Newtonian versus relativistic nonlinear cosmology

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order. Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the post-Newtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume.     

The simplest black holes

A relativistic gravitational theory in (1 + 1) dimensions is presented which exhibits many of the qualitative features of (3 + 1)-dimensional general relativity. The field equations are simple enough for undergraduates to solve yet rich enough in structure to form a useful pedagogical example for exploring the qualitative features of relativistic gravitation. Black hole solutions to the field equations of the theory are derived and its relationship to Newtonian gravity is discussed in detail.     

The Clock Paradox in a Static Homogeneous Gravitational Field1

The gedanken experiment of the clock paradox is solved exactly using the general relativistic equations for a static homogeneous gravitational field. We demonstrate that the general and special relativistic clock paradox solutions are identical and in particular that they are identical for finite acceleration. Practical expressions are obtained for proper time and coordinate time by using the destination distance as the key observable parameter. This solution provides a formal demonstration of the identity between the special and general relativistic clock paradox with finite acceleration and where proper time is assumed to be the same in both formalisms. By solving the equations of motion for a freely falling clock in a static homogeneous field elapsed times are calculated for realistic journeys to the stars. ¹ Both authors contributed equally to this paper.     

Comparison of theoretical and experimental data on the advance of the mercury perihelion: The proper contribution of the gravitational field of the sun to the perihelion advance

An expression for the angular displacement of planets caused by their Newtonian interaction with each other is derived. In some cases, numerical results that follow from it differ from those published earlier (without specifying from where and how they were obtained). As a consequence, the remainder of the relativistic gravitational advance of the Mercury perihelion (after the observational data analysis) exceeds the value that was known before by more than one and a half times and follows from general relativity. However, this remainder value is in a very good agreement with the theoretical value that occurs if i) we acknowledge the material nature of the gravitational field of the Sun and ii) we uncover its physical contribution to the matter energy-momentum tensor in the Hilbert-Einstein equations.     

Gravitation and mass decrease

Consequences in physical theory of assuming the general relativistic time transformation for the de Broglie frequencies of matter, v = E/h = mc²/h, are investigated in this paper. Experimentally it is known that electromagnetic waves from a source in a gravitational field are decreased in frequency, in accordance with the Einstein general relativity time transformation. An extension to de Broglie frequencies implies mass decrease in a gravitational field. Such a decrease gives an otherwise missing energy conservation for some processes; also, a physical alteration is then associated with change in gravitational potential. Further, the general relativity time transformation that is the source of gravitational action in the weak field (Newtonian) approximation then has a physical correlate in the proposed gravitational mass loss. Rotational motion and the associated equivalent gravitational-field mass loss are considered; an essential formal difference between metric (gravitational) mass loss and special relativity mass increase is discussed. For a spherical, nonrotating mass collapsed to its Schwarzschild radius the postulated mass loss is found to give a 25% decrease in the mass acting as origin of an external gravitational field.     

Momentum transfer in gravitation theory

An inductive flux of gravitational momentum in Newtonian theory, and a radiative flux of gravitational momentum in general relativity, while being physically dissimilar quantities, are shown to be formally equivalent by expressing them in terms of a symplectic form on the solution space of the respective field equations. The conditions under which the relativistic flux may be considered inductive are then investigated, and a physical example of how this might occur is given in the setting of a strongly curved space-time.     

Occam’s Razor as a Formal Basis for a Physical Theory

We introduce the principle of Occam's Razor in a form that can be used as a basis for economical formulations of physics. This allows us to explain the general structure of the Lagrangian for a composite physical system, as well as some other artificial postulates behind the variational formulations of physical laws. As an example, we derive Hamilton's principle of stationary action together with the Lagrangians for the cases of Newtonian mechanics, relativistic mechanics and a relativistic particle in an external gravitational field.     

Classical Gravitational Interactions and Gravitational Lorentz Force

In quantum gauge theory of gravity, the gravitational field is represented by gravitational gauge field. The field strength of gravitational gauge field has both gravitoelectric component and gravitomagnetic component. In classical level, gauge theory of gravity gives classical Newtonian gravitational interactions in a relativistic form. Besides, it gives gravitational Lorentz force, which is the gravitational force on a moving object in gravitomagnetic field. The direction of gravitational Lorentz force is not the same as that of classical gravitational Newtonian force. Effects of gravitational Lorentz force should be detectable, and these effects can be used to discriminate gravitomagnetic field from ordinary electromagnetic magnetic field.     

