45 research outputs found
Explicit Riemannian manifolds with unexpectedly behaving center of mass
The (relativistic) center of mass of an asymptotically flat Riemannian
manifold is often defined by certain surface integral expressions evaluated
along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and
Misner (ADM). There are also what we call 'abstract' definitions of the center
of mass in terms of a foliation near infinity itself, going back to the
constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give
rise to surface integral expressions when equipped with suitable systems of
coordinates. We discuss subtle asymptotic convergence issues regarding the ADM-
and the coordinate expressions related to the CMC-center of mass. In
particular, we give explicit examples demonstrating that both can diverge -- in
a setting where Einstein's equation is satisfied. We also give explicit
examples of the same asymptotic order of decay with prescribed mass and center
of mass. We illustrate both phenomena by providing analogous examples in
Newtonian gravity. Our examples conflict with some results in the literature.Comment: examples with prescribed mass and center of mass included; asymptotic
decay described in more detail; references update
Uniqueness of photon spheres in electro-vacuum spacetimes
In a recent paper, the authors established the uniqueness of photon spheres
in static vacuum asymptotically flat spacetimes by adapting Bunting and
Masood-ul-Alam's proof of static vacuum black hole uniqueness. Here, we
establish uniqueness of suitably defined sub-extremal photon spheres in static
electro-vacuum asymptotically flat spacetimes by adapting the argument of
Masood-ul-Alam. As a consequence of our result, we can rule out the existence
of electrostatic configurations involving multiple "very compact" electrically
charged bodies and sub-extremal black holes.Comment: 16 pages. This paper extends the photon sphere uniqueness result
obtained in arXiv:1504.05804 from the vacuum to the electro-vacuum setting.
While the general proof method is similar, a number of new nontrivial issues
aris
A universal inequality between angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter
We prove that for sub-extremal axisymmetric and stationary black holes with
arbitrary surrounding matter the inequality holds, where is the
angular momentum and the horizon area of the black hole.Comment: 8 page
A flow approach to Bartnik's static metric extension conjecture in axisymmetry
We investigate Bartnik's static metric extension conjecture under the
additional assumption of axisymmetry of both the given Bartnik data and the
desired static extensions. To do so, we suggest a geometric flow approach,
coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to
the Einstein vacuum equations. The elliptic Weyl-Papapetrou system becomes a
free boundary value problem in our approach. We study this new flow and the
coupled flow--free boundary value problem numerically and find axisymmetric
static extensions for axisymmetric Bartnik data in many situations, including
near round spheres in spatial Schwarzschild of positive mass.Comment: 60 pages, 13 figures. Expanded Section 3.3 to address longtime
existence and uniqueness of solutions to the linearised flow equations. To
appear in Pure and Applied Mathematics Quarterly, special issue in honour of
Robert Bartni
On the center of mass of asymptotically hyperbolic initial data sets
We define the (total) center of mass for suitably asymptotically hyperbolic
time-slices of asymptotically anti-de Sitter spacetimes in general relativity.
We do so in analogy to the picture that has been consolidated for the (total)
center of mass of suitably asymptotically Euclidean time-slices of
asymptotically Minkowskian spacetimes (isolated systems). In particular, we
unite -- an altered version of -- the approach based on Hamiltonian charges
with an approach based on CMC-foliations near infinity. The newly defined
center of mass transforms appropriately under changes of the asymptotic
coordinates and evolves in the direction of an appropriately defined linear
momentum under the Einstein evolution equations
Coordinates are messy -- not only in General Relativity
The coordinate freedom of General Relativity makes it challenging to find
mathematically rigorous and physically sound definitions for physical
quantities such as the center of mass of an isolated gravitating system. We
will argue that a similar phenomenon occurs in Newtonian Gravity once one
ahistorically drops the restriction that one should only work in Cartesian
coordinates when studying Newtonian Gravity. This will also shed light on the
nature of the challenge of defining the center of mass in General Relativity.
Relatedly, we will give explicit examples of asymptotically Euclidean
relativistic initial data sets which do not satisfy the Regge--Teitelboim
parity conditions often used to achieve a satisfactory definition of center of
mass. These originate in our joint work with Jan Metzger. This will require
appealing to Bartnik's asymptotic harmonic coordinates.Comment: This article will appear in Gravity, Cosmology, and Astrophysics -- A
Journey of Exploration and Discovery with Female Pioneers (editors Betti
Hartmann, Jutta Kunz), Springer 2023,
https://link.springer.com/book/978303142097