Showing papers on "Introduction to the mathematics of general relativity published in 2015"
TL;DR: In this paper, the authors considered the sector of Horndeski's gravity characterized by the coupling between the kinetic scalar field term and the Einstein tensor and numerically constructed neutron star configurations where the external geometry is identical to the Schwarzschild metric but the interior structure is considerably different from standard general relativity.
Abstract: We consider the sector of Horndeski's gravity characterized by the coupling between the kinetic scalar field term and the Einstein tensor. We numerically construct neutron star configurations where the external geometry is identical to the Schwarzschild metric but the interior structure is considerably different from standard general relativity. We constrain the only parameter of this model from the requirement that compact configurations exist, and we argue that solutions less compact than neutron stars, such as white dwarfs, are also supported. Therefore, our model provides an explicit modification of general relativity that is astrophysically viable and does not conflict with Solar System tests.
143 citations
TL;DR: In this article, a generalization of the results obtained in the context of general relativity generalize to Lanczos-Lovelock gravity in a straightforward but non-trivial manner.
Abstract: The deep connection between gravitational dynamics and horizon thermodynamics leads to several intriguing features in general relativity. In this chapter we provide a generalization of several of such results to Lanczos-Lovelock gravity. To our expectation it turns out that most of the results obtained in the context of general relativity generalize to Lanczos-Lovelock gravity in a straightforward but non-trivial manner. First, we provide an alternative and more general derivation of the connection between Noether charge for a specific time evolution vector field and gravitational heat density of the boundary surface. Taking a cue from this, we have introduced naturally defined four-momentum current associated with gravity and matter energy momentum tensor for both Lanczos-Lovelock Lagrangian. Then, we consider the concepts of Noether charge for null boundaries in Lanczos-Lovelock gravity by providing a direct generalization of previous results derived in the context of general relativity. Further we have shown that gravitational field equations for arbitrary static and spherically symmetric spacetimes with horizon can be written as a thermodynamic identity in the near horizon limit, transcending general relativity.
88 citations
TL;DR: In this article, the relationship between general relativity and the theory of Einstein algebras was examined and it was shown that according to a formal criterion for theoretical equivalence recently proposed by Halvorson (2012, 2015) and Weatherall (2015), the two are equivalent theories.
Abstract: In this paper, we examine the relationship between general relativity and the theory of Einstein algebras. We show that according to a formal criterion for theoretical equivalence recently proposed by Halvorson (2012, 2015) and Weatherall (2015), the two are equivalent theories.
55 citations
TL;DR: The evolution of BH binaries in vacuum spacetimes constitutes the two-body problem in general relativity as mentioned in this paper, and the solution of this problem in the framework of the Einstein field equations is a substantially more complex exercise than that of the dynamics of two point masses in Newtonian gravity, but also presents us with a wealth of new exciting physics.
Abstract: The evolution of black-hole (BH) binaries in vacuum spacetimes constitutes the two-body problem in general relativity. The solution of this problem in the framework of the Einstein field equations is a substantially more complex exercise than that of the dynamics of two point masses in Newtonian gravity, but it also presents us with a wealth of new exciting physics. Numerical methods are likely to be the only way to compute the dynamics of BH systems in the fully nonlinear regime and have been pursued since the 1960s, culminating in dramatic breakthroughs in 2005. Here we review the methodology and the developments that finally gave us a solution of this fundamental problem of Einstein's theory and discuss the breakthroughs' implications for the wide range of contemporary BH physics.
46 citations
TL;DR: An overview of the contributions numerical relativity has made to our understanding of strong field gravity can be found in the book "General Relativity and Gravitation: A Centennial Perspective", commemorating the 100th anniversary of general relativity.
Abstract: This article is an overview of the contributions numerical relativity has made to our understanding of strong field gravity, to be published in the book "General Relativity and Gravitation: A Centennial Perspective", commemorating the 100th anniversary of general relativity.
41 citations
TL;DR: In this article, the authors define a procedure by which observers can measure a type of special-relativistic linear and angular momentum $(P^a, J^{ab})$ at a point in a curved spacetime using only the spacetime geometry in a neighborhood of that point.
Abstract: We define a procedure by which observers can measure a type of special-relativistic linear and angular momentum $(P^a, J^{ab})$ at a point in a curved spacetime using only the spacetime geometry in a neighborhood of that point. The method is chosen to yield the conventional results in stationary spacetimes near future null infinity.
