88,715 research outputs found

### A new proof of Birkhoff's theorem

Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation
reduces to an equation conformally related to the field equation for
2-dimensional gravity following from the Lagrangian L = R^(1/3).
Solutions for 2-dimensional gravity always possess a local isometry because
the traceless part of its Ricci tensor identically vanishes. Combining both
facts, we get a new proof of Birkhoff's theorem; contrary to other proofs, no
coordinates must be introduced.
The SO(m)-spherically symmetric solutions of the (m+1)-dimensional Einstein
equation can be found by considering L = R^(1/m) in two dimensions. This yields
several generalizations of Birkhoff's theorem in an arbitrary number of
dimensions, and to an arbitrary signature of the metric.Comment: 17 pages, LaTeX, no figures, Grav. and Cosm. in prin

### The Newtonian limit of fourth-order gravity

The weak-field slow-motion limit of fourth-order gravity will be discussed.Comment: 5 pages, LaTe

### On the Space of 3-dimensional Homogeneous Riemannian Manifolds

We answer the following question: Let l, m, n be arbitrary real numbers. Does
there exist a 3-dimensional homogeneous Riemannian manifold whose eigenvalues
of the Ricci tensor are just l, m and n ?Comment: 2 pages, LaTeX, reprinted from Proc. Conf. Brno (1995

### Linear energy bounds for Heisenberg spin systems

Recently obtained results on linear energy bounds are generalized to
arbitrary spin quantum numbers and coupling schemes. Thereby the class of
so-called independent magnon states, for which the relative ground-state
property can be rigorously established, is considerably enlarged. We still
require that the matrix of exchange parameters has constant row sums, but this
can be achieved by means of a suitable gauge and need not be considered as a
physical restriction

### Non-trivial Solutions of the Bach Equation Exist

We show that solutions of the Bach equation exist which are not conformal
Einstein spaces.Comment: 3 pages, LaTeX, no figur

### A new conformal duality of spherically symmetric space-times

A contribution linear in r to the gravitational potential can be created by a
suitable conformal duality transformation: the conformal factor is 1/(1+r)^2
and r will be replaced by r/(1+r), where r is the Schwarzschild radial
coordinate. Thus, every spherically symmetric solution of conformal Weyl
gravity is conformally related to an Einstein space. This result finally
resolves a long controversy about this topic.
As a byproduct, we present an example of a spherically symmetric Einstein
space which is a limit of a sequence of Schwarzschild-de Sitter space-times but
which fails to be expressable in Schwarzschild coordinates. This example also
resolves a long controversy.Comment: 11 pages, LaTeX, no figure

### Motion of a thin spherically symmetric Shell of Dust in the Schwarzschild field

The equation of motion announced in the title was already deduced for the
cases the inner metric being flat and the shell being negligibly small (test
matter), using surface layers and geodesic trajectories resp. Here we derive
the general equation of motion and solve it in closed form for the case of
parabolic motion. Especially the motion near the horizon and near the
singularity are examined.Comment: Reprinted from: 10th International Conference on General Relativity
and Gravitation, Padova (Italy) July 4 - 9, 1983. Eds.: B. Bertotti, F. de
Felice, A. Pascolini, Contributed papers Vol. 1, Roma (1983) page 339-34

### Perihelion advance for orbits with large eccentricities in the Schwarzschild black hole

We deduce a new formula for the perihelion advance of a test particle in the
Schwarzschild black hole by applying a newly developed non-linear
transformation within the Schwarzschild space-time. By this transformation we
are able to apply the well-known formula valid in the weak-field approximation
near infinity also to trajectories in the strong-field regime near the horizon
of the black hole.Comment: 22 pages, new results added at the end of scts. 4 and 5, accepted for
Phys. Rev.

### On Ellis' programme within fourth order gravity

For the non-tachyonic curvature squared action we show that the expanding
Bianchi-type I models tend to the dust-filled Einstein-de Sitter model for t
tending to infinity if the metric is averaged over the typical oscillation
period. Applying a conformal equivalence between curvature squared action and a
minimally coupled scalar field (which holds for all dimensions > 2) the problem
is solved by discussing a massive scalar field in an anisotropic cosmological
model.Comment: 9 pages, LaTeX, no figur

### The metric in the superspace of Riemannian metrics and its relation to gravity

The space of all Riemannian metrics is infinite-dimensional. Nevertheless a
great deal of usual Riemannian geometry can be carried over. The superspace of
all Riemannian metrics shall be endowed with a class of Riemannian metrics;
their curvature and invariance properties are discussed. Just one of this class
has the property to bring the lagrangian of General Relativity into the form of
a classical particle's motion. The signature of the superspace metric depends
in a non-trivial manner on the signature of the original metric, we derive the
corresponding formula. Our approach is a local one: the essence is a metric in
the space of all symmetric rank-two tensors, and then the space becomes a
warped product of the real line with an Einstein space.Comment: 10 pages, LaTeX, reprinted from Proc. Conf. Diff. Geom. Appl., Brno,
Czechoslovakia 1989, WSPC Singapore, Eds. J. Janyska, D. Krupk

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