Gravitational Case Research Articles

From higher-spin gauge interactions to Compton amplitudes for root-Kerr

We develop massive higher-spin theory as a framework for describing dynamics of rotating compact objects, such as Kerr black holes. In this paper, we explore gauge interactions up to quartic order and corresponding Compton amplitudes of higher-spin massive objects coupled to electromagnetism and Yang-Mills theory. Their classical counterparts are known as root-Kerr gauge-theory solutions, whose amplitudes are closely related to those of Kerr black holes. We use three distinct approaches: (i) massive higher-spin gauge symmetry to introduce cubic interactions for all spins and the quartic interactions up to spin 3, which is implemented both off shell and via Ward identities; (ii) a chiral higher-spin approach to construct quartic Lagrangians with correct degrees of freedom to all spins; (iii) on-shell functional patterns before and after taking the classical limit to constrain the Compton amplitudes. As final results, we arrive at simple local formulae for the candidate root-Kerr Compton amplitudes both in the quantum regime and classical limit, to all orders in spin. This is a precursor to the gravitational Kerr case, which is presented in a follow-up paper.

Infrared finite scattering theory: scattering states and representations of the BMS group

Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an “out” state with memory which gives rise to infrared divergences in the standard S-matrix. To obtain an infrared-finite scattering theory, one must suitably include states with memory. However, except in the case of QED with massive charged particles, asymptotic states with memory that have finite energy and angular momentum have not been constructed for more general theories (e.g. massless QED, Yang-Mills and quantum gravity). To this end, we construct direct-integral representations over the “Lorentz orbit” of a given memory and classify all “orbit space representations” that have well-defined energy and angular momentum. We thereby provide an explicit construction of a large supply of physical states with memory as well as the explicit action of the BMS charges all states. The construction of such states is a key step toward the formulation of an infrared-finite scattering theory. While we primarily focus on the quantum gravitational case, we outline how the methods presented in this paper can be applied to obtain representations of the Poincaré group with memory for more general quantum field theories.

About Some Applications of Mathematics

In this paper, a new mass-velocity relation is derived. The orbit of planetary motion is deduced based on this new relation and the Newtonian laws of motion along with the concept of proper time. The derivation does not use the tensor concept, but the relativistic orbit is obtained as a first approximation of the modified Newtonian theory. Next, we discuss the possibility of introducing a field dependent metric for the EM field by considering the observation that E2 - c2 B2 or E2 - B2 with c = 1 is approximately an invariant quantity for EM Field [1-3]. It is possible to extend the metric for the gravitational case also [4]. The discussion of anomalous characteristic of Lorentz Transformation (LT) and the introduction of a non-linear transformation connecting Local Space-time coordinates (ct,x) with proper system is continued.

Field Dependant Metric for Gravitational/EM Fields and a NonLinear Transformation for Local to Proper Space-Time World

In this paper, the introduction of a metric for Gravitational Field is examined based on [4]; this can be extended to EM Fields also by change of the parameters involved (ϵ1 , μ1 ) to (ϵ2 , μ2 ) We know that E2 ‒ c2 B2 or E2 ‒B2 with is an invariant quantity for the EM Fields which can be extended to gravitational fields, as done in [1]. We discuss the metric for the gravitational case and extended to the EM fields as well. The discussion of anomalous characteristic of Lorentz Transformation (LT) and the introduction of a non-linear transformation connecting Local Space-time coordinates (ct,x) with proper system (cτ, xτ ) is continued.

The effects of general relativity on close-in radial-velocity-detected exosystems

