Space-time Symmetries Research Articles

The Symmetry of Primitive Root in Number Theory

Symmetry refers to the exact same characteristics on both sides of an object or figure on a central axis or plane. In science and daily life, symmetrical things give people a sense of dignity and contain the beauty of balance and stability. It includes concepts such as space symmetry, time symmetry and group theory. In this article, the author explores an interesting phenomenon of primitive roots in number theory–symmetry, by symmetry, all primitive roots can be quickly solved.

Where Charged Sectors are Localizable: A Viewpoint from Covariant Cohomology

Given a Haag–Kastler net on a globally hyperbolic spacetime, one can consider a family of regions where quantum charges are supposed to be localized. Assuming that the net fulfils certain minimal properties (factoriality of the global observable algebra and relative Haag duality), we give a geometric criterion that the given family must fulfil to have a superselection structure with charges localized on its regions. Our criterion is fulfilled by all the families used in the theory of sectors (double cones, spacelike cones, diamonds, hypercones). In order to take account of eventual spacetime symmetries, our superselection structures are constructed in terms of covariant charge transporters, a novel cohomological approach generalizing that introduced by J. E. Roberts. In the case of hypercones, with the forward light cone as an ambient spacetime, we obtain a superselection structure with Bose–Fermi parastatistics and particle-antiparticle conjugation. It could constitute a candidate for a different description of the charged sectors introduced by Buchholz and Roberts for theories including massless particles.

The Algebra and Calculus of Stochastically Perturbed Spacetime with Classical and Quantum Applications

We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents.

Spontaneous and Explicit Spacetime Symmetry Breaking in Einstein–Cartan Theory with Background Fields

Explicit and spontaneous breaking of spacetime symmetry under diffeomorphisms, local translations, and local Lorentz transformations due to the presence of fixed background fields is examined in Einstein–Cartan theory. In particular, the roles of torsion and violation of local translation invariance are highlighted. The nature of the types of background fields that can arise and how they cause spacetime symmetry breaking is discussed. With explicit breaking, potential no-go results are known to exist, which if not evaded lead to inconsistencies between the Bianchi identities, Noether identities, and the equations of motion. These are examined in detail, and the effects of nondynamical backgrounds and explicit breaking on the energy–momentum tensor when torsion is present are discussed as well. Examples illustrating various features of both explicit and spontaneous breaking of local translations are presented and compared to the case of diffeomorphism breaking.

Hyperbolic Rotation with Euclidean Angle Illuminates Space Time Spinors

In this article it will be shown that by introducing a hyperbolic rotation as function of a Euclidean rotation angle, all possible Lorentz group spacetime rotations can be performed by Euclidean rotation parameters. This unification in Lorentz group rotations parameters (all Euclidean) allows for the calculation of the volume of a spacetime region bounded by the light cone of a past event, i.e., a causality-volume with the shape of a spacetime three-sphere. As will be calculated, the surface area of the spacetime three-sphere is the volume of our 3D space. Additionally, the unification in Lorentz group rotation parameters allows for the derivation of a Dirac spinor purely from the spacetime three-sphere symmetries. This derivation from the spacetime symmetries of a geometrical object yields the same result as solving the Dirac equation with quantum mechanical eigenvalue eigenvector complex matrix calculations.

Floquet gap-dependent topological classifications from color-decorated frequency lattices with space-time symmetries

Floquet gap-dependent topological classifications from color-decorated frequency lattices with space-time symmetries

Projective spacetime symmetry of spacetime crystals

Wigner’s seminal work on the Poincaré group revealed one of the fundamental principles of quantum theory: symmetry groups are projectively represented. The condensed-matter counterparts of the Poincaré group could be the spacetime groups of periodically driven crystals or spacetime crystals featuring spacetime periodicity. In this study, we establish the general theory of projective spacetime symmetry algebras of spacetime crystals and reveal their intrinsic connections to gauge structures. As important applications, we exhaustively classify (1,1)D projective symmetry algebras and systematically construct spacetime lattice models for them all. Additionally, we present three consequences of projective spacetime symmetry that surpass ordinary theory: the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows. Our work provides both theoretical and experimental foundations to explore novel physics protected by projective spacetime symmetry of spacetime crystals.

