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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., Comment: Corrected typos

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., Comment: 28 pages, 7 figures

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., Comment: Final Version

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., Comment: 9 pages, 6 figures. Final version published in Classical and Quantum Gravity

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., Comment: 12 pages, 2 figures, to be published in Classical and Quantum Gravity

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.

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., Comment: To appear in Physical Review D

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., Comment: This version (accepted for publication) contains some minor corrections and additional references

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., Comment: 20 pages, 9 figures

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., Comment: 12 Pages, The paper has been accepted for publication in IEEE Transaction on Information Technology in Biomedicine, 2011

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., Comment: accepted for publication in IEEE Transactions on Signal Processing

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., Comment: 11 pages, 6 figures, submitted to PRD

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., Comment: This version is contains more thorough presentations and discussions of the key results. To be published in PRD. (8 pages, no figures)

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., Comment: 25 pages, 11 Figures, submitted to IEEE Transactions on Communications, June 1, 2010

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., Comment: submitted to IEEE Transactions on Wireless Communications (25 pages)

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., Comment: Minor typos corrected and references updated. To be published in PRD

The BSSN and standard gauge equations are written in covariant form with respect to spatial coordinate transformations. The BSSN variables are defined as tensors with no density weights. This allows us to evolve a given set of initial data using two different coordinate systems and to relate the results using the familiar tensor transformation rules. Two variants of the covariant equations are considered. These differ from one another in the way that the determinant of the conformal metric is evolved., Comment: 6 pages, no figures. Notation has been improved and typos have been corrected

The ADM Hamiltonian formulation of general relativity with prescribed lapse and shift is a weakly hyperbolic system of partial differential equations. In general weakly hyperbolic systems are not mathematically well posed. For well posedness, the theory should be reformulated so that the complete system, evolution equations plus gauge conditions, is (at least) strongly hyperbolic. Traditionally, reformulation has been carried out at the level of equations of motion. This typically destroys the variational and Hamiltonian structures of the theory. Here I show that one can extend the ADM formalism to (i) incorporate the gauge conditions as dynamical equations and (ii) affect the hyperbolicity of the complete system, all while maintaining a Hamiltonian description. The extended ADM formulation is used to obtain a strongly hyperbolic Hamiltonian description of Einstein's theory that is generally covariant under spatial diffeomorphisms and time reparametrizations, and has physical characteristics. The extended Hamiltonian formulation with 1+log slicing and gamma--driver shift conditions is weakly hyperbolic., Comment: This version contains minor corrections and clarifications. The format has been changed to conform with IOP style

The moving puncture method is analyzed for a single, non-spinning black hole. It is shown that the puncture region is not resolved by current numerical codes. As a result, the geometry near the puncture appears to evolve to an infinitely long cylinder of finite areal radius. The puncture itself actually remains at spacelike infinity throughout the evolution. In the limit of infinite resolution the data never become stationary. However, at any reasonable finite resolution the grid points closest to the puncture are rapidly drawn into the black hole interior by the Gamma-driver shift condition. The data can then evolve to a stationary state. These results suggest that the moving puncture technique should be viewed as a type of "natural excision"., Comment: This is the final version to be published in PRD. One important change: the phrase "excision by under resolution" is replaced by the more appropriate phrase "natural excision"

A new technique is presented for modifying the Einstein evolution equations off the constraint hypersurface. With this approach the evolution equations for the constraints can be specified freely. The equations of motion for the gravitational field variables are modified by the addition of terms that are linear and nonlocal in the constraints. These terms are obtained from solutions of the linearized Einstein constraints., Comment: 4 pages, 1 figure, uses REVTeX

In this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement. Our code uses truncation error estimates to adaptively refine the grid as part of the solution process. The presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work. Code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement., Comment: 12 pages, 9 figures, Modified in response to reviewer suggestions, added figure, added references. Accepted for publication in J. Comp. Phys

We solve for single distorted black hole initial data using the puncture method, where the Hamiltonian constraint is written as an elliptic equation in R^3 for the nonsingular part of the metric conformal factor. With this approach we can generate isometric and non--isometric black hole data. For the isometric case, our data are directly comparable to those obtained by Bernstein et al., who impose isometry boundary conditions at the black hole throat. Our numerical simulations are performed using a parallel multigrid elliptic equation solver with adaptive mesh refinement. Mesh refinement allows us to use high resolution around the black hole while keeping the grid boundaries far away in the asymptotic region., Comment: This is the final version to be published in Phys. Rev. D. It includes minor revisions and some additional tests

We present an algorithm for treating mesh refinement interfaces in numerical relativity. We detail the behavior of the solution near such interfaces located in the strong field regions of dynamical black hole spacetimes, with particular attention to the convergence properties of the simulations. In our applications of this technique to the evolution of puncture initial data with vanishing shift, we demonstrate that it is possible to simultaneously maintain second order convergence near the puncture and extend the outer boundary beyond 100M, thereby approaching the asymptotically flat region in which boundary condition problems are less difficult and wave extraction is meaningful., Comment: 18 pages, 12 figures. Minor changes, final PRD version

We study the propagation of waves across fixed mesh refinement boundaries in linear and nonlinear model equations in 1--D and 2--D, and in the 3--D Einstein equations of general relativity. We demonstrate that using linear interpolation to set the data in guard cells leads to the production of reflected waves at the refinement boundaries. Implementing quadratic interpolation to fill the guard cells suppresses these spurious signals., Comment: 46 pages, 18 figures, Submitted to Journal of Computational Physics

