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Recently, theoretical studies on the bumblebee gravity model, a nonminimally-coupled vector-tensor theory that violates the Lorentz symmetry, have flourished, with a simultaneous increase in the utilization of observations to impose constraints. The static spherical solutions of neutron stars (NSs) in the bumblebee theory are calculated comprehensively in this work. These solutions with different coupling constants reveal a rich theoretical landscape for NSs, including vectorized NSs and NSs with finite radii but divergent masses. With these solutions, preliminary constraints on the asymptotic vector field values are obtained through restrictions on the stellar radius.

As a key science project of the Square Kilometre Array (SKA), the discovery and timing observations of radio pulsars in the Galactic Center would provide high-precision measurements of the spacetime around the supermassive black hole, Sagittarius A* (Sgr A*), and initiate novel tests of general relativity. The spin of Sgr A* could be measured with a relative error of $\lesssim 1\%$ by timing one pulsar with timing precision that is achievable for the SKA. However, the real measurements depend on the discovery of a pulsar in a very compact orbit, $P_b\lesssim0.5\,{\rm yr}$. Here for the first time we propose and investigate the possibility of probing the spin of Sgr A* with two or more pulsars that are in orbits with larger orbital periods, $P_b\sim 2- 5\,{\rm yr}$, which represents a more realistic situation from population estimates. We develop a novel method for directly determining the spin of Sgr A* from the timing observables of two pulsars and it can be readily extended for combining more pulsars. With extensive mock data simulations, we show that combining a second pulsar improves the spin measurement by $2-3$ orders of magnitude in some situations, which is comparable to timing a pulsar in a very tight orbit.

Incorporating first-order QED effects, we explore the shadows of Kerr-Newman black holes with a magnetic charge through the numerical backward ray-tracing method. Our investigation accounts for both the direct influence of the electromagnetic field on light rays and the distortion of the background spacetime metric due to QED corrections. We notice that the area of the shadow increases with the QED effect, mainly due to the fact that the photons move more slowly in the effective medium and become easier to be trapped by the black hole.

We explore matter fields containing Gauss-Bonnet correction terms within the framework of renormalizable quantum field theory. By revising the gauge model with a charged scalar multiplier and two sets of fermion families within a flat universe model using torsion, we introduce Gauss-Bonnet corrections into the action to investigate field equations within the context of supersymmetric mixed inflationary models. After analytically computing the modified gauge boson field equations through the incorporation of Gauss-Bonnet theory into the fundamental field equations, we derive the torsion characteristics and energy-momentum tensor properties of a flat universe. Our analysis can extend to more specific Gauss-Bonnet correction models, enabling the derivation of gravitational characteristic equations with practical applications. This streamlines Gauss-Bonnet models, assesses the model's sensitivity to correction parameters, and explores high-sensitivity models with observational significance. Furthermore, in this derivation process, we identify that Gauss-Bonnet correction terms can directly impact the Hubble constant through Einstein's equations, indicating the potential for verifying Gauss-Bonnet theory through astronomical observations.

Timing a pulsar orbiting around Sagittarius A* (Sgr A*) can provide us with a unique opportunity of testing gravity theories. We investigate the detectability of a vector charge carried by the Sgr A* black hole (BH) in the bumblebee gravity model with simulated future pulsar timing observations. The spacetime of a bumblebee BH introduces characteristic changes to the orbital dynamics of the pulsar and the light propagation of radio signals. Assuming a timing precision of 1 ms, our simulation shows that a 5-yr observation of a pulsar with an orbital period $P_b\sim 0.5\,{\rm yr}$ and an orbital eccentricity $e\sim 0.8$ can probe a vector charge-to-mass ratio as small as $Q/M\sim 10^{-3}$, which is much more stringent than the current constraint from the Event Horizon Telescope (EHT) observations, and comparable to the prospective constraint from extreme mass-ratio inspirals with the Laser Interferometer Space Antenna (LISA).

