is actually present at t ≫ 1 and becomes more evident for larger l. In addition, for the results that have smaller l's, their amplitudes are larger when the asymptotic character of |Rnl(t, r * )| clearly appears. Finally, the farther away an observer is set, the fewer oscillations are perceived; however, from our particular fixed set of parameters, the best spot to observe the wiggles of the emitted spacetime waves is close to r * ≃ α. [ABSTRACT FROM AUTHOR]

We consider the existence and Hölder continuity conditions for the higher-order derivative of self-intersection local time for u(t, x), where u (t , x) = u 1 (t , x) , ... , u d (t , x) is the solution to a system of non-linear stochastic heat equations driven by a d-dimensional space-time white noise. Moreover, we study the cases of intersection local time and collision local time. [ABSTRACT FROM AUTHOR]

This paper provides a new geometric framework to describe non-conservative field theories with explicit dependence on the space–time coordinates by combining the k-cosymplectic and k-contact formulations. This geometric framework, the k-cocontact geometry, permits the development of Hamiltonian and Lagrangian formalisms for these field theories. We also compare this new formulation in the autonomous case with the previous k-contact formalism. To illustrate the theory, we study the nonlinear damped wave equation with external time-dependent forcing. [ABSTRACT FROM AUTHOR]

We analyze a massive spinor field satisfying the Dirac equation in the universal covering space of two-dimensional anti-de Sitter space. In order to obtain well-defined dynamics for the classical field despite the lack of global-hyperbolicity of the spacetime, we impose a suitable set of boundary conditions that render the spatial component of the Dirac operator self-adjoint. Then, we find which of the solution spaces obtained by imposing the self-adjoint boundary conditions are invariant under the action of the isometry group of the spacetime manifold, namely, the universal covering group of S L (2 , R). The invariant solution spaces are then identified with unitary irreducible representations of this group using the classification given by Pukánszky [Math. Ann. 156, 96–143 (1964)]. We determine which of these correspond to invariant positive- or negative-frequency subspaces and, thus, result in a vacuum state invariant under the isometry group after canonical quantization. Additionally, we examine the invariant theories obtained from the self-adjoint boundary conditions, which result in a non-invariant vacuum state, identifying the unitary representation this state belongs to. [ABSTRACT FROM AUTHOR]

The precision of optical atomic clocks is approaching a regime where they resolve gravitational time dilation on smaller scales than their own extensions. Hence, an accurate description of quantum clocks has to take their spatial extension into account. In this article, as a first step toward a fully relativistic description of extended quantum clocks, we investigate a quantized version of Einstein's light clock fixed at a constant distance from a large massive object like the Earth. The model consists of a quantum light field in a one-dimensional cavity in Schwarzschild spacetime, where the distance between the mirrors is fixed by a rigid rod. By comparing a vertical and a horizontal clock, we propose an operational way to define the clock time when the clock resolves gravitational time dilation on scales smaller than its extension. In particular, we show that the time measured by the vertical light clock is equivalent to the proper time defined at its center. We also derive fundamental bounds on the precision of these clocks for measurements of proper time and the Schwarzschild radius. [ABSTRACT FROM AUTHOR]