Fermions with SU(1, n) spacetime symmetry

We construct theories of free fermions in (2n − 1)-dimensions with SU(1, n) spacetime symmetry from the null reduction of fermions on a 2n-dimensional Ω-deformed Minkowski background for n = 2 and n = 3. These play a role in the 5d SU(1, 3)-invariant theories that are conjectured to offer a full description of certain 6d superconformal field theories. We find the (2n − 1)-dimensional manifestation of the supersymmetry of a free 2n-dimensional boson-fermion system, which we use to fix the fermion two-point functions. It is then shown that the full 2n-dimensional two-point function can be recovered through resummation. Limits of the theories are considered, and it is observed that both Galilean and Carrollian field theories appear in different regimes. We confirm that the correlation functions obey the SU(1, n) Ward identities and the representations of the fermions under this group are discussed.

Possible ambient kinematics

In a seminal paper, Bacry and Lévy–Leblond classified kinematical algebras, a class of Lie algebras encoding the symmetries of spacetime. Homogeneous spacetimes (infinitesimally, Klein pairs) associated with these possible kinematics can be partitioned into four families—riemannian, lorentzian, galilean, and carrollian—based on the type of invariant metric structure they admit. In this work, we classify possible ambient kinematics—defined as extensions of kinematical algebras by a scalar ideal—as well as their associated Klein pairs. Kinematical Klein pairs arising as quotient space along the extra scalar ideal are said to admit a lift into the corresponding ambient Klein pair. While all non-galilean Klein pairs admit a unique—trivial and torsionfree—higher-dimensional lift, galilean Klein pairs are constructively shown to admit lifts into two distinct families of ambient Klein pairs. The first family includes the bargmann algebra as well as its curved/torsional avatars while the second family is novel and generically allows lifts into torsional ambient spaces. We further comment on the relation between these two families and the maximally symmetric family of leibnizian Klein pairs.

The Tully-Fisher law and dark matter effects derived via modified symmetries

In any physical system, when we move from short to large scales, new spacetime symmetries emerge which help us to simplify the dynamics of the system. In this letter we demonstrate that certain variations on the symmetries of general relativity at large scales generate the effects equivalent to dark matter ones. In particular, we reproduce the Tully-Fisher law, consistent with the predictions proposed by MOND. Additionally, we demonstrate that the dark matter effects derived in this way are consistent with the predictions suggested by MOND, without modifying gravity.

On the rigidity of cosmological space-times

In this paper, we analyse a family of geometrically well-behaved cosmological space-times (V^{n+1},g), which are foliated by intrinsically isotropic space-like hypersurfaces {M_t}_{tin mathbb {R}}, which are orthogonal to a family of co-moving observers defined by a global time-like vector field U. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity spaces (M_t,g_t) associated with U can have as sectional curvature a sign-changing function k(t). Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of (M_t,g_t) for each t. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular, SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

Space-time reflection symmetry in the Jones formalism in optics

Space-time reflection symmetry in the Jones formalism in optics

Bicrossproduct vs. twist quantum symmetries in noncommutative geometries: the case of ϱ-Minkowski

We discuss the quantum Poincaré symmetries of the ϱ-Minkowski spacetime, a space characterised by an angular form of noncommutativity. We show that it is possible to give them both a bicrossproduct and a Drinfel’d twist structure. We also obtain a new noncommutative ⋆-product, which is cyclic with respect to the standard integral measure.

Energy conditions for non-timelike thin shells

We study energy conditions for non-timelike thin shells in arbitrary dimensions. It is shown that the induced energy–momentum tensor on a shell Σ is of the Hawking–Ellis type I if Σ is spacelike and either of type I, II, or III if Σ is null. Then, we derive simple equivalent representations of the standard energy conditions for . In particular, on a spacelike shell or on a null shell with non-vanishing surface current, inevitably violates the dominant energy condition. If the surface pressure on the null shell is vanishing in addition, is of type III and violates all the standard energy conditions. Those fully general results are obtained without imposing a spacetime symmetry and can be used in any theory of gravity. Lastly, several applications of the main results are presented in general relativity in four dimensions.

Boson star with parity-odd symmetry in wormhole spacetime

In this paper, we revisit the model of bosonic matter in the form of a free complex scalar field with a nontrivial wormhole spacetime topology supported by a free phantom field. We obtain a new type of boson star with wormhole solutions, in which the complex scalar field possess full parity-odd symmetry with respect to the two asymptotically flat spacetime regions. When the size of the throat is small, the behavior of boson stars with wormhole approaches that of boson stars. When the size of the throat is intermediate, the typical spiraling dependence of the mass and the particle number on the frequency of the boson stars is replaced by a loop structure. However, as the size becomes relatively large, the loop structure will also disappear. In particular, The complex scalar field could form two boson stars with opposite phase differences with respect to the two spacetime regions in the limit of vanishing throat size. We analyze the properties of this new type of boson stars with wormhole and further show that the wormhole spacetime geometry.