Future observations with next-generation large-area radio telescopes are expected to discover radio pulsars (PSRs) closely orbiting around Sagittarius~A* (Sgr~A*), the supermassive black hole (SMBH) dwelling at our Galactic Center (GC). Such a system can provide a unique laboratory for testing General Relativity (GR), as well as the astrophysics around the GC. In this paper, we provide a numerical timing model for PSR-SMBH systems based on the post-Newtonian (PN) equation of motion, and use it to explore the prospects of measuring the black hole (BH) properties with pulsar timing. We further consider the perturbation caused by the dark matter (DM) distribution around Sgr~A*, and the possibility of constraining DM models with PSR-SMBH systems. Assuming a 5-year observation of a normal pulsar in an eccentric ($e=0.8$) orbit with an orbital period $P_b = 0.5\,$yr, we find that -- with weekly recorded times of arrival (TOAs) and a timing precision of 1 ms -- the power-law index of DM density distribution near the GC can be constrained to about 20%. Such a measurement is comparable to those measurements at the Galactic length scale but can reveal small-scale properties of the DM.

Soft gamma-ray repeaters (SGRs) are widely understood as slowly rotating isolated neutron stars. Their generally large spin-down rates, high magnetic fields, and strong outburst energies render them different from ordinary pulsars. In a few giant flares (GFs) and short bursts of SGRs, high-confidence quasi-periodic oscillations (QPOs) were observed. Although remaining an open question, many theoretical studies suggest that the torsional oscillations caused by starquakes could explain QPOs. Motivated by this scenario, we systematically investigate torsional oscillation frequencies based on the strangeon-star (SS) model with various values of harmonic indices and overtones. To characterize the strong-repulsive interaction at short distances and the non-relativistic nature of strangeons, a phenomenological Lennard-Jones model is adopted. We show that, attributing to the large shear modulus of SSs, our results explain well the high-frequency QPOs ($\gtrsim 150\,\mathrm{Hz}$) during the GFs. The low-frequency QPOs ($\lesssim 150\,\mathrm{Hz}$) can also be interpreted when the ocean-crust interface modes are included. We also discuss possible effects of the magnetic field on the torsional mode frequencies. Considering realistic models with general-relativistic corrections and magnetic fields, we further calculate torsional oscillation frequencies for quark stars. We show that it would be difficult for quark stars to explain all QPOs in GFs. Our work advances the understanding of the nature of QPOs and magnetar asteroseismology.

We develop a consistent approach to calculate the moment of inertia (MOI) for axisymmetric neutron stars (NSs) in the Lorentz-violating Standard-Model Extension (SME) framework. To our knowledge, this is the first relativistic MOI calculation for axisymmetric NSs in a Lorentz-violating gravity theory other than deformed, rotating NSs in the General Relativity. Under Lorentz violation, there is a specific direction in the spacetime and NSs get stretched or compressed along that direction. When a NS is spinning stationarily along this direction, a conserved angular momentum and the concept of MOI are well defined. In the SME framework, we calculate the partial differential equation governing the rotation and solve it numerically with the finite element method to get the MOI for axisymmetric NSs caused by Lorentz violation. Besides, we study an approximate case where the correction to the MOI is regarded solely from the deformation of the NS and compare it with its counterpart in the Newtonian gravity. Our formalism and the numerical method can be extended to other theories of gravity for static axisymmetric NSs.

In this work, we study the images of a Kerr black hole (BH) immersed in uniform magnetic fields, illuminated by the synchrotron radiation of charged particles in the jet. We particularly focus on the spontaneously vortical motions (SVMs) of charged particles in the jet region and investigate the polarized images of electromagnetic radiations from the trajectories along SVMs. We notice that there is a critical value $\omega_c$ for charged particle released at a given initial position and subjected an outward force, and once $|qB_0/m|=|\omega_B|>|\omega_c|$ charged particles can move along SVMs in the jet region. We obtain the polarized images of the electromagnetic radiations from the trajectories along SVMs. Our simplified model suggests that the SVM radiations can act as the light source to illuminate the BH and form a photon ring structure.

