12 research outputs found
Rigid motion revisited: rigid quasilocal frames
We introduce the notion of a rigid quasilocal frame (RQF) as a geometrically
natural way to define a "system" in general relativity. An RQF is defined as a
two-parameter family of timelike worldlines comprising the worldtube boundary
of the history of a finite spatial volume, with the rigidity conditions that
the congruence of worldlines is expansion-free (constant size) and shear-free
(constant shape). This definition of a system is anticipated to yield simple,
exact geometrical insights into the problem of motion in general relativity. It
begins by answering the questions what is in motion (a rigid two-dimensional
system boundary), and what motions of this rigid boundary are possible. Nearly
a century ago Herglotz and Noether showed that a three-parameter family of
timelike worldlines in Minkowski space satisfying Born's 1909 rigidity
conditions has only three degrees of freedom instead of the six we are familiar
with from Newtonian mechanics. We argue that in fact we can implement Born's
notion of rigid motion in both flat spacetime (this paper) and arbitrary curved
spacetimes containing sources (subsequent papers) - with precisely the expected
three translational and three rotational degrees of freedom - provided the
system is defined quasilocally as the two-dimensional set of points comprising
the boundary of a finite spatial volume, rather than the three-dimensional set
of points within the volume.Comment: 10 pages (two column), 24 pages (preprint), 1 figur
On the degrees of freedom of a semi-Riemannian metric
A semi-Riemannian metric in a n-manifold has n(n-1)/2 degrees of freedom,
i.e. as many as the number of components of a differential 2-form. We prove
that any semi-Riemannian metric can be obtained as a deformation of a constant
curvature metric, this deformation being parametrized by a 2-for
Quasi-local rotating black holes in higher dimension: geometry
With a help of a generalized Raychaudhuri equation non-expanding null
surfaces are studied in arbitrarily dimensional case. The definition and basic
properties of non-expanding and isolated horizons known in the literature in
the 4 and 3 dimensional cases are generalized. A local description of horizon's
geometry is provided. The Zeroth Law of black hole thermodynamics is derived.
The constraints have a similar structure to that of the 4 dimensional spacetime
case. The geometry of a vacuum isolated horizon is determined by the induced
metric and the rotation 1-form potential, local generalizations of the area and
the angular momentum typically used in the stationary black hole solutions
case.Comment: 32 pages, RevTex
Reference frames and rigid motions in relativity: Applications
The concept of rigid reference frame and of constricted spatial metric, given
in the previous work [\emph{Class. Quantum Grav.} {\bf 21}, 3067,(2004)] are
here applied to some specific space-times: In particular, the rigid rotating
disc with constant angular velocity in Minkowski space-time is analyzed, a new
approach to the Ehrenfest paradox is given as well as a new explanation of the
Sagnac effect. Finally the anisotropy of the speed of light and its measurable
consequences in a reference frame co-moving with the Earth are discussed.Comment: 13 pages, 1 figur