Measurement of Quantum Fluctuations in Geometry

A particular form for the quantum indeterminacy of relative spacetime position of events is derived from the limits of measurement possible with Planck wavelength radiation. The indeterminacy predicts fluctuations from a classically defined geometry in the form of ``holographic noise'' whose spatial character, absolute normalization, and spectrum are predicted with no parameters. The noise has a distinctive transverse spatial shear signature, and a flat power spectral density given by the Planck time. An interferometer signal displays noise due to the uncertainty of relative positions of reflection events. The noise corresponds to an accumulation of phase offset with time that mimics a random walk of those optical elements that change the orientation of a wavefront. It only appears in measurements that compare transverse positions, and does not appear at all in purely radial position measurements. A lower bound on holographic noise follows from a covariant upper bound on gravitational entropy. The predicted holographic noise spectrum is estimated to be comparable to measured noise in the currently operating interferometer GEO600. Because of its transverse character, holographic noise is reduced relative to gravitational wave effects in other interferometer designs, such as LIGO, where beam power is much less in the beamsplitter than in the arms.Comment: 7 pages, 2 figures, LaTeX. Extensive rewrite of original version, including more detailed analysis. Main result is the same but the estimate of noise in strain units for GEO600, showing 1/f behavior at low f and flat at high f, is improved. To appear in Phys. Rev.

The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in \RR^2 through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two $C^0$ normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between $C^0$ tori located at the same energy levels. By means of the {\em impact map} we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the {\em scattering map}, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnol'd diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring

Indeterminacy of Holographic Quantum Geometry

An effective theory based on wave optics is used to describe indeterminacy of position in holographic spacetime with a UV cutoff at the Planck scale. Wavefunctions describing spacetime positions are modeled as complex disturbances of quasi-monochromatic radiation. It is shown that the product of standard deviations of two position wavefunctions in the plane of a holographic light sheet is equal to the product of their normal separation and the Planck length. For macroscopically separated positions the transverse uncertainty is much larger than the Planck length, and is predicted to be observable as a "holographic noise" in relative position with a distinctive shear spatial character, and an absolutely normalized frequency spectrum with no parameters once the fundamental wavelength is fixed from the theory of gravitational thermodynamics. The spectrum of holographic noise is estimated for the GEO600 interferometric gravitational-wave detector, and is shown to approximately account for currently unexplained noise between about 300 and 1400Hz. In a holographic world, this result directly and precisely measures the fundamental minimum interval of time.Comment: 4 pages, LaTeX. Considerably shortened from earlier version. Conclusions are unchanged. Submitted to PR

Holographic Geometry and Noise in Matrix Theory

Using Matrix Theory as a concrete example of a fundamental holographic theory, we show that the emergent macroscopic spacetime displays a new macroscopic quantum structure, holographic geometry, and a new observable phenomenon, holographic noise, with phenomenology similar to that previously derived on the basis of a quasi-monochromatic wave theory. Traces of matrix operators on a light sheet with a compact dimension of size $R$ are interpreted as transverse position operators for macroscopic bodies. An effective quantum wave equation for spacetime is derived from the Matrix Hamiltonian. Its solutions display eigenmodes that connect longitudinal separation and transverse position operators on macroscopic scales. Measurements of transverse relative positions of macroscopically separated bodies, such as signals in Michelson interferometers, are shown to display holographic nonlocality, indeterminacy and noise, whose properties can be predicted with no parameters except $R$. Similar results are derived using a detailed scattering calculation of the matrix wavefunction. Current experimental technology will allow a definitive and precise test or validation of this interpretation of holographic fundamental theories. In the latter case, they will yield a direct measurement of $R$ independent of the gravitational definition of the Planck length, and a direct measurement of the total number of degrees of freedom.Comment: 19 pages, 2 figures; v2: factors of Planck mass written explicitly, typos correcte

Fast matrix treatment of 3-D radiative transfer in vegetation canopies: SPARTACUS-Vegetation 1.1

A fast scheme is described to compute the 3-D interaction of solar radiation with vegetation canopies. The canopy is split in the horizontal plane into one clear region and one or more vegetated regions, and the two-stream equations are used for each, but with additional terms representing lateral exchange of radiation between regions that are proportional to the area of the interface between them. The resulting coupled set of ordinary differential equations is solved using the matrix-exponential method. The scheme is compared to solar Monte Carlo calculations for idealized scenes from the RAMI4PILPS intercomparison project, for open forest canopies and shrublands both with and without snow on the ground. Agreement is good in both the visible and infrared: for the cases compared, the root-mean-squared difference in reflectance, transmittance and canopy absorptance is 0.020, 0.038 and 0.033, respectively. The technique has potential application to weather and climate modelling

