Showing papers on "Introduction to the mathematics of general relativity published in 2002"

Gravity: An Introduction to Einstein's General Relativity

Abstract: Einstein's theory of general relativity is a cornerstone of modern physics. It also touches upon a wealth of topics that students find fascinating – black holes, warped spacetime, gravitational waves, and cosmology. Now reissued by Cambridge University Press, this ground-breaking text helped to bring general relativity into the undergraduate curriculum, making it accessible to virtually all physics majors. One of the pioneers of the 'physics-first' approach to the subject, renowned relativist James B. Hartle, recognized that there is typically not enough time in a short introductory course for the traditional, mathematics-first, approach. In this text, he provides a fluent and accessible physics-first introduction to general relativity that begins with the essential physical applications and uses a minimum of new mathematics. This market-leading text is ideal for a one-semester course for undergraduates, with only introductory mechanics as a prerequisite.

Static charged perfect fluid spheres in general relativity

Abstract: Interior perfect fluid solutions for the Reissner-Nordstr\"om metric are studied on the basis of a new classification scheme. It specifies which two of the characteristics of the fluid are given functions and accordingly picks up one of the three main field equations, the other two being universal. General formulas are found for charged de Sitter solutions, the case of a constant energy component of the energy-momentum tensor, the case of known pressure (including charged dust), and the case of a linear equation of state. Explicit new global solutions, mainly in elementary functions, are given as illustrations. The known solutions are briefly reviewed and corrected.

Harmonic coordinate method for simulating generic singularities

Abstract: This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.

Theorems on Existence and Global Dynamics for the Einstein Equations.

Abstract: This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.

Magnetic strings in anti-de Sitter general relativity

Abstract: We obtain spacetimes generated by static and rotating magnetic string sources in Einstein relativity with negative cosmological constant (Λ > 0). Since the spacetime is asymptotically a cylindrical anti-de Sitter spacetime, we will be able to calculate the mass, momentum and electric charge of the solutions. We find two families of solutions, one with longitudinal magnetic field and the other with angular magnetic field. The source for the longitudinal magnetic field can be interpreted as composed of a system of two symmetric and superposed electrically charged lines with one of the electrically charged lines being at rest and the other spinning. The angular magnetic field solution can be similarly interpreted as composed of charged lines but now one is at rest and the other has a velocity along the axis. This solution cannot be extended down to the origin.

The principle of symmetric criticality in general relativity

Abstract: We consider a version of Palais' principle of symmetric criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. Given a group action on a space of fields, PSC asserts that for any group-invariant Lagrangian, the equations obtained by restriction of Euler–Lagrange equations to group-invariant fields are equivalent to the Euler–Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric and show that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. These results are illustrated with a variety of examples from general relativity.

An example relevant to the Kretschmann-Einstein debate

Abstract: We cast the flat space theory of a scalar field in generally covariant form by introducing an auxiliary field $\lambda$ The resulting theory is couched in terms of an action integral $S$, and all the fields (the scalar, the spacetime metric, and $\lambda$) are dynamical in the sense of being varied freely in $S$ Conservation of energy-momentum emerges as a formal consequence of diffeomorphism invariance, in close analogy with the situation in ordinary general relativity

A radiation scalar for numerical relativity.

Abstract: This Letter describes a scalar curvature invariant for general relativity with a certain, distinctive feature. While many such invariants exist, this one vanishes in regions of space-time which can be said unambiguously to contain no gravitational radiation. In more general regions which incontrovertibly support nontrivial radiation fields, it can be used to extract local, coordinate-independent information partially characterizing that radiation. While a clear, physical interpretation is possible only in such radiation zones, a simple algorithm can be given to extend the definition smoothly to generic regions of space-time.

Illustrating stability properties of numerical relativity in electrodynamics

Abstract: We show that a reformulation of the Arnowitt-Deser-Misner equations in general relativity, which has dramatically improved the stability properties of numerical implementations, has a direct analogue in classical electrodynamics. We numerically integrate both the original and the revised versions of Maxwell's equations, and show that their distinct numerical behavior reflects the properties found in linearized general relativity. Our results shed further light on the stability properties of general relativity, illustrate them in a very transparent context, and may provide a useful framework for further improvement of numerical schemes.

Numerical evolution of axisymmetric, isolated systems in general relativity

Abstract: We describe in this article a new code for evolving axisymmetric isolated systems in general relativity. Such systems are described by asymptotically flat space-times which have the property that they admit a conformal extension. We are working directly in the extended ``conformal'' manifold and solve numerically Friedrich's conformal field equations, which state that Einstein's equations hold in the physical space-time. Because of the compactness of the conformal space-time the entire space-time can be calculated on a finite numerical grid. We describe in detail the numerical scheme, especially the treatment of the axisymmetry and the boundary.

