arXiv subjects

J. David Brown

Publications and source records attributed to J. David Brown.

At least 19 recordsLinked to original sources

arXiv2026

Work Done by Sliding Friction

Friction does work when two objects in contact slide past one another. This process is studied using a simple numerical model consisting of two flexible asperites, one for each object. The force of friction is modeled as a conservative electric force that can be either attractive or repulsive. Because the asperites are flexible, not rigid, the conservative friction force generates thermal energy and can decrease or increase the objects' translational kinetic energies. The concept of pseudowork plays an important role throughout the analysis. The difference between work and pseudowork is equal to the change in internal energy. The numerical model is used to illuminate the reasons why work is typically greater than pseudowork. For a real physical system, this is the source of thermal energy. The processes discussed here fall into two categories: forced sliding and free sliding. With forced sliding, applied forces keep the objects moving with constant velocities. With free sliding, the only force on one of the objects (in the direction of motion) is friction.

physics.ed-ph↗

arXiv2025

Encounter between an extended hyperelastic body and a Schwarzschild black hole with quadrupole-order effects

We model the general relativistic interaction of a small hyperelastic sphere with a Schwarzschild black hole as it follows an initially marginally-bound orbit through a close encounter. While the interaction reveals effects that are encoded by the Mathisson-Papapetrou-Dixon (MPD) multipolar equations through quadrupole order, the calculation is made using an independent general relativistic finite element scheme that we described earlier (Phys.~Rev.~D 108(8):084020, October 2023). The finite element calculation is done in Schwarzschild coordinates, following a large and scalable number of mass elements in interaction with each other through elastic forces derived from a potential energy function and with the spacetime geometry. After the fact, we analyze the dynamics using a local Fermi coordinate system, computing (1) the deviation of the center of mass of the body relative to the initial marginally-bound orbit, (2) changes in orbital and spin angular momenta, and (3) the decrease in orbital energy and accompanying deposition of energy into internal elastic dynamics. The interaction leads to the capture of the small body into a highly eccentric orbit ($e \simeq 0.99998$ in a sample calculation).

gr-qc↗

arXiv2023

Extended body dynamics in general relativity: hyperelastic models

We present a numerical framework for modeling extended hyperelastic bodies based on a Lagrangian formulation of general relativistic elasticity theory. We use finite element methods to discretize the body, then use the semi--discrete action to derive ordinary differential equations of motion for the discrete nodes. The nodes are evolved in time using fourth--order Runge--Kutta. We validate our code against the normal modes of oscillation of a hyperelastic sphere, which are known analytically in the limit of small (linear), slow (Newtonian) oscillations. The algorithm displays second order convergence. This numerical framework can be used to obtain the orbital motion and internal dynamics of a hyperelastic body of any shape, for any spacetime metric, and for varying hyperelastic energy models.

gr-qc↗

arXiv2023

Singular Lagrangians and the Dirac--Bergmann Algorithm in Classical Mechanics

Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular. That is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption insures that (i) Lagrange's equations can be solved for the accelerations as functions of coordinates and velocities, and (ii) the definition of the conjugate momenta can be inverted for the velocities as functions of coordinates and momenta. This assumption, however, is unnecessarily restrictive -- there are interesting classical dynamical systems with singular Lagrangians. The algorithm for analyzing such systems was developed by Dirac and Bergmann in the 1950's. After a brief review of the Dirac--Bergmann algorithm, several physical examples are constructed from familiar components: point masses connected by massless springs, rods, cords and pulleys. The algorithm is also used to develop an initial value formulation of systems with holonomic constraints.

physics.class-ph↗

arXiv2022

Generalized geodesic deviation in de Sitter spacetime

The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally treated as the same order in smallness as the separation itself. This assumption is discussed in various papers, but is typically not recognized in textbooks. Relaxing this assumption leads to the generalized geodesic deviation equation (GGDE). We demonstrate the distinction between the GDE and the GGDE by computing the relative acceleration between timelike geodesics in two-dimensional de Sitter spacetime. We do this by considering a fiducial geodesic and a secondary geodesic (both timelike) that cross with nonzero speed. These geodesics are spanned by a spacelike geodesic, whose tangent evaluated at the fiducial geodesic defines the separation. The second derivative of the separation describes the relative acceleration between the fiducial and secondary geodesics. Near the crossing point, where the separation between the timelike geodesics is small but their rates of separation can be large, we show that the GGDE holds but the GDE fails to apply.

gr-qc↗

arXiv2022

Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac-Bergmann Algorithm

The Dirac-Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories -- general relativity, electromagnetism, Yang Mills, string theory, etc. The Dirac-Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac-Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this paper I analyze a finite-dimensional system that exhibits all of the major steps in the algorithm. The system includes primary and secondary constraints, first and second class constraints, restrictions on Lagrange multipliers, and both physical and gauge degrees of freedom. This relatively simple system provides a platform for discussing the Dirac conjecture, constructing Dirac brackets, and applying gauge conditions.

