62 research outputs found
Rotating dust solutions of Einstein's equations with 3-dimensional symmetry groups, Part 3: All Killing fields linearly independent of u^{\alpha} and w^{\alpha}
This is the third and last part of a series of 3 papers. Using the same
method and the same coordinates as in parts 1 and 2, rotating dust solutions of
Einstein's equations are investigated that possess 3-dimensional symmetry
groups, under the assumption that each of the Killing vectors is linearly
independent of velocity and rotation at every point
of the spacetime region under consideration. The Killing fields are found and
the Killing equations are solved for the components of the metric tensor in
every case that arises. No progress was made with the Einstein equations in any
of the cases, and no previously known solutions were identified. A brief
overview of literature on solutions with rotating sources is given.Comment: One missing piece, signaled after eq. (10.7), is added after (10.21).
List of corrections: In (3.7) wrong subscript in vorticity; In (3.10) wrong
subscript in last term of g_{23}; In (4.23) wrong formulae for g_{12} and
g_{22}; In (7.17) missing factor in velocity; In (7.18) one wrong factor in
g_{22}; In (10.9) factor in vorticity; In (10.15) - (10.20) y_0 = 0; In
(10.20) wrong second term in y. The rewriting typos did not influence result
Rotating perfect fluid sources of the NUT metric
Locally rotationally symmetric perfect fluid solutions of Einstein's
gravitational equations are matched along the hypersurface of vanishing
pressure with the NUT metric. These rigidly rotating fluids are interpreted as
sources for the vacuum exterior which consists only of a stationary region of
the Taub-NUT space-time. The solution of the matching conditions leaves
generally three parameters in the global solution. Examples of perfect fluid
sources are discussed.Comment: 8 pages, late
Non-Abelian pp-waves in D=4 supergravity theories
The non-Abelian plane waves, first found in flat spacetime by Coleman and
subsequently generalized to give pp-waves in Einstein-Yang-Mills theory, are
shown to be 1/2 supersymmetric solutions of a wide variety of N=1 supergravity
theories coupled to scalar and vector multiplets, including the theory of SU(2)
Yang-Mills coupled to an axion \sigma and dilaton \phi recently obtained as the
reduction to four-dimensions of the six-dimensional Salam-Sezgin model. In this
latter case they provide the most general supersymmetric solution. Passing to
the Riemannian formulation of this theory we show that the most general
supersymmetric solution may be constructed starting from a self-dual Yang-Mills
connection on a self-dual metric and solving a Poisson equation for e^\phi. We
also present the generalization of these solutions to non-Abelian AdS pp-waves
which allow a negative cosmological constant and preserve 1/4 of supersymmetry.Comment: Latex, 1+12 page
Dirichlet Boundary Value Problems of the Ernst Equation
We demonstrate how the solution to an exterior Dirichlet boundary value
problem of the axisymmetric, stationary Einstein equations can be found in
terms of generalized solutions of the Backlund type. The proof that this
generalization procedure is valid is given, which also proves conjectures about
earlier representations of the gravitational field corresponding to rotating
disks of dust in terms of Backlund type solutions.Comment: 22 pages, to appear in Phys. Rev. D, Correction of a misprint in
equation (4
Differentially rotating disks of dust: Arbitrary rotation law
In this paper, solutions to the Ernst equation are investigated that depend
on two real analytic functions defined on the interval [0,1]. These solutions
are introduced by a suitable limiting process of Backlund transformations
applied to seed solutions of the Weyl class. It turns out that this class of
solutions contains the general relativistic gravitational field of an arbitrary
differentially rotating disk of dust, for which a continuous transition to some
Newtonian disk exists. It will be shown how for given boundary conditions (i.
e. proper surface mass density or angular velocity of the disk) the
gravitational field can be approximated in terms of the above solutions.
Furthermore, particular examples will be discussed, including disks with a
realistic profile for the angular velocity and more exotic disks possessing two
spatially separated ergoregions.Comment: 23 pages, 3 figures, submitted to 'General Relativity and
Gravitation
Ellipsoidal shapes in general relativity: general definitions and an application
A generalization of the notion of ellipsoids to curved Riemannian spaces is
given and the possibility to use it in describing the shapes of rotating bodies
in general relativity is examined. As an illustrative example, stationary,
axisymmetric perfect-fluid spacetimes with a so-called confocal inside
ellipsoidal symmetry are investigated in detail under the assumption that the
4-velocity of the fluid is parallel to a time-like Killing vector field. A
class of perfect-fluid metrics representing interior NUT-spacetimes is obtained
along with a vacuum solution with a non-zero cosmological constant.Comment: Latex, 22 pages, Revised version accepted in Class. Quantum. Grav.,
references adde
Yang's gravitational theory
Yang's pure space equations (C.N. Yang, Phys. Rev. Lett. v.33, p.445 (1974))
generalize Einstein's gravitational equations, while coming from gauge theory.
We study these equations from a number of vantage points: summarizing the work
done previously, comparing them with the Einstein equations and investigating
their properties. In particular, the initial value problem is discussed and a
number of results are presented for these equations with common energy-momentum
tensors.Comment: 28 pages, to appear in Gen. Rel. Gra
Null dust in canonical gravity
We present the Lagrangian and Hamiltonian framework which incorporates null
dust as a source into canonical gravity. Null dust is a generalized Lagrangian
system which is described by six Clebsch potentials of its four-velocity Pfaff
form. The Dirac--ADM decomposition splits these into three canonical
coordinates (the comoving coordinates of the dust) and their conjugate momenta
(appropriate projections of four-velocity). Unlike ordinary dust of massive
particles, null dust therefore has three rather than four degrees of freedom
per space point. These are evolved by a Hamiltonian which is a linear
combination of energy and momentum densities of the dust. The energy density is
the norm of the momentum density with respect to the spatial metric. The
coupling to geometry is achieved by adding these densities to the gravitational
super-Hamiltonian and supermomentum. This leads to appropriate Hamiltonian and
momentum constraints in the phase space of the system. The constraints can be
rewritten in two alternative forms in which they generate a true Lie algebra.
The Dirac constraint quantization of the system is formally accomplished by
imposing the new constraints as quantum operator restrictions on state
functionals. We compare the canonical schemes for null and ordinary dust and
emhasize their differences.Comment: 25 pages, REVTEX, no figure
Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry
In the search for vacuum solutions, with or without a cosmological constant,
of the Einstein field equations of Petrov type N with twisting principal null
directions, the CR structures to describe the parameter space for a congruence
of such null vectors provide a very useful tool. A work of Hill, Lewandowski
and Nurowski has given a good foundation for this, reducing the field equations
to a set of differential equations for two functions, one real, one complex, of
three variables. Under the assumption of the existence of one Killing vector,
the (infinite-dimensional) classical symmetries of those equations are
determined and group-invariant solutions are considered. This results in a
single ODE of the third order which may easily be reduced to one of the second
order. A one-parameter class of power series solutions, g(w), of this
second-order equation is realized, holomorphic in a neighborhood of the origin
and behaving asymptotically as a simple quadratic function plus lower-order
terms for large values of w, which constitutes new solutions of the twisting
type N problem. The solution found by Leroy, and also by Nurowski, is shown to
be a special case in this class. Cartan's method for determining equivalence of
CR manifolds is used to show that this class is indeed much more general.
In addition, for a special choice of a parameter, this ODE may be integrated
once, to provide a first-order Abel equation. It can also determine new
solutions to the field equations although no general solution has yet been
found for it.Comment: 28 page
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