Showing papers on "Introduction to the mathematics of general relativity published in 1970"
TL;DR: In this article, the Bianchi identity is shown to imply that the Misner-Sharp-Hernandez mass function is an integral of two combinations of Einstein's equations for any energymomentum tensor and that mass energy flow is conservative.
Abstract: The mass‐energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner‐Sharp‐Hernandez mass function is an integral of two combinations of Einstein's equations for any energy‐momentum tensor and that mass‐energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass‐function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative‐mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point‐mass solution.
207 citations
TL;DR: In this article, the equations of hydrodynamics in the 2 ½-post-Newtonian approximation to general relativity are derived, which is also the approximation in which terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance.
Abstract: In this paper the equations of hydrodynamics in the 2½-post-Newtonian approximation to general relativity are derived. In this approximation all terms of O(c-5) are retained consistently with Einstein's field equations; it is also the approximation in which terms representing the reaction of the fluid to the emission of gravitational radiation by the system first make their appearance. The paper is in four parts. In Part I (by S. C.) the lowest-order terms in the metric coefficients are derived which are consequences of the imposition of the Sommerfeld radiation-condition at infinity. It is shown (following an early investigation of Trautman) that these terms are of O(c-5) in g00, of O(c-6) in g0α, and of O(c-5) in gαβ. Unique expressions are obtained for these terms. They are found to be purely of Newtonian origin. In Part II (by S. C. and F. P. E.) the equations of motion governing the fluid in the 2½-post-Newtonian approximation are derived. In addition to the coefficients already determined, these equations depend on a knowledge of the term of O(c-7) in goo. This term is determined by an explicit appeal to the field equation. It is further shown that this approximation brings no change to the density (c2ρμ0√-g) and the linear momentum (πα) that are conserved in the second post-Newtonian approximation. In Part III (by S. C.) it is shown that the terms of O(c-5) in the equations of motion contribute principally to the dissipation of the energy and the angular momentum conserved in the second post-Newtonian approximation. The rates of dissipation of energy and of angular momentum that are predicted are in exact agreement with the expectations of the linearized theory of gravitational radiation. Finally, in Part IV (by S. C. and F. P. E.) the energy, θ00-c2ρμ0√-g, to be associated with the 2½-post-Newtonian approximation is derived by evaluating the (0, 0)-component of the Landau-Lifshitz complex and the conserved density in the 3½-post-Newtonian approximation.
162 citations
01 Mar 1970
62 citations
TL;DR: In this article, exterior and interior solutions of Einstein's equations are given for fluid moving with the speed of light and having a superposed spin, the spin is microscopic and does not refer to the rotation of world lines, which are straight.
Abstract: Exterior and interior solutions of Einstein's equations are given for fluid moving with the speed of light and having a superposed spin. The spin is microscopic and does not refer to the rotation of world lines, which are straight. A strange feature is that the exterior solution is in every case locally isometric to an exterior solution for a non-spinning null fluid.
61 citations
TL;DR: In this article, it was shown that if space-time is nonempty at one time, it will be non-empty at all times provided that the energy momentum tensor of the matter satisfies a physically reasonable condition.
Abstract: It is shown that in classical general relativity, if space-time is nonempty at one time, it will be nonempty at all times provided that the energy momentum tensor of the matter satisfies a physically reasonable condition. The apparent contradiction with the quantum predictions for the creation and annihilation of matter particles by gravitons is discussed and is shown to arise from the lack of a good energy momentum operator for the matter in an unquantised curved space-time metric.
56 citations
TL;DR: In this article, it was shown that under some circumstances instead of an indefinite gravitational collapse there is a minimum of the volume and a bouncing back in the case of a stationary cluster of particles moving under the influence of the gravitational field produced by all of them together.
Abstract: In an ingenious way rotation (but no angular momentum) has been introduced in the case of spherical symmetry by Einstein, who has considered a stationary cluster of particles moving freely under the influence of the gravitational field produced by all of them together. The aim of the present work is to extend his idea to the non-static case, and it seems that under some circumstances instead of an indefinite gravitational collapse there is a minimum of the volume and a bouncing back.