Tests of gravity are important to the development of our understanding of gravitation and spacetime. Binary pulsars provide a superb playground for testing gravity theories. In this chapter we pedagogically review the basics behind pulsar observations and pulsar timing. We illustrate various recent strong-field tests of the general relativity (GR) from the Hulse-Taylor pulsar PSR B1913+16, the double pulsar PSR J0737$-$3039, and the triple pulsar PSR J0337+1715. We also overview the inner structure of neutron stars (NSs) that may influence some gravity tests, and have used the scalar-tensor gravity and massive gravity theories as examples to demonstrate the usefulness of pulsar timing in constraining specific modified gravity theories. Outlooks to new radio telescopes for pulsar timing and synergies with other strong-field gravity tests are also presented.

In this note we revisit the emergent conformal symmetry in the near-ring region of warped spacetime. In particular, we propose a novel construction of the emergent near-ring $sl(2,R)_{\text{QNM}}$ symmetry. We show that each eikonal QNM family falls into one highest-weight representation of this algebra, and $sl(2,R)_{\text{QNM}}$ can be related to the near-ring isometry group $sl(2,R)_{\text{ISO}}$ in a simple way. Furthermore we find that the coherent state space of $sl(2,R)_{\text{QNM}}$ can be identified with the phase space of the photon ring.

In this work, we study circular motions of charged particles and their polarized images around the Kerr black hole immersed in a weak magnetic field. We pay special attention to the case that both the magnetic field and the charge-to-mass ratio are not big, thus the effective potential along the radial motion reduce to a cubic form approximately so that we can express the radius of the innermost stable circular orbit analytically in terms of the energy and angular momentum of charged particles. Moreover, we computed the polarized synchrotron radiations of these particles and obtained the polarized images semi-analytically for various spins, observational angles and prograde and retrograde orbits. In particular, We find that these parameters have significant impacts on the polarization rotation and the magnitude of the polarization flux.

The discovery of radio pulsars (PSRs) around the supermassive black hole (SMBH) in our Galactic Center (GC), Sagittarius A* (Sgr A*), will have significant implications for tests of gravity. In this paper, we predict restrictions on the parameters of the Yukawa gravity by timing a pulsar around Sgr A* with a variety of orbital parameters. Based on a realistic timing accuracy of the times of arrival (TOAs), $\sigma_{\rm TOA}=100\,\mu{\rm s}$, and using a number of 960 TOAs in a 20-yr observation, our numerical simulations show that the PSR-SMBH system will improve current tests of the Yukawa gravity when the range of the Yukawa interaction varies between $10^{1}$-$10^{4}\,{\rm AU}$, and it can limit the graviton mass to be $m_g \lesssim 10^{-24}\,{\rm eV}/c^2$.

In this work, we derive two formulas encoding the polarization direction and luminosity of synchrotron radiations from the moving electrons in curved spacetime under the geometric optics approximation. As an application, we further study the polarized images of synchrotron radiations from electron sources in Schwarzschild black hole spacetime with a vertical and uniform magnetic field. In particular, by focusing on the circular orbits of electrons on the equatorial plane, we show the polarized images of the synchrotron radiations from these orbits for different observational angles and discuss the variations of the polarization directions concerning the angles.

Neutron stars (NSs) in scalar-tensor theories of gravitation with the phenomenon of spontaneous scalarization can develop significant deviations from general relativity. Cases with a massless scalar were studied widely. Here we compare the NS scalarizations in the Damour--Esposito-Farèse theory, the Mendes-Ortiz theory, and the $\xi$-theory with a massive scalar field. Numerical solutions for slowly rotating NSs are obtained. They are used to construct the X-ray pulse profiles of a pair of extended hot spots on the surface of NSs. We also calculate the tidal deformability for NSs with spontaneous scalarization which is done for the first time with a massive scalar field. We show the universal relation between the moment of inertia and the tidal deformability. The X-ray pulse profiles, the tidal deformability, and the universal relation may help to constrain the massive scalar-tensor theories in X-ray and gravitational-wave observations of NSs, including the Neutron star Interior Composition Explorer (NICER) satellite, Square Kilometre Array (SKA) telescope, and LIGO/Virgo/KAGRA laser interferometers.

