Relativistic Newtonian Dynamics Research Articles

Testing Relativistic Time Dilation beyond the Weak-Field Post-Newtonian Approximation

In General Relativity, the gravitational field of a spherically symmetric non-rotating body is described by the Schwarzschild metric. This metric is invariant under time reversal, which implies that the power series expansion of the time dilation contains only even powers of v / c . The weak-field post-Newtonian approximation defines the relativistic time dilation of order ϵ (or of order ( v / c ) 2 ) of the small parameter. The next non-zero term of the time dilation is expected to be of order ϵ 2 , which is impossible to measure with current technology. The new model presented here, called Relativistic Newtonian Dynamics, describes the field with respect to the coordinate system of a far-removed observer. The resulting metric preserves the symmetries of the problem and satisfies Einstein’s field equations, but predicts an additional term of order ϵ 3 / 2 for the time dilation. This term will cause an additional periodic time delay for clocks in eccentric orbits. The analysis of the gravitational redshift data from the Galileo satellites in eccentric orbits indicates that, by performing an improved satellite mission, it would be possible to test this additional time delay. This would reveal which of the coordinate systems and which of the above metrics are real. In addition to the increase of accuracy of the time dilation predictions, such an experiment could determine whether the metric of a spherically symmetric body is time reversible and whether the speed of light propagating toward the gravitating body is the same as the speed propagating away from it. More accurate time dilation and one-way speed of light formulas are important for astronomical research and for global positioning systems.

Relativistic Gravitation Based on Symmetry

We present a Relativistic Newtonian Dynamics ( R N D ) for motion of objects in a gravitational field generated by a moving source. As in General Relativity ( G R ), we assume that objects move by a geodesic with respect to some metric, which is defined by the field. This metric is defined on flat lab spacetime and is derived using only symmetry, the fact that the field propagates with the speed of light, and the Newtonian limit. For a field of a single source, the influenced direction of the field at spacetime point x is defined as the direction from x to the to the position of the source at the retarded time. The metric depends only on this direction and the strength of the field at x. We show that for a static source, the R N D metric is of the same form as the Whitehead metric, and the Schwarzschild metric in Eddington–Finkelstein coordinates. Motion predicted under this model passes all classical tests of G R . Moreover, in this model, the total time for a round trip of light is as predicted by G R , but velocities of light and object and time dilation differ from the G R predictions. For example, light rays propagating toward the massive object do not slow down. The new time dilation prediction could be observed by measuring the relativistic redshift for stars near a black hole and for sungrazing comets. Terrestrial experiments to test speed of light predictions and the relativistic redshift are proposed. The R N D model is similar to Whitehead’s gravitation model for a static field, but its proposed extension to the non-static case is different. This extension uses a complex four-potential description of fields propagating with the speed of light.

Geometrization of Newtonian Dynamics

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 conservative force. We introduce the relativity of spacetime : an object lives in its own spacetime, whose geometry is determined by all of the forces affecting it. We also introduce the Generalized Principle of Inertia which unifies Newton’s first and second laws and states that: An inanimate object moves freely, that is, with zero acceleration, in its own spacetime. We derive the metric of an object’s spacetime in two ways. The first way uses conservation of energy to derive a Newtonian metric. 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 (RND) for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. In the case of the gravitational field of a static, spherically symmetric mass distribution, this metric turns out to be the Schwarzschild metric. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity. In the second way, we obtain the RND metric directly, without first obtaining a Newtonian metric. Instead of conservation of energy, we use conservation of angular momentum, a carefully defined Newtonian limit and Tangherlini’s condition. The non-static case is handled by applying Lorentz covariance to the static case.

A geometric relativistic dynamics under any conservative force

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.

Relativistic Newtonian dynamics

A new Relativistic Newtonian Dynamics (RND) for motion under a conservative force capable to describe non-classical behavior in astronomy is proposed. The rotor experiments using Mössbauer spectroscopy with synchrotron radiation, described in the paper, indicate the influence of non-gravitational acceleration or potential energy on time. Similarly, the observed precession of Mercury and the periastron advance of binaries can be explained by the influence of gravitational potential energy on spacetime. The proposed RND incorporates the influence of potential energy on spacetime in Newton’s dynamics. The effect of this influence on time intervals, space increments and velocities is described explicitly by the use of the concept of escape trajectory. For an attracting conservative static potential we derived the RND energy conservation and the dynamics equation for motion of objects with non-zero mass and for massless particles. These equations are subsequently simplified for motion under a central force. Without the need to curve spacetime, this model predicts accurately the four non-classical observations in astronomy used to test the General Relativity.

Gravitational deflection in relativistic Newtonian dynamics

In a recent series of papers, the authors introduced a new Relativistic Newtonian Dynamics (RND) and tested its validity by the accurate prediction of the gravitational time dilation, the anomalous precession of Mercury, the periastron advance of any binary and the Shapiro time delay. This dynamics incorporates the influence of potential energy on spacetime in Newtonian dynamics and, unlike Einstein's General Relativity, treats gravity as a force without the need to curve spacetime. In this paper, this dynamics is applied to derive the gravitational deflection of both objects with non-zero mass and of massless particles passing the strong gravitating field of a massive body. Equations for the trajectory and the resulting analytical expressions for the deflection angle, in terms of the distance and velocity at the point of closest approach to the massive object, were derived in both cases. It is shown that with a carefully defined limit, the trajectory of a massless particle is the limiting case of that of an object with non-zero mass. In the “weak” deflection limit, the derived expression for the deflection angle of a massless particle (photon) reproduces the experimentally tested Einstein's formula for weak gravitational lensing of a light ray, thereby providing another test for the validity of the RND.

Relativistic Newtonian dynamics for objects and particles

Relativistic Newtonian Dynamics (RND) was introduced in a series of recent papers by the author, in partial cooperation with J. M. Steiner. RND was capable of describing non-classical behavior of motion under a central attracting force. RND incorporates the influence of potential energy on spacetime in Newtonian dynamics, treating gravity as a force in flat spacetime. It was shown that this dynamics predicts accurately gravitational time dilation, the anomalous precession of Mercury and the periastron advance of any binary. In this paper the model is further refined and extended to describe also the motion of both objects with non-zero mass and massless particles, under a conservative attracting force. It is shown that for any conservative force a properly defined energy is conserved on the trajectories and if this force is central, the angular momentum is also preserved. An RND equation of motion is derived for motion under a conservative force. As an application, it is shown that RND predicts accurately also the Shapiro time delay —the fourth test of GR.

Predicting the relativistic periastron advance of a binary without curving spacetime

Relativistic Newtonian dynamics, the simple model used previously for predicting accurately the anomalous precession of Mercury, is now applied to predict the periastron advance of a binary. The classical treatment of a binary as a two-body problem is modified to account for the influence of the gravitational potential on spacetime. Without curving spacetime, the model predicts the identical equation for the relativistic periastron advance as the post-Newtonian approximation of the general relativity formalism thereby providing further substantiation of this model.

Relativistic Newtonian Dynamics under a central force

Planck's formula and General Relativity indicate that potential energy influences spacetime. Using Einstein's Equivalence Principle and an extension of his Clock Hypothesis, an explicit description of this influence is derived. We present a new relativity model by incorporating the influence of the potential energy on spacetime in Newton's dynamics for motion under a central force. This model extends the model used by Friedman and Steiner (EPL, 113 (2016) 39001) to obtain the exact precession of Mercury without curving spacetime. We also present a solution of this model for a hydrogen-like atom, which explains the reason for a probabilistic description.