21 research outputs found
The local and global geometrical aspects of the twin paradox in static spacetimes: I. Three spherically symmetric spacetimes
We investigate local and global properties of timelike geodesics in three
static spherically symmetric spacetimes. These properties are of its own
mathematical relevance and provide a solution of the physical `twin paradox'
problem. The latter means that we focus our studies on the search of the
longest timelike geodesics between two given points. Due to problems with
solving the geodesic deviation equation we restrict our investigations to
radial and circular (if exist) geodesics. On these curves we find general
Jacobi vector fields, determine by means of them sequences of conjugate points
and with the aid of the comoving coordinate system and the spherical symmetry
we determine the cut points. These notions identify segments of radial and
circular gepdesics which are locally or globally of maximal length. In de
Sitter spacetime all geodesics are globally maximal. In CAdS and
Bertotti--Robinson spacetimes the radial geodesics which infinitely many times
oscillate between antipodal points in the space contain infinite number of
equally separated conjugate points and there are no other cut points. Yet in
these two spacetimes each outgoing or ingoing radial geodesic which does not
cross the centre is globally of maximal length. Circular geodesics exist only
in CAdS spacetime and contain an infinite sequence of equally separated
conjugate points. The geodesic curves which intersect the circular ones at
these points may either belong to the two-surface or lie outside
it.Comment: 27 pages, 0 figures, typos corrected, version published in APP
Jacobi fields, conjugate points and cut points on timelike geodesics in special spacetimes
Several physical problems such as the `twin paradox' in curved spacetimes
have purely geometrical nature and may be reduced to studying properties of
bundles of timelike geodesics. The paper is a general introduction to
systematic investigations of the geodesic structure of physically relevant
spacetimes. The investigations are focussed on the search of locally and
globally maximal timelike geodesics. The method of dealing with the local
problem is in a sense algorithmic and is based on the geodesic deviation
equation. Yet the search for globally maximal geodesics is non-algorithmic and
cannot be treated analytically by solving a differential equation. Here one
must apply a mixture of methods: spacetime symmetries (we have effectively
employed the spherical symmetry), the use of the comoving coordinates adapted
to the given congruence of timelike geodesics and the conjugate points on these
geodesics. All these methods have been effectively applied in both the local
and global problems in a number of simple and important spacetimes and their
outcomes have already been published in three papers. Our approach shows that
even in Schwarzschild spacetime (as well as in other static spherically
symetric ones) one can find a new unexpected geometrical feature: instead of
one there are three different infinite sets of conjugate points on each stable
circular timelike geodesic curve. Due to problems with solving differential
equations we are dealing solely with radial and circular geodesics.Comment: A revised and expanded version, self-contained and written in an
expository style. 36 pages, 0 figures. A substantially abridged version
appeared in Acta Physica Polonica
The local and global geometrical aspects of the twin paradox in static spacetimes: II. Reissner--Nordstr\"{o}m and ultrastatic metrics
This is a consecutive paper on the timelike geodesic structure of static
spherically symmetric spacetimes. First we show that for a stable circular
orbit (if it exists) in any of these spacetimes all the infinitesimally close
to it timelike geodesics constructed with the aid of the general geodesic
deviation vector have the same length between a pair of conjugate points. In
Reissner--Nordstr\"{o}m black hole metric we explicitly find the Jacobi fields
on the radial geodesics and show that they are locally (and globally) maximal
curves between any pair of their points outside the outer horizon. If a radial
and circular geodesics in R--N metric have common endpoints, the radial one is
longer. If a static spherically symmetric spacetime is ultrastatic, its
gravitational field exerts no force on a free particle which may stay at rest;
the free particle in motion has a constant velocity (in this sense the motion
is uniform) and its total energy always exceeds the rest energy, i.~e.~it has
no gravitational energy. Previously the absence of the gravitational force has
been known only for the global Barriola--Vilenkin monopole. In the spacetime of
the monopole we explicitly find all timelike geodesics, the Jacobi fields on
them and the condition under which a generic geodesic may have conjugate
points
Every timelike geodesic in anti--de Sitter spacetime is a circle of the same radius
We refine and analytically prove an old proposition due to Calabi and Markus
on the shape of timelike geodesics of anti--de Sitter space in the ambient flat
space. We prove that each timelike geodesic forms in the ambient space a circle
of the radius determined by , lying on a Euclidean two--plane. Then we
outline an alternative proof for . We also make a comment on the shape
of timelike geodesics in de Sitter space.Comment: An expanded version of the work published in International Journal of
Modern Physics D. 8 pages, 0 figure
Anisotropic Inflation from Extra Dimensions
Vacuum multidimensional cosmological models with internal spaces being
compact -dimensional Lie group manifolds are considered. Products of
3-spheres and manifold (a novelty in cosmology) are studied. It turns
out that the dynamical evolution of the internal space drives an accelerated
expansion of the external world (power law inflation). This generic solution
(attractor in a phase space) is determined by the Lie group space without any
fine tuning or arbitrary inflaton potentials. Matter in the four dimensions
appears in the form of a number of scalar fields representing anisotropic scale
factors for the internal space. Along the attractor solution the volume of the
internal space grows logarithmically in time. This simple and natural model
should be completed by mechanisms terminating the inflationary evolution and
transforming the geometric scalar fields into ordinary particles.Comment: LaTeX, 11 pages, 5 figures available via fax on request to
[email protected], submitted to Phys. Lett.