45 citations
TL;DR: In this article, the Einstein tensors of metrics having a 3-parameter group of isometries with 2-dimensional non-null orbits were studied in order to obtain algebraic conditions guaranteeing an additional normal Killing vector.
Abstract: The Einstein tensors of metrics having a 3-parameter group of (global) isometries with 2-dimensional non-null orbitsG3(2,s/t) are studied in order to obtainalgebraic conditions guaranteeing an additional normal Killing vector. It is shown that Einstein spaces withG3(2,s/t) allow aG4. A critical review of some of the literature on Birkhoff's theorem and its generalizations is given.
39 citations
TL;DR: The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed in this paper.
Abstract: The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3‐parameter isometry group with 2‐dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebanski and Stachel.
36 citations
TL;DR: In this article, it was shown that all solutions of the Einstein field equations which satisfy (1) and exhibit local rotational symmetry, necessarily satisfy (2) and (3).
Abstract: Solutions of the Einstein field equations are considered subject to the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the Weyl tensor is algebraically special, (3) the corresponding repeated principal null congruence is geodesic and shearfree. If in addition, the repeated principal null congruence is non-expanding, it follows that the twist of this congruence must be non-zero (for a physically reasonable fluid). The general line element subject to this additional restriction is derived. Furthermore, it is shown that all solutions of the Einstein field equations which satisfy (1) and exhibit local rotational symmetry, necessarily satisfy (2) and (3).
34 citations
Book•
01 Jan 1970
TL;DR: In this article, the authors present a history of the history of Einstein's geometry as a branch of physics, including the development of the theory of Superspace and its application in general physics.
Abstract: Soluble Models of Quantum Gravitation.- The Quantization Program for General Relativity.- Particles and Geometry.- The Sandwich Conjecture.- Classical and Quantum Dynamics of a Closed Universe.- Post-Newtonian Methods and Conservation Laws.- Relativistic Boltzmann Theory and the Grad Method of Moments.- A Lemma on the Einstein-Liouville Equations.- Gravitational Radiation Experiments.- General Relativity Experiments Using Low Temperature Techniques.- Contribution to the History of Einstein's Geometry as a Branch of Physics.- Gravitational Radiation Damping.- The Nature of the Schwarzschild Singularity.- Singularities.- Energy-Momentum of Radiating Systems.- The Theory of Superspace.- Spacetime as a Sheaf of Geodesics in Superspace.- Author Index.
TL;DR: In this paper, a generalLorentz-Covariant calculus for the Ricci calculus and of the spinor calculus is presented, which implies the equivalence principle of space-time covariance and allows the geometrisation of gravitational fields according toEinstein's principle of equivalence.
Abstract: The principle of general relativity means the principle of generalLorentz-covariance of the physical equations in the language of tetrads and metrical spinors. A generalLorentz-Covariant calculus and the generalLorentz-covariant generalisations of the Ricci calculus and of the spinor calculus are given. The generalLorentz-covariant representation implies theEinstein principle of space-time covariance and allows the geometrisation of gravitational fields according toEinstein's principle of equivalence.
TL;DR: In this paper, the quaternion representation of general relativity derived in a previous study is applied to the problem of planetary motion, and it is found that while the outward appearance of the geodesic equation is the same in this formalism as it is in the conventional Einstein formulation, when the derivatives are expressed with respect to the differential increment in four-space, ds, the special frame of reference in which this equation is recast in terms of derivatives with respectto the time changes (i equations of motion) are somewhat different in the two formulations.
