This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

/pdf/the-motion-of-point-particles-in-curved-spacetime-2hw7vyn00m.pdf

The Motion of Point Particles in Curved Spacetime

A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the position and orientation of the matter fields. In this manner all properties of the background spacetime are removed from physics and what remains are a set of ‘intrinsic’ relations between physical fields. For a wide range of phenomena, including all present experimental tests, the theory reproduces the predictions of general relativity. Differences do emerge, however, through the first–order nature of the equations and the global properties of the gauge fields and through the relationship with quantum theory. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle and consistency with the minimal coupling procedure selects an action which is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin–torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time–dependent, spherically symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These equations are then applied to the study of cosmology and to the formation and properties of black holes. Insistence on finding global solutions, together with the first–order nature of the equations, leads to a new understanding of the role played by time reversal. This alters the physical picture of the properties of a horizon around a black hole. The existence of global solutions enables one to discuss the properties of field lines inside the horizon due to a point charge held outside it. The Dirac equation is studied in a black hole background and provides a quick (though ultimately unsound) derivation of the Hawking temperature. Some applications to cosmology are also discussed and a study of the Dirac equation in a cosmological background reveals that the only models consistent with homogeneity are spatially flat. It is emphasized throughout that the description of gravity in terms of gauge fields, rather than spacetime geometry, leads to many simple and powerful physical insights. The language of ‘geometric algebra’ best expresses the physical and mathematical content of the theory and is employed throughout. Methods for translating the equations into other languages (tensor and spinor calculus) are given in appendices.

Gravity, gauge theories and geometric algebra

After a brief consideration of the high-speed scattering of two point charges we thoroughly discuss high-speed scattering for a charged particle by a fixed mass and of two uncharged particles of comparable masses. We use perturbation technique over Minkowski spacetime in the de Donder gauge and solve the field equations and the resulting equations of motion (which take the reaction of the particles' quasistatic self-field into account) by iteration. The obtained energy-momentum conservation laws allow the computation of second-order corrections for the scattering angle and the cross section. The asymptotic structure of the far-field indicates synchrotron radiation (electromagnetic and gravitational, respectively) which causes an energy loss whose reaction on the motion is briefly considered in the low-velocity limit including bound motion. (For neutral particles this is a third-order effect).

High-Speed Scattering of Charged and Uncharged Particles in General Relativity

* Preface * Stachel: Fresnel's (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies * Katzir: Poincare's Relativistic Theory of Gravitation * Renn: Standing on the Shoulders of a Dwarf: General Relativity-A Triumph of Einstein and Grossman's Erroneous Entwurf Theory * Renn: Before the Riemann Tensor: The Emergence of Einstein's Double Strategy * Norton: A Conjecture on Einstein, the Independent Reality of Spacetime Coordinate Systems and the Disaster of 1913 * Lehner: Einstein and the Principle of General Relativity, 1916-1921 * Kennefick: Einstein and the Problem of Motion: A Small Clue * Brading: A Note on General Relativity, Energy Conservation, and Noether's Theorems * Rynasiewicz: Weyl vs. Reichenbach on Lichtgeometrie * Gale: Dingle and de Sitter Against the Metaphysicians, or Two Ways to Keep Modern Cosmology Physical * Kragh: George Gamow and the 'Factual Approach' to Relativistic Cosmology * Sanchez-Ron: George McVittie, The Uncompromising Empiricist * Smeenk: False Vacuum: Early Universe Cosmology and the Development of Inflation * Majer and Sauer: Hilbert's 'World Equations' and His Vision of a Unified Science * Wunsch: Einstein, Kaluza and the Fifth Dimension * Goenner: Unified Field Theory: Early History and Interplay between Mathematics and Physics * Mattingly: Is Quantum Gravity Necessary? * Wazeck: Einstein in the Daily Press: A Glimpse into the Gehrcke Papers * Goldberg: Syracuse: 1949-1952 * Newman: A Biased and PersonalDescription of GR at Syracuse University, 1951-1961

/pdf/the-universe-of-general-relativity-3tghvxl25k.pdf

The Universe of General Relativity

We establish the existence of stable and metastable stationary black hole bound states at finite temperature and chemical potentials in global and planar four-dimensional asymptotically anti-de Sitter space We determine a number of features of their holographic duals and argue they represent structural glasses We map out their thermodynamic landscape in the probe approximation, and show their relaxation dynamics exhibits logarithmic aging, with aging rates determined by the distribution of barriers

Holographic Vitrification

In a recent paper, Prof. Whittaker has discussed the effect, according to the general theory of relativity, of gravitation on electromagnetic phenomena. In particular, he has discussed electrostatics in gravitational fields of two kinds, namely (i) the field due to a single gravitating mass, in which case space-time has the metric discovered by Schwarzschild, and (ii) a limiting case of this, called a quasi-uniform field, in which the gravitational force is, in the neighbourhood of the origin, uniform. Whittaker’s general method, so far as electrostatical problems were concerned, was to solve the partial differential equation satisfied by the electrostatic potential in terms of generalised harmonic functions, and then, from these, to build up other solutions. In this way, he succeeded in finding an algebraic expression which represents the potential of a single electron in the quasi-uniform field; he did not, however, obtain a corresponding algebraic expression for the potential of an electron in the Schwarzschild field.

E. T. Copson

Papers

On electrostatics in a gravitational field