Relativity

The following sections are included:

• Acknowledgements

• Foreword

• Preface

• Contents

In this chapter we present the theory of special relativity that fixes the structure of spacetime without gravitation. The subject has been discussed in many texts, but our approach here is less technical and goes along the original intuitive lines of Einstein. The chapter starts with the postulates of special relativity, namely the principles of relativity and the constancy of the speed of light. This is then followed by discussing the basic concept of coordinate systems, and particularly the inertial system. Simultaneity, an essential notion in special relativity theory, is subsequently analyzed. These basic concepts are then followed by the Galilean transformation and group, and the Lorentz transformation and group. Consequences of the Lorentz transformation are then drawn. A four-dimensional formulation of spacetime, following Minkowski, is subsequently given. The light cone structure, an important description of spacetime, is then given. The discussion on special relativity theory is concluded by giving the formula that relates the mass, energy and momentum, along with the introduction of the energy-momentum four-vector. Our presentation will be along the original lines of Einstein's theory rather than dealing with technicalities, in order to emphasize the deep analogy of special relativity to the cosmological special relativity given in Chapter 2.

In this chapter we present the cosmological special relativity theory along the lines of Einstein's special relativity. While Einstein's special relativity deals with the continuum of space and time, cosmological special relativity deals with distances and velocities of celestial bodies. The role of the propagation of light will consequently be replaced by Hubble's law, with negligible gravitation, and in the next chapters with gravitation. The relative quantity here is the cosmic time as compared to the velocity in Einstein's theory. In other words, cosmic times are assumed to be relative in this theory. The chapter starts by reviewing the present day status of cosmology. The postulates of the theory are given and the notion of cosmic frame is introduced. Spacevelocity and relative cosmic time are subsequently discussed.The inadequacy of the classical transformation in physics is discussed and a comparison of the Universe expansion to the light propagation is given. The transformation between spacevelocity coordinates at different cosmic times, called the cosmological transformation, is derived and its physical interpretation is given. Consequences of the cosmological transformation are then drawn. The inflation of the Universe at the very early stage is discussed in details and is shown to be of the order of magnitude similar to that first predicted by Guth. Also discussed is the temperature of the Universe. It is shown that in the Universe there is a minimal acceleration, always in the direction of the motion of particles. An interesting formula that relates the cosmological redshift to the osmic time without any extra parameters is given. This is like the Doppler shift that is well known in physics. The theory is then extended into five dimensions by adding the time coordinate, so as to make it spacetime and velocity theory but still without gravitation. This fact demands a special treatment for Maxwell's equations which are now extended into five dimensions by adding to them a new interaction, the Higgs interaction. The generalized Maxwell's equations in five dimensions are written down explicitly in the standard three-dimensional notation, and it is shown how the Higgs interaction is unified with the electromagnetic theory. Kinematical topics in the Universe, such as velocity, acceleration and cosmic distances, are then discussed in details and some formulas are obtained within this theory and compared with those in special relativity theory.

In this chapter a short review of general relativity theory is given. We begin the discussion with a brief review of Riemannian geometry, including transformation of coordinates, covariant and contravariant vectors, tensors, the metric tensor, Christoffel symbols, covariant differentiation, the Riemann, Einstein and Ricci tensors, geodesics, and the Bianchi identities. This followed by a description of the physical foundations of general relativity. These are the principle of equivalence and the principle of general covariance. A detailed description is given of the Eötvös experiment and its new version, the Dicke experiment that improves its accuracy. The gravitational field equations of Einstein are then derived in a tensorial form. The basis of these equations is the Newtonian field equations and this fact is shown in details. The Schwarzschild solution of Einstein's field equations is derived. Although this is the simplest solution of the Einstein's field equations, its astrophysical importance is very high since it leads to the concept of black hole. Experimental verification of general relativity is subsequently given. The chapter is then concluded with a review of the problem of motion in general relativity theory. This problem is unique in general relativity theory since the equations of motion are consequence of the field equations. The Bianchi identities play an important role in the singularities at the locations of particles. The Einstein-Infeld-Hoffmann equations of motion, which are post-Newtonian, are derived. These equations include nonlinear terms that lead to the description of the perihelion advance of planets.

