The equations of motion in spacetime endowed with stationary metric of general relativity and the equivalent gravitational scalar generalized potential of special relativity

This paper shows that gravitational results of general relativity (GR) can be reached by using special relativity (SR) via a SR Lagrangian that derives from the corresponding GR time dilation and vice versa. It also presents a new SR gravitational central scalar generalized potential V=V(r,r.,ϕ.), where r is the distance from the center of gravity and r.,ϕ. are the radial and angular velocity, respectively. This is associated with the Schwarzschild GR time dilation from where a SR scalar generalized potential is obtained, which is exactly equivalent to the Schwarzschild metric. Thus, the Precession of Mercury’s Perihelion, the Gravitational Deflection of Light, the Shapiro time delay, the Gravitational Red Shift, etc., are explained with the use of SR only. The techniques used in this paper can be applied to any GR spacetime metric, Teleparallel Gravity, etc., in order to obtain the corresponding SR gravitational scalar generalized potential and vice versa. Thus, the case study of Newtonian Gravitational Potential according to SR leads to the corresponding non-Riemannian metric of GR. Finally, it is shown that the mainstream consideration of the Gravitational Red Shift contains two approximations, which are valid in weak gravitational fields only.

The OPTIS satellite – improved tests of Special and General Relativity

The development of Geometric theories of gravitation and the application of the Dynamics of General Relativity (GR) is the mainstream approach of gravitational field. Besides, the Generalized Special Relativity (GSR) contains the fundamental parameter (ξ I) of Theories of Physics (TPs). Thus, it expresses at the same time Newtonian Physics (NPs) for ξ I→ 0 and Special Relativity (SR) for ξ I=1. Moreover, the weak Equivalence Principle (EP) in the context of GSR, has the interpretation: m G=m, where m G and m are the gravitational mass and the inertial rest mass, respectively. In this paper, we bridge GR with GSR. This is achieved, by using a GSR-Lagrangian, which contains the corresponding GR-proper time. Thus, we obtain a new central scalar GSR-gravitational generalized potential V=V(k,l,r,r_dot,ϕ _dot), where k=k(ξ I), l=l(ξ I), r is the distance from the center of gravity and r_dot, ϕ _dot are the radial and angular velocity, respectively. The replacement k=1 and l=ξ I 2 makes the above GSR-potential equivalent to the original Schwarzschild Metric (SM). Thus, it explains the Precession of Mercury’s Perihelion (PMP), Gravitational Deflection of Light (GDL), Gravitational Red Shift (GRS) etc, by using SR and/or NPs. The procedure described in this paper can be applied to any other GR-spacetime metric, in order to find out the corresponding GSR-gravitational potential. So, we also use the GR-proper time of the 3rd Generalized Schwarzschild Metric (3GSM) and we obtain the central scalar GSR-gravitational potential V=V(a,k,l,r_dot,ϕ _dot), where a=a(r). The combination of the above with MOND interpolating functions, or distributions of Dark Matter (DM) in galaxies, provides the functions corresponding a=a(r). Thus, we obtain a new GSR-Gravitational field, which explains the PMP, GDL, GRS as well as the Rotation Curves in Galaxies, eliminating the corresponding DM.

Siva’s theories explained the necessity of new theory for description of the Universe, space ,time ,space-time and matter. It explained the formation of ‘space time continuum’ in terms of ‘Films of the universe’ and an effect of consciousness associated to living things. Thus it is required to bring consciousness in to physical laws and transformations. The relation between physical world and consciousness has been analyzed clearly and explained that consciousness, if we interpret in physics, must be an inertial frame of reference which can be transformed in to inertial frames defined by ‘Special Theory of Relativity’. It is possible only by changing the signal velocity from ‘c’ to ‘c√2’ . Thus the ‘Special Theory of Relativity’ has been modified and named as ‘Super theory of Relativity’. The relativistic factor for it is also calculated as [1+(v2/c2)]1/2 where v = vo [{1- (vo2/c2)}]-1/2 here vo is its absolute velocity .The necessity to adopt a new signal velocity which is greater than that of light has been discussed and the ‘Principle of Relativity’ and ‘Principle of simultaneity’ which are basics for transformation has been applied to interpret it in terms of relativity. It has been concluded that velocity of light is a part of signal velocity and photon will have rest mass. It says that the observable velocity is a result of absolute velocity multiplied by relativistic factor for ‘Super Theory of Relativity’. Thus infinite signal transformation is introduced for transformation between Inertial frames of reference. Infinite signal velocity will explain the ‘Quantum entanglement’ in terms of transformation of physical laws from one frame to another as explained in ‘Special & General Theories of Relativity’.

