360 research outputs found
Spacetime Slices and Surfaces of Revolution
Under certain conditions, a -dimensional slice of a
spherically symmetric black hole spacetime can be equivariantly embedded in
-dimensional Minkowski space. The embedding depends on a real parameter
that corresponds physically to the surface gravity of the black hole
horizon.
Under conditions that turn out to be closely related, a real surface that
possesses rotational symmetry can be equivariantly embedded in 3-dimensional
Euclidean space. The embedding does not obviously depend on a parameter.
However, the Gaussian curvature is given by a simple formula: If the metric is
written , then
\K_g=-{1/2}\phi''(r).
This note shows that metrics and occur in dual pairs, and that
the embeddings described above are orthogonal facets of a single phenomenon. In
particular, the metrics and their respective embeddings differ by a Wick
rotation that preserves the ambient symmetry.
Consequently, the embedding of depends on a real parameter. The ambient
space is not smooth, and is inversely proportional to the cone angle
at the axis of rotation. Further, the Gaussian curvature of is given
by a simple formula that seems not to be widely known.Comment: 15 pages, added reference
Strong-field tidal distortions of rotating black holes: III. Embeddings in hyperbolic 3-space
In previous work, we developed tools for quantifying the tidal distortion of
a black hole's event horizon due to an orbiting companion. These tools use
techniques which require large mass ratios (companion mass much smaller
than black hole mass ), but can be used for arbitrary bound orbits, and for
any black hole spin. We also showed how to visualize these distorted black
holes by embedding their horizons in a global Euclidean 3-space,
. Such visualizations illustrate interesting and important
information about horizon dynamics. Unfortunately, we could not visualize black
holes with spin parameter : such holes cannot
be globally embedded into . In this paper, we overcome this
difficulty by showing how to embed the horizons of tidally distorted Kerr black
holes in a hyperbolic 3-space, . We use black hole perturbation
theory to compute the Gaussian curvatures of tidally distorted event horizons,
from which we build a two-dimensional metric of their distorted horizons. We
develop a numerical method for embedding the tidally distorted horizons in
. As an application, we give a sequence of embeddings into
of a tidally interacting black hole with spin . A
small amplitude, high frequency oscillation seen in previous work shows up
particularly clearly in these embeddings.Comment: 10 pages, 6 figure
Extremal -invariant eigenvalues of the Laplacian of -invariant metrics
The study of extremal properties of the spectrum often involves restricting
the metrics under consideration. Motivated by the work of Abreu and Freitas in
the case of the sphere endowed with -invariant metrics, we consider
the subsequence of the spectrum of a Riemannian manifold
which corresponds to metrics and functions invariant under the action of a
compact Lie group . If has dimension at least 1, we show that the
functional admits no extremal metric under volume-preserving
-invariant deformations. If, moreover, has dimension at least three,
then the functional is unbounded when restricted to any conformal
class of -invariant metrics of fixed volume. As a special case of this, we
can consider the standard O(n)-action on ; however, if we also require the
metric to be induced by an embedding of in , we get an
optimal upper bound on .Comment: To appear in Mathematische Zeitschrif
Global embedding of the Kerr black hole event horizon into hyperbolic 3-space
An explicit global and unique isometric embedding into hyperbolic 3-space,
H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is
given. In particular, this allows the embedding into H^3 of surfaces of
revolution having negative, but finite, Gaussian curvature at smooth fixed
points of the U(1) isometry. As an example, we exhibit the global embedding of
the Kerr-Newman event horizon into H^3, for arbitrary values of the angular
momentum. For this example, considering a quotient of H^3 by the Picard group,
we show that the hyperbolic embedding fits in a fundamental domain of the group
up to a slightly larger value of the angular momentum than the limit for which
a global embedding into Euclidean 3-space is possible. An embedding of the
double-Kerr event horizon is also presented, as an example of an embedding
which cannot be made global.Comment: 16 pages, 13 figure
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