Aims. The detection of the first exoplanet around a solar-type star revealed the existence of close-in planets. Several of these close-in planets are part of multi-planet systems. For systems detected via the radial velocity (RV) method, we lack information on the mutual inclination of the orbital planes. The aim of this work is to study the long-term stability of RV-detected two-planet systems with close-in planets and identify possible three-dimensional configurations for these systems that are compatible with observations. To do so, we focused on the protective mechanism of the Lidov-Kozai (LK) secular resonance and studied the effects of general relativity (GR) on long-term evolution. Methods. By means of an analytical study based on a high-order secular Hamiltonian expansion in the eccentricities and inclinations, we first identified ranges of values for the orbital and mutual inclinations that are compatible with the presence of the LK resonance in the purely gravitational case. Then, adding the secular contribution of the relativistic corrections exerted by the central star on the inner planet, namely the advance of its pericenter precession, we analysed the outcomes of the two sets of simulations. We compared our results to analytical estimates to determine the importance of GR effects. Results. We find that for the majority of the systems considered, GR strongly affects the dynamics of the system and, most of the time, voids the LK resonance, as observed for GJ 649, GJ 832, HD 187123, HD 190360, HD 217107, and HD 47186. The long-term stability of these systems is then possible whatever the mutual inclination of the orbits. On the contrary, for GJ 682, HD 11964, HD 147018, and HD 9446, the LK resonant region in the parameter space of the orbital and mutual inclinations is left (almost) unchanged when GR effects are considered, and consequently their long-term stability is only possible if the mutual inclination of the orbits is low or if the systems are in the LK regime with a high mutual inclination.

Constrain the time variation of the gravitational constant via the propagation of gravitational waves

The gravitational constant variation means the breakdown of the strong equivalence principle. As the cornerstone of general relativity, the validity of general relativity can be examined by studying the gravitational constant variation. Such variations have the potential to affect both the generation and propagation of gravitational waves. Here, our focus lies on the effect of gravitational constant variation specifically on the propagation of gravitational waves. In a previous paper (An et al., 2023 [13]) we have studied such effect based on a Maxwell-like equation. That work admits two drawbacks. The assumption that describing gravitational waves with Maxwell-like equation is not solid. And more such treatment can only deal with space dependent gravitational constant. In the current paper we will remedy these two issues. We employ two analytical methods, namely based on the Fierz-Pauli action and the perturbation of Einstein-Hilbert action around Minkowski spacetime, both leading to the same gravitational wave equation. By solving this equation, we find the effects of gravitational constant variation on gravitational wave propagation. The result is consistent with previous investigations based on Maxwell-like equations for gravitational waves for space dependent gravitational constant cases. In addition we can constrain the time variation of the gravitational constant via the propagation of gravitational waves based on GW170817 data. And more, we find that small variations in the gravitational constant result in an amplitude correction at the leading order and a phase correction at the sub-leading order for gravitational waves. These results provide valuable insights for probing gravitational constant variation and can be directly applied to gravitational wave data analysis.

Wave scattering event shapes at high energies

We study the space and properties of global and local observables for radiation emitted in the scattering of a massive scalar field in gauge and gravitational plane-wave backgrounds, in both the quantum and classical theory. We first compute the radiated momentum and angular momentum flow, demonstrating that they are good local observables determined by the amplitude and phase of the waveform. We then focus on the corresponding global observables, which in the gravitational case requires dealing with the collinear divergence of the gravitational Compton cross-section. We show using the KLN theorem that we can obtain an infrared-finite cross-section only by summing over forward scattering diagrams; this suggests dressing the initial state in the direction collinear to the plane wave in order to be able to compute observables integrated over the celestial sphere. Finally, we explore the high-energy behaviour of our observables. We find that classical global observables generically exhibit a power-law mass divergence in electrodynamics and a logarithmic mass divergence in gravity, even when radiation reaction is included. We then show explicitly how this is consistently resolved in the full quantum theory.

Two-body problem in curved spacetime: exploring gravitational wave transient cases

