Journal articles: 'Newtonian, post-newtonian limit'

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De Laurentis, Mariafelicia, and Antonio Jesus Lopez-Revelles. "Newtonian, Post-Newtonian and Parametrized Post-Newtonian limits of f(R, 𝒢) gravity." International Journal of Geometric Methods in Modern Physics 11, no. 10 (November 2014): 1450082. http://dx.doi.org/10.1142/s0219887814500820.

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We discuss in detail the weak field limit of f(R, 𝒢) gravity taking into account analytic functions of the Ricci scalar R and the Gauss–Bonnet invariant 𝒢. Specifically, we develop, in metric formalism, the Newtonian, Post-Newtonian (PN) and Parametrized Post-Newtonian (PPN) limits starting from general f(R, 𝒢) Lagrangian. The special cases of f(R) and f(𝒢) gravities are considered. In the case of the Newtonian limit of f(R, 𝒢) gravity, a general solution in terms of Green's functions is achieved.

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CIUFOLINI, IGNAZIO. "NEW CLASS OF METRIC THEORIES OF GRAVITY NOT DESCRIBED BY THE PARAMETRIZED POST-NEWTONIAN (PPN) FORMALISM." International Journal of Modern Physics A 06, no. 30 (December 20, 1991): 5511–32. http://dx.doi.org/10.1142/s0217751x91002604.

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After an introduction to theories of gravity alternative to general relativity, metric theories (Sec. 1) and the parametrized post-Newtonian (PPN) formalism (Sec. 2), we define a new class of metric theories of gravity (Sec. 3). It turns out that the post-Newtonian approximation of these new theories is not described by the PPN formalism (Sec. 4); in fact, in the limit of weak field and slow motions, the post-Newtonian expression of the metric tensor contains an, a priori, infinite set of new terms and correspondingly an, a priori, infinite set of new PPN parameters. As a consequence, the parametrized post-Newtonian formulas describing the classical relativistic tests should include these new parameters, and therefore the experimental values of the classical relativistic effects should not be used to put limits only on the standard ten PPN parameters. Finally, we note that a subset of this new class of theories has the same post-Newtonian limit and value of the PPN parameters as general relativity, and therefore is automatically in agreement with the classical general-relativistic tests (Sec. 4, theory III).

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Bahamonde, Sebastian, Konstantinos F. Dialektopoulos, Manuel Hohmann, and Jackson Levi Said. "Post-Newtonian limit of teleparallel Horndeski gravity." Classical and Quantum Gravity 38, no. 2 (December 16, 2020): 025006. http://dx.doi.org/10.1088/1361-6382/abc441.

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Gonzalez-Espinoza, Manuel, Giovanni Otalora, Lucila Kraiselburd, and Susana Landau. "Parametrized post-Newtonian formalism in higher-order Teleparallel Gravity." Journal of Cosmology and Astroparticle Physics 2022, no. 05 (May 1, 2022): 010. http://dx.doi.org/10.1088/1475-7516/2022/05/010.

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Abstract We study the parametrized post-Newtonian (PPN) limit of higher-derivative-torsion Modified Teleparallel Gravity. We start from the covariant formulation of modified Teleparallel Gravity by restoring the spin connection of the theory. Then, we perform the post-Newtonian expansion of the tetrad field around the Minkowski background and find the perturbed field equations. We compute the PPN metric for the higher-order Teleparallel Gravity theories which allows us to show that at the post-Newtonian limit this more general class of theories are fully conservative and indistinguishable from General Relativity . In this way, we extend the results that were already found for F(T) gravity in previous works. Furthermore, our calculations reveal the importance of considering a second post-Newtonian (2PN) order approximation or a parametrized post-Newtonian cosmology (PPNC) framework where additional perturbative modes coming from general modifications of Teleparallel Gravity could lead to new observable imprints.

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Sasaki, Misao. "Post-Newtonian approximation in the test particle limit." Banach Center Publications 41, no. 2 (1997): 75–83. http://dx.doi.org/10.4064/-41-2-75-83.

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Dyadina, Polina, and Sofya Labazova. "The post-Newtonian limit of hybrid f(R)-gravity." EPJ Web of Conferences 191 (2018): 07009. http://dx.doi.org/10.1051/epjconf/201819107009.

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The post-Newtonian formalism is developed for hybrid f(R)-gravity. The model is tested in the weak field regime of the Solar System. Using scalartensor representation of hybrid f(R)-gravity we impose restrictions on the parameters of the theory and show that hybrid f(R)-gravity is not ruled out by the observations in the weak field limit.

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Kol, Barak, Michele Levi, and Michael Smolkin. "Comparing space+time decompositions in the post-Newtonian limit." Classical and Quantum Gravity 28, no. 14 (June 9, 2011): 145021. http://dx.doi.org/10.1088/0264-9381/28/14/145021.

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Li, Weijun, Bo Yang, Cunliang Ma, Xia Zhou, Zhongwen Feng, and Guansheng He. "Periastron precession due to a Janis–Newman–Winicour wormhole in the weak field limit." Modern Physics Letters A 36, no. 22 (July 20, 2021): 2150164. http://dx.doi.org/10.1142/s0217732321501649.

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The precession effect of periastron for a massive test particle in the spacetime of a Janis–Newman–Winicour wormhole is studied in the weak field limit. Based on the metric of this static and spherically symmetric wormhole in harmonic coordinates, we derive the second post-Newtonian dynamics of the particle. The second-order orbital precession of periastron is then obtained via a post-Newtonian iterative technique under the Wagoner–Will–Epstein–Haugan representation. Our result is found to be consistent with the classical precession effect when the asymptotic scalar charge is dropped.

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Alfaro, Jorge, and Pablo González. "δ Gravity: Dark Sector, Post-Newtonian Limit and Schwarzschild Solution." Universe 5, no. 5 (April 26, 2019): 96. http://dx.doi.org/10.3390/universe5050096.

