Showing papers on "Introduction to the mathematics of general relativity published in 2011"

Plebański formulation of general relativity: a practical introduction

Abstract: We give a pedagogical introduction into an old, but unfortunately not commonly known formulation of GR in terms of self-dual two-forms due to in particular Jerzy Plebanski. Our presentation is rather explicit in that we show how the familiar textbook solutions: Schwarzschild, Volkoff–Oppenheimer, as well as those describing the Newtonian limit, a gravitational wave and the homogeneous isotropic Universe can be obtained within this formalism. Our description shows how Plebanski formulation gives quite an economical alternative to the usual metric and frame-based schemes for deriving Einstein equations.

Covariant formulation of the post-1-Newtonian approximation to general relativity

Abstract: We derive a coordinate-independent formulation of the post-1-Newtonian approximation to general relativity. This formulation is a generalization of the Newton-Cartan geometric formulation of Newtonian gravity. It involves several fields and a connection, but no spacetime metric at the fundamental level. We show that the usual coordinate-dependent equations of post-Newtonian gravity are recovered when one specializes to asymptotically flat spacetimes and to appropriate classes of coordinates.

Some consequences of born's reciprocal relativity in phase-spaces

Abstract: We explore some novel consequences of Born's reciprocal relativity theory in flat phase-space and generalize the theory to the curved space–time scenario. We provide, in particular, six specific results resulting from Born's reciprocal relativity and which are not present in special relativity. These are: momentum-dependent time delay in the emission and detection of photons; energy-dependent notion of locality; superluminal behavior; relative rotation of photon trajectories due to the aberration of light; invariance of areas-cells in phase-space and modified dispersion relations. We finalize by constructing a Born reciprocal general relativity theory in curved space–time which requires the introduction of a complex Hermitian metric, torsion and nonmetricity. The latter procedure can be extended to the curved phase-space scenario.

A kinetic theory of diffusion in general relativity with cosmological scalar field

Abstract: A new model to describe the dynamics of particles undergoing diffusion in general relativity is proposed. The evolution of the particle system is described by a Fokker-Planck equation without friction on the tangent bundle of spacetime. It is shown that the energy-momentum tensor for this matter model is not divergence-free, which makes it inconsistent to couple the Fokker-Planck equation to the Einstein equations. This problem can be solved by postulating the existence of additional matter fields in spacetime or by modifying the Einstein equations. The case of a cosmological scalar field term added to the left hand side of the Einstein equations is studied in some details. For the simplest cosmological model, namely the flat Robertson-Walker spacetime, it is shown that, depending on the initial value of the cosmological scalar field, which can be identified with the present observed value of the cosmological constant, either unlimited expansion or the formation of a singularity in finite time will occur in the future. Future collapse into a singularity also takes place for a suitable small but positive present value of the cosmological constant, in contrast to the standard diffusion-free scenario.

The initial-boundary value problem in general relativity

Abstract: In this article we summarize what is known about the initial-boundary value problem for general relativity and discuss present problems related to it.

Variational Principles In General Relativity

Abstract: In these lectures we shall derive the Einstein field and the equations of motion for uncharged and charged self- gravitating fluids from variational principles. We shall also see how singular hyper-surfaces (shock waves) and the equations governing their behavior may be treated by means of these principles. In addition we shall show how the “second variation” problem is related to the discussion of the stability of the solutions of the Einstein field equations.

Equations of Motion in General Relativity

Abstract: 1. Introduction 2. Foundations of the Post Newtonian Approximation 3. The Third Post Newtonian Approximation 4. Two-Body Problem in General Relativity 5. Small Black Holes: Geometrical Preliminaries 6. Small Charged Black Holes: Equations of Motion 7. Gravitational Physics of Few Body Systems

Revisiting the classical electron model in general relativity

Abstract: Motivated by earlier studies (Tiwari et al. in Astrophys. Space Sci. 182:105, 1984; Herrera and Varela in Phys. Lett. 189:11, 1994), we model electron as a spherically symmetric charged perfect fluid distribution of matter. The existing model is extended assuming a matter source that is characterized by quadratic equation of state in the context of general theory of relativity. For the suitable choices of the parameters, our charged fluid models almost satisfy the physical properties of electron.