Abstract: In this paper, the quaternion representation of general relativity derived in a previous study is applied to the problem of planetary motion It is found that while the outward appearance of the geodesic equation is the same in this formalism as it is in the conventional Einstein formulation, when the derivatives are expressed with respect to the differential increment in four-space, ds, the special frame of reference in which this equation is recast in terms of derivatives with respect to the time changes (ie equations of motion) are somewhat different in the two formulations The reason has to do with the fact that 1) in this theory scalar invariant ds obeys the algebra of a quaternion-number field while in the Einstein formulation it belongs to a real-number field and 2) the bound states of planets are described here by quaternion-field variables that depend on the time parameter in a phase factor while there is no time dependence at all in the metric-tensor formulation for this physical situation As a result of this alteration in the description of a planetary orbit, it is found that the angular momentum,Lq,as compared with the angular momentum in the Einstein and Newtonian theories is as follows:Lq:LE:LN=mK exp [−2γ/r]:mK(1−2γ/r)1/2:mK, whereK is a constant that characterizes a particular orbit,m is the planetary mass andγ is the Schwarzschild radius On the other hand, it is found that to the order of perturbation that is required to compare with the observations of the anomalous part of the perihelion precession of Mercury’s orbit, the theoretical prediction that comes from this analysis is in numerical agreement with its prediction from Einstein’s theory and with the data Based on this analysis, it is suggested that future experimentation involving controlled artificial-satellite orbits could possibly be used to differentiate between the predictions of this theory and those of Einstein’s tensor formulation and the classical theory
TL;DR: In this paper, simple theorems and relations for charged- dust distributions in general relativity were presented, and some simple relations for the distribution of charged-dust distributions were established.
Abstract: The paper presents some simple theorems and relations for charged- dust distributions in general relativity.
01 Jan 1970
TL;DR: In this article, a new foundation of general Relativity has been proposed for which a new postulate had to be added for which I have chosen the derivability of the equations of motion from the field equations.
Abstract: About a year ago in a short note1 I outlined a “New Foundation of General Relativity.” My main point was that the usual postulational approach to general relativity (GR) does not distinguish between the Nordstrom-Einstein-Fokker (NEF) scalar theory and Einstein’s tensor theory. Thus a new postulate had to be added for which I have chosen the “derivability of the equations of motion from the field equations.” Later, I remembered having seen a review (invited talk) of Einstein on the “Present Status of the Problem of Gravitation” which he delivered at the “85. Naturforscherversammlung zu Wien,” in 1913 before a joint meeting of the Divisions of Physics, Mathematics, and Astronomy and which was published in the same year in the “Physikalische Zeitschrift ”together with the lively discussion which Einstein’s talk generated.2 In his review, Einstein also tried to distinguish between Nordstrom’s second scalar theory (which in 1914 he and Fokker brought into a generally covariant form) and his own (preliminary) tensor theory. I noticed, with some surprise, that Einstein’s argument for favoring his tensor theory was not correct. Nevertheless, his profound intuition led him to the right theory. As Goethe in his “Faust” said,“Der Mensch in seinem dunklen Drange ist sich des rechten Weges wohl bewusst.” Applied to our case, “Einstein in his obscure drive was conscious of the right pathway.”
TL;DR: The principles of relativity are assertions about the structure of physical laws, whose validity or nonvalidity can only be empirically confirmed or falsified as mentioned in this paper, and the weakest forms of those principles are the so-called global propositions, which furnish statements as to which operations, assumed to be performed simultaneously throughout the whole universe, have no influence upon the physical events.
Abstract: The principles of relativity are assertions about the structure of physical laws, whose validity or nonvalidity can only be empirically confirmed or falsified. The weakest forms of those principles are the so-calledglobal propositions. They furnish statements as to which operations—assumed to be performed simultaneously throughout the whole universe—have no influence upon the physical events. Much stronger principles are those of alocal nature. These assert that the physical properties of a system do not change, when the relation of the system is altered vis-a-vis the universe at large. On formulating these local principles, we presuppose either that it is possible to eliminate any influence of the environment or that the influence can be compensated as in the case of universal forces (e.g., gravitational) which can principally not be removed. Still weaker, however, are those formulations of the relativity principles which postulate relativity only for infinitesimally small space-time domains or regions. This distinction yields clarification of all discussions about existence and meaning of a general relativity principle. Such an analysis was already performed by Einstein and Abraham in 1912.
TL;DR: In this article, the behavior of asymptotically flat gravitational fields in the framework of general relativity is studied by the use of tetrad formalism, and the form of the peeling theorem in the above-mentioned coordinates for an arbitrary null tetrad is derived.