We consider isotropic and monochromatic photon emissions from equatorial emitters moving along future-directed timelike geodesics in the near-horizon extremal Kerr (NHEK) and near-horizon near-extremal Kerr (near-NHEK) regions, to asymptotic infinity. We obtain numerical results for the photon escaping probability (PEP) and derive analytical expressions for the maximum observable blueshift (MOB) of the escaping photons, both depending on the emission radius and the emitter's proper motion. In particular, we find that for all anti-plunging or deflecting emitters that can eventually reach to asymptotic infinity, the PEP is greater than $50\%$ while for all plunging emitters the PEP is less than $55\%$, and for the bounded emitters in the (near-)NHEK region, the PEP is always less than $59\%$. In addition, for the emitters on unstable circular orbits in the near-NHEK region, the PEP decreases from $55\%$ to $50\%$ as the orbital radius decreases from the one of the innermost stable circular orbit to the one of the horizon. Furthermore, we show how the orientation of the emitter's motion along the radial or azimuthal direction affects the PEP and the MOB of the emitted photons.

In this work, taking the QED effect into account, we investigate the shadows of the Kerr black holes immersed in uniform magnetic fields through the numerical backward ray-tracing method. We introduce a dimensionless parameter $\Lambda$ to characterize the strength of magnetic fields and studied the influence of magnetic fields on the Kerr black hole shadows for various spins of the black holes and inclination angles of the observers. In particular, we find that the photon "hairs" appear near the left edge of the shadow in the presence of magnetic fields. The photon hairs may be served as a signature of the magnetic fields. We notice that the photon hairs become more evident when the strength of magnetic fields or the spin of the black hole becomes larger. In addition, we study the deformation of the shadows by bringing in quantitative parameters that can describe the position and shape of the shadow edge.

In this work, taking the QED effect into account, we investigate the shadows of the static black hole with magnetic monopoles and neutral black holes in magnetic fields through the numerical backward ray-tracing method. For a static black holes with magnetic monopole, we obtain the relation between the shadow radius and the coupling constant. For neutral black holes in the uniform magnetic fields, we find that the shadow curves deviate very small from the ellipses for equatorial observer, and we read the linear relation between the eccentricity and the coupling constant. For $\theta_o\neq\pi/2$, we find that the shadow curves can be well approximated by ellipses in most cases, except the case that the magnetic field is very strong and the observer sits around the angle $\theta_o=\pi/4$ or $3\pi/4$. Moreover we extend our investigation to a neutral static black hole surrounded with a current loop.

We have performed a precision atomic interferometry experiment on testing the universality of free fall (UFF) considering atoms' spin degree of freedom. Our experiment employs the Bragg atom interferometer with $^{87}$Rb atoms either in hyperfine state $\left| {F = 1,{m_F} = 0} \right\rangle $ or $\left| {F = 2,{m_F} = 0} \right\rangle $, and the wave packets in these two states are diffracted in one pair of Bragg beams alternatively, which can help suppress the common-mode systematic errors. We have obtained an E$\rm{\ddot{o}}$tv$\rm{\ddot{o}}$s ratio $\eta = \left( { 0.9 \pm 2.7} \right) \times {10^{ - 10}}$, and set a new record on the precision with a nearly 5 times improvement. Our experiment gives stronger restrictions on the possible UFF breaking mechanism.