A New Perspective on Relativity: An Odyssey in Non-Euclidean Geometries

Abstract: A Brief History of Relativity, Light, and Gravity Which Geometry? The Origins of Mass Relativity of Thermodynamics The 'General' Theory Short-Circuited Relativity of Hyperbolic Space Nonequivalence of Gravitation and Acceleration Aberration and Radiation Pressure in the Klein and Poincare Models The Inertia of Polarization.

Some Anisotropic Universes in the Presence of Imperfect Fluid Coupling with Spatial Curvature

Abstract: We consider Bianchi VI spacetime, which also can be reduced to Bianchi types VI0-V-III-I. We initially consider the most general form of the energy-momentum tensor which yields anisotropic stress and heat flow. We then derive an energy-momentum tensor that couples with the spatial curvature in a way so as to cancel out the terms that arise due to the spatial curvature in the evolution equations of the Einstein field equations. We obtain exact solutions for the universes indefinitely expanding with constant mean deceleration parameter. The solutions are beriefly discussed for each Bianchi type. The dynamics of the models and fluid are examined briefly, and the models that can approach to isotropy are determined. We conclude that even if the observed universe is almost isotropic, this does not necessarily imply the isotropy of the fluid (e.g., dark energy) affecting the evolution of the universe within the context of general relativity.

The Maxwell and Navier-Stokes that Follow from Einstein Equation in a Spacetime Containing a Killing Vector Field

Abstract: In this paper we are concerned to reveal that any spacetime structure \slg ,D,{\tau}_{[sg] \sslg },\uparrow>, which is a model of a gravitational field in General Relativity generated by an energy-momentum tensor T --- and which contains at least one nontrivial Killing vector field A --- is such that the 2-form field F=dA (where A=[g] \slg (A,)) satisfies a Maxwell like equation --- with a well determined current that contains a term of the superconducting type--- which follows directly from Einstein equation. Moreover, we show that the resulting Maxwell like equations, under an additional condition imposed to the Killing vector field, may be written as a Navier-Stokes like equation as well. As a result, we have a set consisting of Einstein, Maxwell and Navier-Stokes equations that follows sequentially from the first one under precise mathematical conditions and once some identifications about field variables are evinced, as detailed explained throughout the text. We compare and emulate our results with others on the same subject appearing in the literature.

Covariant Renormalizable Anisotropic Theories and Off-Diagonal Einstein-Yang-Mills-Higgs Solutions

Abstract: We use an important decoupling property of gravitational field equations in the general relativity theory and modifications, written with respect to nonholonomic frames with 2+2 spacetime decomposition. This allows us to integrate the Einstein equations (eqs) in very general forms with generic off--diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un--)broken symmetry parameters. We associate families of off-diagonal Einstein manifolds to certain classes of covariant gravity theories which have a nice ultraviolet behavior and seem to be (super) renormalizable in a sense of covariant modifications of Horava-Lifshits gravity. The apparent breaking of Lorentz invariance is present in some "partner" anisotropically induced theories due to nonlinear coupling with effective parametric interactions determined by nonholonomic constraints and generic off-diagonal gravitational and matter fields configurations. Finally, we show how the constructions can be extended to include exact solutions for conjectured covariant reonormalizable models with Einstein-Yang-Mills-Higgs fields.

Shock Wave Interactions in General Relativity and the Emergence of Regularity Singularities

Abstract: We show that the regularity of the gravitational metric tensor cannot be lifted from C to C by any C coordinate transformation in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel’s Theorem [6] which states that such coordinate transformations always exist in a neighborhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of singularity in spacetime, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the spacetime is not locally Minkowskian under any coordinate transformation. In particular, at such singularities, delta function sources in the second derivatives of the gravitational metric tensor exist in all coordinate systems, but due to cancelation, the curvature tensor remains uniformly bounded.

Research of Gravitation in Flat Minkowski Space

Abstract: In this paper it is introduced and studied an alternative theory of gravitation in flat Minkowski space. Using an antisymmetric tensor φ, which is analogous to the tensor of electromagnetic field, a non-linear connection is introduced. It is very convenient for studying the perihelion/periastron shift, deflection of the light rays near the Sun and the frame dragging together with geodetic precession i.e. effects where angles are involved. Although the corresponding results are obtained in rather different way, they are the same as in the General Relativity. The results about the barycenter of two bodies are also the same as in the General Relativity. Comparing the derived equations of motion for the n-body problem with the Einstein-Infeld-Hoffmann equations, it is found that they differ from the EIH equations by Lorentz invariant terms of order c−2.