Abstract: The behaviour of asymptotically flat gravitational fields in the framework of general relativity is studied by the use of tetrad formalism. For this, a system of coordinates u, r, H and 0 is used, such that at spatial infinity u = const. is a null hypersurface and r, 0 and 5 reduce to the usual spherical polar coordinates. A set of four vectors (a tetrad) is also chosen with the only restriction that they are everywhere null. The metric tensor and the four vectors are expanded in inverse powers of r; the rotation coefficients and the tetrad components of the Riemann tensor are then calculated in a similar expansion; and the first two terms in the expansion beyond their values for a flat space are retained. The field equations in these approximations are derived explicitly and their effect on the expansion of the tetrad components of the Riemann tensor is studied; and the total energy and linear momentum are examined. In this paper three main results are derived: (i) the form of the peeling theorem in the above-mentioned coordinates for an arbitrary null tetrad; (ii) the generalized expression for the news function of the field; (iii) a simple criterion for recognizing certain classes of non-radiating fields. 1. INTRODIUCTION During the last decade a great deal of work has been done on asymptotically flat spaces in general relativity. Bondi, van der Burg & Metzner (i962) investigated the case of a gravitational field with an axis and a plane of symmetry while Sachs (i962) investigated the general case of a field with no symmetries. Among their results, two important ones were the mass-loss formula and the so-called peeling theorem. Using a slightly different approach, Newman & Penrose (I962) and Newman & Unti (I962) derived the peeling theorem from different assumptions and analysed in more detail the field at infinity. And finally, Newman & Penrose (I965) discovered ten absolute constants of the field with physical meanings still unknown. All the foregoing investigations were carried out in a system of coordinates
TL;DR: In this paper, a general and clear derivation of the field equations set up by Einstein is given, and it is shown that Misra's criticisms are completely unfounded and that the final results were valid.
Abstract: In 1939 Einstein published a paper with the above title in which he investigated the gravitational field of a spherically symmetric system consisting of a large number of gravitating particles of equal masses moving in concentric circular orbits, randomly oriented in space, under the influence of the field produced by all the particles together. His object was to show that Schwarzschild-like singularities do not exist in cases which have physical reality. In a paper published in 1964 Misra claimed that the field equations set up by Einstein were « mixed up and erroneous » but that Einstein’s final results were valid. It is shown in the present paper that, although Einstein’s paper is extremely confusing and contains some mistakes, Misra’s criticisms are completely unfounded. A general and clear derivation of Einstein’s results is given in this paper.
TL;DR: In this paper, the configuration variables for the gravitational field gmn are assigned arbitrarily on two infinitesimally neighboring spacelike hypersurfaces, and the extent to which a solution of the vacuum Einstein field equations can be found consistent with the given assignment is investigated.
Abstract: The configuration variables for the gravitational field gmn are assigned arbitrarily on two infinitesimally neighboring spacelike hypersurfaces. We then investigate the extent to which a solution of the vacuum Einstein field equations can be found consistent with the given assignment. A local approach, employing Dirac's Hamiltonian formalism, reveals that solutions can be found locally which are nonunique and highly unstable.
TL;DR: In this article, it was shown that there exists no space with Rik = 0, which is a ''true'' field, i.e., Riklm ≠ 0, and which admits an analogous relativistic group.
Abstract: The usual definition of a homogeneous field in general relativity implies a space with Riklm = 0, thus admitting a group of motions isomorphic to the Poincare group. After discussing the symmetry group of the homogeneous field in Newtonian space, we point out that there exists no space with Rik = 0, which is a ``true'' field, i.e., Riklm ≠ 0, and which admits an analogous relativistic group. We then study fields, solutions of Rik = 0, which define spaces that admit a 4‐parameter group of motions locally isomorphic to the groups T1 ⊗ [T2⊗sO(2)] and T1 ⊗ [T2⊗sO(1,1)]. We compare the motion of a test particle in these fields with the motion in the usual homogeneous field.