Gravitational-Wave Transient Catalogues (GWTC) from the LIGO-Virgo-KAGRA collaborations (LVC and LVK) contain almost a hundred gravitational wave (GW) detection cases. We explore them from the perspective of the two-body problem in curved spacetime, starting with the first case, GW150914, which marks the GW discovery []. In this paper, the LVC authors estimated the characteristic (chirp) mass of the binary blackhole system emitted this signal. Their calculation was based on Numerical-Relativity (NR) templates and presumably accounted fully for the non-linearity of GR. The same team later presented an alternative analysis of GW150914 [], using the quadrupole post-Newtonian (PN) approximation of GR. Both analyses gave similar results, despite being based on quite different assumptions about the linearity or non-linearity of the coordinate reference frame near the GW source. Here we revisit the PN-analysis of GW150914 for which we use less noisy input GW frequencies, as we have filtered them by reading them from the time-frequency map of GW150914. As in paper [], our result also agrees with the NR-based chirp mass value published in []. Additionally, we apply the PN-approximation formalism to the rest of the GWTC cases, finding that practically all of their PN-approximated chirp masses coincide with the published NR-based values from GWTC. In our view, this implies that the NR-based theory, which is currently in use for processing GW signals, does not fully account for the difference between the source and detector reference frames because the PN-approximation, which is used for the comparison, does not account for this difference by design, given the flat-spacetime initial assumptions of this approximation. We find that the basis of this issue lies in the source-to-detector coordinate transformation. For example, when obtaining the equation of motion of a coalescing binary system by integrating its energy-momentum tensor and varying the corresponding reduced action functional, the lapse and shift functions are not involved within the Arnowitt-Deser-Misner (ADM) parametrisation scheme, which is typically used for the NR-based calculation of GW waveforms A similar non-involvement of the lapse and shift functions is known to occur in the description of motion of an orbiter around a Schwarzschild blackhole. Here the GR expression for the orbital angular frequency, as seen by a remote observer, coincides with the Keplerian non-relativistic formula until the very last orbits before the plunge phase (although being fully GR-compliant). This non-involvement of the time lapse function renders the source-to-detector coordinate transformation suitable for building GW waveforms corresponding to the detector frame. However, the inverse (detector-to-source) transformation requires the derivatives of GW frequencies to be known in the source reference frame. The lack of this knowledge leads to a systematic error in the estimated chirp masses of GW sources. The corresponding luminosity distances of these sources also turn out to be overestimated.

Similarities and differences between two coincidently gravitational bullet cases

Many cases of gravitational bullets are reported in developed and non-developed countries. However, few papers highlighted these cases in the literature. In our study, we present two cases of gravitational bullets that have an unusual coincidence in the injury characteristics through their ages, and gender, the site of the inlet.

On the Lagrangian structure of transport equations: Relativistic Vlasov systems

We study the Lagrangian structure of relativistic Vlasov systems, such as the relativistic Vlasov‐Poisson and the relativistic quasi‐eletrostatic limit of Vlasov‐Maxwell equations. We show that renormalized solutions of these systems are Lagrangian and that these notions of solution, in fact, coincide. As a consequence, finite‐energy solutions are shown to be transported by a global flow. Moreover, we extend the notion of generalized solution for “effective” densities, and we prove the existence of such solutions. Finally, under a higher integrability assumption of the initial condition, we show that solutions have every energy bounded, even in the gravitational case. These results extend to our setting those recently obtained for the Vlasov‐Poisson system in a series of papers by Ambrosio, Colombo, and Figalli; here, we analyze relativistic systems and also consider the contribution of the magnetic force into the evolution equation.

Cosmology application of the astrophysics originated gravitational wave

We review the current status of the astrophysics-originated gravitational wave astronomy and several cosmology-related science topics of the future spaceborne/ground-based gravitational wave observatory. Specifically, we highlight the gravitational lensing and gravitational wave sirens-related science cases. We first introduce several astrophysical gravitational wave sources, including the compact binary system (stellar black hole binary, and neutron star binary), continuous-wave sources (fast-spinning neutron star, and extreme mass ratio binary system), and the galactic white dwarf binary. Second, we review gravitational wave time-delay strong lensing, lensing magnification, wave-optics and the diffraction effect. Third, we mention the theoretical development of the luminosity distance space distortion effect. Finally, we discuss the electromagnetic counterpart, multi-messenger astronomy, and its cosmological application.

Self-force effects in post-Minkowskian scattering

Abstract We revisit the old problem of the self-force on a particle moving in a weak-field spacetime in the context of renewed interest in two-body gravitational scattering. We analytically calculate the scalar, electromagnetic, and gravitational self-force on a particle moving on a straight-line trajectory at a large distance from a Newtonian star, and use these results to find the associated correction to its motion. In the gravitational case we must also include the matter-mediated force, which acts at the same perturbative order as the gravitational self-force. We further augment the gravitational results with geodesic calculations at second order in the central body mass to determine the full, explicit solution to the two-body gravitational scattering problem at second post-Minkowskian order (2PM). We calculate the momentum transfer (which reproduces Westpfahl’s old result), the change in mechanical angular momentum (which matches the radiative flux recently computed by Damour), and the change in mechanical mass moment (the time-space components of the angular momentum tensor), which has not previously appeared. Besides the new 2PM results of explicit trajectories and all conserved quantities, this work clarifies the role of gravitational self-force in PM scattering theory and provides a foundation for higher-order calculations.