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We present a new kind of model, which we call δ Theories, where standard theories are modified including new fields, motivated by an additional symmetry ( δ symmetry). In previous works, we proved that δ Theories just live at one loop, so the model in a quantum level can be interesting. In the gravitational case, we have δ Gravity, based on two symmetric tensors, g μ ν and g ˜ μ ν , where quantum corrections can be controlled. In this paper, a review of the classical limit of δ Gravity in a Cosmological level will be developed, where we explain the accelerated expansion of the universe without Dark Energy and the rotation velocity of galaxies by the Dark Matter effect. Additionally, we will introduce other phenomenon with δ Gravity like the deflection of the light produced by the sun, the perihelion precession, Black Holes and the Cosmological Inflation.

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Clifton, Timothy, Máximo Bañados, and Constantinos Skordis. "The parameterized post-Newtonian limit of bimetric theories of gravity." Classical and Quantum Gravity 27, no. 23 (November 15, 2010): 235020. http://dx.doi.org/10.1088/0264-9381/27/23/235020.

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Sáez, D. "Parametrized-post-Newtonian limit of the scalar-tetradic theory A." Physical Review D 32, no. 10 (November 15, 1985): 2575–78. http://dx.doi.org/10.1103/physrevd.32.2575.

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Dyadina, P. I., S. P. Labazova, and S. O. Alexeyev. "Post-Newtonian Limit of Hybrid Metric-Palatini f(R)-Gravity." Journal of Experimental and Theoretical Physics 129, no. 5 (November 2019): 838–48. http://dx.doi.org/10.1134/s1063776119110025.

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Goldoni, O., M. F. A. da Silva, R. Chan, and G. Pinheiro. "Vaidya solution in general covariant Hořava–Lifshitz gravity with the minimum coupling and without projectability: Infrared limit." International Journal of Modern Physics D 24, no. 02 (February 2015): 1550021. http://dx.doi.org/10.1142/s0218271815500212.

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In this paper, we have studied nonstationary radiative spherically symmetric spacetime, in general covariant theory (U(1) extension) of the Hořava–Lifshitz gravity with the minimum coupling, in the parameterized post-Newtonian (PPN) approximation, without the projectability condition and in the infrared limit. The Newtonian prepotential φ was assumed null. We have shown that there is not the analog of the Vaidya's solution in the Hořava–Lifshitz Theory with the minimum coupling, as we know in the General Relativity Theory (GRT).

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Friedman, Yaakov, Tzvi Scarr, and Joseph Steiner. "A geometric relativistic dynamics under any conservative force." International Journal of Geometric Methods in Modern Physics 16, no. 01 (January 2019): 1950015. http://dx.doi.org/10.1142/s0219887819500154.

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Riemann’s principle “force equals geometry” provided the basis for Einstein’s General Relativity — the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton’s First Law — the Generalized Principle of Inertia stating that: An inanimate object moves inertially, that is, with constant velocity, in its own spacetime whose geometry is determined by the forces affecting it. Classical Newtonian dynamics is treated within this framework, using a properly defined Newtonian metric with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new Relativistic Newtonian Dynamics for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.

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Roshan, Mahmood, and Fatimah Shojai. "Notes on the post-Newtonian limit of the massive Brans–Dicke theory." Classical and Quantum Gravity 28, no. 14 (June 7, 2011): 145012. http://dx.doi.org/10.1088/0264-9381/28/14/145012.

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Frauendiener, Jörg, and László B. Szabados. "A note on the post-Newtonian limit of quasi-local energy expressions." Classical and Quantum Gravity 28, no. 23 (November 16, 2011): 235009. http://dx.doi.org/10.1088/0264-9381/28/23/235009.

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Lotte, Baljeet Kaur, and Subodha Mishra. "Effect of post-Newtonian-like self-energy, quantum gravity and exchange correlations on Schwarzschild black hole: Application of uncertainty principle." Modern Physics Letters A 35, no. 11 (February 7, 2020): 2050081. http://dx.doi.org/10.1142/s0217732320500819.

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The expressions for the corrected radius and the Hawking temperature of a Schwarzschild black hole are derived by calculating the total energy of a self-gravitating system of N fermions when the corrections to gravitational interaction due to post-Newtonian-like self-energy due to two graviton exchange- and one-loop contribution of quantum gravity effect are included. Since the particles are fermions, the exchange-correlation energy is also included consistently. It is found that though the three corrections are small, the correction due to the exchange-correlation is much more than the other two. The configuration of the many-particle system that we study is possible since it has no Buchdahl limit in the post-Newtonian approximation.

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LIU, DAO-JUN, BIN YANG, and XING-HUA JIN. "A MODIFIED MODEL OF NONLOCALLY CORRECTED GRAVITY: COSMOLOGY AND THE NEWTONIAN LIMIT." Modern Physics Letters A 26, no. 30 (September 28, 2011): 2287–97. http://dx.doi.org/10.1142/s0217732311036553.

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A modified form of nonlocally corrected theory of gravity is investigated in the context of cosmology and the Newtonian limit. This form of nonlocal correction to classic Einstein–Hilbert action can be locally represented by a triple-scalar–tensor theory in which one of the scalar degrees of freedom is phantom-like and the other two are quintessence-like. We show that there exists a stable de Sitter solution for the cosmological dynamics if a suitable form of potential function V(ϕ) (or equivalently, f(R)) is selected. However, no matter what a potential function is selected, there is always an early time repeller solution corresponding to a radiation dominated universe. Besides, the equations for linear scalar perturbations are presented and it is shown that the form of potential function V(ϕ) is stringently constrained by the solar system test, although the post-Newtonian parameter γ is not directly affected by this function.

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Le Tiec, Alexandre, and Luc Blanchet. "The close-limit approximation for black hole binaries with post-Newtonian initial conditions." Classical and Quantum Gravity 27, no. 4 (January 28, 2010): 045008. http://dx.doi.org/10.1088/0264-9381/27/4/045008.

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Roxburgh, Ian W. "Post-Newtonian limit of Finsler space theories of gravity and solar system tests." General Relativity and Gravitation 24, no. 4 (April 1992): 419–31. http://dx.doi.org/10.1007/bf00760417.

