Showing papers on "Introduction to the mathematics of general relativity published in 2014"

Why Einstein did not believe that General Relativity geometrizes gravity

Abstract: I argue that, contrary to folklore, Einstein never really cared for geometrizing the gravitational or (subsequently) the electromagnetic field; indeed, he thought that the very statement that General Relativity geometrizes gravity "is not saying anything at all". Instead, I shall show that Einstein saw the "unification" of inertia and gravity as one of the major achievements of General Relativity. Interestingly, Einstein did not locate this unification in the field equations but in his interpretation of the geodesic equation, the law of motion of test particles.

Conservative form of Boltzmann’s equation in general relativity

Abstract: We derive a conservative form of Boltzmann's equation in general relativity, which is concisely written. Several explicit forms of this equation are written for black-hole spacetime with several coordinate conditions in real spacetime and momentum-space coordinates.

Einstein’s Special Theory of Relativity and the Problems in the Electrodynamics of Moving Bodies That Led Him to It

Abstract: Typesetter: the figures embedded here are lower resolution tif images. The original figures were drawn using a vector graphics program (Canvas 9) and should be printed from vector graphic files, e.g. epsf.

Newton’s third law in the framework of special relativity

Abstract: Newton’s third law states that any action is countered by a reaction of equal magnitude but opposite direction. The total force in a system not affected by external forces is thus zero. However, according to the principles of relativity a signal cannot propagate at speeds exceeding the speed of light. Hence the action cannot be generated at the same time with the reaction due to the relativity of simultaneity, thus the total force cannot be null at a given time. The following analysis provides for a better understanding of the ways natural laws would behave within the framework of Special Relativity, and on how this understanding may be used for practical purposes. It should be emphasized that although momentum can be created in the material part of the system as described in the following work momentum cannot be created in the physical system, hence for any momentum that is acquired by matter an opposite momentum is attributed to the electromagnetic field.

Einstein’s General Relativity and Pure Gravity in a Cosserat and De Sitter-Witten Spacetime Setting as the Explanation of Dark Energy and Cosmic Accelerated Expansion

Abstract: Ordinary energy and dark energy density are determined using a Cosserat-Cartan and killing-Yano reinterpretation of Einstein’s special and general relativity. Thus starting from a maximally symmetric space with 528 killing vector fields corresponding to Witten’s five Branes model in eleven dimensional M-theory we reason that 504 of the 528 are essentially the components of the relevant killing-Yano tensor. In turn this tensor is related to hidden symmetries and torsional coupled stresses of the Cosserat micro-polar space as well as the Einstein-Cartan connection. Proceeding in this way the dark energy density is found to be that of Einstein’s maximal energy mc2 where m is the mass and c is the speed of light multiplied with a Lorentz factor equal to the ratio of the 504 killing-Yano tensor and the 528 states maximally symmetric space. Thus we have E (dark) = mc2 (504/528) = mc2 (21/22) which is about 95.5% of the total maximal energy density in astounding agreement with COBE, WMAP and Planck cosmological measurements as well as the type 1a supernova analysis. Finally theory and results are validated via a related theory based on the degrees of freedom of pure gravity, the theory of nonlocal elasticity as well as ‘t Hooft-Veltman renormalization method.

Critical study and discrimination of different formulations of electromagnetic force density and consequent stress tensors inside matter

Abstract: By examination of the exerted electromagnetic (EM) force on boundary of an object in a few examples, we look into the compatibility of the stress tensors corresponding to different formulas of the EM force density with special relativity. Ampere-Lorentz's formula of the EM force density is physically justifiable in that the electric field and the magnetic flux density act on the densities of the total charges and the total currents, unlike Minkowski's formula which completely excludes the densities of the bounded charges and the bounded currents inside homogeneous media. Abraham's formula is fanciful and devoid of physical meaning. Einstein-Laub's formula seems to include the densities of the total charges and the total currents at first sight, but grouping the bounded charges and the bounded currents into pointlike dipoles erroneously results in the hidden momentum being omitted, hence the error in [Phys. Rev. Lett. 108, 193901 (2012)]. Naturally, the Ampere-Lorentz stress tensor accords with special relativity. The Minkowski sress tensor is also consistent with special relativity. It is worth noting that the mathematical expression of the Minkowski stress tensor can be quite different from the well-known form of this stress tensor in the literature. We show that the Einstein-Laub stress tensor is incompatible with special relativity, and therefore we rebut the Einstein-Laub force density. Since the Abraham momentum density of the EM fields is inherently corresponding to the Einstein-Laub force density [Phys. Rev. Lett. 111, 043602 (2013)], our rebuttal may also shed light on the controversy over the momentum of light.