In this chapter we develop the theory of cosmological general relativity (CGR). It is an extension of cosmological special relativity (CSR), presented in Chapter 2, to curved Riemannian manifold. The transition from CSR to CGR, in principle, is similar, but not identical, to the transition from Einstein's special relativity (ESR) to ordinary general relativity (EGR). In the framework of CGR gravitation is described by a curved four-dimensional Riemannian spacevelocity . CGR incorporates the Hubble-Carmeli constant τ at the outset. This constant describes the Big Bang time and is the analogue of the speed of light in vacuum c in Einstein's relativity theory. The Hubble law is assumed in CGR as a fundamental law. CGR, in essence, extends Hubble's law so as to incorporate gravitation in it; it is actually a distribution theory that relates distances and velocities between galaxies. The theory involves only measured quantities and it takes a picture of the Universe as it is at any instant of time. No cosmological constant is required in this theory. In the four-dimensional space-velocity ds expresses the proper velocity just as in ordinary relativity it expresses the proper time. The null condition ds = 0 describes the expansion of the Universe in this theory just as the null condition ds = 0 in Einstein's theory describes the propagation of light. The energy-momentum tensor in this theory is a modification of the standard one so as to adjust it properly to the cosmological theory. The spherically symmetric metric is presented and the field equations are written down and solved. The equations for the expansion of the Universe and their solutions are given. The numerical value of the constant τ is given. The Tolman metric of an expanding Universe is presented in detail. Kantowski-Sachs metric describing the expanding Universe is given in detail. This is done in the space-velocity manifold. The issue of having any spacetime converted to a spacevelocity is discussed in connection to whether or not the Schwarzschild metric can be considered as an expanding Universe in the manifold of space and velocity.

In this chapter we discuss variety of physical properties of the gravitational field both in the framework of standard general relativity theory and cosmological relativity theory. We first discuss the important law of motion of Newton and derive it from general relativity by taking the combination of the weak gravitational field and the geodesic equation. Along the same lines the Einstein gravitational constant is also determined. It follows that the weak gravitational field equations provide exactly the function that is needed in the geodesic equation, and thus Newton's law of motion is determined. We then discuss the geodesic equation in the framework of cosmology. This has not been done so far in both standard cosmological theory and in cosmological general relativity.

In this chapter cosmological special relativity is extended to five dimensions by adding time to the three spatial dimensions and the velocity of the Hubble expansion. As a consequence of this extension, equations of electrodynamics are considered through the extended skew-symmetric tensor, in which a new field is included along with the electric and magnetic fields. This new field is due to the Higgs interaction associated with the expansion of the Universe. It is unified with the electromagnetic interaction in the frame of cosmology. The field equations are developed in five dimensions. In addition to the well-known Maxwell equations new equations that describe the mix-up of different fields are obtained.

In this chapter cosmological general relativity is extended to five dimensions, the five-dimensional brane world theory. The Bianchi identities and the field equations are developed for the Universe filled up with gravity. The mass density of the Universe is obtained to be a few hydrogen atoms per cubic meter. Velocity is considered as an independent variable, rather than the coordinate time derivative. The expansion of the (spherically-symmetric)Universe is examined, and it is shown that at present time the Universe is accelerating. The Tully-Fisher law is further obtained by considering the geodesic equation in the five-dimensional general relativity theory for a bound spherically symmetric matter distribution. It is thus shown that the assumption of existence of halo dark matter is not necessary to explain the Tully-Fisher relation between the circular velocity of the star, moving around the galactic core, and the mass of the galaxy. The cosmological contribution to the redshift from celestial bodies is considered. The near universal fact of observing redshifts and not blueshifts in the cosmos indicates it is due to cosmological expansion and therefore the matter density Ω < 1. Gravitational redshift from celestial bodies is generally small. Finally, the three classical general relativity tests are verified in cosmological general relativity in the weak field limit.

In this chapter we consider the five-dimensional (5D) extension of cosmological special and general relativity. In this framework a single 5D tensor conservation law can represent the equations of hydrodynamics, unifying the continuity equation and the conservation of the stress-energy tensor.This results in additional terms, which can be interpreted as representing matter creation (or annihilation). Thus, in principle, this picture permits the interpretation of the particle production phenomena as a cosmological effect. The following is based on Gemelli 2006 and 2007 (see references).

We have seen in the previous chapters that the 5D cosmological general relativity theory reproduces all the results that have been successfully tested for Einstein's 4D theory. However the theory, because of its fifth dimension, namely the velocity of the expanding Universe, predicts something different for the propagation of gravity waves on cosmological distance scales. This analysis indicates that gravitational radiation may not propagate as an unattenuated wave where effects of the Hubble expansion are felt. In such cases the energy does not travel over very large length scales but is evanescent and dissipated into the surrounding space as heat. The following is based on Hartnett and Tobar (see Hartnett and Tobar 2006).This chapter uses the linearized gravitational field equations while general relativity is highly nonlinear. To take this further the next step would be to use the exact theory of gravitation to handle this problem.