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Frauendiener, J., and L. B. Szabados. "Corrigendum: A note on the post-Newtonian limit of quasi-local energy expressions." Classical and Quantum Gravity 29, no. 6 (February 27, 2012): 069501. http://dx.doi.org/10.1088/0264-9381/29/6/069501.

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MOFFAT, J. W., and V. T. TOTH. "MODIFIED JORDAN–BRANS–DICKE THEORY WITH SCALAR CURRENT AND THE EDDINGTON–ROBERTSON γ-PARAMETER." International Journal of Modern Physics D 21, no. 12 (November 2012): 1250084. http://dx.doi.org/10.1142/s0218271812500848.

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The Jordan–Brans–Dicke theory of gravitation, which promotes the gravitational constant to a dynamical scalar field, predicts a value for the Eddington–Robertson post-Newtonian parameter γ that is significantly different from the general relativistic value of unity. This contradicts precision solar system measurements that tightly constrain γ around 1. We consider a modification of the theory, in which the scalar field is sourced explicitly by matter. We find that this leads to a modified expression for the γ-parameter. In particular, a specific choice of the scalar current yields γ = 1, just as in general relativity, while the weak equivalence principle is also satisfied. This result has important implications for theories that mimic Jordan–Brans–Dicke theory in the post-Newtonian limit in the solar system, including our scalar–tensor–vector gravity (STVG) modified gravity theory (MOG).

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Mai, La-Su, Hai-Liang Li, and Pierangelo Marcati. "Non-relativistic limit analysis of the Chandrasekhar–Thorne relativistic Euler equations with physical vacuum." Mathematical Models and Methods in Applied Sciences 29, no. 03 (March 2019): 531–79. http://dx.doi.org/10.1142/s0218202519500155.

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Our results provide a first step to make the formal analysis rigorous in terms of [Formula: see text] proposed by Chandrasekhar [S. Chandrasekhar, The post-Newtonian equations of hydrodynamics in general relativity, Astrophys. J. 142 (1965) 1488–1512; S. Chandrasekhar, post-Newtonian equations of hydrodynamics and the stability of gaseous masses in general relativity, Phys. Rev. Lett. 14 (1965) 241–244], motivated by the methods of Einstein, Infeld and Hoffmann, see Thorne [K. S. Thorne, The general-relativistic theory of stellar structure and dynamics, in Proc. Int. School of Physics “Enrico Fermi,” Course XXXV, at Varenna, Italy, July 12–24, 1965, ed. L. Gratton (Academic Press, 1966), pp. 166–280]. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light [Formula: see text] and suitably small time [Formula: see text] we obtain uniform, with respect to [Formula: see text] “a priori” estimates for the local smooth solutions. Moreover, the smooth solutions of the cylindrically symmetric relativistic Euler equations converge to the solutions of the classical compressible Euler equation at the rate of order [Formula: see text].

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Shao, Lijing, Norbert Wex, and Michael Kramer. "New Constraints on Preferred Frame Effects from Binary Pulsars." Proceedings of the International Astronomical Union 8, S291 (August 2012): 496–98. http://dx.doi.org/10.1017/s1743921312024647.

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AbstractPreferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, and . In the limit of a small orbital eccentricity, and contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of and in the strong-field regime. The constraint on also surpasses its counterpart in the weak-field regime.

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Chapman, James, Jin Woo Jang, and Robert M. Strain. "On the Determinant Problem for the Relativistic Boltzmann Equation." Communications in Mathematical Physics 384, no. 3 (May 7, 2021): 1913–43. http://dx.doi.org/10.1007/s00220-021-04101-2.

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AbstractThis article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma [1]. In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum p to the post collisional momentum $$p'$$ p ′ ; specifically we calculate the determinant for $$p\mapsto u = \theta p'+\left( 1-\theta \right) p$$ p ↦ u = θ p ′ + 1 - θ p for $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] . Afterwards we give an upper-bound for this determinant that has no singularity in both p and q variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (1991) [8] and Guo-Strain (2012) [12]. These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.

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Alba, David, and Luca Lusanna. "The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 1. The equations of motion in arbitrary Schwinger time gauges 1This paper is one of three companion papers published in the same issue of Can. J. Phys." Canadian Journal of Physics 90, no. 11 (November 2012): 1017–76. http://dx.doi.org/10.1139/p11-100.

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We study the coupling of N charged scalar particles plus the electromagnetic field to Arnowitt–Deser–Misner (ADM) tetrad gravity and its canonical formulation in asymptotically Minkowskian space–times without super-translations. To regularize the self-energies, both the electric charge and the sign of the energy of the particles are Grassmann-valued. The introduction of the noncovariant radiation gauge allows reformulation of the theory in terms of transverse electromagnetic fields and to extract the generalization of the Coulomb interaction among the particles in the riemannian instantaneous 3-spaces of global noninertial frames, the only ones allowed by the equivalence principle. Then we make the canonical transformation to the York canonical basis, where there is a separation between the inertial (gauge) variables and the tidal ones inside the gravitational field and a special role of the eulerian observers associated with the 3+1 splitting of space–time. The Dirac hamiltonian is weakly equal to the weak ADM energy. The Hamilton equations in Schwinger time gauges are given explicitly. In the York basis they are naturally divided into four sets: (i) the contracted Bianchi identities; (ii) the equations for the inertial gauge variables; (iii) the equations for the tidal ones; and (iv) the equations for matter. Finally, we give the restriction of the Hamilton equations and of the constraints to the family of nonharmonic 3-orthogonal gauges, in which the instantaneous riemannian 3-spaces have a nonfixed trace 3K of the extrinsic curvature but a diagonal 3-metric. The inertial gauge variable 3K (the general-relativistic remnant of the freedom in the clock synchronization convention) gives rise to a negative kinetic term in the weak ADM energy vanishing only in the gauges with 3K = 0: is it relevant for dark energy and back-reaction? In the second paper will appear the linearization of the theory in these nonharmonic 3-orthogonal gauges to obtain hamiltonian post-minkowskian gravity (without post-newtonian approximations) with asymptotic Minkowski background, nonflat instantaneous 3-spaces and no post-newtonian expansion. This will allow the exploration of the inertial effects induced by the York time 3K in nonflat 3-spaces (they do not exist in newtonian gravity) and to check how well dark matter can be explained as an inertial aspect of Einstein’s general relativity: this will be done in a third paper on the post-minkowskian 2-body problem in the absence of the electromagnetic field and on its 0.5 post-newtonian limit.