Mysteries of R ik = 0: A novel paradigm in Einstein's theory of gravitation

Abstract: Despite a century-long effort, a proper energy-stress tensor of the gravitational field, could not have been discovered. Furthermore, it has been discovered recently that the standard formulation of the energy-stress tensor of matter, suffers from various inconsistencies and paradoxes, concluding that the tensor is not consistent with the geometric formulation of gravitation [Astrophys. Space Sci., 2009, 321: 151; Astrophys. Space Sci., 2012, 340: 373]. This perhaps hints that a consistent theory of gravitation should not have any bearing on the energy-stress tensor. It is shown here that the so-called “vacuum” field equations R ik = 0 do not represent an empty spacetime, and the energy, momenta and angular momenta of the gravitational and the matter fields are revealed through the geometry, without including any formulation thereof in the field equations. Though, this novel discovery appears baffling and orthogonal to the usual understanding, is consistent with the observations at all scales, without requiring the hypothetical dark matter, dark energy or inflation. Moreover, the resulting theory circumvents the long-standing problems of the standard cosmology, besides explaining some unexplained puzzles.

The General Theory Of Relativity, Metric Theory Of Relativity And Covariant Theory Of Gravitation. Axiomatization And Critical Analysis

Abstract: The axiomatization of general theory of relativity (GR) is done. The axioms of GR are compared with the axioms of the metric theory of relativity and the covariant theory of gravitation. The need to use the covariant form of the total derivative with respect to the proper time of the invariant quantities, the 4-vectors and tensors is indicated. The definition of the 4-vector of force density in Riemannian spacetime is deduced.

From continuum mechanics to general relativity

Abstract: Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein's theory of general relativity; it is also a much faster way of reaching general relativity than the conventional route. Our approach is simple and natural: we form a very general model and then apply two physical assumptions supported by experimental evidence. This easily reduces our construction to a model equivalent to general relativity. Finally, we suggest a simple way of modifying our theory to investigate non-standard space-time symmetries.

An observer principle for general relativity

Abstract: We give a mathematical uniqueness theorem which in particular shows that symmetric tensors in general relativity are uniquely determined by their monomial functions on the light cone. Thus, for an observer to observe a tensor at an event in general relativity is to contract with the velocity vector of the observer, repeatedly to the rank of the tensor. Thus two symmetric tensors observed to be equal by all observers at a specific event are necessarily equal at that event.

Some electrically charged relativistic stellar models in general relativity

Abstract: Some new families of electrically charged stellar models of ultra-compact star have been studied. With the help of particular form of one of the metric potentials the Einstein–Maxwell field equations in general relativity have been transformed to a system of ordinary differential equations. The interior matter pressure, energy–density, and the adiabatic sound speed are expressed in terms of simple algebraic functions. The constant parameters involved in the solution have been set so that certain physical criteria satisfied. Based on the analytic model developed in the present work, the values of the relevant physical quantities have been calculated by assuming the estimated masses and radii of some well known potential strange star candidates like X-ray pulsar Her X-1, millisecond X-ray pulsar SAX J 1808.4-3658, and 4U 1820-30. The analytical equations of state of the charged matter distribution may play a significant role in the study of the internal structure of highly compact charged stellar objects in general relativity.

