Space-time Symmetries Research Articles

Improved Noether’s theorem for spacetime symmetries

We exploit an ambiguity somewhat hidden in Noether's theorem to derive systematically, for relativistic field theories, the stress-energy tensor's improvement terms that are associated with additional spacetime symmetries beyond translations. We work out explicitly the cases of Lorentz invariance, scale invariance, and full conformal invariance. The main idea is to use, directly in the translation Noether theorem, the fact that these additional symmetries can be thought of as suitably modulated translations. Compared to more standard derivations of the improvement terms, ours (1) unifies all different cases in a single framework, (2) involves no guesswork, (3) yields the desired algebraic properties (symmetry and/or tracelessness) of the stress-energy tensor off-shell, and (4) unifies the translation Noether theorem with those of the additional spacetime symmetries, yielding at the same time both the improved stress-energy tensor and the additional Noether currents.

Vertical Equilibrium Model for Accretion Flow: An Exact General Relativistic Self-consistent Analysis

We provide a complete general relativistic and fluid dynamical treatment to derive an expression for the relativistic accretion rate in curved spacetime. We exploit the Killing symmetry of stationary axisymmetric spacetime to obtain the conserved current, using which we find an integral relation between mass flux and height of the accretion disk. In comparison with the existing different models of thin- and slim-disk approaches our analysis is self-consistent and mathematically rigorous. We reviewed the previous work and surveyed the recent status in this field. Our formula is different from the existing expression in the literature. The difference arises solely due to the rotation parameter of the compact star. The effect becomes important for slim/thick disks and for rapidly spinning compact stars. Thus one needs to alter the formula for Newtonian or spherically symmetric spacetime while studying the accretion flows around a Kerr black hole. Our formula can also be used to connect the variability of the vertical structure of the accretion disk through the luminosity variation of X-ray binaries, and can thereby explain the origin of the hysteresis loop of the spectral states.

Scalar field cosmology via Noether symmetries in energy–momentum squared gravity

This article examines the analytic solutions of isotropic spacetime with the minimal coupling of scalar field and matter in the context of energy–momentum squared gravity. The scalar field includes quintessence and phantom dark energy models. We evaluate solutions of a physical system through the Noether symmetry approach that provide some viable constraints to choose the cosmological models on the basis of current observations. We consider different models of this theory to establish Noether equations, symmetry generators and corresponding conserved quantities. We compute analytic solutions and examine their graphical representation for various cosmological quantities. The behavior of these quantities is found to be consistent with current observations implying that this theory shows expanding behavior of the universe. We find that symmetry generators and their conserved parameters exist in all cases. • Solutions of isotropic spacetime with minimal coupling of scalar field and matter are studied in modified theory. • We evaluate solutions of a physical system through the Noether symmetry approach. • The behavior of physical quantities is found to be consistent with current observations. • We find that symmetry generators and their conserved parameters exist in all cases.

Yangian Symmetry in Five Dimensions.

Quantum gravity in AdS_{7}×S^{4} is dual to a six-dimensional (6D) superconformal field theory, known as the 6D (2,0) theory, which is very challenging to describe because it lacks a conventional Lagrangian description. On the other hand, certain null reductions of the 6D (2,0) theory give rise to 5D Lagrangian theories with SU(1,3) spacetime symmetry, SO(5) R symmetry, and 24 supercharges. This appears to be closely related to the superconformal group of a 3D superconformal Chern-Simons theory known as the Aharony-Bergman-Jafferis-Maldacena theory, which is believed to be integrable in the planar limit, if one exchanges the role of conformal and R symmetry. In this Letter, we construct a representation of the 5D superconformal group using 6D supertwistors and show that it admits an infinite dimensional extension known as Yangian symmetry, which opens up the possibility that these 5D theories are exactly solvable in the planar limit.

Topological classification of gapped/gap-preserving rational space-time crystal systems

The traditional systems researched in condensed matter physics always have spatial translation symmetry. However for space-time crystal systems, the spatial translation symmetry is no longer preserved and the lattice potential have space-time translation symmetry instead. We show that a rational space-time crystal system is equal to a traditional floquet system. Then, we find a way to solve the floquet equation analytically and construct an effective Hamiltonian of the rational space-time crystal system. By this effective Hamiltonian, we obtain the topological classification of gapped and gap-preserving rational space-time crystal systems. Our works reveal the correlation between space-time crystal systems and floquet systems and give a systematic method to explore the properties of rational space-time crystal systems.

