559 research outputs found
Asymptotic Symmetries of Rindler Space at the Horizon and Null Infinity
We investigate the asymptotic symmetries of Rindler space at null infinity
and at the event horizon using both systematic and ad hoc methods. We find that
the approaches that yield infinite-dimensional asymptotic symmetry algebras in
the case of anti-de Sitter and flat spaces only give a finite-dimensional
algebra for Rindler space at null infinity. We calculate the charges
corresponding to these symmetries and confirm that they are finite, conserved,
and integrable, and that the algebra of charges gives a representation of the
asymptotic symmetry algebra. We also use relaxed boundary conditions to find
infinite-dimensional asymptotic symmetry algebras for Rindler space at null
infinity and at the event horizon. We compute the charges corresponding to
these symmetries and confirm that they are finite and integrable. We also
determine sufficient conditions for the charges to be conserved on-shell, and
for the charge algebra to give a representation of the asymptotic symmetry
algebra. In all cases, we find that the central extension of the charge algebra
is trivial.Comment: 37 pages, 4 figures. Version 3: New Section 5 adde
Universal optimality of the and Leech lattices and interpolation formulas
We prove that the root lattice and the Leech lattice are universally
optimal among point configurations in Euclidean spaces of dimensions and
, respectively. In other words, they minimize energy for every potential
function that is a completely monotonic function of squared distance (for
example, inverse power laws or Gaussians), which is a strong form of robustness
not previously known for any configuration in more than one dimension. This
theorem implies their recently shown optimality as sphere packings, and broadly
generalizes it to allow for long-range interactions.
The proof uses sharp linear programming bounds for energy. To construct the
optimal auxiliary functions used to attain these bounds, we prove a new
interpolation theorem, which is of independent interest. It reconstructs a
radial Schwartz function from the values and radial derivatives of and
its Fourier transform at the radii for integers
in and in . To prove this
theorem, we construct an interpolation basis using integral transforms of
quasimodular forms, generalizing Viazovska's work on sphere packing and placing
it in the context of a more conceptual theory.Comment: 95 pages, 6 figure
High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition
This paper presents a novel adaptive-sparse polynomial dimensional
decomposition (PDD) method for stochastic design optimization of complex
systems. The method entails an adaptive-sparse PDD approximation of a
high-dimensional stochastic response for statistical moment and reliability
analyses; a novel integration of the adaptive-sparse PDD approximation and
score functions for estimating the first-order design sensitivities of the
statistical moments and failure probability; and standard gradient-based
optimization algorithms. New analytical formulae are presented for the design
sensitivities that are simultaneously determined along with the moments or the
failure probability. Numerical results stemming from mathematical functions
indicate that the new method provides more computationally efficient design
solutions than the existing methods. Finally, stochastic shape optimization of
a jet engine bracket with 79 variables was performed, demonstrating the power
of the new method to tackle practical engineering problems.Comment: 18 pages, 2 figures, to appear in Sparse Grids and
Applications--Stuttgart 2014, Lecture Notes in Computational Science and
Engineering 109, edited by J. Garcke and D. Pfl\"{u}ger, Springer
International Publishing, 201
Subleading Regge limit from a soft anomalous dimension
Wilson lines capture important features of scattering amplitudes, for example
soft effects relevant for infrared divergences, and the Regge limit. Beyond the
leading power approximation, corrections to the eikonal picture have to be
taken into account. In this paper, we study such corrections in a model of
massive scattering amplitudes in N = 4 super Yang-Mills, in the planar limit,
where the mass is generated through a Higgs mechanism. Using known three-loop
analytic expressions for the scattering amplitude, we find that the first power
suppressed term has a very simple form, equal to a single power law. We propose
that its exponent is governed by the anomalous dimension of a Wilson loop with
a scalar inserted at the cusp, and we provide perturbative evidence for this
proposal. We also analyze other limits of the amplitude and conjecture an exact
formula for a total cross-section at high energies.Comment: 19 pages, several appendices, many figure
Quantum astrometric observables II: time delay in linearized quantum gravity
A clock synchronization thought experiment is modeled by a diffeomorphism
invariant "time delay" observable. In a sense, this observable probes the
causal structure of the ambient Lorentzian spacetime. Thus, upon quantization,
it is sensitive to the long expected smearing of the light cone by vacuum
fluctuations in quantum gravity. After perturbative linearization, its mean and
variance are computed in the Minkowski Fock vacuum of linearized gravity. The
na\"ive divergence of the variance is meaningfully regularized by a length
scale , the physical detector resolution. This is the first time vacuum
fluctuations have been fully taken into account in a similar calculation.
Despite some drawbacks this calculation provides a useful template for the
study of a large class of similar observables in quantum gravity. Due to their
large volume, intermediate calculations were performed using computer algebra
software. The resulting variance scales like , where
is the Planck length and is the distance scale separating the ("lab" and
"probe") clocks. Additionally, the variance depends on the relative velocity of
the lab and the probe, diverging for low velocities. This puzzling behavior may
be due to an oversimplified detector resolution model or a neglected second
order term in the time delay.Comment: 30 pages, 8 figures, revtex4-1; v3: minor updates and corrections,
close to published versio
Asymptotics in the time-dependent Hawking and Unruh effects
In this thesis, we study the Hawking and Unruh effects in time-dependent
situations, as registered by localised spacetimes observers in several
asymptotic situations.
(Full abstract inside document.)Comment: Thesis submitted to the University of Nottingham for the Degree of
Doctor of Philosophy. Thesis supervisor: Dr. Jorma Louko. 230 pages. 20
figure
Fixed-Functionals of three-dimensional Quantum Einstein Gravity
We study the non-perturbative renormalization group flow of f(R)-gravity in
three-dimensional Asymptotically Safe Quantum Einstein Gravity. Within the
conformally reduced approximation, we derive an exact partial differential
equation governing the RG-scale dependence of the function f(R). This equation
is shown to possess two isolated and one continuous one-parameter family of
scale-independent, regular solutions which constitute the natural
generalization of RG fixed points to the realm of infinite-dimensional theory
spaces. All solutions are bounded from below and give rise to positive definite
kinetic terms. Moreover, they admit either one or two UV-relevant deformations,
indicating that the corresponding UV-critical hypersurfaces remain finite
dimensional despite the inclusion of an infinite number of coupling constants.
The impact of our findings on the gravitational Asymptotic Safety program and
its connection to new massive gravity is briefly discussed.Comment: 34 pages, 14 figure
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