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Sobolenko, M., P. Berczik, and R. Spurzem. "Merging timescale for the supermassive black hole binary in interacting galaxy NGC 6240." Astronomy & Astrophysics 652 (August 2021): A134. http://dx.doi.org/10.1051/0004-6361/202039859.

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Context. One of the mechanisms leading to the creation of a supermassive black hole (SMBH) is the so-called hierarchical merging scenario. Central SMBHs at the final phase of interacting and coalescing host-galaxies are observed as SMBH binary (SMBHB) candidates at different separations from hundreds of parsecs to megaparsecs. Aims. Today one of the strongest SMBHB candidates is the ultraluminous infrared galaxy NGC 6240 which was spatially and spectroscopically resolved in X-rays by Chandra. Dynamical calculation of central SMBHBs merging in a dense stellar environment allows us to retrace their evolution from kiloparsec to megaparsec scales. The main goal of our dynamical modeling was to reach the final, gravitational wave emission regime for the model BHs. Methods. We present direct N-body simulations with up to one million particles and relativistic post-Newtonian corrections for the SMBH particles up to 3.5 post-Newtonian terms. Results. Generally speaking, the set of initial physical conditions can strongly affect our merging time estimations. However, within a certain range of our parameters, we do not find any strong correlation between merging time and BH-to-BH mass or BH-to-bulge mass ratios. Varying the numerical parameters (like particle number – N) does not significantly change the merging time limits. From our 20 models, we find an upper limit on the merging time for central SMBHBs of less than ∼55 Myr. This precise number is only valid for our combination of initial mass ratios. Conclusions. Further detailed research of rare dual and multiple BHs in dense stellar environments (based on observational data) could clarify the dynamical co-evolution of central BHs and their host-galaxies.

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Klioner, S. A., E. Gerlach, and M. H. Soffel. "Relativistic aspects of rotational motion of celestial bodies." Proceedings of the International Astronomical Union 5, S261 (April 2009): 112–23. http://dx.doi.org/10.1017/s174392130999024x.

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AbstractRelativistic modelling of rotational motion of extended bodies represents one of the most complicated problems of Applied Relativity. The relativistic reference systems of IAU (2000) give a suitable theoretical framework for such a modelling. Recent developments in the post-Newtonian theory of Earth rotation in the limit of rigidly rotating multipoles are reported below. All components of the theory are summarized and the results are demonstrated. The experience with the relativistic Earth rotation theory can be directly applied to model the rotational motion of other celestial bodies. The high-precision theories of rotation of the Moon, Mars and Mercury can be expected to be of interest in the near future.

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Sobolenko, Margaryta, Peter Berczik, and Rainer Spurzem. "Large scale direct galaxy collision simulations with central supermassive binary black holes." Proceedings of the International Astronomical Union 10, S312 (August 2014): 105–8. http://dx.doi.org/10.1017/s1743921315007620.

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AbstractWe present a set of, large scale direct N-body simulations of the galaxy collision with the central Supermassive Black Hole Binary (SMBHB) system. Based on our simulations which include the accurate Post Newtonian (PN) relativistic dynamical corrections we can estimated the merging time for the real astrophysical object. Each galaxy initially was represented as a set of particles (up to N=500k) with Plummer distribution. The SMBHBs system is described using the two special high mass, i.e. “relativistic”, particles. The interaction between these two particles have an extra PN correction terms (up to 3.5PN). Merging time upper limit was obtained for the closely interacting galaxy system NGC 6240.

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Chrobok, Thoralf, and Horst-Heino von Borzeszkowski. "Relativistic Theory of Irreversible Thermodynamics for Multi-Component Fluids and Its Post-Newtonian Limit in Relation to Classical Extended Thermodynamics." Journal of Non-Equilibrium Thermodynamics 45, no. 2 (April 26, 2020): 133–53. http://dx.doi.org/10.1515/jnet-2019-0069.

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AbstractFirst, the special-relativistic Theory of Irreversible Processes for a multi-component fluid is formulated. It is based on (i) the balance equations of the particle number and the energy-momentum for the total system (i. e., the mixture of the components) as well as the sub-systems (i. e., the components) and (ii) the dissipation inequality and the Gibbs equation for the mixture. In order to allow for reactions between the single components, in contrast to the total system, the sub-systems are assumed to be open, which means that their particle number and energy-momentum are not constrained by conservation laws. Without making any assumptions on the thermodynamic behavior of the interacting components, one arrives at a thermodynamic description of the mixture showing now heat conduction and viscosity. In particular, this makes it possible to calculate the entropy production and, thus, to identify thermodynamic currents and forces. In a second part, the post-Newtonian limit of this theory is calculated to show that for the mixture there result relations known from classical Extended Thermodynamics that partly are corrected by entrainment terms. The mathematical origin and physical consequences of these terms are discussed.

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HE, G., and W. LIN. "ROTO-TRANSLATIONAL EFFECTS ON DEFLECTION OF LIGHT AND PARTICLE BY MOVING KERR BLACK HOLE." International Journal of Modern Physics D 23, no. 04 (March 18, 2014): 1450031. http://dx.doi.org/10.1142/s021827181450031x.

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Velocity effects in first-order Schwarzschild deflection of light and particles have been explored in the previous literature. In this paper, we investigate the roto-translational-motion induced deflection by one moving Kerr black hole with an arbitrary, but constant speed. It is shown that the coupling between the effects of the rotation and the translational motion always exists for both light and particles. The contribution of the roto-translational deflection to the total bending angle is discussed in detail. This ratio takes upper limit for light and it decreases monotonically with increasing translational velocity for a massive particle. For a given translational velocity of black hole, this ratio increases with the particle's velocity. In addition, the post-Newtonian dynamics of the photon and particle is also presented.