We also explore the extent to which spatially separated observers can compare the values of angular momentum that they measure and find consistent results. We define a generalization of parallel transport along curves which gives a prescription for transporting values of angular momentum along curves that yields the correct result in special relativity. If observers use this prescription, then they will find that the angular momenta they measure are observer dependent, because of the effects of spacetime curvature. The observer dependence can be quantified by a kind of generalized holonomy. We show that bursts of gravitational waves with memory generically give rise to a nontrivial generalized holonomy: there is, in this context, a close relation between the observer dependence of angular momentum and the gravitational-wave memory effect.
34 citations
TL;DR: The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem as mentioned in this paper, which is the basis for our work.
Abstract: The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem. The purpose of this article is to describe the background to and impact of this achievement, as well as the result itself. In some respects, the idea of viewing the field equations of general relativity as a system of evolution equations goes back to Einstein himself; in an argument justifying that gravitational waves propagate at the speed of light, Einstein used a special choice of coordinates to derive a system of wave equations for the linear perturbations on a Minkowski background. Over the following decades, Hilbert, de Donder, Lanczos, Darmois and many others worked to put Einstein's ideas on a more solid footing. In fact, the issue of local uniqueness (giving a rigorous justification for the statement that the speed of propagation of the gravitational field is bounded by that of light) was already settled in the 1930s by the work of Stellmacher. However, the first person to demonstrate both local existence and uniqueness in a setting in which the notion of finite speed of propagation makes sense was Yvonne Choquet-Bruhat. In this sense, her work lays the foundation for the formulation of Einstein's equations as an initial value problem. Following a description of the results of Choquet-Bruhat, we discuss the development of three research topics that have their origin in her work. The first one is local existence. One reason for addressing it is that it is at the heart of the original paper. Moreover, it is still an active and important research field, connected to the problem of characterizing the asymptotic behaviour of solutions that blow up in finite time. As a second topic, we turn to the questions of global uniqueness and strong cosmic censorship. These questions are of fundamental importance to anyone interested in justifying that the Cauchy problem makes sense globally. They are also closely related to the issue of singularities in general relativity. Finally, we discuss the topic of stability of solutions to Einstein's equations. This is not only an important and active area of research, it is also one that only became meaningful thanks to the work of Yvonne Choquet-Bruhat.
33 citations
TL;DR: In this paper, a class of modified gravity theories that deform general relativity in a way that breaks time reversal invariance and, very mildly, locality are described. But the algebra of constraints, local physical degrees of freedom, and their linearized equations of motion are unchanged, yet observable effects may be present on cosmological scales, which have implications for the early history of the universe.
Abstract: We describe a class of modified gravity theories that deform general relativity in a way that breaks time reversal invariance and, very mildly, locality. The algebra of constraints, local physical degrees of freedom, and their linearized equations of motion, are unchanged, yet observable effects may be present on cosmological scales, which have implications for the early history of the universe. This is achieved in the Hamiltonian framework, in a way that requires the constant mean curvature gauge conditions and is, hence, inspired by shape dynamics.
25 citations
TL;DR: In this paper, the energy distribution associated with a static spherically symmetric non-singular phantom black hole metric in general relativity was calculated in quasi-Cartesian coordinates.
Abstract: We calculate the energy distribution associated with a static spherically symmetric non-singular phantom black hole metric in Einstein's prescription in general relativity. As required for the Einstein energy-momentum complex, we perform the calculations in quasi-Cartesian coordinates. We also calculate the momentum components and obtain a zero value, as expected from the geometry of the metric.
25 citations
Book•
21 Apr 2015
TL;DR: In this paper, the authors present the foundations of singular Semi-Riemannian Geometry and Singular General Relativity based on the author's research, and an extension of differential geometry and of Einstein's equation to singularities is reported.
Abstract: This work presents the foundations of Singular Semi-Riemannian Geometry and Singular General Relativity, based on the author's research. An extension of differential geometry and of Einstein's equation to singularities is reported. Singularities of the form studied here allow a smooth extension of the Einstein field equations, including matter. This applies to the Big-Bang singularity of the FLRW solution. It applies to stationary black holes, in appropriate coordinates (since the standard coordinates are singular at singularity, hiding the smoothness of the metric). In these coordinates, charged black holes have the electromagnetic potential regular everywhere. Implications on Penrose's Weyl curvature hypothesis are presented. In addition, these singularities exhibit a (geo)metric dimensional reduction, which might act as a regulator for the quantum fields, including for quantum gravity, in the UV regime. This opens the perspective of perturbative renormalizability of quantum gravity without modifying General Relativity.
