Gravitational Field Research Articles

Properties of dynamical black hole entropy

We study the first law for non-stationary perturbations of a stationary black hole whose event horizon is a Killing horizon, that relates the first-order change in the mass and angular momentum to the change in the entropy of an arbitrary horizon cross-section. Recently, Hollands, Wald and Zhang [1] have shown that the dynamical black hole entropy that satisfies this first law, for general relativity, is Sdyn = (1 − v∂v)SBH, where v is the affine parameter of the null horizon generators and SBH is the Bekenstein-Hawking entropy, and for general diffeomorphism covariant theories of gravity Sdyn = (1 − v∂v)SWall, where SWall is the Wall entropy. They obtained the first law by applying the Noether charge method to non-stationary perturbations and arbitrary cross-sections. In this formalism, the dynamical black hole entropy is defined as an “improved” Noether charge, which is unambiguous to first order in the perturbation. In the present article we provide a pedagogical derivation of the physical process version of the non-stationary first law for general relativity by integrating the linearised Raychaudhuri equation between two arbitrary horizon cross-sections. Moreover, we generalise the derivation of the first law in [1] to non-minimally coupled matter fields that are smooth on the horizon, using boost weight arguments rather than Killing field arguments, and we relax some of the gauge conditions on the perturbations by allowing for non-zero variations of the horizon Killing field and surface gravity. Finally, for f(Riemann) theories of gravity we show explicitly using Gaussian null coordinates that the improved Noether charge is Sdyn = (1 − v∂v)SWall, which is a non-trivial check of [1].

Towards enhanced elevated-temperature mechanical properties by regulation of eutectic structure in Al-Cu-Si alloys solidified under super-gravity field

High volume fraction of tri-phase eutectic structure is challenging to obtain in ternary alloys through conventional methods. Currently, we reported that super-gravity solidification can be utilized to obtain high-volume fraction Al-Al2Cu-Si tri-phase eutectic structure in Al-Cu-Si alloys. The refinement of the eutectic structure can be detected mainly due to the enhancement of mass transfer, affecting the nucleation rate and growth of eutectic phase in the Al-Cu-Si eutectic system. Interestingly, the refined Al2Cu and Si eutectic structures are stable even up to 300 °C thermal exposure without obvious coarsening, contributing to the enhancement of elevated-temperature mechanical properties (∼268.8±8.8 MPa at 300 °C), which is ∼51.9 % higher than those solidified under normal gravity field (∼177.0±1.9 MPa). The refinement and homogeneous distribution of more Al-Al2Cu-Si tri-phase eutectic structures provide better stress dispersion, favoring crack initiation and propagation within the tri-phase eutectics, resulting in a transition from brittle fracture to ductile fracture mode.

Contribution Analysis of the Addition of Laser Ranging Interferometry on GRACE-FO Gravity Field Estimation under Different Accelerometer Calibration Schemes

To investigate the influence of K-Band microwave ranging (KBR) and Laser Ranging Interferometer (LRI) inter-satellite range-rates from the GRACE-FO mission on the recovery of gravity fields, this paper explores two distinct methods for accelerometer scale estimation, namely diagonal matrix parameterization and full matrix parameterization. Integrating both KBR and LRI data from GRACE-FO into gravity field recovery, we have derived six new time series of monthly gravity field solutions based on two different accelerometer calibration approaches, entitled KBRLRI_Com (incorporating both KBR and LRI data), KBR_Only (using pure KBR data), LRI_Only (solely utilizing LRI data) truncate to degree and order 60 over the period from January 2019 to June 2022. Analyses of KBRLRI_Com, KBR_Only, and LRI_Only reveal the following findings: (1) LRI demonstrates significant advantages over KBR in both diagonal and full scale matrix scenarios, across the time and frequency domains, which shows lower root mean square (RMS) values of the post-fit residuals in the time domain and substantially lower noise at higher frequencies; (2) the comparison between KBRLRI_Com, KBR_Only, and LRI_Only in terms of geoid degree error suggests that KBRLRI_Com agrees well with KBR_Only and LRI_Only at the low degrees (below degree 30), while effectively reducing the high-frequency noise, especially when the diagonal scale matrices are employed; the benefit of KBRLRI_Com solution is mainly attributed to the enhanced quantity of ranging observations. (3) by analyzing the signal and noise over the globe, ocean, Australian continent, and Amazon and Ganges river basins, we find that KBRLRI_Com is highly consistent with KBR_Only and LRI_Only in terms of spatial signals, while KBRLRI_Com exhibits less spatial noise; when employing diagonal and full matrices, the noise of gravity field solutions derived by incorporating LRI data is mitigated by 8.8% and 3.9% over the global ocean compared to KBR_Only, as well as 7.6% and 3.2% in the Australian continent; especially in the diagonal matrix case, the incorporation of LRI measurements for calculating monthly gravity fields results in a more pronounced decrease of spatial noise. (4) in the oceanic region, KBR_Only and LRI_Only have average RMS values of 11.3 cm and 11.6 cm in diagonal matrices, respectively, and 10.2cm and 11.6cm in the full matrix, indicating a generally comparable level of accuracy between the two models.