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Kopeikin, Sergei M., and Carl R. Gwinn. "Sub-Microarcsecond Astrometry and New Horizons in Relativistic Gravitational Physics." International Astronomical Union Colloquium 180 (March 2000): 303–7. http://dx.doi.org/10.1017/s0252921100000439.

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AbstractAttaining the limit of sub-microarcsecond optical resolution will completely revolutionize fundamental astrometry by merging it with relativistic gravitational physics. Beyond the sub-microarcsecond threshold, one will meet in the sky a new population of physical phenomena caused by primordial gravitational waves from the early universe and/or different localized astronomical sources, space-time topological defects, moving gravitational lenses, time variability of gravitational fields of the solar system and binary stars, and many others. Adequate physical interpretation of these yet undetectable sub-microarcsecond phenomena cannot be achieved on the ground of the “standard” post-Newtonian approach (PNA), which is valid only in the near-zone of astronomical objects having a time-dependent gravitational field. We describe a new, post-Minkowskian relativistic approach for modeling astrometric observations having sub-microarcsecond precision and briefly discuss the light-propagation effects caused by gravitational waves and other phenomena related to time-dependent gravitational fields. The domain of applicability of the PNA in relativistic space astrometry is outlined explicitly.

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Will, Clifford M. "The confrontation between general relativity and experiment." Proceedings of the International Astronomical Union 5, S261 (April 2009): 198–99. http://dx.doi.org/10.1017/s174392130999038x.

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AbstractWe review the experimental evidence for Einstein's general relativity. A variety of high precision null experiments confirm the Einstein Equivalence Principle, which underlies the concept that gravitation is synonymous with spacetime geometry, and must be described by a metric theory. Solar system experiments that test the weak-field, post-Newtonian limit of metric theories strongly favor general relativity. Binary pulsars test gravitational-wave damping and aspects of strong-field general relativity. During the coming decades, tests of general relativity in new regimes may be possible. Laser interferometric gravitational-wave observatories on Earth and in space may provide new tests via precise measurements of the properties of gravitational waves. Future efforts using X-ray, infrared, gamma-ray and gravitational-wave astronomy may one day test general relativity in the strong-field regime near black holes and neutron stars.

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BECHOUCHE, PHILIPPE, NORBERT J. MAUSER, and SIGMUND SELBERG. "ON THE ASYMPTOTIC ANALYSIS OF THE DIRAC–MAXWELL SYSTEM IN THE NONRELATIVISTIC LIMIT." Journal of Hyperbolic Differential Equations 02, no. 01 (March 2005): 129–82. http://dx.doi.org/10.1142/s0219891605000415.

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We study the behavior of solutions of the Dirac–Maxwell system (DM) in the nonrelativistic limit c → ∞, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c → ∞, sometimes also called post-Newtonian, yields a Schrödinger–Poisson system, where the spin and magnetic field no longer appear. We prove that DM is locally well-posed for H1 data (for fixed c), and that as c → ∞ the existence time grows at least as fast as log(c), provided the data are uniformly bounded in H1. Moreover, if the datum for the Dirac spinor converges in H1, then the solution of DM converges, modulo a phase correction, in C([0,T];H1) to a solution of a Schrödinger–Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit. The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Klein–Gordon–Maxwell system.

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Bonetti, Matteo, Alberto Sesana, Enrico Barausse, and Francesco Haardt. "Post-Newtonian evolution of massive black hole triplets in galactic nuclei – III. A robust lower limit to the nHz stochastic background of gravitational waves." Monthly Notices of the Royal Astronomical Society 477, no. 2 (April 6, 2018): 2599–612. http://dx.doi.org/10.1093/mnras/sty874.

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Khlopunov, M., and D. V. Gal'tsov. "Gravitational radiation from a binary system in odd-dimensional spacetime." Journal of Cosmology and Astroparticle Physics 2022, no. 04 (April 1, 2022): 014. http://dx.doi.org/10.1088/1475-7516/2022/04/014.

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Abstract We explore possible manifestations of an odd number of extra dimensions in gravitational radiation, which are associated with violation of Huygens' principle in flat odd-dimensional spacetime. Our setup can be regarded as the limit of an infinite compactification radius in ADD model and is not viable as realistic cosmology, but it still may be useful as a simple analytically solvable model catching certain features of more realistic scenarios. The model consists of two point masses moving inside a flat three-dimensional brane, embedded in a five-dimensional Minkowski space and interacting only through a massless scalar field localized on the same brane, while gravitational radiation is emitted into the bulk. This setup avoids the difficulties associated with taking into account the gravitational stresses binding the system, which require the cubic terms in the perturbative gravitational Lagrangian, and permits to limit ourselves to linearized theory. We calculate radiation in a linearized five-dimensional gravity generalizing the Rohrlich-Teitelboim approach to extract the emitted part of the retarded gravitational field. The source term consists of a local contribution from point particles and a non-local contribution from scalar field stresses, which is calculated using the DIRE approach to post-Newtonian expansions. In the nonrelativistic limit, we find an analog of the quadrupole formula containing an integral over the history of the particles' motion preceding the retarded time. We also show that, for an observer on the brane, the radiation contains a third polarization: the breathing mode.

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Szekeres, Peter. "Newtonian and Post-Newtonian Limits of Relativistic Cosmology." General Relativity and Gravitation 32, no. 6 (June 2000): 1025–39. http://dx.doi.org/10.1023/a:1001965526092.

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Sen, Kaustubha, Tetsuya Hashimoto, Tomotsugu Goto, Seong Jin Kim, Bo Han Chen, Daryl Joe D. Santos, Simon C. C. Ho, Alvina Y. L. On, Ting-Yi Lu, and Tiger Y.-Y. Hsiao. "Constraining violations of the weak equivalence principle Using CHIME FRBs." Monthly Notices of the Royal Astronomical Society 509, no. 4 (November 23, 2021): 5636–40. http://dx.doi.org/10.1093/mnras/stab3344.