S-folds and AdS3 flows from the D3-brane

We investigate supersymmetric flows in type IIB supergravity that preserve an SO(2, 2) space-time symmetry and asymptote to AdS5× S5 at both endpoints. The flows are constructed as Janus-type ℝ × AdS3 BPS domain-walls in the effective four-dimensional [SO(1, 1) × SO(6)] ⋉ ℝ12 gauged maximal supergravity describing the massless sector of type IIB supergravity compactified on S1 × S5. The compactification includes an S-duality hyperbolic twist along the S1 which, when combined with an appropriate choice of boundary conditions for the running scalars, generates special flows that develop an S-fold regime at their core, thus enhancing the space-time symmetry there to SO(2, 3). Via the AdS/CFT correspondence, the flows constructed here are conjectured to describe conformal interfaces in a circle compactification of mathcal{N} = 4 SYM4.

Phononic Stiefel-Whitney topology with corner vibrational modes in two-dimensional Xenes and ligand-functionalized derivatives

Two-dimensional (2D) Stiefel-Whitney (SW) insulator is a fragile topological state characterized by the second SW class in the presence of space-time inversion symmetry. So far, SWIs have been proposed in several electronic materials but seldom in phononic systems. Here we recognize that a large class of 2D buckled honeycomb crystals termed Xenes and their ligand-functionalized derivatives realize the nontrivial phononic SW topology. The phononic SWIs are identified by a nonzero second SW number $w_2=1$, associated with gaped edge states and robust topological corner modes. Despite the versatility of electronic topological properties in these materials, the nontrivial phononic SW topology is mainly attributed to the double band inversion between in-plane acoustic and out-of-plane optical bands with opposite parities due to the structural buckling of the honeycomb lattice. Our findings not only reveal an overlooked phononic topological property of 2D Xene-related materials, but also afford abundant readily synthesizable material candidates with simple phononic spectra for further experimental studies of phononic SW topology physics.

C−P−T fractionalization

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity $(-1)^{\rm F}$. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-$(-1)^{\rm F}$ group structures, e.g., Dirac spinor is in a projective representation of $\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$ but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T$^2=(-1)^{\rm F}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

Asymptotically free and safe quantum gravity scenarios consistent with Hubble, laboratory, and inflation scale physics

To find out the possible scenarios for quantum gravity consistent with the observed universe, we numerically investigate the non-perturbative renormalization group equations of a general quadratic gravity theory recently derived by Sen, Wetterich and Yamada (\textit{JHEP} 03 (2022) 130). As boundary conditions, we impose consistency with the Hubble scale and the laboratory scale experiments, and the Starobinsky model of inflation. We find two kinds of trajectories which go to different regime at the trans-Planckian scales: i) a trajectory which flows to the asymptotically free regime, and ii) a trajectory which flows to the asymptotically safe regime. To determine the early-time cosmological scenario, an additional observational data from beyond the homogeneous and isotropic space-time is necessary.

Noether Symmetries and Conservation Laws in Non-Static Plane Symmetric Spacetime

In this paper, we find all nonstatic plane symmetric spacetime metrics whose corresponding Lagrangians possess Noether symmetries. The set of determining equations is analyzed through a Maple algorithm that restricts the metric coefficients to satisfy certain conditions. These restrictions on metric coefficients, while using them to solve the determining equations, give rise to a number of plane symmetric metrics admitting 4-, 5-, 6-, 7-, 8-, 9-, 11-, and 17-dimensional Noether algebras. The Noether theorem is used to find a conserved quantity corresponding to each Noether symmetry. Some physical implications are discussed by finding bounds for different energy conditions for the obtained metrics.