Symmetry Group Analysis for Perfect Fluid Inhomogeneous Cosmological Models in General Relativity

Abstract: In this paper, we have searched the existence of the similarity solution for plane symmetric inhomogeneous cosmological models in general relativity. The matter source consists of perfect fluid with proportionality relation between expansion scalar and shear scalar. For this, Lie group analysis is used to identify the generator (isovector fields) that leave the given system of PDEs (Einstein’s field equations) invariant for the models under consideration. A new class of exact solutions of Einstein’s field equation have been obtained for inhomogeneous space-time. The physical behaviors and geometric aspects of the derived models have been discussed in detail.

Curvature fluctuations on asymptotically de Sitter spacetimes via the semiclassical Einstein's equations

Abstract: It has been proposed recently to consider in the framework of cosmology an extension of the semiclassical Einstein's equations in which the Einstein tensor is considered as a random function. This paradigm yields a hierarchy of equations between the $n$-point functions of the quantum, normal ordered, stress energy-tensor and those associated to the stochastic Einstein tensor. Assuming that the matter content is a conformally coupled massive scalar field on de Sitter spacetime, this framework has been applied to compute the power spectrum of the quantum fluctuations and to show that it is almost scale-invariant. We test the robustness and the range of applicability of this proposal by applying it to a less idealized, but physically motivated, scenario, namely we consider Friedmann-Robertson-Walker spacetimes which behave only asymptotically in the past as a de Sitter spacetime. We show in particular that, under this new assumption and independently from any renormalization freedom, the power spectrum associated to scalar perturbations of the metric behaves consistently with an almost scale-invariant power spectrum.

General relativity exactly described in terms of Newton's laws within curved geometries

Abstract: Many years ago Milne and McCrea showed in their well-known paper that the Hubble expansion occurring in general relativity could be exactly described by the use of Newtonian mechanics. It will be shown that a similar method can be extended to, and used within, curved geometries when Newton's second law is expressed within a four-dimensional curved spacetime. The second law will be shown to yield an equation that is exactly identical to the geodesic equation of motion of general relativity. This in itself yields no new information concerning relativity since the equation is mathematically identical to the relativistic equation. However, when the time in the second law is defined to have a constant direction as effectively occurs in Newtonian mechanics, and no longer acts as a fourth dimension as exists in relativity theory, it separates into a vector equation in a curved three-dimensional space and an additional second scalar equation that describes conservation of energy. It is shown that the curved Newtonian equations of motion define the metric coefficients which occur in the Schwarzschild solution and that they also define its equations of motion. Also, because the curved Newtonian equations developed here use masses as gravitational sources, as occurs in Newtonian mechanics, they make it possible to derive the solution for other kinds of mass distributions and are used here to find the metric equation for a thin mass-rod and the equation of motion for a mass particle orbiting it in its relativistic gravitational field.

Neo-newtonian theories

Abstract: General Relativity is the modern theory of gravitation. It has replaced the newtonian theory in the description of the gravitational phenomena. In spite of the remarkable success of the General Relativity Theory, the newtonian gravitational theory is still largely employed, since General Relativity, in most of the cases, just makes very small corrections to the newtonian predictions. Moreover, the newtonian theory is much simpler, technically and conceptually, when compared to the relativistic theory. In this text, we discuss the possibility of extending the traditional newtonian theory in order to incorporate typical relativistic effects, but keeping the simplicity of the newtonian framework. We denominate these extensions neo-newtonian theories. These theories are discussed mainly in the contexts of cosmology and compact astrophysical objects.

Scalar graviton as dark matter

Abstract: In the report, the theory of unimodular bimode gravity built on principles of unimodular gauge invariance/relativity and general covariance is exposed. Besides the massless tensor graviton of General Relativity, the theory includes an (almost) massless scalar graviton treated as the gravitational dark matter. A spherically symmetric vacuum solution, describing the coherent scalar-graviton field for the soft-core dark halos with the asymptotically flat rotation curves, is demonstrated.