23 citations
TL;DR: In this paper, the authors derived the self-force equation for a small compact object moving on an accelerated world line in a background spacetime which is a solution of the coupled gravitational and scalar field equations of scalar-tensor theory.
Abstract: Motivated by the discovery of floating orbits and the potential to provide extra constraints on alternative theories, in this paper we derive the self-force equation for a small compact object moving on an accelerated world line in a background spacetime which is a solution of the coupled gravitational and scalar field equations of scalar-tensor theory. In the Einstein frame, the coupled field equations governing the perturbations sourced by the particle share the same form as the field equations for perturbations of a scalarvac spacetime, with both falling under the general class of hyperbolic field equations studied by Zimmerman and Poisson. Here, we solve the field equations formally in terms of retarded Green functions, which have explicit representations as Hadamard forms in the neighbourhood of the world line. Using a quasi-local expansion of the Hadamard form, we derive the regular solutions in Fermi normal coordinates according to the Detweiler-Whiting prescription. To compute the equation of motion, we parameterize the world line in terms of a mass and "charge", which we define in terms of the original Jordan frame mass, its derivative, and the parameter which translates the proper time in the Jordan frame to the Einstein frame. These parameters depend on the value of the background scalar field and its self-field corrections. The equation of motion which follows from the regular fields strongly resembles the equation for the self-force acting on a charged, massive particle in a scalarvac geometry of general relativity. Unlike the scalar vacuum scenario, the "charge" parameter in the scalar-tensor self-force equation is time variable and leading to additional local and tail terms. We also provide evolution equations for the world line parameters under the influence of the self-fields.
TL;DR: In this article, it was shown that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation for any Riemannian manifold M with space-like or time-like boundary components.
Abstract: The present paper shows that general relativity in the Arnowitt-Deser-Misner formalism admits a BV-BFV formulation. More precisely, for any $d + 1
ot= 2$ (pseudo-) Riemannian manifold M with space-like or time-like boundary components, the BV data on the bulk induces compatible BFV data on the boundary. As a byproduct, the usual canonical formulation of general relativity is recovered in a straightforward way.
TL;DR: In this paper, the authors studied a class of solutions to Lovelock gravity that is described by the warped product of a two-dimensional Lorentzian metric and an $n$-dimensional Einstein space.
Abstract: This paper studies a class of $D=n+2(\ge 6)$ dimensional solutions to Lovelock gravity that is described by the warped product of a two-dimensional Lorentzian metric and an $n$-dimensional Einstein space. Assuming that the angular part of the stress-energy tensor is proportional to the Einstein metric, it turns out that the Weyl curvature of an Einstein space must obey two kinds of algebraic conditions. We present some exact solutions satisfying these conditions. We further define the quasilocal mass corresponding to the Misner-Sharp mass in general relativity. It is found that the quasilocal mass is constructed out of the Kodama flux and satisfies the unified first law and the monotonicity property under the dominant energy condition. Making use of the quasilocal mass, we show Birkhoff's theorem and address various aspects of dynamical black holes characterized by trapping horizons.
TL;DR: In this paper, a set of first principles is proposed to obtain general relativity in the canonical Hamiltonian framework without presupposing space-time in any way, and the ADM Hamiltonian can be obtained in CMC gauge with arbitrary (finite, nonzero) speed of light and an extra term linear in York time.
Book•
24 Jun 2015
TL;DR: In this paper, the authors present a general introduction to the special Relativity De Sitter Invariant Special Relativity dS/AdS Relativistic Quantum Mechanics dS and AdS General Relativity Hydrogen Atom and Superluminal Neutrino.
Abstract: General Introduction Einstein's Special Relativity De Sitter Invariant Special Relativity dS/AdS Relativistic Quantum Mechanics dS/AdS General Relativity Hydrogen Atom in dS/AdS-Special Relativity Superluminal Neutrino in dS/AdS-Special Relativity Non-relativistic Limit of dS/AdS-Special Relativity
Book•
25 Jun 2015
TL;DR: In this article, it was shown that the Earth's field does not rotate with Earth's rotation, but instead attenuates with distance from the gravitating body, which accounts for the Michelson-Morley null result and predicts a small fringe shift.