Relations between Newtonian mechanics, general relativity, and quantum mechanics

Abstract: When Euclidean coordinate lengths are replaced by the metric lengths of a curved geometry within Newton’s second law of motion, the metric form of the second law can be shown to be identical to the geodesic equation of motion of general relativity. The metric coefficients are contained in the metric lengths and satisfy the field equations of general relativity. Because metric lengths are the physically measured lengths, their use makes it possible to understand general relativity directly in terms of physical quantities such as energy and momentum within a curved space–time. The metric form of the second law contains gravitational effects in exactly the same manner as occurs in relativity. Its mathematical derivation uses vectors rather than tensors, and nongravitational forces can occur in this modified second law without a tensor form. Because quantum mechanics is based on Newtonian concepts of energy and momentum, it is shown that when metric lengths replace coordinate lengths in Dirac’s wave equation, it has a covariant form under a metric transformation of the physically measured distances themselves, rather than a coordinate transformation. Metric transformations are also used to describe the Dirac equation for the gravitational central field in a Schwarzschild metric.

Distributional Sources in General Relativity. Two Point-Like Examples Revisited

Abstract: A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the density , associated to the Einstein tensor of the regularized metric, rather than the Einstein tensor itself, be a distribution and (ii) the regularized metric be a continuous metric with a discontinuous extrinsic curvature across a non-null hypersurface of codimension one. In this paper, the curvature and Einstein tensors of the geometries associated to point sources in the (2 + 1)-dimensional gravity and the Schwarzschild spacetime are considered. In both examples the regularized metrics are continuous regular metrics, as defined by Geroch and Traschen, with well defined distributional curvature tensors at all the intermediate steps of the calculation. The limit in which the support of these curvature tensors tends to the singular region of the original spacetime is studied and the results are contrasted with the ones obtained in previous works.

Exact Bianchi Type-I Cosmological Micro Model in Modified Theory of General Relativity

Abstract: An anisotropic, homogeneous Bianchi type-I cosmological micro model is obtained in Barber's modified theory of general relativity. Some properties of the model are discussed. Further, it is found that this theory leads to Einstein theory as the coupling parameterλ → 0 in micro level (i.e., quantum level) in general.

The methods of gluing manifolds in general relativity

Abstract: Some areas of modern theoretical physics such as modern cosmology contain different manifolds which must be glued together along a common boundary. These boundaries can be spacelike, timelike, or lightlike hypersurfaces. This paper shows how this gluing for different hypersurfaces is possible. Two different approaches are considered and the extent to which these approaches are equivalent are discussed. In particular, we will construct a distributional approach for dynamics of lightlike hypersurfaces in general relativity. Since Einstein’s equations are nonlinear PDEs, for discontinuous metrics such as signature changing metrics, product of distributions are unavoidable. To glue two different manifolds which admit signature change, we consider this problem in the context of Colombeau’s new theory of generalized functions. Some examples are given for clarification.

Einstein's gravitational field

Abstract: There exists some confusion, as evidenced in the literature, regarding the nature of the gravitational field in Einstein's General Theory of Relativity It is argued here the this confusion is a result of a change in interpretation of the gravitational field Einstein identified the existence of gravity with the inertial motion of accelerating bodies (ie bodies in free-fall) whereas contemporary physicists identify the existence of gravity with space-time curvature (ie tidal forces) The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with

Linear Einstein equations and Kerr-Schild maps

Abstract: We prove that given a solution of the Einstein equations gab for the matter field Tab, an autoparallel null vector field la and a solution (lalc, ac) of the linearized Einstein equation on the given background, the Kerr–Schild metric gac + λlalc (λ arbitrary constant) is an exact solution of the Einstein equation for the energy–momentum tensor Tac + λac + λ2l(ac)blb. The mixed form of the Einstein equation for Kerr–Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed spacetime due to Gurses and Gursey.

Computer algebra in general relativity

Abstract: This paper gives a short review of the status and applications of computer algebra systems for calculations in relativity and gravitation.

The Interaction of Gravity with Other Fields

Abstract: We consider the interaction of gravity, as expressed by Einstein's Equations of General Relativity, to other force fields. We describe some recent results, discussing both the mathematics, and the physical interpretations. These results concern both elementary particles, as well as cosmological models. (This paper describes joint work variously done with with F. Finster, N. Kamran, B. Temple, and S.-T. Yau.)