Collision of High Frequency Plane Gravitational and Electromagnetic Waves

We study the head-on collision of linearly polarized, high frequency plane gravitational waves and their electromagnetic counterparts in the Einstein-Maxwell theory. The post-collision space-times are obtained by solving the vacuum Einstein-Maxwell field equations in the geometrical optics approximation. The head-on collisions of all possible pairs of these systems of waves is described and the results are then generalised to non-linearly polarized waves which exhibit the maximum two degrees of freedom of polarization.Comment: Latex file, 17 pages, accepted for publication in International Journal of Modern Physics

A Bayesian Estimate of the Primordial Helium Abundance

We introduce a new statistical method to estimate the primordial helium abundance, Y_p from observed abundances in a sample of galaxies which have experienced stellar helium enrichment. Rather than using linear regression on metal abundance we construct a likelihood function using a Bayesian prior, where the key assumption is that the true helium abundance must always exceed the primordial value. Using a sample of measurements compiled from the literature we find estimates of Y_p between 0.221 and 0.236, depending on the specific subsample and prior adopted, consistent with previous estimates either from a linear extrapolation of the helium abundance with respect to metallicity, or from the helium abundance of the lowest metallicity HII region, I Zw 18. We also find an upper limit which is insensitive to the specific subsample or prior, and estimate a model-independent bound Y_p < 0.243 at 95% confidence, favoring a low cosmic baryon density and a high primordial deuterium abundance. The main uncertainty is not the model of stellar enrichment but possible common systematic biases in the estimate of Y in each individual HII region.Comment: 14 pages, latex, 3 ps figure

Interferometers as Probes of Planckian Quantum Geometry

A theory of position of massive bodies is proposed that results in an observable quantum behavior of geometry at the Planck scale, $t_P$. Departures from classical world lines in flat spacetime are described by Planckian noncommuting operators for position in different directions, as defined by interactions with null waves. The resulting evolution of position wavefunctions in two dimensions displays a new kind of directionally-coherent quantum noise of transverse position. The amplitude of the effect in physical units is predicted with no parameters, by equating the number of degrees of freedom of position wavefunctions on a 2D spacelike surface with the entropy density of a black hole event horizon of the same area. In a region of size $L$, the effect resembles spatially and directionally coherent random transverse shear deformations on timescale $\approx L/c$ with typical amplitude $\approx \sqrt{ct_PL}$. This quantum-geometrical "holographic noise" in position is not describable as fluctuations of a quantized metric, or as any kind of fluctuation, dispersion or propagation effect in quantum fields. In a Michelson interferometer the effect appears as noise that resembles a random Planckian walk of the beamsplitter for durations up to the light crossing time. Signal spectra and correlation functions in interferometers are derived, and predicted to be comparable with the sensitivities of current and planned experiments. It is proposed that nearly co-located Michelson interferometers of laboratory scale, cross-correlated at high frequency, can test the Planckian noise prediction with current technology.Comment: 23 pages, 6 figures, Latex. To appear in Physical Review

Detecting Vanishing Dimensions Via Primordial Gravitational Wave Astronomy

Lower-dimensionality at higher energies has manifold theoretical advantages as recently pointed out. Moreover, it appears that experimental evidence may already exists for it - a statistically significant planar alignment of events with energies higher than TeV has been observed in some earlier cosmic ray experiments. We propose a robust and independent test for this new paradigm. Since (2+1)-dimensional spacetimes have no gravitational degrees of freedom, gravity waves cannot be produced in that epoch. This places a universal maximum frequency at which primordial waves can propagate, marked by the transition between dimensions. We show that this cut-off frequency may be accessible to future gravitational wave detectors such as LISA.Comment: Somewhat expanded version with discussions that could not fit into the PRL version; references adde

Driving Rydberg-Rydberg transitions from a co-planar microwave waveguide

The coherent interaction between ensembles of helium Rydberg atoms and microwave fields in the vicinity of a solid-state co-planar waveguide is reported. Rydberg-Rydberg transitions, at frequencies between 25 GHz and 38 GHz, have been studied for states with principal quantum numbers in the range 30 - 35 by selective electric-field ionization. An experimental apparatus cooled to 100 K was used to reduce effects of blackbody radiation. Inhomogeneous, stray electric fields emanating from the surface of the waveguide have been characterized in frequency- and time-resolved measurements and coherence times of the Rydberg atoms on the order of 250 ns have been determined.Comment: 5 pages, 5 figure