16 research outputs found
Mass Partitions via Equivariant Sections of Stiefel Bundles
We consider a geometric combinatorial problem naturally associated to the
geometric topology of certain spherical space forms. Given a collection of
mass distributions on , the existence of affinely independent
regular -fans, each of which equipartitions each of the measures, can in
many cases be deduced from the existence of a -equivariant
section of the Stiefel bundle over , where
is the Stiefel manifold of all orthonormal -frames in
or , and
is the corresponding unit sphere. For example, the
parallelizability of when , or implies that any
two masses on can be simultaneously bisected by each of
pairwise-orthogonal hyperplanes, while when or 4, the triviality of the
circle bundle over the standard Lens Spaces
yields that for any mass on , there exist a pair of
complex orthogonal regular -fans, each of which equipartitions the mass.Comment: 11 pages, final versio
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans
Parabolic Jets from the Spinning Black Hole in M87
The M87 jet is extensively examined by utilizing general relativistic
magnetohydrodynamic (GRMHD) simulations as well as the steady axisymmetric
force-free electrodynamic (FFE) solution. Quasi-steady funnel jets are obtained
in GRMHD simulations up to the scale of gravitational radius
() for various black hole (BH) spins. As is known, the funnel edge
is approximately determined by the following equipartitions; i) the magnetic
and rest-mass energy densities and ii) the gas and magnetic pressures. Our
numerical results give an additional factor that they follow the outermost
parabolic streamline of the FFE solution, which is anchored to the event
horizon on the equatorial plane. We also identify the matter dominated,
non-relativistic corona/wind play a dynamical role in shaping the funnel jet
into the parabolic geometry. We confirm a quantitative overlap between the
outermost parabolic streamline of the FFE jet and the edge of jet sheath in
VLBI observations at -, suggesting that the
M87 jet is likely powered by the spinning BH. Our GRMHD simulations also
indicate a lateral stratification of the bulk acceleration (i.e., the
spine-sheath structure) as well as an emergence of knotty superluminal
features. The spin characterizes the location of the jet stagnation surface
inside the funnel. We suggest that the limb-brightened feature could be
associated with the nature of the BH-driven jet, if the Doppler beaming is a
dominant factor. Our findings can be examined with (sub-)mm VLBI observations,
giving a clue for the origin of the M87 jet.Comment: 29 pages, 23 figures, accepted for publication in Ap