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ABSTRACT Einstein’s General relativity (GR) is the basis of modern astronomy and astrophysics. Testing the validity of basic assumptions of GR is important. In this work, we test a possible violation of the weak equivalence principle (WEP), i.e. there might be a time lag between photons of different frequencies caused by the effect of gravitational fields if the speeds of photons are slightly different at different frequencies. We use Fast radio bursts (FRBs), which are astronomical transients with millisecond time-scales detected in the radio frequency range. Being at cosmological distances, accumulated time delay of FRBs can be caused by the plasma in between an FRB source and an observer, and by gravitational fields in the path of the signal. We segregate the delay due to dispersion and gravitational field using the post-Newtonian formalism (PPN) parameter Δγ, which defines the space curvature due to gravity by a unit test mass. We did not detect any time delay from FRBs but obtained tight constraints on the upper limit of Δγ. For FRB20181117C with z = 1.83 ± 0.28 and νobs = $676.5\, {\rm MHz}$, the best possible constraint is obtained at log(Δγ) = $-21.58 ^{+0.10}_{-0.12}$ and log(Δγ/rE) = $-21.75 ^{+0.10}_{-0.14}$, respectively, where rE is the energy ratio of two photons of the same FRB signal. This constraint is about one order of magnitude better than the previous constraint obtained with FRBs, and five orders tighter than any constraint obtained using other cosmological sources.

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Boshkayev, K., A. Malybayev, H. Quevedo, G. Nurbakyt, A. Taukenova, and A. Urazalina. "THE CORRESPONDENCE OF THE EREZ-ROSEN SOLUTION WITH THE HARTLE-THORNE SOLUTION IN THE LIMITING CASE OF ~ࡽ AND ~ࡹ૛." PHYSICO-MATHEMATICAL SERIES 5, no. 333 (October 15, 2020): 19–27. http://dx.doi.org/10.32014/2020.2518-1726.78.

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The link between exterior solutions to the Einstein gravitational field equations such as the exact Erez-Rosen metric and approximate Hartle-Thorne metric is established here for the static case in the limit of linear mass quadrupole moment (Q) and second order terms in total mass (M). To this end, the Geroch-Hansen multipole moments are calculated for the Erez-Rosen and Hartle-Thorne solutions in order to find the relationship among the parameters of both metrics. The coordinate transformations are sought in a general form with two unknown functions in the corresponding limit of ~Q and ~M^2. By employing the perturbation theory, the approximate Erez-Rosen metric is written in the same coordinates as the Hartle-Thorne metric. By equating the radial and azimuthal components of the metric tensor of both solutions the sought functions are found in a straightforward way. It is shown that the approximation ~Q and ~M^2, which is used throughout the article, is physical and suitable for solving most problems of celestial mechanics in post-Newtonian physics. This approximation does not require the use of the Zipoy-Voorhees transformation, which is a necessary strict mathematical requirement in the ~Q approximation, i.e. when no other approximations are made. This implies that the explicit form of the coordinate transformations depends entirely on the approximation that is adopted in each particular case. The results obtained here are in agreement with the previous results in the literature and can be applied to different astrophysical goals. The paper pursues not only pure scientific, but also academic purposes and can be used as an auxiliary and additional material to the special courses of general theory of relativity, celestial mechanics and relativistic astrophysics.

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Rannu, K. A., P. I. Dyadina, and S. O. Alexeyev. "Post-Newtonian limits for Randall-Sundrum model." Journal of Physics: Conference Series 600 (April 28, 2015): 012049. http://dx.doi.org/10.1088/1742-6596/600/1/012049.

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Alba, David, and Luca Lusanna. "The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 3. The post-minkowskian N-body problem, its post-newtonian limit in nonharmonic 3-orthogonal gauges and dark matter as an inertial effect 1This paper is one of three companion papers published in the same issue of Can. J. Phys." Canadian Journal of Physics 90, no. 11 (November 2012): 1131–78. http://dx.doi.org/10.1139/p11-102.

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We conclude the study of the post-minkowskian (PM) linearization of ADM tetrad gravity in the York canonical basis for asymptotically minkowskian space–times in the family of nonharmonic 3-orthogonal gauges parametrized by the York time 3K(τ, s) (the inertial gauge variable, not existing in Newton gravity, describing the general relativistic remnant of the freedom in clock synchronization in the definition of the shape of the instantaneous 3-spaces as 3-submanifolds of space–time). As matter we consider only N scalar point particles with a Grassmann regularization of the self-energies and with an ultraviolet cutoff making possible the PM linearization and the evaluation of the PM solution for the gravitational field. We study in detail all the properties of these PM space–times emphasizing their dependence on the gauge variable 3K(1) = (1/Δ)3K(1) (the nonlocal York time): Riemann and Weyl tensors, 3-spaces, time-like and null geodesics, red-shift, and luminosity distance. Then we study the post-newtonian (PN) expansion of the PM equations of motion of the particles. We find that in the two-body case at the 0.5PN order there is a damping (or antidamping) term depending only on 3K(1). This opens the possibility of explaining dark matter in Einstein theory as a relativistic inertial effect: the determination of 3K(1) from the masses and rotation curves of galaxies would give information on how to find a PM extension of the existing PN celestial frame used as an observational convention in the 4-dimensional description of stars and galaxies. Dark matter would describe the difference between the inertial and gravitational masses seen in the non-euclidean 3-spaces, without a violation of their equality in the 4-dimensional space–time as required by the equivalence principle.

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Vázquez-Aceves, Verónica, Lorenz Zwick, Elisa Bortolas, Pedro R. Capelo, Pau Amaro Seoane, Lucio Mayer, and Xian Chen. "Revised event rates for extreme and extremely large mass-ratio inspirals." Monthly Notices of the Royal Astronomical Society 510, no. 2 (December 2, 2021): 2379–90. http://dx.doi.org/10.1093/mnras/stab3485.