Equations of motion, in cylindrical coordinates, for the observed rotation of gases within the gravitational potential of spiral galaxies are derived from Carmeli's 5-Dimensional Cosmological General Relativity theory. A Tully-Fisher type relation results, and rotation curves are reproduced without the need for non-baryonic halo dark matter. Two acceleration regimes are found that are separated by a critical acceleration ≈ 4.75 X 10^-10 m/s².For accelerations larger than the critical value the Newtonian force law applies, but for accelerations less than the critical value the Carmelian regime applies. In the Newtonian regime the accelerations fall off as r^-2, but in the Carmelian regime the accelerations fall off as r^-1. This is new physics, but it fits Milgrom's phenomenological MOND theory. The following is based on the work of Hartnett (Hartnett 2006).

We are now in a position to present the "standard" theory of cosmology that is based on the original Einstein general relativity theory. That means in particular the introduction of the Friedmann Universe. For that one needs to introduce preliminary concepts, such as the deceleration parameter, an important parameter in this approach. We then determine the geometry of the spatial three-dimensional spaces which represent our three-dimensional space, which is assumed to be homogeneous and isotropic. A classification of these three-dimensional spaces into those with positive, negative or zero curvature is then given. Solutions of the Einstein field equations are then presented and related to the three-dimensional subspaces.From that the theories of Friedmann and others follow.

In this chapter we discuss the cosmic time which is shown to be a relative notion in cosmology. We then present the line elements that express the Hubble expansion. The coordinates here are the spatial coordinates x, y, z, and the velocity coordinate v, which are actually what astronomers use in their measurements. Two such line elements are presented: the first is the empty space (no matter exists), and the second including matter in the Universe. These line elements are the comparable to the standard Minkowskian and Friedmann-Robertson-Walker line elements in ordinary general relativity theory. The following is based on Carmeli 2003.

Several key relations are derived for Cosmological General Relativity which are used in standard observational cosmology. These include the luminosity distance, angular size, surface brightness and matter density.Luminosity distance and matter density relations are used to fit type Ia supernova (SNe Ia) data, giving consistent, well behaved fits over a broad range of redshifts 0.1 < 𝑧 < 2. The best fit to the data for the local matter density parameter is Ω_𝑚 = 0.0401±0.0199. Because Ω_𝑚is within the baryonic budget there is no need for any dark matter to account for the SNe Ia redshift luminosity data. From this local density it is determined that the redshift where the Universe expansion transitions from deceleration to acceleration is Because the fitted data covers the range of the predicted transition redshift 𝑧_𝑡, there is no need for any dark energy to account for the expansion rate transition. We conclude that the expansion is now accelerating and that the transition from a closed to an open Universe occurred about 8:54 Gyr ago. The following is based on Hartnett and Oliveira 2007b.

The redshift-distance modulus relation, the Hubble Diagram, derived from Cosmological General Relativity are extended to arbitrarily large redshifts. Numerical methods are employed and a density function is found that results in a valid solution of the field equations at all redshifts. The extension is compared to 302 type Ia supernova data as well as to 69 Gamma-ray burst data. The latter however do not truly represent a ‘standard candle’ as the derived distance modulii are not independent of the cosmology used. Nevertheless the analysis shows a good fit can be achieved without the need to assume the existence of dark matter. The Carmelian theory is also shown to describe a Universe that is always spatially flat. This results from the underlying assumption of the energy density of a cosmological constant Ω_Λ = 1, the result of vacuum energy. The curvature of the Universe is described by a spacevelocity metric where the energy content of the curvature at any epoch is Ω_𝑲 = Ω_Λ – = 1 –Ω where Ω is the matter density of the Universe. Hence the total density is always Ω_𝑲 + Ω = 1. The following is based on Hartnett 2008.

In this chapter homogeneous spaces will be discussed in some details that will include the concepts of Lie derivative and the Killing equation. Following that the Bianchi classification of homogeneous spaces, frequently mentioned in discussing cosmology, will be given.

The following sections are included:

• Appendix A Mathematical Conventions

• Appendix B Integration of the Equation of the Universe Expansion

• Appendix C Spheroidal and Elliptical Galaxy Velocity Dispersion from CGR

• Appendix D Bibliography

• Index