On SO(2, 1) symmetry in general relativity

Abstract: The role of SO(2, 1) symmetry in general relativity is analysed. Cosmological solutions of Einstein field equations invariant with respect to a spacelike Lie algebra r, with 3 ≤ r ≤ 6 and containing so(2, 1) as a subalgebra, are also classified.

A formal approach to machian general relativity

Abstract: We take Mach's principle to mean that the local properties of a test particle should depend on the global properties of the geometry. Using a complex wave-like metric and an appropriate redefinition of the energy-momentum tensor, we show this to be possible in principle within the context of general relativity. We outline implications for higher-dimensional theories of gravity.

Geometrical Aspects of Cosmic Magnetic Fields

Abstract: We discuss how the vector nature of magnetic fields and the geometrical interpretation of gravity introduced by general relativity lead to a special coupling between magnetism and spacetime curvature. This magneto-geometrical interaction effectively transfers the tension properties of the field into the spacetime fabric, triggering a variety of effects with potentially far-reaching implications.

Dynamics of Null Hypersurfaces in General Relativity

Abstract: This paper shows how the structure and dynamics of a lightlike thin shell in general relativity can be obtained from a distributional approach.

General-covariant constraint-free evolution system for Numerical Relativity

Abstract: A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations, when supplemented by suitable coordinate conditions, determine the time evolution of all these variables without any constraint. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints hold true. The extended system is well posed when using the natural extension of either harmonic coordinates or the harmonic slicing condition in normal coordinates

Spherically Symmetric Solutions in Relativistic Astrophysics

Abstract: In this thesis we study classes of static spherically symmetric spacetimes admitting a perfect fluid source, electromagnetic fields and anisotropic pressures. Our intention is to generate exact solutions that model the interior of dense, relativistic stars. We find a sufficient condition for the existence of series solutions to the condition of pressure isotropy for neutral isolated spheres. The existence of a series solution is demonstrated by the method of Frobenius. With the help of MATHEMATICA (Wolfram 1991) we recovered the Tolman VII model for a quadratic gravitational potential, but failed to obtain other known classes of solution. This establishes the weakness, in certain instances, of symbolic manipulation software to extract series solutions from differential equations. For a cubic potential, we obtained a new series solution to the Einstein field equations describing neutral stars. The gravitational and thermodynamic variables are non-singular and continuous. This model also satisfies the important barotropic equation of state p = p(p). Two new exact solutions to the Einstein-Maxwell system, that generalise previous results for uncharged stars, were also found. The first of these generalises the solution of Maharaj and Mkhwanazi (1996), and has well-behaved matter and curvature variables. The second solution reduces to the Durgapal and Bannerji (1983) model in the uncharged limit; this new result may only serve as a toy model for quark stars because of negative energy densities. In both examples we observe that the solutions may be expressed in terms of hypergeometric and elementary functions; this indicates the possibility of unifying isolated solutions under the hypergeometric equation. We also briefly study compact stars with spheroidal geometry, that may be charged or admit anisotropic pressure distributions. The adapted forms of the pressure isotropy condition can be written as a harmonic oscillator equation. Two simple examples are presented.

Energy-Momentum and Angular Momentum Carried by Gravitational Waves in Extended New General Relativity

Abstract: In an extended, new form of general relativity, which is a teleparallel theory of gravity, we examine the energy-momentum and angular momentum carried by gravitational wave radiated from Newtonian point masses in a weak-field approximation. The resulting wave form is identical to the corresponding wave form in general relativity, which is consistent with previous results in teleparallel theory. The expression for the dynamical energy-momentum density is identical to that for the canonical energy-momentum density in general relativity up to leading order terms on the boundary of a large sphere including the gravitational source, and the loss of dynamical energy-momentum, which is the generator of internal translations, is the same as that of the canonical energy-momentum in general relativity. Under certain asymptotic conditions for a non-dynamical Higgs-type field ψ k , the loss of “spin” angular momentum, which is the generator of internal SL(2 ,C ) transformations, is the same as that of angular momentum in general relativity, and the losses of canonical energy-momentum and orbital angular momentum, which constitute the generator of

Non conducting spherically symmetric fluids

Abstract: A class of spherically symmetric spacetimes invariantly defined by a zero flux condition is examined first from a purely geometrical point of view and then physically by way of Einstein's equations for a general fluid decomposition of the energy-momentum tensor. The approach, which allows a formal inversion of Einstein's equations, explains, for example, why spherically symmetric perfect fluids with spatially homogeneous energy density must be shearfree.