Light deflection angle in General Relativity via Daftardar-Jafari method

In this paper, the iterative method suggested by Daftardar and Jafari hereafter called Daftardar-Jafari method (DJM) is applied for studying the deflection of light in General Relativity. For this purpose, a brief review of the nonlinear geodesic equations in the spherical symmetry spacetime and the main ideas of DJM are given. As an illustrative example, the simple case of the Schwarzschild metric is considered for which the approximate solution to the null-geodesic equation and the deflection angle of light are obtained. We also compare the obtained result with some similar results presented earlier in the literature. Â

The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein’s general relativity. We construct a proof of the zeroth law of black hole thermo-dynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove that surface gravity remains constant on its horizon when computed for such stationary black holes, which is the zeroth law. We argue that the constancy of surface gravity on the horizon is related to specific components of the equations of motion in such theories. We further use a specific boost symmetry of the near horizon space-time of the stationary black hole to constrain the off-shell structure of the equations of motion. Our proof for the zeroth law is valid up to arbitrary order in the expansion in the higher derivative couplings.

Rotating cosmologies: classical and quantum

We revisit spatially flat, anisotropic cosmologies within the framework of mini-superspace. Putting special emphasis on the symmetries of the mini-superspace action and on the associated conservation laws, we unveil a new class of rotating cosmologies driven by solid matter. Their rotating is physical, in that it is characterized in an invariant way in terms of a conserved angular momentum. Along the way, we confirm the results of Bartolo et al. regarding the slow decay of anisotropies for solid inflation. We then use our minisuperspace approach as a laboratory to address certain puzzles of quantum cosmology — among these, how to characterize the spacetime symmetries of a quantum state at the level of the wavefunction of the universe. For the case of a solid driven cosmology, this question seems better defined than in more standard cases. Other questions remain unanswered, though; in particular, the general question of how to operate a minisuperspace-like truncation of degrees of freedom that is consistent at the quantum level.

Black hole shadow with soft hairs

Light bending by the strong gravity around the black hole will form the so-called black hole shadow, the shape of which can shed light on the structure of the near-horizon geometry to possibly reveal novel physics of strong gravity and black hole. In this work, we adopt both analytical and ray-tracing methods to study the black hole shadow in the presence of the infrared structure of gravity theory, which manifests the asymptotic symmetries of spacetime as the supertranslation soft hairs of the black hole. Though the black hole metrics with and without the soft hair are related by large gauge transformations, the near horizon geometries relevant for the shape of the shadow are quite different. Moreover, the Hamiltonian for the geodesic seems intrinsically different, i.e., the loss of separability due to the breaking of spherical symmetry by soft hair. By applying ray-tracing computations, we find that the soft hair, although not affecting the shape of the shadow, may change the average size and position of the shadow. Images resulting from soft hair black holes with surrounding accretion flows are also discussed.

2nd order approximate Noether and Lie symmetries of Gibbons–Maeda–Garfinkle–Horowitz–Strominger charged black hole in the Einstein frame

In this paper, approximate Noether and Lie symmetries of 2nd order for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) charged black hole in the Einstein frame are analyzed comprehensively. To explore the approximate Noether symmetries of 2nd order, Noether symmetries of Minkowski spacetime are used which forms a 17 dimensional Lie algebra. It is observed that no new approximate Noether symmetry is obtained at 1st and 2nd order. To examine the 1st and 2nd order approximate Lie symmetries of the GMGHS black hole spacetime, 35 Lie symmetries (exact) of the Minkowski spacetime are used which forms an algebra sl(6, R). It is shown that no new approximate Lie symmetry exists at 1st and 2nd order and only exact 35 symmetries are recouped as trivial approximate Lie symmetries at both orders. Furthermore, no energy rescaling factor is seen in this spacetime.

Discrete spacetime symmetries, second quantization, and inner products in a non-Hermitian Dirac fermionic field theory

We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creation and annihilation operators is just that of a Hermitian theory.

Electromagnetic Fields around Black Holes in Einstein Æther Gravity

Axial symmetry and stationary properties of spacetime allow to find exact analytical solutions of differential equations describing fields and particles in a gravitational background. The present work is mainly devoted to derivation of exact solutions of Maxwell’s equations for magnetic fields generated by current loops around static black holes (BHs) in Einstein-aether gravity based on the spacetime symmetries in both regions: (i) interior and (ii) exterior to the current loop for a proper observer. The spacetime symmetries are applied in separating variables to solve the second order ordinary differential equation for vector potential of electromagnetic field and the equations of motion of test particles around the aether BH. We also study effects of the aether field on innermost stable circular orbits (ISCOs) of the test particles assuming the current loop position is placed there. It is obtained that the ISCO radius, as well as dipole magnetic moment of the current loop decrease with the increase of the aether parameter c14. Moreover, the performed analysis indicates that the aether field causes a decrease in the magnetic field inside and outside the current loop due to the change of its position.

