218 research outputs found
Extremal Problems in Minkowski Space related to Minimal Networks
We solve the following problem of Z. F\"uredi, J. C. Lagarias and F. Morgan
[FLM]: Is there an upper bound polynomial in for the largest cardinality of
a set S of unit vectors in an n-dimensional Minkowski space (or Banach space)
such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n
and that equality holds iff the space is linearly isometric to \ell^n_\infty,
the space with an n-cube as unit ball. We also remark on similar questions
raised in [FLM] that arose out of the study of singularities in
length-minimizing networks in Minkowski spaces.Comment: 6 pages. 11-year old paper. Implicit question in the last sentence
has been answered in Discrete & Computational Geometry 21 (1999) 437-44
General Relativity and Gravitation: A Centennial Perspective
To commemorate the 100th anniversary of general relativity, the International
Society on General Relativity and Gravitation (ISGRG) commissioned a Centennial
Volume, edited by the authors of this article. We jointly wrote introductions
to the four Parts of the Volume which are collected here. Our goal is to
provide a bird's eye view of the advances that have been made especially during
the last 35 years, i.e., since the publication of volumes commemorating
Einstein's 100th birthday. The article also serves as a brief preview of the 12
invited chapters that contain in-depth reviews of these advances. The volume
will be published by Cambridge University Press and released in June 2015 at a
Centennial conference sponsored by ISGRG and the Topical Group of Gravitation
of the American Physical Society.Comment: 37 page
Recommended from our members
Holography beyond AdS/CFT
Physicists have long sought to fully understand how gravity can be fully formulated within a quantum mechanical framework. A promising avenue of research in this direction was born from the idea of holography - that gravitational physics can be recast as a different theory living in fewer dimensions. Evidence of this phenomena was first observed in the arena of black hole physics, where the entropy of a black hole was calculated to scale with its area, not its volume. The advent of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence provided an explicit realization of holography for gravitational theories in AdS space. This brought a flurry of activity dedicated to dissecting this correspondence. Ultimately, however, it will be necessary to move beyond the confines of AdS/CFT in order to understand our universe, as we live in a de Sitter type universe. The research presented in this dissertation attempts to broaden the scope of holgraphic theories to include more phenomenologically relevant universes. To do so, we utilize a top-down approach and take results from AdS/CFT that appear to be general holographic results and see how they can be applied in spacetimes other than AdS. In particular, we take the Ryu-Takayanagi (RT) formula, along with its related results, and investigate what we can learn by applying it to general spacetimes. Doing so naturally forces us to utilize holographic screens, as these are the largest such surfaces where the RT formula can be self-consistently applied. This approach allows us to examine properties of the purported boundary theory for general spacetimes, including the entanglement structure and propagation speeds of excitations dual to bulk excitations. This is done in chapters 2 and 3. In chapters 4 and 5, we use this lens of generalized holography to elucidate the nature of the relationship between entanglement and emergent geometry. Finally, in chapter 6, we revisit one the important underlying assumption that the RT formula is general and demonstrate the validity of this assumption
Dynamic wormholes, anti-trapped surfaces, and energy conditions
Adapting and extending a suggestion due to Page, we define a wormhole throat
to be a marginally anti-trapped surface, that is, a closed two-dimensional
spatial hypersurface such that one of the two future-directed null geodesic
congruences orthogonal to it is just beginning to diverge. Typically a dynamic
wormhole will possess two such throats, corresponding to the two orthogonal
null geodesic congruences, and these two throats will not coincide, (though
they do coalesce into a single throat in the static limit). The divergence
property of the null geodesics at the marginally anti-trapped surface
generalizes the ``flare-out'' condition for an arbitrary wormhole. We derive
theorems regarding violations of the null energy condition (NEC) at and near
these throats and find that, even for wormholes with arbitrary time-dependence,
the violation of the NEC is a generic property of wormhole throats. We also
discuss wormhole throats in the presence of fully antisymmetric torsion and
find that the energy condition violations cannot be dumped into the torsion
degrees of freedom. Finally by means of a concrete example we demonstrate that
even temporary suspension of energy-condition violations is incompatible with
the flare-out property of dynamic throats.Comment: 32 pages in plain LaTex, no figures. Additional text and references
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Path Integral Complexity and Kasner Singularities
We explore properties of path integral complexity in field theories on time dependent backgrounds using its dual description in terms of Hartle-Hawking wavefunctions. In particular, we consider boundary theories with time dependent couplings which are dual to Kasner-AdS metrics in the bulk with a time dependent dilaton. We show that holographic path integral complexity decreases as we approach the singularity, consistent with earlier results from holographic complexity conjectures. Furthermore, we find examples where the complexity becomes universal i.e., independent of the Kasner exponents, but the properties of the path integral tensor networks depend sensitively on this data
Entanglement entropy of black holes
The entanglement entropy is a fundamental quantity which characterizes the
correlations between sub-systems in a larger quantum-mechanical system. For two
sub-systems separated by a surface the entanglement entropy is proportional to
the area of the surface and depends on the UV cutoff which regulates the
short-distance correlations. The geometrical nature of the entanglement entropy
calculation is particularly intriguing when applied to black holes when the
entangling surface is the black hole horizon. I review a variety of aspects of
this calculation: the useful mathematical tools such as the geometry of spaces
with conical singularities and the heat kernel method, the UV divergences in
the entropy and their renormalization, the logarithmic terms in the
entanglement entropy in 4 and 6 dimensions and their relation to the conformal
anomalies. The focus in the review is on the systematic use of the conical
singularity method. The relations to other known approaches such as 't Hooft's
brick wall model and the Euclidean path integral in the optical metric are
discussed in detail. The puzzling behavior of the entanglement entropy due to
fields which non-minimally couple to gravity is emphasized. The holographic
description of the entanglement entropy of the black hole horizon is
illustrated on the two- and four-dimensional examples. Finally, I examine the
possibility to interpret the Bekenstein-Hawking entropy entirely as the
entanglement entropy.Comment: 89 pages; an invited review to be published in Living Reviews in
Relativit
Sets of unit vectors with small subset sums
We say that a family of m {xi}Ιi ε[m]\} vectors in a Banach space X satisfies the k-collapsing condition if the sum of any k of them has norm at most 1. Let C(k, d) denote the maximum cardinality of a k-collapsing family of unit vectors in a d-dimensional Banach space, where the maximum is taken over all spaces of dimension d. Similarly, let CB(k, d) denote the maximum cardinality if we require in addition that the m vectors sum to 0. The case k = 2 was considered by Füredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawlor and Morgan (1994) on geometric shortest networks in smooth finite-dimensional Banach spaces. We show that CB(k, d) = max {k + 1, 2d} for all k, d ≥ 2. The behaviour of C(k, d) is not as simple, and we derive various upper and lower bounds for various ranges of k and d. These include the exact values C(k, d) = max {k + 1, 2d} in certain cases. We use a variety of tools from graph theory, convexity and linear algebra in the proofs: in particular the Hajnal–Szemerédi Theorem, the Brunn– Minkowski inequality, and lower bounds for the rank of a perturbation of the identity matrix
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