Understanding gait alterations: trunk flexion and its effects on walking neuromechanics.

Evolutionary and functional adaptations of morphology and postural tone of the spine and trunk are intrinsically shaped by the field of gravity in which humans move. Gravity also significantly impacts the timing and levels of neuromuscular activation, particularly in foot-support interactions. During step-to-step transitions, the centre of mass velocity must be redirected from downwards to upwards. When walking upright, this redirection is initiated by the trailing leg, propelling the body forward and upward before foot contact of the leading leg, defined as an anticipated transition. In this study, we investigated the neuromechanical adjustments when walking with a bent posture. Twenty adults walked on an instrumented treadmill at 4 km h-1 under normal (upright) conditions and with varying degrees of anterior trunk flexion (10, 20, 30 and 40deg). We recorded lower-limb kinematics, ground reaction forces under each foot, and the electromyography activity of five lower-limb muscles. Our findings indicate that with increasing trunk flexion, there is a lack of these anticipatory step-to-step transitions, and the leading limb performs the redirection after the ground collision. Surprisingly, attenuating distal extensor muscle activity at the end of stance is one of the main impacts of trunk flexion. Our observations may help us to understand the physiological mechanisms and biomechanical regulations underlying our tendency towards an upright posture, as well as possible motor control disturbances in some diseases associated with trunk orientation problems.

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

Combination of altimetry crossovers and Doppler observables for orbit determination and geodetic parameter recovery: Application to Callisto

An accurate knowledge of the orientation, the tidal deformability, and the gravity field of a celestial body is fundamental to provide constraints on its internal structure. These quantities may be retrieved by processing radiometric tracking and altimetry data from a probe in orbit around such body. This paper presents a method to combine altimetry crossovers with two-way Doppler tracking observations at normal equation level, using the Bernese GNSS Software and the pyXover software library. This method was applied to a proposed 200km altitude orbiter around Callisto, a privileged destination for the upcoming phase of Solar System exploration. Enhancing “standard” Doppler tracking with altimetry generally benefited both orbit determination and a joint estimation of the orientation of the north pole and of planetary librations. The retrieval of low-degree gravity field parameters was also improved by the addition of altimetry data. However, the improvements on the estimated parameters were highly dependent on the characteristics of the simulation, e.g., the underlying topography roughness. Overall, combining radioscience with altimetry data accounted for a visible reduction of correlations among estimated parameters, while also allowing for a consistent estimation of the “vertical” Love number h2 along with gravity.

Gravitational spinoptics in a curved space-time

In this paper we discuss propagation of the weak high-frequency gravitational waves in a curved spacetime background. We develop a so-called spinoptics approximation which takes into account interaction of the spin of the field with the curvature of the background metric. This is achieved by modifying the standard geometric optics approximation by including the helicity sensitive terms of the order 1/ω in the eikonal equation. The novelty of the approach developed in this paper is that instead of study of the high-frequency expansion of the equations for the gravitational field perturbations we construct the effective action for the gravitational spinoptics. The gravitational spinoptics equations derived by variation of the effective action correctly reproduce the earlier obtained results. However, the proposed effective action approach is technically more simple and transparent. It allows one to reduce the study of the high-frequency gravitational waves to study classical dynamics of massless particles with internal discrete degree of freedom (helicity). The formalism is covariant and it can be applied for arbitrary vacuum space-time background.

On the Equivalence of Gravitational Waves Formulations in Teleparallel Gravity and General Relativity

Abstract Gravitational waves have become a vital way to test our current understanding of gravity. In this work, we explore the formulations of gravitational waves in two distinct theories of gravity: the Teleparallel Equivalent of General Relativity (TEGR), a subclass of Teleparallel Gravity (TG), and General Relativity (GR). By linearizing the gravitational field equations in each theory, we demonstrate that the formulations of gravitational waves in TEGR and GR are equivalent. This equivalence suggests that the teleparallel formalism can effectively describe gravitational waves, providing a novel perspective on the topic and potentially broadening the scope of gravitational wave research.