TL;DR: In this paper, the motion of incoherent matter and hence of test particles in the presence of fields with an arbitrary energy-momentum tensor is studied. But the equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection.
Abstract: This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.
TL;DR: The magnetic field induced by a rotating mass-shell having at its center a stationary charged sphere was calculated in the framework of linearized general relativity in this article, where the magnetic field was defined as a function of the magnetic energy of the mass.
TL;DR: In this paper, the radarecho-measurations of Shapiro's radar-measure are the experimentum crucis for Einstein's against Newton's theory in a spherical symmetric gravitational field.
Abstract: The gravity theories of Newton and Einstein are giving opposite sentences about the velocity of light in gravitational field. According to the Newtonian theory the velocity v in gravitational field is greater than the velocity c in a field-free space: v > c. According to general relativity theory we have a smaller velocity: v < c. For a spherical symmetric gravitational field Newton's theory gives but Einstein's theory of 1911 gives and general relativity gives . Therefore, the radarecho-measurations of Shapiro are the experimentum crucis for Einstein's against Newton's theory.
TL;DR: In this paper, the scalar tensor theory is cast into canonical form and the constraint equations are obtained from which it is possible to discuss the self-energy problem of the neutral particle and the electron.
Abstract: By first casting the scalar‐tensor theory into canonical form, the constraint equations are obtained from which it is possible to discuss the self‐energy problem of the neutral particle and the electron. With the assumptions of Minkowski space and a constant scalar field as boundary conditions at infinity, it is shown by variational arguments that the inertia of these objects is unchanged from the corresponding general relativity values. Accordingly, if the boundary value of the scalar is altered by a change of matter at infinity, the mass of the electron is unchanged. From this point of view, the scalar‐tensor theory is no more compatible with Mach's principle than is general relativity.
TL;DR: The results of the application of previously developed techniques to the analysis of the Petrov type III solutions to the vacuum Einstein equations are presented in this article, where the procedure involves the computer aided analysis to the extent that the functions uniquely and invariantly generating all local analytic solutions are determined.
Abstract: The results of the application of previously developed techniques to the analysis of the Petrov type III solutions to the vacuum Einstein equations are presented. The procedure involves the computer aided analysis of the Einstein‐Petrov equations to the extent that the functions uniquely and invariantly generating all local analytic solutions are determined. For the case of type III it is shown that, relative to a given fixed point in the manifold, all local analytic solutions are uniquely and invariantly determined by six arbitrary analytic functions of one variable and six others of two variables. These functions, called generating functions, thus provide a representation of all such solutions and may be used for the study of the structure of the family of Einstein empty space metrics.
TL;DR: The resolution of this paradox has an important bearing on the foundations of general relativity because, to resolve it, some authors have taken the extreme position that either general relativity or Mach's principle must be abandoned.
Abstract: THIRRING'S solution for a rotating mass shell is frequently used to illustrate the appearance of centrifugal and Coriolis force in general relativity In the equations of motion of test particles within the shell, terms appear which are of second order in the shell angular velocity, ω These terms are conventionally identified with “centrifugal force”, yet they do bear the relationship to “Coriolis force” that one would expect from Mach's principle The resolution of this paradox has an important bearing on the foundations of general relativity because, to resolve it, some authors have taken the extreme position that either general relativity or Mach's principle must be abandoned In this communication we resolve the paradox without abandoning either
01 Jan 1970
TL;DR: In this paper, the authors describe an experiment to test Einstein's theory of general relativity in a satellite in space by means of a nearly perfect gyroscope, which is uniquely made possible by complete use of a low temperature environment and the properties of superconductors including the use of zero magnetic fields and ultra-sensitive magnetometry.
Abstract: In this paper we will describe an experiment to test Einstein’s theory of general relativity in a satellite in space by means of a nearly perfect gyroscope.1 This experiment is uniquely made possible by complete use of a low temperature environment and the properties of superconductors including the use of zero magnetic fields and ultrasensitive magnetometry. In the last part of the talk we will mention other relativity experiments which make use of low temperature physics and ultra-sensitive magnetometers.