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Robert Beig

Publications and source records attributed to Robert Beig.

34 records · Page 2Linked to original sources
arXiv2007

Relativistic Aberration for Accelerating Observers

We investigate the effects of the aberration of light for a uniformly accelerating observer. The observer we consider is initially at rest with respect to a luminous spherical object--a star, say--and then starts to move away with constant acceleration. The main results we derive are the following: (i) The observer always sees an initial increase of the apparent size of the object; (ii) The apparent size of the object approaches a non-zero value as the proper time of the observer goes to infinity. (iii) There exists a critical value of the acceleration such that the apparent size of the object is always increasing when the acceleration is super-critical. We show that, while (i) is a purely non-relativistic effect, (ii) and (iii) are effects of the relativistic aberration of light and are intimately connected with the Lorentzian geometry of Minkowksi spacetime. Finally, the examples we present illustrate that, while more or less negligible in everyday life, the three effects can be significant in the context of space-flight.

physics.class-ph↗
arXiv2007

Helically symmetric N-particle solutions in scalar gravity

Within a scalar model theory of gravity, where the interaction between particles is given by the half-retarded + half-advanced solution of the scalar wave equation, we consider an N-body problem: we investigate configurations of N particles which form an equilateral N-angle and are in helical motion about their common center. We prove that there exists a unique equilibrium configuration and compute the equilibrium radius explicitly in a post-Newtonian expansion.

gr-qc↗
arXiv2007

On the motion of a compact elastic body

We study the problem of motion of a relativistic, ideal elastic solid with free surface boundary by casting the equations in material form ("Lagrangian coordinates"). By applying a basic theorem due to Koch, we prove short-time existence and uniqueness for solutions close to a trivial solution. This trivial, or natural, solution corresponds to a stress-free body in rigid motion.

gr-qc↗
arXiv2005

Concepts of Hyperbolicity and Relativistic Continuum Mechanics

After a short introduction to the characteristic geometry underlying weakly hyperbolic systems of partial differential equations we review the notion of symmetric hyperbolicity of first-order systems and that of regular hyperbolicity of second-order systems. Numerous examples are provided, mainly taken from nonrelativistic and relativistic continuum mechanics.

gr-qc↗
arXiv2005

CMC-Slicings of Kottler-Schwarzschild-de Sitter Cosmologies

There is constructed, for each member of a one-parameter family of cosmological models, which is obtained from the Kottler-Schwarzschild-de Sitter spacetime by identification under discrete isometries, a slicing by spherically symmetric Cauchy hypersurfaces of constant mean curvature. These slicings are unique up to the action of the static Killing vector. Analytical and numerical results are found as to when different leaves of these slicings do not intersect, i.e. when the slicings form foliations.

gr-qc↗
arXiv2004

Nonrelativistic and Relativistic Continuum Mechanics

There is described a spacetime formulation of both nonrelativistic and relativistic elasticity. Specific attention is devoted to the causal structure of the theories and the availability of local existence theorems for the initial-value problem. Much of the presented material is based on joint work of B.G.Schmidt and the author (in Class.Quantum Grav.\textbf{20} (2003), 889-904).

gr-qc↗
arXiv2004

Bowen-York Tensors

There is derived, for a conformally flat three-space, a family of linear second-order partial differential operators which send vectors into tracefree, symmetric two-tensors. These maps, which are parametrized by conformal Killing vectors on the three-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular these maps send source-free electric fields into TT-tensors. Moreover, if the original vector field is the Coulomb field on $\mathbb{R}^3\backslash \lbrace0\rbrace$, the resulting tensor fields on $\mathbb{R}^3\backslash \lbrace0\rbrace$ are nothing but the family of TT-tensors originally written down by Bowen and York.

gr-qc↗
arXiv2003

Relativistic Elasticity

Relativistic elasticity on an arbitrary spacetime is formulated as a Lagrangian field theory which is covariant under spacetime diffeomorphisms. This theory is the relativistic version of classical elasticity in the hyperelastic, materially frame-indifferent case and, on Minkowski space, reduces to the latter in the non-relativistic limit . The field equations are cast into a first -- order symmetric hyperbolic system. As a consequence one obtains local--in--time existence and uniqueness theorems under various circumstances.

gr-qc↗
arXiv2002

Static, Self-Gravitating Elastic Bodies

There is proved an existence theorem, in the Newtonian theory, for static, self-gravitating bodies composed of elastic material. The theorem covers the case where these bodies are small, but allows them to have arbitrary shape.

gr-qc↗
arXiv2000

The maximal slicing of a Schwarzschild black hole

We describe recent work due to Niall Ó Murchadha and the author (Phys. Rev. D57, 4728 (1998)) on the late time behaviour of the maximal foliation of the extended Schwarzschild geometry which results from evolving a time symmetric slice into the past and future, with time running equally fast at both spatial ends of the manifold. We study the lapse function of this foliation in the limit where proper time-at-infinity goes to ${+\infty}$ or ${-\infty}$ and the slices approach $r=3m/2$.

gr-qc↗
arXiv2000

Time-Independent Gravitational Fields

This article reviews, from a global point of view, rigorous results on time independent spacetimes. Throughout attention is confined to isolated bodies at rest or in uniform rotation in an otherwise empty universe. The discussion starts from first principles and is, as much as possible, self-contained.

gr-qc↗
arXiv2000

Multipole Moments of Static Spacetimes

This paper reviews work, largely due to W. Simon and the author, on multipole theory of static spacetimes. The main purpose is to make this work, which lies at the interface of potential theory, conformal geometry and general relativity, known to mathematicians and to perhaps motivate them to look at the open problems which still remain.

gr-qc↗
arXiv2000

Generalized Bowen--York Initial Data

A class of vacuum initial-data sets is described which are based on certain expressions for the extrinsic curvature first studied and employed by Bowen and York. These expressions play a role for the momentum constraint of general relativity which is analogous to the role played by the Coulomb solution for the Gauss-law constraint of electromagnetism.

gr-qc↗
arXiv1997

On a global conformal invariant of initial data sets

In the present paper a global conformal invariant $Y$ of a closed initial data set is constructed. A spacelike hypersurface $Σ$ in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation ${\cal D}_e$, the so-called real Sen connection, which turns out to be determined completely by the initial data $h_{ab}$ and $χ_{ab}$ induced on $Σ$, and coincides, in the case of vanishing second fundamental form $χ_{ab}$, with the Levi-Civita covariant derivation $D_e$ of the induced metric $h_{ab}$. $Y$ is built from the real Sen connection ${\cal D}_e$ in the similar way as the standard Chern-Simons invariant is built from $D_e$. The number $Y$ is invariant with respect to changes of $h_{ab}$ and $χ_{ab}$ corresponding to conformal rescalings of the spacetime metric. In contrast the quantity $Y$ built from the complex Ashtekar connection is not invariant in this sense. The critical points of our $Y$ are precisely the initial data sets which are locally imbeddable into conformal Minkowski space.

gr-qc↗
arXiv1994

Initial Data for General Relativity with Toroidal Conformal Symmetry

A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no isometries, but a $U(1)\times U(1)$ group of conformal isometries. After decomposing the Lichnerowicz conformal factor in a double Fourier series on the group orbits, the solutions are given in terms of a countable family of uncoupled ODEs on the orbit space.

gr-qc↗