Semiclassical gravity with a conformally covariant field in globally hyperbolic spacetimes

We prove that semiclassical gravity in conformally static, globally hyperbolic spacetimes with a massless, conformally coupled Klein–Gordon field is well posed when viewed as a coupled theory for the dynamical conformal factor of the metric and the Klein–Gordon theory. Namely, it admits unique and stable solutions whenever constrained fourth-order initial data for the conformal factor and suitably defined Hadamard initial data for the Klein–Gordon state are provided on a spacelike Cauchy surface. As no spacetime symmetries are imposed on the conformal factor, the present result implies that, provided constrained initial data exist, there also exist exact solutions to the semiclassical gravity equations beyond the isotropic, homogeneous, or static cases.

Hyperbolic inflationary model with nonzero curvature

We consider a cosmological model consisting of two scalar fields defined in the hyperbolic plane known as hyperbolic inflation. For the background space, we consider a homogeneous and isotropic spacetime with nonzero curvature. We study the asymptotic behaviour of solutions and search for attractors in the expanding regime. We prove that two hyperbolic inflationary stages are stable solutions that can solve the flatness problem and describe acceleration for both open and closed models, and additionally, we obtain a Milne-like attractor solution for the open model. We also investigate the contracting branch obtaining mirror solutions with the opposite dynamical behaviours.

Cobordism and deformation class of the standard model

't Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry $G$ (including spacetime and internal symmetries) in a $d\mathrm{d}$ spacetime are known to be classified by a $d+1\mathrm{d}$ cobordism group ${\mathrm{TP}}_{d+1}(G)$, whose group generator is a $d+1\mathrm{d}$ cobordism invariant written as an invertible topological field theory (iTFT) with a partition function ${\mathbf{Z}}_{d+1}$. It has recently been proposed that the deformation class of QFTs is specified by its symmetry $G$ and an iTFT ${\mathbf{Z}}_{d+1}$. Seemingly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. In this work, we ask which cobordism class and deformation class control the 4d standard model (SM) of ungauged or gauged $(\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1))/{\mathbb{Z}}_{q}$ group for $q=1$, 2, 3, 6 with a continuous or discrete baryon minus lepton $(\mathbf{B}\ensuremath{-}\mathbf{L})$-like symmetry. We show that the answer contains some combination of 5d iTFTs; two $\mathbb{Z}$ classes associated with $(\mathbf{B}\ensuremath{-}\mathbf{L}{)}^{3}$ and $(\mathbf{B}\ensuremath{-}\mathbf{L})\ensuremath{-}{(\text{gravity})}^{2}$ 4d perturbative local anomalies, a ${\mathbb{Z}}_{16}$ class Atiyah-Patodi-Singer $\ensuremath{\eta}$ invariant, a ${\mathbb{Z}}_{2}$ class Stiefel-Whitney ${w}_{2}{w}_{3}$ invariant associated with 4d nonperturbative global anomalies, and additional ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ global anomalies involving higher symmetries whose charged objects are Wilson electric or 't Hooft magnetic line operators. Out of the multiple infinite $\mathbb{Z}$ classes of local anomalies and 24576 classes of global anomalies, we pin down the deformation class of the SM labeled by (${N}_{f},{n}_{{\ensuremath{\nu}}_{R}},\text{ }{\mathrm{p}}^{\ensuremath{'}},q$), the family number, the total ``right-handed sterile'' neutrino number, the magnetic monopole datum, and the mod $q$ relation. We show that grand unification such as Georgi-Glashow $su(5)$, Pati-Salam $su(4)\ifmmode\times\else\texttimes\fi{}su(2)\ifmmode\times\else\texttimes\fi{}su(2)$, Barr's flipped $u(5)$, and the familiar or modified $so(n)$ models of Spin($n$) gauge group, e.g., with $n=10$, 18 can all reside in an appropriate SM deformation class. We show that ultra unification, which replaces some of sterile neutrinos with new exotic gapped/gapless sectors (e.g., topological or conformal field theory) or gravitational sectors with topological origins via cobordism constraints, also resides in an SM deformation class. Neighbor quantum phases near SM or their phase transitions, and neighbor gapless quantum critical regions naturally exhibit beyond SM phenomena.