Dynamical instability is shown to occur in differentially rotating polytropes with N = 3.33 and $T/|W| \gtrsim 0.14$. This instability has a strong m=1 mode, although the m=2, 3, and 4 modes also appear. Such instability may allow a centrifugally-hung core to begin collapsing to neutron star densities on a dynamical timescale. The gravitational radiation emitted by such unstable cores may be detectable with advanced ground-based detectors, such as LIGO II. If the instability occurs in a supermassive star, it may produce gravitational radiation detectable by the space-based detector LISA., Comment: 5 pages, 4 figures; submitted to The Astrophysical Journal Letters

This work closes certain gaps in the literature on material reference systems in general relativity. It is shown that perfect fluids are a special case of DeWitt's relativistic elastic media and that the velocity--potential formalism for perfect fluids can be interpreted as describing a perfect fluid coupled to a fleet of clocks. A Hamiltonian analysis of the elastic media with clocks is carried out and the constraints that arise when the system is coupled to gravity are studied. When the Hamiltonian constraint is resolved with respect to the clock momentum, the resulting true Hamiltonian is found to be a functional only of the gravitational variables. The true Hamiltonian is explicitly displayed when the medium is dust, and is shown to depend on the detailed construction of the clocks., Comment: 18 pages, ReVTeX

It is shown that in the instanton approximation the rate of creation of black holes is always enhanced by a factor of the exponential of the black hole entropy relative to the rate of creation of compact matter distributions (stars). This result holds for any generally covariant theory of gravitational and matter fields that can be expressed in Hamiltonian form. It generalizes the result obtained previously for the pair creation of magnetically charged black holes by a magnetic field in Einstein--Maxwell theory. The particular example of pair creation of electrically charged black holes by an electric field in Einstein--Maxwell theory is discussed in detail., Comment: (12 pages, ReVTeX) Revised version of "Pair Creation of Electrically Charged Black Holes". New section shows that the BH pair creation rate is enhanced by a factor $\exp(BH entropy)$ for any Hamiltonian gravity + matter theory

The connection is established between two different action principles for perfect fluids in the context of general relativity. For one of these actions, $S$, the fluid four--velocity is expressed as a sum of products of scalar fields and their gradients (the velocity--potential representation). For the other action, ${\bar S}$, the fluid four--velocity is proportional to the totally antisymmetric product of gradients of the fluid Lagrangian coordinates. The relationship between $S$ and ${\bar S}$ is established by expressing $S$ in Hamiltonian form and identifying certain canonical coordinates as ignorable. Elimination of these coordinates and their conjugates yields the action ${\bar S}$. The key step in the analysis is a point canonical transformation in which all tensor fields on space are expressed in terms of the Lagrangian coordinate system supplied by the fluid. The canonical transformation is of interest in its own right. It can be applied to any physical system that includes a material medium described by Lagrangian coordinates. The result is a Hamiltonian description of the system in which the momentum constraint is trivial., Comment: (29 pages, REVTEX) (\end{document} statement has been added. That's all.)

The authors have recently proposed a ``microcanonical functional integral" representation of the density of quantum states of the gravitational field. The phase of this real--time functional integral is determined by a ``microcanonical" or Jacobi action, the extrema of which are classical solutions at fixed total energy, not at fixed total time interval as in Hamilton's action. This approach is fully general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper we describe how to obtain Jacobi's action for general relativity. We evaluate it for a certain complex metric associated with a rotating black hole and discuss the relation of the result to the density of states and to the entropy of the black hole. (Dedicated to Yvonne Choquet-Bruhat in honor of her retirement.), Comment: 12 pages, plain tex

The authors have introduced recently a ``microcanonical functional integral" which yields directly the density of states as a function of energy. The phase of the functional integral is Jacobi's action, the extrema of which are classical solutions at a given energy. This approach is general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper, however, we ignore gravity and illustrate the use of Jacobi's action by computing the density of states for a nonrelativistic harmonic oscillator. (Festschrift for Dieter Brill), Comment: 10 pages, plain tex. (This replaces the previous version which consisted of less than half of the manuscript.)

The gravitational field in a spatially finite region is described as a microcanonical system. The density of states $\nu$ is expressed formally as a functional integral over Lorentzian metrics and is a functional of the geometrical boundary data that are fixed in the corresponding action. These boundary data are the thermodynamical extensive variables, including the energy and angular momentum of the system. When the boundary data are chosen such that the system is described semiclassically by {\it any} real stationary axisymmetric black hole, then in this same approximation $\ln\nu$ is shown to equal 1/4 the area of the black hole event horizon. The canonical and grand canonical partition functions are obtained by integral transforms of $\nu$ that lead to "imaginary time" functional integrals. A general form of the first law of thermodynamics for stationary black holes is derived. For the simpler case of nonrelativistic mechanics, the density of states is expressed as a real-time functional integral and then used to deduce Feynman's imaginary-time functional integral for the canonical partition function., Comment: 29 pages, plain TeX

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on ${}^3B$, the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate ${}^3B$. The integral of the energy density over such a two-surface $B$ is the quasilocal energy associated with a spacelike three-surface $\Sigma$ whose intersection with ${}^3B$ is the boundary $B$. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary $B$ as a surface embedded in $\Sigma$. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on ${}^3B$ in the direction orthogonal to $B$. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on ${}^3B$. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding., Comment: 35 pages, plain TeX. Two figures available upon request (jdbrown@math.ncsu.edu)