When action is not least for orbits in general relativity

Abstract: In Newtonian mechanics, the action for a true trajectory between two spacetime events A and B is a minimum if the final event B occurs before the kinetic focus of the initial event A; otherwise, the action is a saddle point. We give a simplified proof of an analogous theorem in differential geometry for worldlines in a curved spacetime. We locate the kinetic focus for orbits in a Schwarzschild field in the lowest-order post-Newtonian approximation and show that the kinetic focus is shifted beyond its Newtonian value of one angular cycle by a fractional amount of order O(v2/c2).

A note on the coordinate freedom in describing the motion of particles in general relativity

Abstract: Our previous work developed a framework for treating the motion of a small body in general relativity, based on a one-parameter-family of solutions to Einstein's equation. Here we give an analysis of the coordinate freedom allowed within this framework, as is needed to determine the form of the equations of motion when they are expressed in general gauges.

Scalar-tensor cosmologies with a potential in the general relativity limit

Abstract: We consider Friedmann-Lemaitre-Robertson-Walker flat cosmological models in the framework of general Jordan frame scalar-tensor theories of gravity with arbitrary coupling functions, in the era when the energy density of the scalar potential dominates over the energy density of ordinary matter. To study the regime suggested by the local weak field tests (i.e. close to the so-called limit of general relativity) we propose a nonlinear approximation scheme, solve for the phase trajectories, and provide a complete classification of possible solutions. We argue that the topology of phase trajectories in the nonlinear approximation is representative of those of the full system, and thus can tell for which scalar-tensor models general relativity functions as an attractor. To the classes of models which asymptotically approach general relativity we give the solutions also in cosmological time and conclude with some observational implications.

Information transmittal, Newton's law of gravitation, and tensor approach to general relativity

Abstract: The special relativity considered in [A Einstein, Zur Elektrodynamik der bewegte Korper Ann Physik, 17 (1905) 891-921] is based on the concept of finite speed of information transmittal by the available signals (rays of light) It is demonstrated that the same concept applies to Newton's law of universal gravitation since the magnitude of distances between attracting masses can be physically defined (carried, accounted in acting forces of gravity) only by signals (physical processes) propagating at finite velocities It follows that the speed of propagation of gravity is finite The linear transformations of special relativity are applied to Newton's law of gravitation to take into account the relativistic effects of information transmittal in a field of central forces of attraction Relativistic representations of Newton's law are obtained with respect to the center of gravity exposing illusory effects that appear at high velocities It is verified that in atomic physics the effect of Newtonian gravitation on the motion of elementary particles at high velocities is negligible also in relativistic consideration Computational methods are developed to measure the intensity of gravitation at a distant space-time location using a body that travels in space, emitting uniform pulses of light that are received by the observer at a different space-time location It is demonstrated that the tensor approach to the general relativity and the united theory of space, time and gravitation in which the geometrical properties (metric) of the four-dimensional space-time continuum depend on the distribution of gravitating masses in space and their motion represent a transformed Lorentz invariant with a new type of inertia in the field of forces changing in space and time Real physical processes evolve according to the forces represented in the tensor form by this invariant which is equivalent to the coordinate-free local invariant of relativistic dynamics that defines the field and the motion of a body whose velocities and accelerations can be measured by relativistic identification methods at a point, time and direction of interest The results open new avenues for research in the general relativity and can be used for software development, field measurements and experimental studies in application to distant or fast moving systems

What if ... General Relativity is not the theory

Abstract: The nature of gravity is fundamental to understand the scaffolding of the Universe and its evolution. Einstein's general theory of relativity has been scrutinized for over ninety five years and shown to describe accurately all phenomena from the solar system to the Universe. However, this success is achieved in the case of the largest scales provided one admits contributions to energy-momentum tensor involving dark components such as dark energy and dark matter. Moreover, the theory has well known shortcomings, such as the problem of singularities, the cosmological constant problem and the well known initial conditions problems for the cosmological description. Furthermore, general relativity also does not fit the well known procedures that allow for the quantization of the other fundamental interactions. In this discussion we briefly review the experimental bounds on the foundational principles of general relativity, and present three recent proposals to extend general relativity or, at least, to regard it under different perspectives.