Classical Gravitational Interactions and Gravitational Lorentz Force

In quantum gauge theory of gravity, the gravitational field is represented by gravitational gauge field.The field strength of gravitational gauge field has both gravitoelectric component and gravitomagnetic component. In classical level, gauge theory of gravity gives classical Newtonian gravitational interactions in a relativistic form. Besides,it gives gravitational Lorentz force, which is the gravitational force on a moving object in gravitomagnetic field The direction of gravitational Lorentz force is not the same as that of classical gravitational Newtonian force. Effects of gravitational Lorentz force should be detectable, and these effects can be used to discriminate gravitomagnetic field from ordinary electromagnetic magnetic field.     

Covariant Newtonian and relativistic dynamics of (magneto)-elastic solid model for neutron star crust

This work develops the dynamics of a perfectly elastic solid model for application to the outer crust of a magnetised neutron star. Particular attention is given to the Noether identities responsible for energy-momentum conservation, using a formulation that is fully covariant, not only (as is usual) in a fully relativistic treatment but also (sacrificing accuracy and elegance for economy of degrees of gravitational freedom) in the technically more complicated case of the Newtonian limit. The results are used to obtain explicit (relativistic and Newtonian) formulae for the propagation speeds of generalised (Alfven type) magneto-elastic perturbation modes.     

Tidal dynamics of relativistic flows near black holes

We point out novel consequences of general relativity involving tidal dynamics of ultrarelativistic relative motion. Specifically, we use the generalized Jacobi equation and its extension to study the force‐free dynamics of relativistic flows near a massive rotating source. We show that along the rotation axis of the gravitational source, relativistic tidal effects strongly decelerate an initially ultrarelativistic flow with respect to the ambient medium, contrary to Newtonian expectations. Moreover, an initially ultrarelativistic flow perpendicular to the axis of rotation is strongly accelerated by the relativistic tidal forces. The astrophysical implications of these results for jets and ultrahigh energy cosmic rays are briefly mentioned.     

Anisotropic Homogeneous Cosmologies in the Post-Newtonian Approximation

In this paper we explore how far the post-Newtonian theory, [9] goes in overcoming the difficulties associated with anisotropic homogeneous cosmologies in the Newtonian approximation. It will be shown that, unlike in the Newtonian case, the cosmological equations of the post-Newtonian approximation are much more in the spirit of general relativity with regard to the nine Bianchi types and issues of singularities.The situations of vanishing rotation and vanishing shear are treated separately. The homogeneous Bianchi I model is considered as an example of a rotation-free cosmology with anisotropy. It is found in the Newtonian approximation that there are arbitrary functions that need to be given for all time if the initial value problem is to be well-posed, while in the post-Newtonian case there is no such need. For the general case of a perfect fluid only the post-Newtonian theory can satisfactorily describe the effects of pressure. This is in accordance with findings in [7] where the post-Newtonian approximation was applied to homogeneous cosmologies.For a shear-free anisotropic homogeneous cosmology the Newtonian theory of Heckmann and Schücking, [2] is explored. Comparisons with its relativistic and post-Newtonian counterparts are made. In the Newtonian theory solutions exist to which there are no analogues in general relativity. The post-Newtonian approximation may provide a way out.     

Interaction between gravity and moving superconductors

We propose a unified phenomenological theory to investigate the interaction between arbitrarily moving superconductors and gravitational fields including the Newtonian gravity, gravitational waves, vector transverse gravitoelectric fields, and vector gravitomagnetic fields. In the limit of weak field and low velocity, the expressions for the induced electromagnetic and gravitational fields in the interior of a moving superconductor are obtained. The Meissner effect, London moment, DeWitt effect, effects of gravitational wave on a superconductor, and induced electric fields in the interior of a freely vibrating superconductor are recovered from these two expressions. We demonstrate that the weak equivalence principle is valid in superconductivity, that Newtonian gravity and gravitational waves will penetrate either a moving superconductor or a superconductor at rest, and that a superconductor at rest cannot shield either vector gravitomagnetic fields or vector transverse gravitoelectric fields.     