gr-qc↗

arXiv2021

Elasticity Theory in General Relativity

The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress-energy-momentum tensor for a hyperelastic body are derived from the gauge-invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau-Lifshitz radar metric, which is the relativistic version of the right Cauchy-Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola-Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

gr-qc↗

arXiv2017

Continuous body dynamics and the Mathisson-Papapetrou-Dixon equations

We show that an effective particle Lagrangian yields the Mathisson-Papapetrou-Dixon (MPD) equations. The spin of the effective particle is defined without any reference to a fixed body frame or angular velocity variable. We then demonstrate that a continuous body, defined by a congruence of world lines and described by a general action, can be rewritten as an effective particle. We analyze the gauge freedom of the body and show that a natural center of mass condition is related to a spin supplementary condition.

gr-qc↗

arXiv2012

Numerical simulations with a first order BSSN formulation of Einstein's field equations

We present a new fully first order strongly hyperbolic representation of the BSSN formulation of Einstein's equations with optional constraint damping terms. We describe the characteristic fields of the system, discuss its hyperbolicity properties, and present two numerical implementations and simulations: one using finite differences, adaptive mesh refinement and in particular binary black holes, and another one using the discontinuous Galerkin method in spherical symmetry. The results of this paper constitute a first step in an effort to combine the robustness of BSSN evolutions with very high accuracy numerical techniques, such as spectral collocation multi-domain or discontinuous Galerkin methods.

gr-qc↗

arXiv2011

Generalized Harmonic Equations in 3+1 Form

The generalized harmonic equations of general relativity are written in 3+1 form. The result is a system of partial differential equations with first order time and second order space derivatives for the spatial metric, extrinsic curvature, lapse function and shift vector, plus fields that represent the time derivatives of the lapse and shift. This allows for a direct comparison between the generalized harmonic and the Arnowitt-Deser-Misner formulations. The 3+1 generalized harmonic equations are also written in terms of conformal variables and compared to the Baumgarte-Shapiro-Shibata-Nakamura equations with moving puncture gauge conditions.

gr-qc↗

arXiv2011

Action Principle for the Generalized Harmonic Formulation of General Relativity

An action principle for the generalized harmonic formulation of general relativity is presented. The action is a functional of the spacetime metric and the gauge source vector. An action principle for the Z4 formulation of general relativity has been proposed recently by Bona, Bona--Casas and Palenzuela (BBP). The relationship between the generalized harmonic action and the BBP action is discussed in detail.

gr-qc↗

arXiv2011

Oppenheimer-Snyder Collapse in Moving-Puncture Coordinates

Moving-puncture coordinates are commonly used in numerical simulations of black holes. Their properties for vacuum Schwarzschild black holes have been analyzed in a number of studies. The behavior of moving-puncture coordinates in spacetimes containing matter, however, is less well understood. In this paper we explore the behavior of these coordinates for Oppenheimer-Snyder collapse, i.e., the collapse of a uniform density, pressureless sphere of dust initially at rest to a black hole. Oppenheimer-Snyder collapse provides a stringent test of the singularity-avoiding properties of moving-puncture coordinates, since the singularity can form more quickly than it would for matter with pressure. Our results include analytical expressions for the matter density, lapse function, and mean curvature at early times, as well as interesting limits for later times. We also carry out numerical simulations to obtain the full solution and these show that, even in the absence of pressure, moving-puncture coordinates are able to avoid the singularity. At late times the geometry settles down to a trumpet slice of a vacuum black hole.

gr-qc↗

arXiv2011

Energy Efficiency and Reliability in Wireless Biomedical Implant Systems

The use of wireless implant technology requires correct delivery of the vital physiological signs of the patient along with the energy management in power-constrained devices. Toward these goals, we present an augmentation protocol for the physical layer of the Medical Implant Communications Service (MICS) with focus on the energy efficiency of deployed devices over the MICS frequency band. The present protocol uses the rateless code with the Frequency Shift Keying (FSK) modulation scheme to overcome the reliability and power cost concerns in tiny implantable sensors due to the considerable attenuation of propagated signals across the human body. In addition, the protocol allows a fast start-up time for the transceiver circuitry. The main advantage of using rateless codes is to provide an inherent adaptive duty-cycling for power management, due to the flexibility of the rateless code rate. Analytical results demonstrate that an 80% energy saving is achievable with the proposed protocol when compared to the IEEE 802.15.4 physical layer standard with the same structure used for wireless sensor networks. Numerical results show that the optimized rateless coded FSK is more energy efficient than that of the uncoded FSK scheme for deep tissue (e.g., digestive endoscopy) applications, where the optimization is performed over modulation and coding parameters.