Abstract: Petr Beckmann's Einstein Plus Two (1987). [2] Beckmann's assumption was that the luminiferous medium, which Michelson failed to detect in 1887, is the local gravitational field, which attenuates with distance from the gravitating body. Overwhelmingly, we are in the Earth's field, which does not rotate with the Earth’s rotation. This accounts for the Michelson-Morley null result and predicts an east-west light speed difference and with it a small fringe shift. An “ether” denser near the sun predicts the bending of light rays by Fermat's Principle, and the gravitational red shift. Einstein's equation accounting for Mercury's orbit was published by Paul Gerber, 17 years before general relativity. Both Sagnac (1913) and Michelson-Gale (1924) showed a fringe shift, but were disqualified as tests of SRT because they involved rotating (non-inertial) reference frames. GPS is said to validate special relativity because relativistic adjustments are entered into the orbiting clocks and would not synchronize without them. But the corrections do not refer clock motion to the observer, as relativity requires, but to the non-rotating Earth centered, inertial reference frame. It is a preferred reference frame — not allowed by SRT. The same criticism applies to the Hafele-Keating experiment (1972), in which atomic clocks flown around the world showed an east-west time difference. After 1916, Einstein restored a “gravitational ether,” indistinguishable from Beckmann's, but played it down. The book concludes that general relativity gives the right results by a roundabout method. SRT has been falsified, unless rescued by the claim that all experiments on the surface of a rotating globe are non-inertial.
Posted Content•
TL;DR: In this paper, the authors studied the up and down structure of permutations using their decomposition into so-called atomic permutations and showed that the quantum fluctuations of the stress-energy tensor induce an almost scale-invariant power spectrum for the perturbation potential and that nonGaussianties arise naturally.
Abstract: The subject of this thesis is the coupling of quantum fields to a classical gravitational background in a semiclassical fashion. It contains a thorough introduction into quantum field theory on curved spacetime with a focus on the stress-energy tensor and the semiclassical Einstein equation. Basic notions of differential geometry, topology, functional and microlocal analysis, causality and general relativity will be summarised, and the algebraic approach to QFT on curved spacetime will be reviewed. Apart from these foundations, the original research of the author and his collaborators will be presented: Together with Fewster, the author studied the up and down structure of permutations using their decomposition into so-called atomic permutations. The relevance of these results to this thesis is their application in the calculation of the moments of quadratic quantum fields. In a work with Pinamonti, the author showed the local and global existence of solutions to the semiclassical Einstein equation in flat cosmological spacetimes coupled to a scalar field by solving simultaneously for the quantum state and the Hubble function in an integral-functional equation. The theorem is proved with a fixed-point theorem using the continuous functional differentiability and boundedness of the integral kernel of the integral-functional equation. In another work with Pinamonti the author proposed an extension of the semiclassical Einstein equations which couples the moments of a stochastic Einstein tensor to the moments of the quantum stress-energy tensor. In a toy model of a Newtonianly perturbed exponentially expanding spacetime it is shown that the quantum fluctuations of the stress-energy tensor induce an almost scale-invariant power spectrum for the perturbation potential and that non-Gaussianties arise naturally.
01 Jan 2015
TL;DR: In this article, a detailed study of quasi-conformal curvature tensor for the spacetime of gen- eral relativity has been made and the existence of Killing and conformal Killing vector fields has been established.
Abstract: A detailed study of quasi-conformal curvature tensor for the spacetime of gen- eral relativity has been made. The spacetimes satisfying Einstein field equations with vanish- ing quasi-conformal curvature tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing quasi-conformal curvature tensor have also been considered. The divergence of quasi-conformal curvature tensor is studied in the setting of perfect fluid with the derivation of many physical results.
TL;DR: In this article, Brans-Dicke-Maxwell type vacuum solutions are considered for a static cylindrically symmetric spacetime in arbitrary dimensions and exact solutions are obtained by directly solving the field equations for the case where an azimuthal magnetic field is present.
Abstract: In this paper, Brans-Dicke-Maxwell type vacuum solutions are considered for a static cylindrically symmetric spacetime in arbitrary dimensions. Exact solutions are obtained by directly solving the field equations for the case where an azimuthal magnetic field is present. Other configurations such as axial magnetic field case can be obtained by suitably relabeling the coordinates. We have also considered conformally related “Einstein frame” to relate the solutions we have obtained with the dilaton-Maxwell type solutions that exist in the literature. We see that for a special case the general solution we present reduces to dilaton-Melvin spacetime. The general relativistic limit of these solutions is also discussed and we found that this limit is different from the four dimensional case.
Posted Content•
TL;DR: In this paper, it was shown that the true tensor of the gravitational field and time is equivalent even under conditions of the general theory of relativity, even under the assumption of the theory of general relativity.