Vertical Displacements and Sea‐Level Changes in Eastern North America Driven by Glacial Isostatic Adjustment: An Ensemble Modeling Approach

AbstractGlacial isostatic adjustment (GIA) describes the response of the solid Earth, oceans, and gravitational field to the spatio‐temporal evolution of ice sheets during a glacial cycle. Present‐day vertical displacements and sea‐level changes vary throughout eastern North America in response to the melting of the Laurentide Ice Sheet following the Last Glacial Maximum. We use the open‐source software SELEN4.0 (a SealEveL EquatioN solver) to investigate the influence of GIA on vertical land motions and sea‐level changes in eastern North America. Further, we evaluate the uncertainties associated with the lithospheric thickness and viscosity structure using an ensemble modeling approach (129,956 total simulations). We identify the best‐fitting rheological profiles by comparing modeled vertical displacements to vertical velocity rates derived from Global Positioning System (GPS). We find a general pattern of subsidence (causing accelerated relative sea‐level rise) in the eastern United States region and uplift (causing relative sea‐level fall) in the eastern Canada region consistent with previous studies for two tested ice sheet models (ICE‐6G(VM5a) and ICE‐7G(VM7)). Overall, we find lower rates of modeled vertical displacement using ICE‐6G(VM5a) compared with ICE‐7G(VM7), which produces lower residuals when compared with the GPS‐derived vertical velocity rates. Our ensemble analysis identifies adjustments to the nominal VM5a and VM7 viscosity models that improve fits to the GPS‐imaged vertical velocity rates throughout eastern North America and on the North American Atlantic Coast. The differences in our best‐fitting models for inland versus coastal regions highlight the importance of exploring lateral viscosity variations for GIA modeling throughout North America and elsewhere.

Fully metallic geodesic lenses as analog electromagnetic models of static and spherically symmetric gravitational fields

Fully metallic geodesic lenses as analog electromagnetic models of static and spherically symmetric gravitational fields

Tectonics of Cauvery basin (India) in onshore and offshore portions

Morphometric studies coupled with geophysical investigations from petroliferous basins provide valuable information about tectonics of the regions. We deduce six morphologic parameters from the River Cauvery's designated (sub)watersheds and perform dendrogram clustering analysis. The main channels of the (sub)watersheds showcase steeper gradient and meandering nature with less concavity. On the other hand, the distributaries portray lesser gradient and straight nature with greater concavity. From the offshore region of the Cauvery basin, we perform gravity modeling. The results show variations in gravity values from low to high, -150 to 50 mGal probably because of the basement ridge-depression features. This anomaly orientation indicates the extension of the sub-basin in the region. A strong NE-SW gravity trend possibly suggests a high-density plug at mid-crust. The low gravity values near the Vedaranyam region indicate low-density rocks within the basement. Places of high gravity anomaly show a decrease in bathymetry depth and vice versa. This indicates isostatic compensation of the Moho geometry, possibly due to crustal thinning in the offshore areas. Isostatic regional gravity fields are manifested by Moho relief and can have implications on the lithospheric mechanical strengths. Bathymetric studies indicate multi-directional slopes. Relatively flat areas can indicate depocenters. E-W trending lineaments mapped on the Precambrian basement from a portion of the study area is presumably a manifestation of Cauvery Shear zone, which also persists in the offshore region.

Dynamic changes of gravity field before the Luding MS6.8 earthquake and its crustal material migration characteristics

On September 5, 2022, an earthquake of magnitude MS6.8 occurred in Luding County, Sichuan Province. This earthquake occurred at the key part of the southeast-clockwise extrusion of material on the eastern margin of the Qinghai–Xizang Plateau, the Y-shaped confluence of the Xianshuihe, Longmenshan and Anninghe fault zones. In this study, the three-dimensional dynamic crustal density changes in the earthquake area are obtained by the typical gravity change data from 2019 to 2022 before the earthquake and gravity inversion by growing bodies. The results indicate that gravity changes presented an obvious four-quadrant and gradient belt distribution in the Luding area before the earthquake. The three-dimensional density horizontal slices show that small density changes occurred at the epicenter in the mid-to-upper crust between 2019.9 - 2020.9 and 2019.9–2021.9. At the same time, the surrounding areas exhibited a positive and negative quadrant distribution. These observations indicate that the source region was likely in a stable locked state, with locking-in shear forces oriented in the NW and NE directions. From 2021.9 to 2022.8, the epicentral region showed negative density changes, indicating that the source region was in the expansion stage, approaching a near-seismic state. The three-dimensional density vertical slices reveal a southeastward migration of positive and negative densities near the epicenter and on the western of the Xianshuihe Fault Zone, indicating that the material is flowing out to the southeast. The observed local negative density changes at the epicenter along the Longmenshan Fault Zone are likely associated with the NE-oriented extensional stress shown by the seismic source mechanism. The above results can provide a basis for interpreting pre-earthquake gravity and density changes, thereby contributing to the advancement of earthquake precursor theory.

