13,224 research outputs found
Troesch complexes and extensions of strict polynomial functors
We develop a new approach of extension calculus in the category of strict
polynomial functors, based on Troesch complexes. We obtain new short elementary
proofs of numerous classical Ext-computations as well as new results.
In particular, we get a cohomological version of the `fundamental theorems'
from classical invariant invariant theory for GL_n for n big enough (and we
give a conjecture for smaller values of n).
We also study the `twisting spectral sequence' E^{s,t}(F,G,r) converging to
the extension groups Ext^*(F^{(r)}, G^{(r)}) between the twisted functors
F^{(r)} and G^{(r)}. Many classical Ext-computations simply amount to the
collapsing of this spectral sequence at the second page (for lacunary reasons),
and it is also a convenient tool to study the effect of the Frobenius twist on
Ext-groups. We prove many cases of collapsing, and we conjecture collapsing is
a general fact.Comment: Revised version, 46 pages. Mathematics unchanged. Typos corrected,
Appendix 9 on Troesch complexes improve
Black Hole Thermodynamics
The discovery in the early 1970s that black holes radiate as black bodies has
radically affected our understanding of general relativity, and offered us some
early hints about the nature of quantum gravity. In this chapter I will review
the discovery of black hole thermodynamics and summarize the many independent
ways of obtaining the thermodynamic and (perhaps) statistical mechanical
properties of black holes. I will then describe some of the remaining puzzles,
including the nature of the quantum microstates, the problem of universality,
and the information loss paradox.Comment: Invited review article. A few parts based on an earlier review,
arXiv:0807.4520. To appear in Int. J. Mod. Phys. D and in "One Hundred Years
of General Relativity: Cosmology and Gravity," edited by Wei-Tou Ni (World
Scientific, Singapore, 2015). v2: added references and appendi
Relativistic bias in neutrino cosmologies
Halos and galaxies are tracers of the underlying dark matter structures.
While their bias is well understood in the case of a simple Universe composed
dominantly of dark matter, the relation becomes more complex in the presence of
massive neutrinos. Indeed massive neutrinos introduce rich dynamics in the
process of structure formation leading to scale-dependent bias. We study this
process from the perspective of general relativity employing a simple spherical
collapse model. We find a characteristic signature at the neutrino
free-streaming scale in addition to a large-scale feature from general
relativity. The scale-dependent halo bias opposes the suppression in the matter
distribution due to neutrino free-streaming and leads to corrections of a few
percent in the halo power spectrum. It is not only sensitive to the sum of the
neutrino-masses, but respond to the individual masses. Accurate models for the
neutrino bias are a crucial ingredient for the future data analysis and play an
important role in constraining the neutrino masses.Comment: 20 pages, 10 figure
Magnetohydrodynamic Simulations of A Rotating Massive Star Collapsing to A Black Hole
We perform two-dimensional, axisymmetric, magnetohydrodynamic simulations of
the collapse of a rotating star of 40 Msun and in the light of the collapsar
model of gamma-ray burst. Considering two distributions of angular momentum, up
to \sim 10^{17} cm^2/s, and the uniform vertical magnetic field, we investigate
the formation of an accretion disk around a black hole and the jet production
near the hole. After material reaches to the black hole with the high angular
momentum, the disk is formed inside a surface of weak shock. The disk becomes
in a quasi-steady state for stars whose magnetic field is less than 10^{10} G
before the collapse. We find that the jet can be driven by the magnetic fields
even if the central core does not rotate as rapidly as previously assumed and
outer layers of the star has sufficiently high angular momentum. The magnetic
fields are chiefly amplified inside the disk due to the compression and the
wrapping of the field. The fields inside the disk propagate to the polar region
along the inner boundary near the black hole through the Alfv{\'e}n wave, and
eventually drive the jet. The quasi-steady disk is not an advection-dominated
disk but a neutrino cooling-dominated one. Mass accretion rates in the disks
are greater than 0.01 Msun/sec with large fluctuations. The disk is transparent
for neutrinos. The dense part of the disk, which locates near the hole, emits
neutrino efficiently at a constant rate of < 8 \times 10^{51} erg/s. The
neutrino luminosity is much smaller than those from supernovae after the
neutrino burst.Comment: 42 pages, accepted for publication in the Astrophysical Journal. A
paper with higher-resolution figures available at
http://www.ec.knct.ac.jp/~fujimoto/collapsar/mhd-color.pd
The black hole final state
We propose that in quantum gravity one needs to impose a final state boundary
condition at black hole singularities. This resolves the apparent contradiction
between string theory and semiclassical arguments over whether black hole
evaporation is unitary.Comment: 17 pages, harvmac, 1 figure, v2: comment about interactions and
references adde
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