Abstract: In general, the modification of our understanding of space and time undergone through Einstein's relativity theory is indeed a profound one But even Einstein's relativity theory does not give satisfactory answers to a lot of questions One of these questions is the problem of the ‘true’ tensor of the gravitational field The purpose of this publication is to provide some new and basic fundamental insights by the proof that the gravitational field and time is equivalent even under conditions of the general theory of relativity
TL;DR: The connection between general relativity and cosmology is discussed in this article, where the authors provide a historical overview of the connection between cosmology and general relativity, two areas whose development has been closely intertwined.
Abstract: This year marks the hundredth anniversary of Einstein's 1915 landmark paper "Die Feldgleichungen der Gravitation" in which the field equations of general relativity were correctly formulated for the first time, thus rendering general relativity a complete theory. Over the subsequent hundred years physicists and astronomers have struggled with uncovering the consequences and applications of these equations. This contribution, which was written as an introduction to six chapters dealing with the connection between general relativity and cosmology that will appear in the two-volume book "One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity," endeavors to provide a historical overview of the connection between general relativity and cosmology, two areas whose development has been closely intertwined.
Posted Content•
TL;DR: In this paper, the authors highlight and resolve what they take to be three common misconceptions in general relativity, relating to the interpretation of the weak equivalence principle and the relationship between gravity and inertia, and the connection between gravitational redshift results and spacetime curvature.
Abstract: We highlight and resolve what we take to be three common misconceptions in general relativity, relating to (a) the interpretation of the weak equivalence principle and the relationship between gravity and inertia; (b) the connection between gravitational redshift results and spacetime curvature; and (c) the strong equivalence principle and the local recovery of special relativity in curved, dynamical spacetime.
TL;DR: In this article, the authors revisited the analysis made by Hwang and Noh [JCAP 1310] aiming the construction of a Newtonian set of equations incorporating pressure effects typical of the General Relativity theory.
Abstract: We revisit the analysis made by Hwang and Noh [JCAP 1310 (2013)] aiming the construction of a Newtonian set of equations incorporating pressure effects typical of the General Relativity theory. We explicitly derive the Hwang-Noh equations, comparing them with similar computations found in the literature. Then, we investigate $i)$ the cosmological expansion, $ii)$ linear cosmological perturbations theory and $iii)$ stellar equilibrium by using the new set of equations and comparing the results with those coming from the usual Newtonian theory, from the Neo-Newtonian theory and from the General Relativity theory. We show that the predictions for the background evolution of the Universe are deeply changed with respect to the General Relativity theory: the acceleration of the Universe is achieved with positive pressure. On the other hand, the behaviour of small cosmological perturbations reproduces the one found in the relativistic context, even if only at small scales. We argue that this last result may open new possibilities for numerical simulations for structure formation in the Universe. Finally, the properties of neutron stars are qualitatively reproduced by Hwang-Noh equations, but the upper mass limit is at least one order of magnitude higher than the one obtained in General Relativity.
TL;DR: In this paper, the authors consider a general scalar-tensor theory of gravity and review briefly different forms it can be presented (different conformal frames and scalar field parametrizations).
Abstract: We consider a general scalar-tensor theory of gravity and review briefly different forms it can be presented (different conformal frames and scalar field parametrizations). We investigate the conditions under which its field equations and the parametrized post-Newtonian parameters coincide with those of general relativity. We demonstrate that these so-called limits of general relativity are independent of the parametrization of the scalar field, although the transformation between scalar fields may be singular at the corresponding value of the scalar field. In particular, the limit of general relativity can equivalently be determined and investigated in the commonly used Jordan and Einstein frames.
01 Jan 2015
TL;DR: In this article, the authors present a theory of gravitation known as isogravitation which allows a unified treatment of generally inhomogeneous and anisotropic, exterior and interior gravitational problems by jointly achieving a clear compatibility with 20th century sciences, thanks to the verification of the Lorentz-Poincar-Santilli isosymmetry, as well as the replacement of the Riemannian curvature with the covering notion of isoflatness.