Time evolution of scalar-tensor cosmologies in the general relativity limit

Abstract: We consider Friedmann-Lemaitre-Robertson-Walker flat cosmological models in the framework of general Jordan frame scalar-tensor theories of gravity in two different cases: in the dust matter dominated era and in the potential dominated era. Motivated by the local weak field constraints and by cosmological observations, we develop and use an approximation scheme for the regime close to the so-called limit of general relativity. The ensuing nonlinear approximate equations for the scalar field can be solved analytically in cosmological time in both cases. We find criteria for the functions ω and V characterizing a scalar-tensor theory, to determine whether the theory does or does not possess solutions converging to general relativity asymptotically in time. The converging solutions can be subsumed under two principal classes: exponentional or polynomial convergence, and damped oscillations around general relativity. The classes of scalar-tensor theories of gravity which contain these types of solutions and s...

Cosmological Models and Singularities in General Relativity

Abstract: This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situation ...

Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained

Abstract: Vectors.- Differential calculus.- Tangent vectors.- Curvilinear coordinate systems.- The metric tensor.- The Christoffel symbols.- Covariant differentiation.- Geodesics.- Curvature.- Conservation laws of classical mechanics.- Einstein's field equations.- Einstein's theory of spacetime and gravitation.- Some applications.- Relativistic universe models.

Einstein’s Equations

Abstract: In this chapter we discuss Einstein’s gravitational equations, which state that the presence of matter and energy creates curvature in spacetime, via $${G}_{jk} = 8\pi \kappa {T}_{jk},$$ (0.1) where Gjk = Ricjk − (1∕2)Sgjk is the Einstein tensor, Tjk is the stress-energy tensor due to the presence of matter, and κ is a positive constant. In 1 we introduce this equation and relate it to previous discussions of stress-energy tensors and their relation to equations of motion. We recall various stationary action principles that give rise to equations of motion and show that (0.1) itself results from adding a term proportional to the scalar curvature of spacetime to standard Lagrangians and considering variations of the metric tensor.

Maxwell and Navier-Stokes Equations Equivalent to Einstein Equation

Abstract: In this paper we are concerned to reveal that any spacetime structure \slg ,D,{\tau}_{[sg] \sslg },\uparrow>, which is a model of a gravitational field in General Relativity generated by an energy-momentum tensor T --- and which contains at least one nontrivial Killing vector field A --- is such that the 2-form field F=dA (where A=[g] \slg (A,)) satisfies a Maxwell like equation --- with a well determined current that contains a term of the superconducting type--- which follows directly from Einstein equation. Moreover, we show that the resulting Maxwell like equations, under an additional condition imposed to the Killing vector field, may be written as a Navier-Stokes like equation as well. As a result, we have a set consisting of Einstein, Maxwell and Navier-Stokes equations that follows sequentially from the first one under precise mathematical conditions and once some identifications about field variables are evinced, as detailed explained throughout the text. We compare and emulate our results with others on the same subject appearing in the literature.

Static, cylindrical symmetry in general relativity and vacuum energy

Abstract: Static, Cylindrical Symmetry in General Relativity and Vacuum Energy. (April 2011) Cynthia Trendafilova Department of Mathematics Department of Physics Texas A&M University Research Advisor: Dr. Stephen Fulling Department of Mathematics In the first section of my research, in analogy with the standard derivation of the spherically symmetric Schwarzschild solution of the Einstein field equations, I find all static, cylindrically symmetric solutions of the Einstein equations for vacuum. These include not only the well known cone solution, which is locally flat, but others in which the metric coefficients are powers of the radial coordinate and the space-time is curved. These solutions appear in the literature, but in different forms, corresponding to different definitions of the radial coordinate. I find expressions for transforming between these different metric forms and examine some special points of interest. I then examine some special cases of non-vacuum solutions of the equations as well. Because all the vacuum solutions are singular on the axis, I match them to interior solutions with nonvanishing energy density and pressure. In addition to the well known cosmic string solution joining on to the cone, we find some numerical solutions that join on to the other exterior solutions. I then consider only a static, flat, cylindrically symmetric space-time. I calculate the components of the stress-energy tensor in terms of the cylinder kernel and its derivatives. The cylinder kernel in cylindrical coordinates has been previously calculated and can be used to find the energy density and pressure on various cylindrical boundaries; future work will include