Mach's principle and Newtonian mechanics

The union of Mach's principle and Newtonian mechanics gives rise to Relational Mechanics. We find that the characteristics of the revised mechanics are: (1) freedom from any reference to absolute space; (2) the identity of inertial and gravitational mass; (3) the relative acceleration of a body in a gravitational field dependent on the mass of the body. All these results are valid in the context of a Newtonian mechanics which is being developed in the center-of-mass system of all the particles. The conservation of linear momentum, energy, angular momentum are expressed in relational terms, i.e., no reference is made to absolute space. Relational Mechanics is a classical relativistic theory which can be formulated to satisfy Einsteinian relativistic requirements. The Hamiltonian formalism for Relational Mechanics is discussed. Preliminary report Bull. Am. Phys. Soc.14, 15 (1969)     

Pseudo-schwarzschild spherical accretion as a classical black hole analogue

We demonstrate that a spherical accretion onto astrophysical black holes, under the influence of Newtonian or various post-Newtonian pseudo-Schwarzschild gravitational potentials, may constitute a concrete example of classical analogue gravity naturally found in the Universe. We analytically calculate the corresponding analogue Hawking temperature as a function of the minimum number of physical parameters governing the accretion flow. We study both the polytropic and the isothermal accretion. We show that unlike in a general relativistic spherical accretion, analogue white hole solutions can never be obtained in such post-Newtonian systems. We also show that an isothermal spherical accretion is a remarkably simple example in which the only one information–the temperature of the fluid, is sufficient to completely describe an analogue gravity system. For both types of accretion, the analogue Hawking temperature may become higher than the usual Hawking temperature. However, the analogue Hawking temperature for accreting astrophysical black holes is considerably lower compared with the temperature of the accreting fluid.     

Equilibrium of a high-current channel in the general theory of relativity

The condition for equilibrium of a high-current channel taking account of both electromagnetic and gravitational interactions of the charges with an arbitrary drift-to-light velocity ratio is derived from the equations of Einstein’s general theory of relativity. The relative motion appearing between the electron and ion subsystems as a result of the current flow gives rise to an additional gravitational attraction between these subsystems. This is a relativistic effect that cannot be obtained in the Newtonian approximation. Zh. éksp. Teor. Fiz. 112, 385–399 (August 1997)     

Electrical Conductivity in General Relativity

The general relativistic kinetic theory including the effect of a stationary gravitational field is applied to the electromagnetic transport processes in conductors. Then it is applied to derive the general relativistic Ohm's law where the gravitomagnetic terms are incorporated. The total electric charge quantity and charge distribution inside conductors carrying conduction current in some relativistic cases are considered. The general relativistic Ohm's law is applied to predict new gravitomagnetic and gyroscopic effects which can, in principle, be used to detect the Lense-Thirring and rotational fields.     

The Newtonian analogue of the relativistic Oppenheimer-Snyder solution

The gravitational collapse of a spherically symmetric mass of incoherent matter (dust) is investigated within the framework of the Newtonian theory and it is shown that the results are identical to the relativistic results obtained by Oppenheimer and Snyder.     

Exact relativistic models of conformastatic charged dust thick disks

We construct relativistic models of charged dust thick disks for a particular conformastatic spacetime through a Miyamoto–Nagai transformation used in Newtonian gravity to model disk like galaxies. Two simple families of thick disk models and a family of thick annular disks based on the field of an extreme Reissner–Nordström black hole and a Morgan–Morgan-like metric are considered. The electrogeodesic motion of test particles around the structures are analyzed. Also the stability of the particles against radial perturbation is studied using an extension of the Rayleigh criteria of stability of a fluid in rest in a gravitational field. The models built satisfy all the energy conditions.     

Relativistic particle dynamics in external gravitational fields

Using the Riemann metric for event space, which leads to Newtonian mechanics at nonrelativistic velocities and not necessarily weak gravitational fields, the dynamics of relativistic particles in external gravitational fields are considered. Trajectories, laws of motion, and equations of light rays are found in homogeneous and Newtonian fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 56–61, October, 1977.In conclusion the authors thank Yu. G. Pavlenko for his evaluation of the results and valuable advice.