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ABSTRACT One of the main targets of the Laser Interferometer Space Antenna (LISA) is the detection of extreme mass-ratio inspirals (EMRIs) and extremely large mass-ratio inspirals (X-MRIs). Their orbits are expected to be highly eccentric and relativistic when entering the LISA band. Under these circumstances, the inspiral time-scale given by Peters’ formula loses precision and the shift of the last-stable orbit (LSO) caused by the massive black hole spin could influence the event rates estimate. We re-derive EMRIs and X-MRIs event rates by implementing two different versions of a Kerr loss-cone angle that includes the shift in the LSO, and a corrected version of Peters’ time-scale that accounts for eccentricity evolution, 1.5 post-Newtonian hereditary fluxes, and spin-orbit coupling. The main findings of our study are summarized as follows: (1) implementing a Kerr loss-cone changes the event rates by a factor ranging between 0.9 and 1.1; (2) the high-eccentricity limit of Peters’ formula offers a reliable inspiral time-scale for EMRIs and X-MRIs, resulting in an event-rate estimate that deviates by a factor of about 0.9–3 when compared to event rates computed with the corrected version of Peters’ time-scale and the usual loss-cone definition. (3) Event-rate estimates for systems with a wide range of eccentricities should be revised. Peters’ formula overestimates the inspiral rates of highly eccentric systems by a factor of about 8–30 compared to the corrected values. Besides, for e0 ≲ 0.8, implementing the corrected version of Peters’ formula is necessary to obtain accurate estimates.

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Liu, K., L. Guillemot, A. G. Istrate, L. Shao, T. M. Tauris, N. Wex, J. Antoniadis, et al. "A revisit of PSR J1909−3744 with 15-yr high-precision timing." Monthly Notices of the Royal Astronomical Society 499, no. 2 (September 30, 2020): 2276–91. http://dx.doi.org/10.1093/mnras/staa2993.

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ABSTRACT We report on a high-precision timing analysis and an astrophysical study of the binary millisecond pulsar, PSR J1909−3744, motivated by the accumulation of data with well improved quality over the past decade. Using 15 yr of observations with the Nançay Radio Telescope, we achieve a timing precision of approximately 100 ns. We verify our timing results by using both broad-band and sub-band template matching methods to create the pulse time-of-arrivals. Compared with previous studies, we improve the measurement precision of secular changes in orbital period and projected semimajor axis. We show that these variations are both dominated by the relative motion between the pulsar system and the Solar system barycentre. Additionally, we identified four possible solutions to the ascending node of the pulsar orbit, and measured a precise kinetic distance of the system. Using our timing measurements and published optical observations, we investigate the binary history of this system using the stellar evolution code mesa, and discuss solutions based on detailed WD cooling at the edge of the WD age dichotomy paradigm. We determine the 3D velocity of the system and show that it has been undergoing a highly eccentric orbit around the centre of our Galaxy. Furthermore, we set up a constraint over dipolar gravitational radiation with the system, which is complementary to previous studies given the mass of the pulsar. We also obtain a new limit on the parametrized post-Newtonian parameter, $\left| \hat\alpha_1 \right|$ < 2.1 × 10−5 at 95 per cent confidence level, which is fractionally better than previous best published value and achieved with a more concrete method.

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Kehagias, Alexandros A. "Post-Newtonian limits and gravitational radiation from the superstring effective action." Nuclear Physics B 423, no. 1 (July 1994): 305–26. http://dx.doi.org/10.1016/0550-3213(94)90574-6.

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Noureen, I., Usman-ul-Haq, and S. A. Mardan. "Impact of f(R,T) gravity in evolution of charged viscous fluids." International Journal of Modern Physics D 30, no. 04 (February 9, 2021): 2150027. http://dx.doi.org/10.1142/s0218271821500279.

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In this work, the evolution of spherically symmetric charged anisotropic viscous fluids is discussed in framework of [Formula: see text] gravity. In order to conduct the analysis, modified Einstein Maxwell field equations are constructed. Nonzero divergence of modified energy momentum tensor is taken that implicates dynamical equations. The perturbation scheme is applied to dynamical equations for stability analysis. The stability analysis is carried out in Newtonian and post-Newtonian limits. It is observed that charge, fluid distribution, electromagnetic field, viscosity and mass of the celestial objects greatly affect the collapsing process as well as stability of stars.

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IORIO, LORENZO. "PERSPECTIVES ON MEASURING THE PPN PARAMETERS β AND γ IN EARTH'S GRAVITATIONAL FIELD TO HIGH ACCURACY WITH CHAMP/GRACE MODELS." International Journal of Modern Physics D 17, no. 05 (May 2008): 815–29. http://dx.doi.org/10.1142/s0218271808012516.

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The current bounds on the PPN parameters β and γ are on the order of 10-4–10-5. Many missions aimed at improving such limits by several orders of magnitude have been proposed, including LATOR, ASTROD, BepiColombo and GAIA. They involve the use of various spacecraft, to be launched along interplanetary trajectories, for measuring the post-Newtonian effects induced by solar gravity on the propagation of electromagnetic waves. In this paper, we investigate the requirements needed to measure the combination ν = (2 + 2γ - β)/3 entering the post-Newtonian Einstein pericenter precession [Formula: see text] of a test particle to an accuracy on the order of ~10-5 with a pair of drag-free spacecraft in Earth's gravitational field. It turns out that the latest gravity models from the dedicated CHAMP and GRACE missions would allow the reduction of the systematic errors of gravitational origin to just this demanding level of accuracy. With regard to the non-gravitational errors, the spectral noise density of the required drag-free sensors amounts to 10-8–10-9 cm s-2 Hz-½ over very low frequencies. Although not yet attainable with present technologies, such a level of compensation is much less demanding than those required for LISA, for instance. As a by-product, an independent measurement of the post-Newtonian gravitomagnetic Lense–Thirring effect with a ~1% accuracy will be possible as well. The forthcoming Earth gravity models from CHAMP and GRACE will further reduce the systematic bias of gravitational origin in both such tests.

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Abuter, R., A. Amorim, N. Anugu, M. Bauböck, M. Benisty, J. P. Berger, N. Blind, et al. "Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole." Astronomy & Astrophysics 615 (July 2018): L15. http://dx.doi.org/10.1051/0004-6361/201833718.

