12 research outputs found
Finding Principal Null Direction for Numerical Relativists
We present a new method for finding principal null directions (PNDs). Because
our method assumes as input the intrinsic metric and extrinsic curvature of a
spacelike hypersurface, it should be particularly useful to numerical
relativists. We illustrate our method by finding the PNDs of the
Kastor-Traschen spacetimes, which contain arbitrarily many black holes in
a de Sitter back-ground.Comment: 10 pages, LaTeX style, WU-AP/38/93. Figures are available (hard
copies) upon requests [[email protected] (H.Shinkai)
Purely gravito-magnetic vacuum space-times
It is shown that there are no vacuum space-times (with or without
cosmological constant) for which the Weyl-tensor is purely gravito-magnetic
with respect to a normal and timelike congruence of observers.Comment: 4 page
Axistationary perfect fluids -- a tetrad approach
Stationary axisymmetric perfect fluid space-times are investigated using the
curvature description of geometries. Attention is focused on space-times with a
vanishing electric part of the Weyl tensor. It is shown that the only
incompressible axistationary magnetic perfect fluid is the interior
Schwarzschild solution. The existence of a rigidly rotating perfect fluid,
generalizing the interior Schwarzschild metric is proven. Theorems are stated
on Petrov types and electric/magnetic Weyl tensors.Comment: 12 page
Geometric Interpretation of the Mixed Invariants of the Riemann Spinor
Mixed invariants are used to classify the Riemann spinor in the case of
Einstein-Maxwell fields and perfect fluids. In the Einstein-Maxwell case these
mixed invariants provide information as to the relative orientation of the
gravitational and electromagnetic principal null directions. Consideration of
the perfect fluid case leads to some results about the behaviour of the
Bel-Robinson tensor regarded as a quartic form on unit timelike vectors.Comment: 31 pages, AMS-LaTe
Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension
We consider time reversal transformations to obtain twofold orthogonal
splittings of any tensor on a Lorentzian space of arbitrary dimension n.
Applied to the Weyl tensor of a spacetime, this leads to a definition of its
electric and magnetic parts relative to an observer (i.e., a unit timelike
vector field u), in any n. We study the cases where one of these parts vanishes
in particular, i.e., purely electric (PE) or magnetic (PM) spacetimes. We
generalize several results from four to higher dimensions and discuss new
features of higher dimensions. We prove that the only permitted Weyl types are
G, I_i and D, and discuss the possible relation of u with the WANDs; we provide
invariant conditions that characterize PE/PM spacetimes, such as Bel-Debever
criteria, or constraints on scalar invariants, and connect the PE/PM parts to
the kinematic quantities of u; we present conditions under which direct product
spacetimes (and certain warps) are PE/PM, which enables us to construct
explicit examples. In particular, it is also shown that all static spacetimes
are necessarily PE, while stationary spacetimes (e.g., spinning black holes)
are in general neither PE nor PM. Ample classes of PE spacetimes exist, but PM
solutions are elusive, and we prove that PM Einstein spacetimes of type D do
not exist, for any n. Finally, we derive corresponding results for the
electric/magnetic parts of the Riemann tensor. This also leads to first
examples of PM spacetimes in higher dimensions. We also note in passing that
PE/PM Weyl tensors provide examples of minimal tensors, and we make the
connection hereof with the recently proved alignment theorem. This in turn
sheds new light on classification of the Weyl tensors based on null alignment,
providing a further invariant characterization that distinguishes the types
G/I/D from the types II/III/N.Comment: 43 pages. v2: new proposition 4.10; some text reshuffled (former sec.
2 is now an appendix); references added; some footnotes cancelled, others
incorporated into the main text; some typos fixed and a few more minor
changes mad
On an alignment condition of the weyl tensor
We generalize an alignment condition of the Weyl tensor given by Barnes and Rowlingson. The alignment condition is then applied to Petrov type D perfect fluid spacetimes. In particular, purely magnetic, Petrov type D, shear-free perfect fluids are shown to be locally rotationally symmetric.<br /
Complex windmill transformation producing new purely magnetic fluids
Minimal complex windmill transformations of G2IB(ii) spacetimes (admitting a two-dimensional Abelian group of motions of the so-called Wainwright B(ii) class) are defined and the compatibility with a purely magnetic Weyl tensor is investigated. It is shown that the transformed spacetimes cannot be perfect fluids or purely magnetic Einstein spaces. We then determine which purely magnetic perfect fluids (PMpfs) can be windmill-transformed into purely magnetic anisotropic fluids (PMafs). Assuming separation of variables, complete integration produces two, algebraically general, G2I-B(ii) PMpfs: a solution with zero 4-acceleration vector and spatial energy–density gradient, previously found by the authors, and a new solution in terms of Kummer's functions, where these vectors are aligned and non-zero. The associated windmill PMafs are rotating but non-expanding. Finally, an attempt to relate the spacetimes to each other by a simple procedure leads to a G2I-B(ii) one-parameter PMaf generalization of the previously found metric