Einstein was known for often presenting "gedanken" or thought experiments (Miller, 1999). This paper is such an exercise. Time dilation in Special Relativity is based on the derived value of γ (a scalar value) (Einstein, 1905). γ will be calculated as a function of relativistic kinetic energy, allowing time dilation to become a function of relativistic kinetic energy. With the new methodology constructed, a "gedanken” experiment is considered. Can a time dilation function be derived, using Newtonian gravitational potential energy in the same manor kinetic energy was used in Special Relativity? This paper carries out the derivation and compares the results with General Relativity's Schwarzschild solution. The what if thought experiment provides a first order accuracy to GR's Schwarzschild solution.

This new edition of Hans Stephani's book is a welcome addition to the many texts now available to the student wishing to study relativity. A new feature, as compared with earlier editions, is a substantial introductory section on special relativity amounting to almost a quarter of the total length of the book. This forms part I and covers all the usual topics one might expect, including particle mechanics, the formulae for aberration and the Doppler effect, tensors in Minkowski spacetime, Maxwell's equations and the energy-momentum tensor for the electromagnetic field, the equations of motion for charged point particles and their fields (with a brief discussion of radiation reaction and runaway solutions) and the energy-momentum tensor for a perfect fluid. But there is also a useful section on the algebraic classification of the electromagnetic field using the apparatus of null tetrads and self-dual bivectors. An unusual feature is a section on pole-dipole charged particles.General relativity is introduced and developed in parts II to VII of the book, which follow closely the text of the 1982 English edition. In part II, the reasons for regarding spacetime as a four-dimensional Riemannian manifold with a metric of Lorentz signature are clearly explained. There follows a concise treatment of the tensor calculus including the covariant derivative, parallel and Fermi-Walker transport, the Lie derivative and the properties of the Riemann tensor. Invariant forms for line, surface and volume integrals are discussed and Stokes' theorem is stated without proof. There is a final section on how physical laws within special relativity may be generalized to curved spacetime.Part III introduces the reader to the Einstein field equations. Schwarzschild's exterior solution and Birkoff's theorem are derived, as are the usual formulae for the perihelion advance for planetary orbits, the deflection of light rays, the gravitational red shift and the geodesic precession of a top. Gravitational lensing is briefly discussed. This part ends with a derivation of the Schwarzschild interior metric for a source with constant rest mass density.The linearized field equations and the question as to whether their solution for a time varying system gives reliable information about solutions to the full nonlinear equations are discussed in part IV. The Landau-Lifshitz pseudo energy-momentum tensor is used to obtain the standard formula for the outward flow of energy due to quadrupole radiation from a bounded system. While the limitations of this approach are emphasized by the author, particularly in regard to the question of whether gravitational waves lead to energy transfer, it is surprising that no mention is made of the important work by Bondi and others in the 1960s which used the nonlinear equations to show that such radiating systems must lose mass. The final sections contain standard treatments of plane wave solutions of both the linearized and nonlinear theory, and a discussion of the Cauchy problem for the vacuum equations.Part V contains a clear account of the Petrov classification of the Weyl tensor (using the same technique as that used earlier in the algebraic classification of the Maxwell field), and of Killing vectors (including a brief introduction to the Bianchi classification of groups of motion) and their relation to conservation laws. Gravitational collapse and black holes are discussed in part VI, which covers such topics as the Kruskal extension of the Schwarzschild metric, the critical mass of a star, the gravitational collapse of a sphere of dust and a brief discussion of the properties of the Kerr metric. Also included in this new edition is a short but useful section on the problem of quantization, of quantum field theory in a given curved spacetime and on the Hawking effect, and on the conformal structure of infinity leading to a definition of asymptotic flatness. In part VII on cosmology, after showing how the cosmological principle leads to the Robertson-Walker metric, the various Friedmann models are derived and discussed. The final sections discuss a Bianchi type I model and the Gödel universe.For its size (at about four hundred pages) the book covers a great deal of material, and it does this very clearly and well. There are some misprints, but almost all of these are of an obvious kind and easily corrected. At the end of most sections there are some well chosen exercises (perhaps there could have been more of these) and useful suggestions for further reading. Students who work through the book carefully, completing the details for those calculations and proofs where these are outlined in the text, will learn a great deal about relativity and be well placed for further study in the subject. While it is probably too advanced in its scope for the average final year (UK) undergraduate course, I would recommend it as a reference book for further reading at this level or as a text for study by a beginning graduate student.