cs.IT↗

arXiv2010

On the Energy Efficiency of LT Codes in Proactive Wireless Sensor Networks

This paper presents an in-depth analysis on the energy efficiency of Luby Transform (LT) codes with Frequency Shift Keying (FSK) modulation in a Wireless Sensor Network (WSN) over Rayleigh fading channels with pathloss. We describe a proactive system model according to a flexible duty-cycling mechanism utilized in practical sensor apparatus. The present analysis is based on realistic parameters including the effect of channel bandwidth used in the IEEE 802.15.4 standard, active mode duration and computation energy. A comprehensive analysis, supported by some simulation studies on the probability mass function of the LT code rate and coding gain, shows that among uncoded FSK and various classical channel coding schemes, the optimized LT coded FSK is the most energy-efficient scheme for distance d greater than the pre-determined threshold level d_T , where the optimization is performed over coding and modulation parameters. In addition, although the optimized uncoded FSK outperforms coded schemes for d < d_T , the energy gap between LT coded and uncoded FSK is negligible for d < d_T compared to the other coded schemes. These results come from the flexibility of the LT code to adjust its rate to suit instantaneous channel conditions, and suggest that LT codes are beneficial in practical low-power WSNs with dynamic position sensor nodes.

cs.IT↗

arXiv2010

Trumpet slices of the Schwarzschild-Tangherlini spacetime

We study families of time-independent maximal and 1+log foliations of the Schwarzschild-Tangherlini spacetime, the spherically-symmetric vacuum black hole solution in D spacetime dimensions, for D >= 4. We identify special members of these families for which the spatial slices display a trumpet geometry. Using a generalization of the 1+log slicing condition that is parametrized by a constant n we recover the results of Nakao, Abe, Yoshino and Shibata in the limit of maximal slicing. We also construct a numerical code that evolves the BSSN equations for D=5 in spherical symmetry using moving-puncture coordinates, and demonstrate that these simulations settle down to the trumpet solutions.

gr-qc↗

arXiv2010

Adaptive Demodulation in Differentially Coherent Phase Systems: Design and Performance Analysis

Adaptive Demodulation (ADM) is a newly proposed rate-adaptive system which operates without requiring Channel State Information (CSI) at the transmitter (unlike adaptive modulation) by using adaptive decision region boundaries at the receiver and encoding the data with a rateless code. This paper addresses the design and performance of an ADM scheme for two common differentially coherent schemes: M-DPSK (M-ary Differential Phase Shift Keying) and M-DAPSK (M-ary Differential Amplitude and Phase Shift Keying) operating over AWGN and Rayleigh fading channels. The optimal method for determining the most reliable bits for a given differential detection scheme is presented. In addition, simple (near-optimal) implementations are provided for recovering the most reliable bits from a received pair of differentially encoded symbols for systems using 16-DPSK and 16- DAPSK. The new receivers offer the advantages of a rate-adaptive system, without requiring CSI at the transmitter and a coherent phase reference at the receiver. Bit error analysis for the ADM system in both cases is presented along with numerical results of the spectral efficiency for the rate-adaptive systems operating over a Rayleigh fading channel.

cs.IT↗

arXiv2009

On the Energy Efficiency of LT Codes in Proactive Wireless Sensor Networks

This paper presents the first in-depth analysis on the energy efficiency of LT codes with Non Coherent M-ary Frequency Shift Keying (NC-MFSK), known as green modulation [1], in a proactive Wireless Sensor Network (WSN) over Rayleigh flat-fading channels with path-loss. We describe the proactive system model according to a pre-determined time-based process utilized in practical sensor nodes. The present analysis is based on realistic parameters including the effect of channel bandwidth used in the IEEE 802.15.4 standard, and the active mode duration. A comprehensive analysis, supported by some simulation studies on the probability mass function of the LT code rate and coding gain, shows that among uncoded NC-MFSK and various classical channel coding schemes, the optimized LT coded NC-MFSK is the most energy-efficient scheme for distance $d$ greater than the pre-determined threshold level $d_T$, where the optimization is performed over coding and modulation parameters. In addition, although uncoded NC-MFSK outperforms coded schemes for $d < d_T$, the energy gap between LT coded and uncoded NC-MFSK is negligible for $d < d_T$ compared to the other coded schemes. These results come from the flexibility of the LT code to adjust its rate to suit instantaneous channel conditions, and suggest that LT codes are beneficial in practical low-power WSNs with dynamic position sensor nodes.

cs.IT↗

arXiv2009

Probing the puncture for black hole simulations

With the puncture method for black hole simulations, the second infinity of a wormhole geometry is compactified to a single "puncture point" on the computational grid. The region surrounding the puncture quickly evolves to a trumpet geometry. The computational grid covers only a portion of the trumpet throat. It ends at a boundary whose location depends on resolution. This raises the possibility that perturbations in the trumpet geometry could propagate down the trumpet throat, reflect from the puncture boundary, and return to the black hole exterior with a resolution--dependent time delay. Such pathological behavior is not observed. This is explained by the observation that some perturbative modes propagate in the conformal geometry, others propagate in the physical geometry. The puncture boundary exists only in the physical geometry. The modes that propagate in the physical geometry are always directed away from the computational domain at the puncture boundary. The finite difference stencils ensure that these modes are advected through the boundary with no coupling to the modes that propagate in the conformal geometry. These results are supported by numerical experiments with a code that evolves spherically symmetric gravitational fields with standard Cartesian finite difference stencils. The code uses the Baumgarte--Shapiro--Shibata--Nakamura formulation of Einstein's equations with 1+log slicing and gamma--driver shift conditions.

gr-qc↗