Armenian Theory of Special Relativity

Abstract: By using the principle of relativity (first postulate), together with new defined nature of the universal speed (our second postulate) and homogeneity of time-space (our third postulate), we derive the most general transformation equations of relativity in one dimensional space. According to our new second postulate, the universal (not limited) speed c in Armenian Theory of Special Relativity is not the actual speed of light but it is the speed of time which is the same in all inertial systems. Our third postulate: the homogeneity of time-space is necessary to furnish linear transformation equations. We also state that there is no need to postulate the isotropy of time-space. Our article is the accumulation of all efforts from physicists to fix the Lorentz transformation equations and build correct and more general transformation equations of relativity which obey the rules of logic and fundamental group laws without internal philosophical and physical inconsistencies.

Reducing space-time to binary information

Abstract: We present a new description of discrete space-time in 1+1 dimensions in terms of a set of elementary geometrical units that represent its independent classical degrees of freedom. This is achieved by means of a binary encoding that is ergodic in the class of space-time manifolds respecting coordinate invariance of general relativity. Space-time fluctuations can be represented in a classical lattice gas model whose Boltzmann weights are constructed with the discretized form of the Einstein–Hilbert action. Within this framework, it is possible to compute basic quantities such as the Ricci curvature tensor and the Einstein equations, and to evaluate the path integral of discrete gravity. The description as a lattice gas model also provides a novel way of quantization and, at the same time, to quantum simulation of fluctuating space-time.

Einstein, Schwarzschild, the Perihelion Motion of Mercury and the Rotating Disk Story

Abstract: On November 18, 1915 Einstein reported to the Prussian Academy that the perihelion motion of Mercury is explained by his new General Theory of Relativity: Einstein found approximate solutions to his November 11, 1915 field equations. Einstein's field equations cannot be solved in the general case, but can be solved in particular situations. The first to offer such an exact solution was Karl Schwarzschild. Schwarzschild found one line element, which satisfied the conditions imposed by Einstein on the gravitational field of the sun, as well as Einstein's field equations from the November 18, 1915 paper. On December 22, 1915 Schwarzschild told Einstein that he reworked the calculation in his November 18 1915 paper of the Mercury perihelion. Subsequently Schwarzschild sent Einstein a manuscript, in which he derived his exact solution of Einstein's field equations. On January 13, 1916, Einstein delivered Schwarzschild's paper before the Prussian Academy, and a month later the paper was published. In March 1916 Einstein submitted to the Annalen der Physik a review article on the general theory of relativity. The paper was published two months later, in May 1916. The 1916 review article was written after Schwarzschild had found the complete exact solution to Einstein's November 18, 1915 field equations. Einstein preferred in his 1916 paper to write his November 18, 1915 approximate solution upon Schwarzschild exact solution (and coordinate singularity therein).

New Solutions of Gravitational Collapse in General Relativity and in the Newtonian Limit

Abstract: We discuss the Oppenheimer-Snyder-Datt (OSD) solution from a new perspective, introduce a completely new formulation of the problem exclusively in external Schwarzschild space-time (ESM) and present a new treatment of the singularities in this new formulation. We also give a new Newtonian approximation of the problem. Furthermore, we present new numerical solutions of the modified OSD-model and of the ball-to-ball-collapse with 4 different numerical methods.

Did Einstein "Nostrify" Hilbert's Final Form of the Field Equations for General Relativity?

Abstract: Einstein's biographer Albrecht F\"olsing explained: Einstein presented his field equations on November 25, 1915, but six days earlier, on November 20, Hilbert had derived the identical field equations for which Einstein had been searching such a long time. On November 18 Hilbert had sent Einstein a letter with a certain draft, and F\"olsing asked about this possible draft: "Could Einstein, casting his eye over this paper, have discovered the term which was still lacking in his own equations, and thus 'nostrified' Hilbert?" Historical evidence support a scenario according to which Einstein discovered his final field equations by "casting his eye over" his own previous works. In November 4, 1915 Einstein wrote the components of the gravitational field and showed that a material point in a gravitational field moves on a geodesic line in space-time, the equation of which is written in terms of the Christoffel symbols. Einstein found it advantageous to use for the components of the gravitational field the Christoffel symbols. Einstein had already basically possessed the field equations in 1912, but had not recognized the formal importance of the Christoffel symbols as the components of the gravitational field. Einstein probably found the final form of the generally covariant field equations by manipulating his own (November 4, 1915) equations. Other historians' findings seem to support the scenario according to which Einstein did not "nostrify" Hilbert.