A Student's Guide to Einstein's Major Papers

The mainstream approach of gravitational field is the development of Geometric theories of gravitation and the application of the Dynamics of General Relativity (GR). Besides, the Generalized Special Relativity (GSR) contains the fundamental parameter (ξI) of Theories of Physics (TPs). Thus, it expresses at the same time Newtonian Physics (NPs) for ξI→0 and Einstein Relativity Theory (ERT) for ξI=1. Moreover, the Equivalence Principle (EP) in the context of GSR, has two possible interpretations: mG = m (1), or mG=γ(ξI,β)m (2), where β=υ/c and mG, m, γ are the gravitational mass, inertial rest mass and Lorentz γ-factor, respectively. In this paper we initially present a new central scalar potential V=V(k,r), where k=k(ξI) and r is the distance from the center of gravity. We demand that ‘this new GSR gravitational field in accordance with EP (1), gives the same precession of Mercury’s orbit as Schwarzschild Metric (SM) does’ and we obtain k=6-ξI2. This emerges Einsteinian SR-horizon at r=5rS, while NPs extends the horizon at six Schwarzschild radius (6rS).We can also explain the Gravitational Red Shift (GRS), if only the proposed GSR Gravitational field strength g=g(k,r) is combined with EP (2). We modify the aforementioned central scalar potential as V=V(h,k,r), where h=h(r). The combination of the above with MOND interpolating functions, or distributions of Dark Matter (DM) in galaxies, provides six different functions h=h(r). Thus, we obtain a new GSR central Gravitational field strength g=g(h,k,r), which not only explains the Precession of Mercury’s Perihelion, but also the Rotation Curves in Galaxies, eliminating Dark Matter.

We present a covariant field-theoretical model of gravitation based on a structural analogy with classical electrodynamics. The theory employs a four-potential formulation and introduces dynamic gravitational fields, including a gravitomagnetic component analogous to the magnetic field in Maxwell’s theory. The resulting equations of motion contain a Lorentz-type force term and a velocity-dependent mass density. While distinct from general relativity (GR) in its foundations, the theory reproduces all five classical tests of GR: Mercury’s perihelion precession, gravitational redshift, the Shapiro time delay, gravitational light deflection and relativistic time dilation as confirmed by the Hafele–Keating experiment. These results emerge directly from the field equations and motion dynamics, without invoking spacetime curvature. The model is consistent with Newtonian gravity in the appropriate limit and maintains compatibility with special relativity and conservation principles. It thus provides a logically coherent and empirically consistent field-based description of gravity that aligns with known physics while offering an alternative conceptual framework.

