139 research outputs found
A Simplified Mathematical Model for the Formation of Null Singularities Inside Black Holes II
We study a simple system of two hyperbolic semi-linear equations, inspired by
the Einstein equations. The system, which was introduced in gr-qc/0612136, is a
model for singularity formation inside black holes. We show for a particular
case of the equations that the system demonstrates a finite time blowup. The
singularity that is formed is a null singularity. Then we show that in this
particular case the singularity has features that are analogous to known
features of models of black-hole interiors - which describe the inner-horizon
instability. Our simple system may provide insight into the formation of null
singularities inside spinning or charged black holes.Comment: 25 pages, 10 figure
A Toy Model for Topology Change Transitions: Role of Curvature Corrections
We consider properties of near-critical solutions describing a test static
axisymmetric D-dimensional brane interacting with a bulk N-dimensional black
hole (N>D). We focus our attention on the effects connected with curvature
corrections to the brane action. Namely, we demonstrate that the second order
phase transition in such a system is modified and becomes first order. We
discuss possible consequences of these results for merger transitions between
caged black holes and black strings.Comment: 11 pages, 9 figures, v2: published versio
A Simplified Mathematical Model for the Formation of Null Singularities Inside Black Holes I - Basic Formulation and a Conjecture
Einstein's equations are known to lead to the formation of black holes and
spacetime singularities. This appears to be a manifestation of the mathematical
phenomenon of finite-time blowup: a formation of singularities from regular
initial data. We present a simple hyperbolic system of two semi-linear
equations inspired by the Einstein equations. We explore a class of solutions
to this system which are analogous to static black-hole models. These solutions
exhibit a black-hole structure with a finite-time blowup on a characteristic
line mimicking the null inner horizon of spinning or charged black holes. We
conjecture that this behavior - namely black-hole formation with blow-up on a
characteristic line - is a generic feature of our semi-linear system. Our
simple system may provide insight into the formation of null singularities
inside spinning or charged black holes in the full system of Einstein
equations.Comment: 39 pages, 3 figures, extended versio
Separability and Vertex Ordering of Graphs
Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family
The Noether charge entropy in anti-deSitter space and its field theory dual
We express the Noether charge entropy density of a black brane in
anti-deSitter space in terms of local operators in the anti-deSitter space
bulk. We find that Wald's expression for the Noether charge entropy needs to be
modified away from the horizon by an additional term that vanishes on the
horizon. We then determine the field theory dual of the Noether charge entropy
for theories that asymptote to Einstein theory. We do so by calculating the
value of the entropy density at the anti-deSitter space boundary and applying
the standard rules of the AdS/CFT correspondence. We interpret the variation of
the entropy density operator from the horizon to the boundary as due to the
renormalization of the effective gravitational couplings as they flow from the
ultra-violet to the infra-red. We discuss the cases of Einstein-Hilbert theory
and f(R) theories in detail and make general comments about more complicated
cases.Comment: 21 pages; v2: minor corrections, results unchanged; v3: typos
correcte
Evaluating the Wald Entropy from two-derivative terms in quadratic actions
We evaluate the Wald Noether charge entropy for a black hole in generalized
theories of gravity. Expanding the Lagrangian to second order in gravitational
perturbations, we show that contributions to the entropy density originate only
from the coefficients of two-derivative terms. The same considerations are
extended to include matter fields and to show that arbitrary powers of matter
fields and their symmetrized covariant derivatives cannot contribute to the
entropy density. We also explain how to use the linearized gravitational field
equation rather than quadratic actions to obtain the same results. Several
explicit examples are presented that allow us to clarify subtle points in the
derivation and application of our method
Wald's entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling
The Bekenstein-Hawking entropy of black holes in Einstein's theory of gravity
is equal to a quarter of the horizon area in units of Newton's constant. Wald
has proposed that in general theories of gravity the entropy of stationary
black holes with bifurcate Killing horizons is a Noether charge which is in
general different from the Bekenstein-Hawking entropy. We show that the Noether
charge entropy is equal to a quarter of the horizon area in units of the
effective gravitational coupling on the horizon defined by the coefficient of
the kinetic term of specific graviton polarizations on the horizon. We present
several explicit examples of static spherically symmetric black holes.Comment: 20 pages ; added clarifications, explanations, new section on the
choice of polarizations, results unchanged; replaced with published versio
Long-Range Acoustic Interactions in Insect Swarms: An Adaptive Gravity Model
The collective motion of groups of animals emerges from the net effect of the interactions between individual members of the group. In many cases, such as birds, fish, or ungulates, these interactions are mediated by sensory stimuli that predominantly arise from nearby neighbors. But not all stimuli in animal groups are short range. Here, we consider mating swarms of midges, which are thought to interact primarily via long-range acoustic stimuli. We exploit the similarity in form between the decay of acoustic and gravitational sources to build a model for swarm behavior. By accounting for the adaptive nature of the midges\u27 acoustic sensing, we show that our \u27adaptive gravity\u27 model makes mean-field predictions that agree well with experimental observations of laboratory swarms. Our results highlight the role of sensory mechanisms and interaction range in collective animal behavior. Additionally, the adaptive interactions that we present here open a new class of equations of motion, which may appear in other biological contexts
Long-range Acoustic Interactions in Insect Swarms: An Adaptive Gravity Model
The collective motion of groups of animals emerges from the net effect of the
interactions between individual members of the group. In many cases, such as
birds, fish, or ungulates, these interactions are mediated by sensory stimuli
that predominantly arise from nearby neighbors. But not all stimuli in animal
groups are short range. Here, we consider mating swarms of midges, which
interact primarily via long-range acoustic stimuli. We exploit the similarity
in form between the decay of acoustic and gravitational sources to build a
model for swarm behavior. By accounting for the adaptive nature of the midges'
acoustic sensing, we show that our "adaptive gravity" model makes mean-field
predictions that agree well with experimental observations of laboratory
swarms. Our results highlight the role of sensory mechanisms and interaction
range in collective animal behavior. The adaptive interactions that we present
here open a new class of equations of motion, which may appear in other
biological contexts.Comment: 25 pages, 15 figure
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