Resonance and surfactant effects on the interaction of a rising bubble and a sedimenting particle in an oscillating fluid

This paper considers the interaction of a rising bubble and a sedimenting fine particle in a liquid with a soluble surfactant subjected to the gravity field, forced flow, and vibrations. The frequency of vibrations is close to the resonance value. The particle motion trajectories are calculated, taking into account its inertia, fluid viscosity, Basset force, buoyancy force, and the mean force acting on the particle due to the inhomogeneity of the pulsating field. The collision efficiency is evaluated, and the cross sections of the sedimenting particles captured by the rising bubble are found. It is demonstrated that even at low vibration intensity but in the presence of a weak surfactant effect, a significant increase in the probability of particle–bubble collision and in the capture cross section can be achieved. The obtained results can be used to intensify the flotation processes by ultrasonic treatment.

Accounting for the expansion of the universe using an energy/momentum model to construct the space-time metric

Background The success of the theories of special and general relativity in describing localised phenomena, such as objects undergoing high speed motion or located in gravitational fields, needs no further elaboration. However, when applied to the evolution of the universe several problems arise which can require an additional model, e.g., inflation during the early expansion, and adjustments to parameters to account for phenomena such as the late-time acceleration of the universe. Methods Focusing on the difference between the ways in which space and time are measured, this paper shows that there are two paths which allow the equations of special relativity to be produced from the same basic postulates. Results Both the standard theory and the energy/momentum, or dynamic model, utilise the Minkowski metric, but with different coordinate systems. The dynamic model transforms Cartesian coordinates into an Euclidean form by multiplying the coordinates by functions of γ (=(1–ν 2)c 2)-1⁄2). When utilising these coordinates, the relativistic equations are unchanged for local phenomena such as the Lorentz coordinate transformation and the energy/momentum equation for high-velocity objects. Conclusions However, the derived coordinates alter the perceived overall structure of the universe in a manner that, for the simplest model under this system, allows the reproduction of observed cosmological features, such as the intrinsic flatness of the universe and the apparent late-time acceleration of its expansion, without the need of additional models or changes in parameter values.

Electromagnetic field in an expanding universe

An expanding universe filled with a massive electromagnetic field is studied. The electromagnetic field (photon) is found to acquire mass during an inflationary period. The photon mass varies with the Hubble parameter as m γ = 3ℏ H/c 2, where H, ℏ and c are the Hubble parameter, Planck constant, and speed of light. A leftover magnetic field since the inflationary era permeates the space between galaxies and interstellar media. The strength of the electromagnetic field changes with the scale factor, E ∝ a −3. The photon gains mass when interacting with the gravitational field while remaining massless in flat space. The massive electromagnetic field adds energy density to the Universe proportional to m 2.

A Quantization for the Toy Model of Black Hole and a Suggestion to the Unification Theory

In my previous paper about the correlations between quantum mechanics and the four different natural forces, I suggested that there may be a fifth force or even the existence of a new particle. However, it remains a mystery to physicists, even after nearly a hundred years, that they cannot unify these four natural forces. This may be because there are indeed too many variables for them to consider. Also, there is a possibility that quantum mechanics may coexist with quantum field theory. In the present paper, this author proposes that there may be a misconception in the computational equation for relative time or gaps in the system for measuring time between quantum mechanics and general relativity. Hence, one may still not be able to unify these natural forces. This author suggests that we may need to rewrite parts of quantum mechanics – the Schrödinger equation – or even the general relativity equation. Additionally, this author proposes that there may be a bridge equation converting between quantum mechanics and general relativity. Moreover, this author has employed the non-trivial zeta zeros to simulate the black hole or the so-called black hole toy model. In practice, there may be an electromagnetic field surrounding the boundary of the black hole, as well as the existence of a continuum along the boundary contour. This author hopes that, in such a case, we may decode those high-frequency electromagnetic waves into useful information and take a further step toward verifying Stephen Hawking’s famous theory on black hole radiation and information entropy. In fact, this author has used the HKLam statistical model theory to express the electromagnetic field energy-stress tensor (with the possibility of quantization) to analogically establish a quantized model for the Einstein Gravitational Field Equation. Hence, the problem of quantum gravity may then be solved. Last but not least, this author also expects that humanity may finally find a way to unify quantum mechanics and general relativity through modifications to the current quantum gravity theory, such as my proposed bridge-converting equation, etc.