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The highly elliptical, 16-year-period orbit of the star S2 around the massive black hole candidate Sgr A✻ is a sensitive probe of the gravitational field in the Galactic centre. Near pericentre at 120 AU ≈ 1400 Schwarzschild radii, the star has an orbital speed of ≈7650 km s−1, such that the first-order effects of Special and General Relativity have now become detectable with current capabilities. Over the past 26 years, we have monitored the radial velocity and motion on the sky of S2, mainly with the SINFONI and NACO adaptive optics instruments on the ESO Very Large Telescope, and since 2016 and leading up to the pericentre approach in May 2018, with the four-telescope interferometric beam-combiner instrument GRAVITY. From data up to and including pericentre, we robustly detect the combined gravitational redshift and relativistic transverse Doppler effect for S2 of z = Δλ / λ ≈ 200 km s−1/c with different statistical analysis methods. When parameterising the post-Newtonian contribution from these effects by a factor f , with f = 0 and f = 1 corresponding to the Newtonian and general relativistic limits, respectively, we find from posterior fitting with different weighting schemes f = 0.90 ± 0.09|stat ± 0.15|sys. The S2 data are inconsistent with pure Newtonian dynamics.

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Valtonen, M. J., S. Mikkola, D. Merritt, A. Gopakumar, H. J. Lehto, T. Hyvönen, H. Rampadarath, R. Saunders, M. Basta, and R. Hudec. "Black holes in active galactic nuclei." Proceedings of the International Astronomical Union 5, S261 (April 2009): 260–68. http://dx.doi.org/10.1017/s1743921309990482.

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AbstractSupermassive black holes are common in centers of galaxies. Among the active galaxies, quasars are the most extreme, and their black hole masses range as high as to 6⋅1010M⊙. Binary black holes are of special interest but so far OJ287 is the only confirmed case with known orbital elements. In OJ287, the binary nature is confirmed by periodic radiation pulses. The period is twelve years with two pulses per period. The last four pulses have been correctly predicted with the accuracy of few weeks, the latest in 2007 with the accuracy of one day. This accuracy is high enough that one may test the higher order terms in the Post Newtonian approximation to General Relativity. The precession rate per period is 39°.1 ± 0°.1, by far the largest rate in any known binary, and the (1.83 ± 0.01)⋅1010M⊙primary is among the dozen biggest black holes known. We will discuss the various Post Newtonian terms and their effect on the orbit solution. The over 100 year data base of optical variations in OJ287 puts limits on these terms and thus tests the ability of Einstein's General Relativity to describe, for the first time, dynamic binary black hole spacetime in the strong field regime. The quadrupole-moment contributions to the equations of motion allows us to constrain the ‘no-hair’ parameter to be 1.0 ± 0.3 which supports the black hole no-hair theorem within the achievable precision.

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Zakharov, Alexander. "Constraints on alternative theories of gravity with observations of the Galactic Center." EPJ Web of Conferences 191 (2018): 01010. http://dx.doi.org/10.1051/epjconf/201819101010.

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To evaluate a potential usually one analyzes trajectories of test particles. For the Galactic Center case astronomers use bright stars or photons, so there are two basic observational techniques to investigate a gravitational potential, namely, (a) monitoring the orbits of bright stars near the Galactic Center as it is going on with 10m Keck twin and four 8m VLT telescopes equipped with adaptive optics facilities (in addition, recently the IR interferometer GRAVITY started to operate with VLT); (b) measuring the size and shape of shadows around black hole with VLBI-technique using telescopes operating in mm-band. At the moment, one can use a small relativistic correction approach for stellar orbit analysis, however, in the future the approximation will not be precise enough due to enormous progress of observational facilities and recently the GRAVITY team found that the first post-Newtonian correction has to be taken into account for the gravitational redshift in the S2 star orbit case. Meanwhile for smallest structure analysis in VLBI observations one really needs a strong gravitational field approximation. We discuss results of observations and their interpretations. In spite of great efforts there is a very slow progress to resolve dark matter (DM) and dark energy (DE) puzzles and in these circumstances in last years a number of alternative theories of gravity have been proposed. Parameters of these theories could be effectively constrained with of observations of the Galactic Center. We show some cases of alternative theories of gravity where their parameters are constrained with observations, in particular, we consider massive theory of gravity. We choose the alternative theory of gravity since there is a significant activity in this field and in the last years theorists demonstrated an opportunity to create such theories without ghosts, on the other hand, recently, the joint LIGO & Virgo team presented an upper limit on graviton mass such as mg< 1:2 × 10-22eV [1] analyzing gravitational wave signal in their first paper where they reported about the discovery of gravitational waves from binary black holes as it was suggested by C. Will [2]. So, the authors concluded that their observational data do not indicate a significant deviation from classical general relativity. We show that an analysis of bright star trajectories could estimate a graviton mass with a commensurable accuracy in comparison with an approach used in gravitational wave observations and the estimates obtained with these two approaches are consistent. Therefore, such an analysis gives an opportunity to treat observations of bright stars near the Galactic Center as a useful tool to obtain constraints on the fundamental gravity law. We showed that in the future graviton mass estimates obtained with analysis of trajectories of bright stars would be better than current LIGO bounds on the value, therefore, based on a potential reconstruction at the Galactic Center we obtain bounds on a graviton mass and these bounds are comparable with LIGO constraints. Analyzing size of shadows around the supermassive black hole at the Galactic Center (or/and in the center of M87) one could constrain parameters of different alternative theories of gravity as well.

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Thompson, M. C. "Rapidly Rotating Core-Collapse Models." Publications of the Astronomical Society of Australia 6, no. 2 (1985): 214–16. http://dx.doi.org/10.1017/s1323358000018130.

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AbstractVery few (if any at all) three dimensional models of the final evolutionary stages of a star’s life have appeared in the literature. Such models may be important if the stellar core maintains sufficient rotational energy during it’s lifetime so that when collapse finally occurs, the increase of rotational energy to gravitational energy, may lead to a non-axisymmetric instability.A sequence of core collapse models with decreasing rotation rate is considered. These models were calculated using a three dimensional, post-Newtonian, hydrodynamical, numerical code. The results show that for high precollapse rotational energies the core can become unstable resulting in the formation of what resemble ‘spiral arms’. Unfortunately, because of limits on computer time, the calculations had to be discontinued shortly after this development occurred.

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Journal articles on the topic 'Newtonian, post-newtonian limit'

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