Conservative and radiative dynamics in classical relativistic scattering and bound systems

As recent work continues to demonstrate, the study of relativistic scattering processes leads to valuable insights and computational tools applicable to the relativistic bound-orbit two-body problem. This is particularly relevant in the post-Minkowskian approach to the gravitational two-body problem, where the field has only recently reached a full description of certain physical observables for scattering orbits, including radiative effects, at the third post-Minkowskian (3PM) order. As an historically instructive simpler example, we consider here the analogous problem in electromagnetism in flat spacetime. We compute the changes in linear momentum of each particle and the total radiated linear momentum, in the relativistic classical scattering of two point-charges, at sixth order in the charges (analogous to 3PM order in gravity). We accomplish this here via direct iteration of the classical equations of motion, while making comparisons where possible to results from quantum scattering amplitudes, with the aim of contributing to the elucidation of conceptual issues and scalability on both sides. We also discuss further extensions to radiative quantities of recently established relations, which analytically continue certain observables from the scattering regime to the regime of bound orbits, applicable for both the electromagnetic and gravitational cases.

Quantum correlations across the horizon in acoustic and gravitational black holes

We investigate, within the framework of quantum field theory in curved space, the correlations across the horizon of a black hole in order to highlight the particle-partner pair creation mechanism at the origin of Hawking radiation. The analysis concerns both acoustic black holes, formed by Bose-Einstein condensates, and gravitational black holes. More precisely, we have considered a typical acoustic black hole metric with two asymptotic homogeneous regions and the Schwarzschild metric as describing a gravitational black hole. By considering equal-time correlation functions, we find a striking disagreement between the two cases: the expected characteristic peak centered along the trajectories of the Hawking particles and their partners seems to appear only for the acoustic black hole and not for the gravitational Schwarzschild one. The reason for that is the existence of a quantum atmosphere displaced from the horizon as the locus of origin of Hawking radiation together, and this is the crucial aspect, with the presence of a central singularity in the gravitational case swallowing everything is trapped inside the horizon. Correlations, however, are not absent in the gravitational case; to see them, one simply has to consider correlation functions at unequal times, which indeed display the expected peak.

Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces

We consider Backus’s problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth’s surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field’s intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is quite difficult, because the resolving operator related to the linearized problem is generally unbounded. Indeed, existence results for Backus’s problem in this framework are not present in the literature. In this work, we locally solve the geomagnetic version of Backus’s problem in the axially symmetric case. In mathematical terms, we show the existence of harmonic functions in the exterior of a sphere, with given (boundary) field’s intensity sufficiently close to that of a dipole and which have the same axial symmetry of a dipole. We also show that unique solutions can be selected by prescribing the average of the potential on the equatorial circle of the sphere. We obtain those solutions as series of spherical harmonics. The functional framework entails the use of fractional Sobolev Hilbert spaces on the sphere, endowed with a spectral norm. A crucial ingredient is the algebra structure of suitable subspaces.

Plane fronted limit of spherical electromagnetic and gravitational waves

We demonstrate how plane fronted waves with colliding wave fronts are the asymptotic limit of spherical electromagnetic and gravitational waves. In the case of the electromagnetic waves we utilize Bateman's representation of radiative solutions of Maxwell's vacuum field equations. The gravitational case involves a novel form of the radiative Robinson--Trautman solutions of Einstein's vacuum field equations.

