94 research outputs found
On the Ultrarelativistic Limit of General Relativity
As is well-known, Newton's gravitational theory can be formulated as a
four-dimensional space-time theory and follows as singular limit from
Einstein's theory, if the velocity of light tends to the infinity. Here
'singular' stands for the fact, that the limiting geometrical structure differs
from a regular Riemannian space-time. Geometrically, the transition Einstein to
Newton can be viewed as an 'opening' of the light cones. This picture suggests
that there might be other singular limits of Einstein's theory: Let all light
cones shrink and ultimately become part of a congruence of singular world
lines. The limiting structure may be considered as a nullhypersurface embedded
in a five-dimensional spacetime. While the velocity of light tends to zero
here, all other velocities tend to the velocity of light. Thus one may speak of
an ultrarelativistic limit of General Relativity. The resulting theory is as
simple as Newton's gravitational theory, with the basic difference, that
Newton's elliptic differential equation is replaced by essentially ordinary
differential equations, with derivatives tangent to the generators of the
singular congruence. The Galilei group is replaced by the Carroll group
introduced by L\'evy-Leblond. We suggest to study near ultrarelativistic
situations with a perturbational approach starting from the singular structure,
similar to post-Newtonian expansions in the case.Comment: 9 pages, Latex, submitted to Acta Physica Polonica (Jadwisin
Conference Proceedings
Race for the Kerr field
Roy P. Kerr has discovered his celebrated metric 45 years ago, yet the
problem to find a generalization of the Schwarzschild metric for a rotating
mass was faced much earlier. Lense and Thirring, Bach, Andress, Akeley, Lewis,
van Stockum and others have tried to solve it or to find an approximative
solution at least. In particular Achilles Papapetrou, from 1952 to 1961 in
Berlin, was interested in an exact solution. He directed the author in the late
autumn of 1959 to work on the problem. Why did these pre-Kerr attempts fail?
Comments based on personal reminiscences and old notes.Comment: 16 page
Self-consistent solutions for low-frequency gravitational background radiation
We study in a Brill-Hartle type of approximation the back-reaction of a
superposition of linear gravitational waves in an Einstein-de Sitter background
up to the second order in the small wave amplitudes . The wave
amplitudes are assumed to form a homogeneous and isotropic stochastic process.
No restriction for the wavelengths is assumed. The effective stress-energy
tensor is calculated in terms of the correlation functions of
the process. We discuss in particular a situation where is the
dominant excitation of the background metric. Apart from the Tolman radiation
universe, a solution with the scale factor of the de Sitter universe exists
with as effective equation of state.Comment: 3 pages, Latex (requires mprocl.sty). To appear in the Proceedings of
the Eighth Marcel Grossmann meeting (Jerusalem, June 22-27, 1977
Revisiting the Light Cone of the Goedel Universe
The structure of a light cone in the Goedel universe is studied. We derive
the intrinsic cone metric, calculate the rotation coefficients of the ray
congruence forming the cone, determine local differential invariants up to
second order, describe the crossover (keel) singularities and give a first
discussion of its focal points. Contrary to many rotation coefficients, some
inner differential invariants attain simple finite standard values at focal
singularities.Comment: 23 pages, 3 figures, iopar
Post-Newtonian extension of the Newton-Cartan theory
The theory obtained as a singular limit of General Relativity, if the
reciprocal velocity of light is assumed to tend to zero, is known to be not
exactly the Newton-Cartan theory, but a slight extension of this theory. It
involves not only a Coriolis force field, which is natural in this theory
(although not original Newtonian), but also a scalar field which governs the
relation between Newtons time and relativistic proper time. Both fields are or
can be reduced to harmonic functions, and must therefore be constants, if
suitable global conditions are imposed. We assume this reduction of
Newton-Cartan to Newton`s original theory as starting point and ask for a
consistent post-Newtonian extension and for possible differences to usual
post-Minkowskian approximation methods, as developed, for example, by
Chandrasekhar. It is shown, that both post-Newtonian frameworks are formally
equivalent, as far as the field equations and the equations of motion for a
hydrodynamical fluid are concerned.Comment: 13 pages, LaTex, to appear in Class. Quantum Gra
The lightcone of G\"odel-like spacetimes
A study of the lightcone of the G\"odel universe is extended to the so-called
G\"odel-like spacetimes. This family of highly symmetric 4-D Lorentzian spaces
is defined by metrics of the form ,
together with the requirement of spacetime homogeneity, and includes the
G\"odel metric. The quasi-periodic refocussing of cone generators with
startling lens properties, discovered by Ozsv\'{a}th and Sch\"ucking for the
lightcone of a plane gravitational wave and also found in the G\"odel universe,
is a feature of the whole G\"odel family. We discuss geometrical properties of
caustics and show that (a) the focal surfaces are two-dimensional null surfaces
generated by non-geodesic null curves and (b) intrinsic differential invariants
of the cone attain finite values at caustic subsets.Comment: 19 pages, 1 figur
Position and frequency shifts induced by massive modes of the gravitational wave background in alternative gravity
Alternative theories of gravity predict the presence of massive scalar,
vector, and tensor gravitational wave modes in addition to the standard
massless spin~2 graviton of general relativity. The deflection and frequency
shift effects on light from distant sources propagating through a stochastic
background of gravitational waves, containing such modes, differ from their
counterparts in general relativity. Such effects are considered as a possible
signature for alternative gravity in attempts to detect deviations from
Einstein's gravity by astrophysical means.Comment: 9 pages, 1 figur
Laser Interferometric Detectors of Gravitational Waves
A laser interferometric detector of gravitational waves is studied and a
complete solution (to first order in the metric perturbation) of the coupled
Einstein-Maxwell equations with appropriate boundary conditions for the light
beams is determined. The phase shift, the light deflection and the rotation of
the polarization axis induced by gravitational waves are computed. The results
are compared with previous literature, and are shown to hold also for detectors
which are large in comparison with the gravitational wavelength.Comment: 13 pages, LaTe
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