Abstract: In this paper, we hope to initiate a scientific debate on some of the widely ignored criticisms of Einstein gravitation by Einstein himself as well as by others, with particular reference to the lack of clear compatibility of general relativity with special relativity, the interior gravitational problem, electrodynamics, quantum mechanics and grand unifications. We show that a resolution of the historical doubts can be apparently achieved via the use of the novel isomathematics and related isogeometries. We then show that the resulting theory of gravitation, known as isogravitation, allows a unified treatment of generally inhomogeneous and anisotropic, exterior and interior gravitational problems by jointly achieving a clear compatibility with 20th century sciences, thanks to the verification of the Lorentz-Poincar-Santilli isosymmetry, as well as the replacement of the Riemannian curvature with the covering notion of isoflatness. We then present, apparently for the first time, the isogravitational isoaxioms characterized by the infinite family of isotopies of Einstein axioms for special relativity, which are applicable to both exterior and interior isogravitational problems. We finally show, also for the first time, the apparent compatibility of the isogravitational; isoaxioms with current knowledge on black holes and other gravitational conditions in the expectation of due scientific process for the final resolution of the historical doubts on general relativity.
13 May 2015
TL;DR: In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion of any external field considered as a noninertial system of reference.
Abstract: In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field In other words, a noninertial frame of reference is equivalent to a certain gravitational field This is known as the principle of equivalence From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed
TL;DR: In this article, it was shown that the fundamental principles of the theory of General Relativity do not correspond to those of the gravity theory of Riemann's spacetimes.
Abstract: While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say, Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993) [1]. Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996) [2]. So, during a time, the declaration of quantum theorists: “I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation.” (Hawking and Penrose, 1996) [2] seemed to solve the problem, but recently achieved with the help of the tightly and collectively synchronized clocks in orbit frontally contradicts fundamental assumptions of the theory of Relativity. These observations are in disagree from predictions of the theory of Relativity. (Hatch, 2004a, 2004b, 2007) [3,4,5]. The mathematical model was developed first by Grossmann who presented it, in 1913, as the mathematical part of the Entwurf theory, still referred to a curved Minkowski spacetime. Einstein completed the mathematical model, in 1915, formulated for Riemann´s spacetimes. In this paper, we present as of General Relativity currently remains only the mathematical model, darkened with the results of Hatch and, course, we conclude that a Einstein´s gravity theory does not exist.
TL;DR: In this article, the spherically symmetric static problem of the General Theory of Relativity (GTR) has been analyzed and three possible forms of the solution are derived and analyzed.
Abstract: The paper is concerned with analysis of various forms of solution for the spherically symmetric static problem of the General Theory of Relativity (GTR). The problem under consideration is reduced to three equations for the components of the Einstein tensor which include three components of the metric tensor. However, only two of these three equations are mutually independent which means that the solution is not unique. Three possible forms of the solution are derived and analyzed in paper. One of these solutions is the traditional singular Schwarzchild solution, whereas two other solutions are not singular.
Journal Article•
TL;DR: In this article, the relation of geometrical and physical quantities in the Newtonian theory of gravitation, general relativity and Lorentz-invariant gravitation theory (LIGT), and also the physical meaning of the metric tensor and the space-time interval in the Euclidean, pseudo-Euclidean and pseudo-Riemannian spaces is investigated.
Abstract: This article is devoted to analysis of the relation of geometrical and physical quantities in the Newtonian theory of gravitation, general relativity and Lorentz-invariant gravitation theory (LIGT), and also to clarification of the physical meaning of the metric tensor and the space-time interval in the Euclidean, pseudo-Euclidean and pseudo-Riemannian spaces. The succession of the use of geometric concepts in these three theories is shown. It is shown that the math expression of interval is mutually uniquely associated with physical equations of elementary particles and LIGT. It is also shown that in LIGT the metric tensor has the physical meaning of the scale factor, defined by means of the Lorentz-invariant transformations. Evidence are given of that the metric tensor in general relativity should have the same meaning.
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TL;DR: Highlights of the twenty-odd-year relationship between Einstein and Hilbert are reviewed in this article, including the encounter that never took place (1912) when Einstein declined Hilbert's invitation to Gottingen, the fateful encounter (1915-1916) leading to a dispute over the final formulation of general relativity, the tragic-comic encounter (1920-1929) over editorship of the Annalen der Mathematik leading to what Einstein called "The battle of the Frogs and Mice", and L'envoi (1932) Einstein's
Abstract: Highlights of the twenty-odd-year relationship between Einstein and Hilbert are reviewed: the encounter that never took place (1912) when Einstein declined Hilbert’s invitation to Gottingen; the fateful encounter (1915–1916) leading to a dispute over the final formulation of general relativity; the tragic-comic encounter (1928–1929) over editorship of the Annalen der Mathematik leading to what Einstein called “The battle of the Frogs and Mice”; L’envoi (1932) Einstein’s final letter of congratulations to Hilbert on his 70th birthday.