Showing papers on "Introduction to the mathematics of general relativity published in 2018"

An optimal transport formulation of the Einstein equations of general relativity

Abstract: The goal of the paper is to give an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

General Relativity as a Hybrid theory: The Genesis of Einstein's work on the problem of motion

Abstract: In this paper I describe the genesis of Einstein's early work on the problem of motion in general relativity (GR): the question of whether the motion of matter subject to gravity can be derived directly from the Einstein field equations. In addressing this question, Einstein himself always preferred the vacuum approach to the problem: the attempt to derive geodesic motion of matter from the vacuum Einstein equations. The paper first investigates why Einstein was so skeptical of the energy-momentum tensor and its role in GR. Drawing on hitherto unknown correspondence between Einstein and George Yuri Rainich, I then show step by step how his work on the vacuum approach came about, and how his quest for a unified field theory informed his interpretation of GR. I show that Einstein saw GR as a hybrid theory from very early on: fundamental and correct as far as gravity was concerned but phenomenological and effective in how it accounted for matter. As a result, Einstein saw energy-momentum tensors and singularities in GR as placeholders for a theory of matter not yet delivered. The reason he preferred singularities was that he hoped that their mathematical treatment would give a hint as to the sought after theory of matter, a theory that would do justice to quantum features of matter.

Scalar Fields in Numerical General Relativity

Abstract: Einstein's field equation of General Relativity (GR) has been known for over 100 years, yet it remains challenging to solve analytically in strongly relativistic regimes, particularly where there is a lack of a priori symmetry. Numerical Relativity (NR) - the evolution of the Einstein Equations using a computer - is now a relatively mature tool which enables such cases to be explored. In this thesis, a description is given of the development and application of a new Numerical Relativity code, GRChombo. GRChombo uses the standard BSSN formalism, incorporating full adaptive mesh refinement (AMR) and massive parallelism via the Message Passing Interface (MPI). The AMR capability permits the study of physics which has previously been computationally infeasible in a full 3+1 setting. The functionality of the code is described, its performance characteristics are demonstrated, and it is shown that it can stably and accurately evolve standard spacetimes such as black hole mergers. We use GRChombo to study the effects of inhomogeneous initial conditions on the robustness of small and large field inflationary models. and investigate the critical behaviour which occurs in the collapse of both spherically symmetric and asymmetric scalar field bubbles.

On the universality of Maxwell’s equations

Abstract: Einstein’s theory of relativity is based on the Principle of Equivalence, Hilbert’s on invariant theory and the calculus of variations. The two paradigms are not equivalent. Using the universality of Maxwell’s equations, Hilbert’s variational method is used to determine the energy–momentum tensor uniquely, and to show that general relativity can be formulated on the basis of Maxwellian, rather than specific physical force fields. A unified field theory is proved in which the Maxwellian force fields are all on an equal footing, distinct from the geometric field.

Conservation law of energy-momentum in general relativity

Abstract: We explain the necessity of application of semi-metric in general relativity. A detailed discussion on the energy-momentum conservation in the general relativity is presented using the mathematical tool of semi-metric. By means of the general covariant spacetime translation transformation, the most general covariant conservation law of energy-momentum is obtained, which is valid for any coordinates and overcomes the flaws of the expressions of Einstain, Landau and Moller.