Global navigation satellite systems (GNSS ) use accurate, stable atomic clocks in satellites and on the ground to provide world-wide position, velocity, and time to millions of users. Orbiting clocks have gravitational and motional frequency shifts that are so large that, without carefully accounting for numerous relativistic effects, the systems would not work. The basis for navigation using GNSS , founded on special and general relativity, includes relativistic principles, concepts and effects such as the constancy of the speed of light, relativity of synchronization, coordinate time, proper time, time dilation, the Sagnac effect, the weak equivalence principle, and gravitational frequency shifts. Additional small relativistic effects such as the coordinate slowing of light speed and the effects of tidal potentials from the moon and the sun may need to be accounted for in the future. Examples of new navigation systems that are being developed and deployed are the European GALILEO system and the Chinese BEIDOU system; these will greatly widen the impact of GNSS. This chapter discusses applications of relativistic concepts in GNSS.

The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide world-wide position and time determination. These clocks have gravitational and motional frequency shifts which are so large that, without carefully accounting for numerous relativistic effects, the system would not work. This paper discusses the conceptual basis, founded on special and general relativity, for navigation using GPS. Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization. Experimental tests of relativity obtained with a GPS receiver aboard the TOPEX/POSEIDON satellite will be discussed. Recently frequency jumps arising from satellite orbit adjustments have been identified as relativistic effects. These will be explained and some interesting applications of GPS will be discussed.

In the year 1905, Albert Einstein presented his theory of Special Relativity and revolutionized our understanding about matter and energy by telling that they were the same. He derived the famous E = mc 2 formula from his realization that space and time have the same status. This realization, in turn, was complimented by his assumption that the speed at which light travels in vacuum is constant in any reference frame. Now what is this reference frame? It is not difficult to visualize the concept of reference frame. Imagine that you are traveling in a train and you throw a ball to a co-passenger in the same direction the train was running. The average speed of the ball should be the distance between you and your co-passenger divided by the time taken by the ball to reach your co-passenger. However, if someone outside the train watches it, one would find that the distance traveled by the ball is the sum of the distance traveled by the train during the time you throw the ball and your co-passenger receives it and the distance traveled by the ball from you to your co-passenger. Consequently, the outside observer would find a higher speed of the ball. The most important thing to notice here is that the time taken by the ball to move from one point to the other will appear to be the same to you and to the outside observer although the distance traveled by the ball would differ. You and your co-passenger measure the speed of the ball inside the train which is one reference frame attached with the train. The outside observer measures the speed of the ball in another reference frame attached to the Earth. One reference frame is moving at a constant speed with respect to another reference frame. If another train passes you with a different speed as compared to the speed of your train and a passenger inside that train measures the speed of the ball, he or she will derive a different speed of the ball. So, the speed of the ball is “relative” to the reference frame wherein it is measured. Einstein considered that light would not follow this rule; the speed of light would remain the same in all reference frames. In order to describe any cosmic event, one has to consider both space and time together. Einstein’s Special Theory of Relativity becomes significant only when the speed of an object is comparable to the speed of light. But nothing can exceed the speed of light. Now, if the speed of the ball or the train varies; i.e., if the ball or the train accelerates or retards, Special Theory of Relativity does not apply. For an accelerating reference frame, General Theory of Relativity has to be invoked. The contraction in length and the dilation of time is special relativistic effects whereas the acceleration due to gravitation is a general relativistic effect. Therefore, Special Relativity is a particular case of General Relativity. According to General Theory of Relativity, one can cancel the effect of gravitation locally by moving in an accelerating reference frame such as a free falling lift or aircraft. But it cannot be canceled globally. We know that the universe is expanding. All the galaxies are receding from each other. Since the rate of expansion of the universe changes, the dynamic of the universe is described by General Theory of Relativity.

This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.

This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.

Generalized special relativity is one of the most promising models that is found to cure many defects of special relativity. These defects include un capability of explaining gravitational red shift and satisfaction of the Newtonian limit. Despite these successes generalized special relativity suffers from noticeable setbacks. First of all its way of derivation make it restricted to weak fields only. Although recent derivation make it applicable to fields other than the weak and gravitational field, but this derivation needs to be strengthed. In this study a light clock is used to derive a useful expression for time in General special relativity. These expressions are typical to that derived before by using curved space. Time, but they are not restricted gravitational and weak fields. The expressions for time hold for all fields.