Test-field limit of metric nonlinear gravity theories
In the framework of alternative metric gravity theories, it has been shown by
several authors that a generic Lagrangian depending on the Riemann tensor
describes a theory with 8 degrees of freedom (which reduce to 3 for f(R)
Lagrangians depending only on the curvature scalar). This result is often
related to a reformulation of the fourth-order equations for the metric into a
set of second-order equations for a multiplet of fields, including a massive
scalar field and a massive spin-2 field. In this article we investigate an
issue which does not seem to have been addressed so far: in ordinary
general-relativistic field theories, all fundamental fields (i.e. fields with
definite spin and mass) reduce to test fields in some appropriate limit of the
model, where they cease to act as sources for the metric curvature. In this
limit, each of the fundamental fields can be excited from its ground state
independently from the others. The question is: does higher-derivative gravity
admit a test-field limit for its fundamental fields? It is easy to show that
for a f(R) theory the test-field limit does exist; then, we consider the case
of Lagrangians quadratically depending on the full Ricci tensor. We show that
the constraint binding together the scalar field and the massive spin-2 field
does not disappear in the limit where they should be expected to act as test
fields, except for a particular choice of the Lagrangian, which cause the
scalar field to disappear (reducing to 7 DOF). We finally consider the addition
of an arbitrary function of the quadratic invariant of the Weyl tensor and show
that the resulting model still lacks a proper test-field limit. We argue that
the lack of a test-field limit for the fundamental fields may constitute a
serious drawback of the full 8 DOF higher-order gravity models, which is not
encountered in the restricted 7 DOF or 3 DOF cases.Comment: Title and abstract modified to make the content of the paper more
clear and readabl
On the twin paradox in static spacetimes: I. Schwarzschild metric
Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider
the twin paradox in static spacetimes. According to a well known theorem in
Lorentzian geometry the longest timelike worldline between two given points is
the unique geodesic line without points conjugate to the initial point on the
segment joining the two points. We calculate the proper times for static twins,
for twins moving on a circular orbit (if it is a geodesic) around a centre of
symmetry and for twins travelling on outgoing and ingoing radial timelike
geodesics. We show that the twins on the radial geodesic worldlines are always
the oldest ones and we explicitly find the conjugate points (if they exist)
outside the relevant segments. As it is of its own mathematical interest, we
find general Jacobi vector fields on the geodesic lines under consideration. In
the first part of the work we investigate Schwarzschild geometry.Comment: 18 pages, paper accepted for publication in Gen. Rel. Gra
Metric gravity theories and cosmology:II. Stability of a ground state in f(R) theories
A fundamental criterion of viability of any gravity theory is existence of a
stable ground-state solution being either Minkowski, dS or AdS space. Stability
of the ground state is independent of which frame is physical. In general, a
given theory has multiple ground states and splits into independent physical
sectors. All metric gravity theories with the Lagrangian being a function of
Ricci tensor are dynamically equivalent to Einstein gravity with a source and
this allows us to study the stability problem using methods developed in GR. We
apply these methods to f(R) theories. As is shown in 13 cases of Lagrangians
the stability criterion works simply and effectively whenever the curvature of
the ground state is determined. An infinite number of gravity theories have a
stable ground state and further viability criteria are necessary.Comment: A modified and expanded version of a second part of the paper which
previously appeared as gr-qc/0702097v1. The first, modified part is now
published as gr-qc/0702097v2 and as a separate paper in Class. Qu. Grav. The
present paper matches the published versio
Nonlinear massive spin-two field generated by higher derivative gravity
We present a systematic exposition of the Lagrangian field theory for the
massive spin-two field generated in higher-derivative gravity. It has been
noticed by various authors that this nonlinear field overcomes the well known
inconsistency of the theory for a linear massive spin-two field interacting
with Einstein's gravity. Starting from a Lagrangian quadratically depending on
the Ricci tensor of the metric, we explore the two possible second-order
pictures usually called "(Helmholtz-)Jordan frame" and "Einstein frame". In
spite of their mathematical equivalence, the two frames have different
structural properties: in Einstein frame, the spin-two field is minimally
coupled to gravity, while in the other frame it is necessarily coupled to the
curvature, without a separate kinetic term. We prove that the theory admits a
unique and linearly stable ground state solution, and that the equations of
motion are consistent, showing that these results can be obtained independently
in either frame. The full equations of motion and the energy-momentum tensor
for the spin--two field in Einstein frame are given, and a simple but
nontrivial exact solution to these equations is found. The comparison of the
energy-momentum tensors for the spin-two field in the two frames suggests that
the Einstein frame is physically more acceptable. We point out that the
energy-momentum tensor generated by the Lagrangian of the linearized theory is
unrelated to the corresponding tensor of the full theory. It is then argued
that the ghost-like nature of the nonlinear spin-two field, found long ago in
the linear approximation, may not be so harmful to classical stability issues,
as has been expected