We investigated Lorentz violation through anisotropy of gravity using a worldwide array of 12 superconducting gravimeters. The Lorentz-violating signal is extracted from the difference between measured gravity and a tidal model. At the level of sensitivity we reach, ocean tides start to play an important role. However, most models available that include ocean tides are empirically based on measured gravity data, which may contain Lorentz-violating signal. In this work we used an ocean tides included tidal model derived from first principles to extract Lorentz-violating signal for the first time. We have bounded space-space components of gravitational Lorentz violation in the minimal standard model extension (SME) up to the order of $10^{-10}$, one order of magnitude improved relative to previous atom-interferometer tests.

A new four-dimensional black hole solution of Einstein-Born-Infeld-Yang-Mills theory is constructed, several degenerated forms of the black hole solution are presented. The related thermodynamical quantities are calculated, with which the first law of thermodynamics is checked to be satisfied. Identifying the cosmological constant as pressure of the system, the phase transition behaviors of the black hole in the extended phase space are studied.

We analyze the near horizon conformal symmetry for black hole solutions in gravity with a conformally coupled scalar field using the method proposed by Majhi and Padmanabhan recently. It is shown that the entropy of the black holes of the form $\mathrm{d}s^2 = - f(r)\mathrm{d}t^2 + \mathrm{d}r^2/f(r)+...$ agrees with Wald entropy. This result is different from previous result obtained by M. Natsuume, T. Okamura and M. Sato using the canonical Hamiltonian formalism, which claims a discrepancy from Wald entropy.

We explore the implications of the requirement of Galilean invariance for classical point particle lagrangians, in which the space is not assumed to be flat to begin with. We show that for the free, time-independent lagrangian, this requirement is equivalent to the existence of gradient Killing vectors on space, which is in turn equivalent to the condition that the space is a direct product, which is totally flat in the Galilean invariant direction. We then consider more general cases and see that there is no simple generalisation to these cases.

In this paper we first propose a framework of structure-preserving submersions, which generalises the concept of a Riemannian submersion, and dualises the concept of subgeometry, or "structure-preserving immersions". The emphasis of our approach is on making precise the free variables and the degree of freedom in a given system, thus making the messy calculations in such problems more bearable and, more importantly, algorithmic. In particular, we derive the degrees of freedom of Riemannian submersions and of Weyl submersions. Then we apply our framework to the study of relativistic dissipationless flow and shear-free flows, obtaining generalisations of the classical Herglotz-Noether theorem to conformally flat spacetime in all dimensions and a partial result of Ellis conjecture to all dimensions.

For physical theories, the degree of arbitrariness of a system is of great importance, and is often closely linked to the concept of degree of freedom, and for most systems this number is far from obvious. In this paper we present an easy to apply algorithm for calculating the degree of arbitrariness for large classes of systems formulated in the language of moving frames, by mere manipulation of the indices of the differential invariants. We then give several examples illustrating our procedure, including a derivation of the degree of arbitrariness of solutions of the Einstein equation in arbitrary dimensions, which is vastly simpler than previous calculations, the degree of arbitrariness of gauge theories of arbitrary groups under Yang-Mills type equations, for which until now only specific cases have been calculated by rather messy means, and the degree of arbitrariness of relativistic rigid flow with or without additional constraints, which has not been derived before. Finally we give the proof of our algorithm.

This paper reviews the concepts and assumptions of rigid flow in relativistic fluid mech- anics, particularly the generalisation of the classical Herglotz-Noether theorem, that are relevant to the fluid approximation of the AdS-CFT dual of large rotating black-holes used by Bhattacharyya et al. We then give a brief outline of the recently found proof the generalised theorem.

In this paper we give a new proof, valid for all dimensions, of the classical Herglotz-Noether theorem that all rotational shear-free and expansion-free flows (rotational Born-rigid flows) in Minkowski spacetime are generated by Killing vector fields (isometric flows). This is aimed as an illustration of a general framework for working with problems that can be described as a structure-preserving submersion, which we will describe in a subsequent paper.