A Hierarchy of Spacetime Symmetries: Holes to Heraclitus

A Hierarchy of Spacetime Symmetries: Holes to Heraclitus

Analytical solutions of spherical structures with relativistic corrections

This paper analyzes the characteristic of a non-static sphere along with anisotropic fluid distribution in the background of modified f(mathcal {G}) theory. Conformal Killing vector is a productive constraint for computing reliable results for modified field equations. The occurrence of conformal Killing vector indicates the existence of symmetries in spacetime and it permits us to choose the coordinates that reduce the number of independent variables. Subsequently, for different conformal Killing vector choices, we obtain several types of precise analytical solutions for both non-dissipative and dissipative systems. We compute the matching conditions in the context of f(mathcal {G}) gravity. In addition to this, we apply specific constraints to the matching conditions in an attempt to determine the significant results. Further, we proceed our investigation by utilizing quasi-homologous condition and vanishing complexity factor condition. Finally, we summarize all the important results which may help to understand the properties of astrophysical objects.

A description of classical and quantum cosmology for a single scalar field torsion gravity

In the background of homogeneous and isotropic flat FLRW spacetime, both classical and quantum cosmologies have been studied for teleparallel dark energy (DE) model. Using Noether symmetry analysis, not only the symmetry vector but also the coupling function in the Lagrangian and the potential of the scalar field have been determined. Also symmetry analysis identifies a cyclic variable in the Lagrangian along the symmetry vector and as a result the Lagrangian simplifies to a great extent so that classical solution is obtained. Subsequently, in quantum cosmology Wheeler–DeWitt (WD) equation has been constructed and the quantum version of the conserved momenta corresponding to Noether symmetry identifies the periodic part of the wave function of the universe and as a result the WD equation becomes solvable. Finally, quantum description shows finite nonzero probability at the classical big bang singularity.

Stiefel-Whitney topological charges in a three-dimensional acoustic nodal-line crystal

Band topology of materials describes the extent Bloch wavefunctions are twisted in momentum space. Such descriptions rely on a set of topological invariants, generally referred to as topological charges, which form a characteristic class in the mathematical structure of fiber bundles associated with the Bloch wavefunctions. For example, the celebrated Chern number and its variants belong to the Chern class, characterizing topological charges for complex Bloch wavefunctions. Nevertheless, under the space-time inversion symmetry, Bloch wavefunctions can be purely real in the entire momentum space; consequently, their topological classification does not fall into the Chern class, but requires another characteristic class known as the Stiefel-Whitney class. Here, in a three-dimensional acoustic crystal, we demonstrate a topological nodal-line semimetal that is characterized by a doublet of topological charges, the first and second Stiefel-Whitney numbers, simultaneously. Such a doubly charged nodal line gives rise to a doubled bulk-boundary correspondence—while the first Stiefel-Whitney number induces ordinary drumhead states of the nodal line, the second Stiefel-Whitney number supports hinge Fermi arc states at odd inversion-related pairs of hinges. These results experimentally validate the two Stiefel-Whitney topological charges and demonstrate their unique bulk-boundary correspondence in a physical system.

Isotropic compact stellar model in Rastall's gravitational theory

Is Rastall's theory of gravity equivalent to Einstein's general relativity theory? This question has sparked a significant debate, prompting researchers to delve into the topic. To investigate further, we apply Rastall's theory's field equation to a spacetime characterized by spherical symmetry. This leads us to encounter a system of non-linear differential equations that is overdetermined. To address this, we make assumptions about the specific form of the metric potential's temporal component, denoted as gtt. Additionally, we impose constraints to eliminate the anisotropic condition, resulting in a vanishing effect. These steps allow us to determine the form of grr and ultimately achieve an isotropic spacetime. Furthermore, our investigation focuses on the potential of obtaining a set of parameters that align with the observed behavior of pulsars. To achieve this, we employ junction conditions to match the interior spacetime with the exterior Schwarzschild configuration, thereby constraining the model's relevant constants. Subsequently, we employ the pulsar SAXJ1748.9−2021, characterized by a measured mass of M=1.81±0.31,M⊙ and a radius of R=11.7±1.7 km, to numerically explore the physical properties of the model. Stability is assessed using the Tolman-Oppenheimer-Volkoff equation and the adiabatic index. Our findings suggest that Rastall's parameter, a key distinction of Rastall's theory from Einstein's general relativity, can play a crucial role in forming a realistic, compact object consistent with observational data. Furthermore, we verify the model's validity by comparing it with various observed masses and radii of different pulsars, ensuring a satisfactory fit between the model proposed in this study and the observed data.