Anisotropic model for dark matter halo with Hernquist density profile

Using the observed flat galactic rotation curve feature and taking into account the presence of anisotropic dark matter with a Hernquist density profile, the space-time geometry of the halo region in galaxies is derived. The gravitational field inside the halo is attractive in nature, and the resultant space-time is flat. We find that our solution affirms the existence of stable circular orbits in the dark halo region. The expression for the equation of state parameter indicates that the matter within the dark halo is non-exotic in nature. An analysis is also conducted on several other aspects of the solution.

Строение центральной части палеопротерозойского Лосевского террейна по данным плотностного моделирования

The results of detailed three-dimensional density modeling of the upper crust of the central part of the Losevsky terrane located in the northern part of the Voronezh crystal massif are considered. The distribution of absolute crustal density values of to a depth of 16 km was obtained as a result of the inversion of local anomalies of the gravitational field in the Bouguer reduction. The solution of the inverse problem of gravimetry was implemented within the framework of the initial model, which was formed using digital geological and geophysical data from the study area: the regional density model of the East European Platform lithosphere and its corresponding regional gravity field; the field of local gravity anomalies, representing the difference between the observed and regional fields of the lithosphere model; generalized information on the density of sedimentary cover rocks and the crystal basement; the thickness scheme of the «gravity-active» layer, obtained on the basis of statistical analysis of the anomalous field; geological and topographic information. Quantitative interpretation of the gravimetry data allowed us to obtain fundamentally new information about the geological structure of the upper crust of the study area. The reliability of the obtained results is confirmed by the consistency of a priori data and the correspondence of the total model (local plus regional) and observed gravitational fields.

Semi-analytic approximate time-optimal asteroid landing with dimensionality reduction shooting

Pinpoint landing on asteroid is very challenging due to the uncertainties of the gravitational field, which highlights the urgent need for autonomous landing algorithms under such uncertainties. Some well-known guidance algorithms, such as APDG and real-time convex programming, either ignore thrust boundary constraints or are severely time consuming. In addition, these algorithms have the same difficulty in analyzing landing errors due to model uncertainty propagation, making the reliability of the algorithms in actual landings questionable. To address these challenges, we propose a landing framework that combines model identification, trajectory analytical solution and closed-loop corrections to achieve near-optimal real-time landing control. Firstly, we simplified the landing scenario, focusing on reducing the dimension of shooting variables to enable rapid trajectory calculation. Secondly, we derived error propagation equations and established criteria for trajectory replanning based on landing error prediction. Finally, we validated our approach with practical examples of the Shoemaker probe landing on Eros 433. Through real-time landing error estimation, our proposed framework enables spacecraft to achieve near time-optimal land within a given error threshold.

Energy of gravitational radiation and the background energy of the space-time

We address the issue of gravitational radiation in the context of the Bondi-Sachs space-time, and consider the expression for the gravitational energy of the radiation obtained in the realm of the teleparallel equivalent of general relativity (TEGR). This expression is independent of the radial distance (i.e., of powers of 1/r) and depends exclusively on the functions c(u,θ,ϕ) and d(u,θ,ϕ), which yield the news functions (u is the retarded time, u=t−r). We investigate the mathematical and physical features of this energy expression in the simpler framework of axial symmetry. Once a burst of gravitational radiation takes place in a self gravitating system, that leads to a loss of the Bondi mass, gravitational radiation is emitted throughout the whole space-time. The existence and presence of this radiation in the background structure of the space-time is consistent with the analysis developed by Papapetrou, and Hallidy and Janis, who found no proof that a gravitational system that emits a burst of gravitational radiation is preceded and followed by two stationary gravitational field configurations, namely, it seems that it is impossible for a gravitational system, which is initially stationary, to return to a stationary state after emitting a burst of axially symmetric gravitational radiation, in which case the space-time is not even asymptotically stationary. Therefore, it is plausible that the gravitational energy of radiation is present in the background structure of the space-time, and this is the energy predicted in the TEGR. This analysis leads us to conjecture that the noise detected in the large terrestrial gravitational wave observatories is intrinsically related to the background gravitational radiation.