Non‐Singular Calculation of Geomagnetic Vectors and Geomagnetic Gradient Tensors

AbstractSome spherical harmonic expressions of gravitational and geomagnetic field elements will become infinite when computation point approaches polar regions, as the sine function of the geocentric co‐latitude contained in the denominator tends to be zero. Currently, this singularity problem has been solved for gravitational field case, however, it remains unsolved for geomagnetic vectors (GVs) and geomagnetic gradient tensors (GGTs). Because the latter use Schmidt semi‐normalized associated Legendre function (SNALF), which is different from fully‐normalized associated Legendre function used in the former. To overcome this singularity problem, we derive new non‐singular expressions of the first‐ and second‐order derivatives of Schmidt SNALF (order m equals to 0 or 1), and the corresponding two kinds of spherical harmonic polynomials. When the non‐singular expressions are applied to the traditional formulae of GVs and GGTs, more practical expressions of GVs and GGTs with non‐singularity are formulated by refining the cases that the order m equals to 0, 1, 2 and other values. Furthermore, to provide flexible calculation strategies for Schmidt SNALF, we derive four kinds of recursive formulae, including the standard forward row recursion, the standard forward column recursion, the cross degree and order recursion, and the Belikov recursion. The standard formula to check the calculation accuracy of various recursive formulae of SNALF are also given. Besides, we demonstrate the effectiveness and reliability of the new derived non‐singular expressions of GVs and GGTs and analyze the computation speed and stability of the four recursive formulae by extensive numerical experiments.

Gravitational and gravitoscalar thermodynamics

Gravitational thermodynamics and gravitoscalar thermodynamics with S2 × ℝ boundary geometry are investigated through the partition function, assuming that all Euclidean saddle point geometries contribute to the path integral and dominant ones are in the B3 × S1 or S2 × Disc topology sector. In the first part, I concentrate on the purely gravitational case with or without a cosmological constant and show there exists a new type of saddle point geometry, which I call the “bag of gold(BG) instanton,” only for the Λ > 0 case. Because of this existence, thermodynamical stability of the system and the entropy bound are absent for Λ > 0, these being universal properties for Λ ≤ 0. In the second part, I investigate the thermodynamical properties of a gravity-scalar system with a φ2 potential. I show that when Λ ≤ 0 and the boundary value of scalar field Jφ is below some value, then the entropy bound and thermodynamical stability do exist. When either condition on the parameters does not hold, however, thermodynamical stability is (partially) broken. The properties of the system and the relation between BG instantons and the breakdown are discussed in detail.

Poynting vector, super-Poynting vector, and principal observers in electromagnetism and general relativity

In electromagnetism, the concept of Poynting vector as measured by an observer is well known. A mathematical analogue in general relativity is the super-Poynting vector of the Weyl tensor. Observers for which the (super-)Poynting vector vanishes are called principal. When, at a given point, the electromagnetic field is non-null, or the gravitational field is of Weyl–Petrov type I or D, principal observers instantaneously passing through that point always exist. We survey characterizations of such observers and study their relation to arbitrary observers. In the non-null electromagnetic case it is known that, given any observer, there is a principal observer which moves relative to the first in the direction of his Poynting vector. Replacing Poynting by super-Poynting yields a possible gravitational analogue; we show that this analogy indeed holds for any observer when the Petrov type is D, but only for a one-dimensional variety of observers when the Petrov type is I. We provide algorithms to obtain the principal observers directly from the electric and magnetic fields (in the electromagnetic case) or electric and magnetic parts of the Weyl tensor (in the gravitational case) relative to an arbitrary observer. It is found that in Petrov type D doubly aligned non-null Einstein–Maxwell fields (which include all classical charged black hole solutions) the Poynting and super-Poynting vectors are aligned, at each point and for each observer, and the principal observers coincide. Our results are illustrated in simple examples.

Deflection angle with electromagnetic interaction and gravitational-electromagnetic dual lensing

The trajectory deflection and gravitational-electromagnetic dual lensing (GEL) of charged signal in general charged static and spherically symmetric spacetimes are considered in this work. We show that the perturbative approach previously developed for neutral particles can be extended to the electromagnetic interaction case. The deflection angle still takes a (quasi-)series form and the finite distance effect of both the source and observer can be taken into account. Comparing to pure gravitational case, the apparent angles of the images in the GEL, their magnifications and time delay all receive the electromagnetic corrections starting from the first non-trivial order. The sign and relative size of the leading corrections are determined by ∼Q/M q/E where M, Q, q, E are the spacetime mass and charge, and signal particle charge and energy respectively. It is found that for qQ > 0 (or < 0), the electromagnetic interaction will decrease (or increase) the deflection angle, and in GEL the impact parameters, apparent angles, magnifications and total travel time for each image. The time delay is increased for small β and qQ > 0, and otherwise always increased regardless the sign of qQ. The results are then applied to the deflection and GEL of charged protons in cosmic rays in Reissner-Nordstrom, charged dilaton and charged Horndeski spacetimes.