On a general Relativity Extension

Abstract: The careful analysis of the duality properties of Riemann's curvature tensor points to possibility of extension of Einstein's General Relativity to the nonabelian Yang-Mills theory The motion equations of the theory are Yang-Mills' equations for the curvature tensor Einstein's equations (with cosmological term to appear as an integration constant) are contained in the theory proposed New is that now gravitational field is not exceptionally determined by matter energy-momentum but can possess its own non-Einsteinian dynamics(vacuum fluctuations, self-interaction)which is generally an attribute of nonabelian gauge field The gravitational equations proper due to either matter energy-momentum or vacuum fluctuations are side conditions imposed on the Riemann tensor, like self-duality conditions

Applications of embedding theory in higher dimensional general relativity.

Abstract: The study of embeddings is applicable and significant to higher dimensional theories of our universe, high-energy physics and classical general relativity. In this thesis we investigate local and global isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Theorems have been established that guarantee the existence of such embeddings. However, most known explicit results concern embedded spaces with relatively simple Ricci curvature. We consider the fourdimensional gravitational field of a global monopole, a simple non-vacuum space with a more complicated Ricci tensor, which is of theoretical interest in its own right, and occurs as a limit in Einstein-Gauss-Bonnet Kaluza-Klein black holes, and we obtain an exact solution for its embedding into Minkowski space. Our local embedding space can be used to construct global embedding spaces, including a globally flat space and several types of cosmic strings. We present an analysis of the result and comment on its significance in the context of induced matter theory and the Einstein-Gauss-Bonnet gravity scenario where it can be viewed as a local embedding into a Kaluza-Klein black hole. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate which embedded spaces admit bulks of a specific type. We show that the general Schwarzschild-de Sitter spacetime and the Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space with a particular metric form, and we discuss their five-dimensional solutions. Furthermore, we determine that the only spherically symmetric spacetime in retarded time coordinates that can be embedded into a particular Einstein bulk is the general Vaidya-de Sitter solution with constant mass. These analyses help to provide insight to the general embedding problem. We also consider the conformal Killing geometry of a five-dimensional Einstein space that embeds a static spherically symmetric spacetime, and we show how the Killing geometry of the embedded space is inherited by its bulk. The study of embedding properties such as these enables a deeper mathematical understanding of higher dimensional cosmological models and is also of physical interest as conformal symmetries encode conservation laws.

The Two Extended Bodies Problem in General Relativity within the post-Newtonian Approximation

Abstract: The dynamics of extended bodies is a fundamental problem in any gravitational theory. In the case of General Relativity, this problem is under study since the theory was published. Several methods have been developed and different approaches are avalaible in the literature to interpret the relativistic contributions in the motion under gravity influence. The main goal in this thesis is to study a general method to face the equation of motion for extended bodies in General Relativity. We started with a proposal in the Newtonian theory, which consists in a multipolar expansion for the gravitational potentials as a function of the mass density moments and other physical variables as the stress tensor. The methodology give us the equation of motion for an isolated and self-gravitating system of extended bodies in Newtonian mechanics. A geometrical approach to get the equation of motion is also used for the Newtonian problems, it allows us to extend the methodology to General Relativity. In General Relativity, some new concepts are necessary: world tube, world line, generalized ideas of momentum, angular momentum, torque and center of mass are introduced in a general context. General expressions for the equation of motion in the case of extended bodies are written without any restriction. In order to gain some physical understanding, we compute the Papapetrous's equation of motion for a test extended body in a static and isotropic metric. Finally, we study a system of two extended bodies in the post-Newtonian approximation. We define the mass multipole moments and momentum from the gravitational potentials (metric functions) to the first post-Newtonian order. We follow the Landau-Liftshitz formalism to find out the equation of motion for the moments and applying the standard coordinate transformation for this theory, we write the traslational equation of motion.