Traversable Geometric Dark Energy Wormholes Constrained by Astrophysical Observations

Total Page:16

File Type:pdf, Size:1020Kb

Traversable Geometric Dark Energy Wormholes Constrained by Astrophysical Observations Eur. Phys. J. C (2016) 76:484 DOI 10.1140/epjc/s10052-016-4321-4 Regular Article - Theoretical Physics Traversable geometric dark energy wormholes constrained by astrophysical observations Deng Wang1,a, Xin-he Meng2,3,b 1 Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China 2 Department of Physics, Nankai University, Tianjin 300071, China 3 State Key Lab of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100080, China Received: 2 February 2016 / Accepted: 18 August 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In this paper, we introduce the astrophysical 1 Introduction observations into the wormhole research. We investigate the evolution behavior of the dark energy equation of state Modern astronomical observations with increasing evidence, parameter ω by constraining the dark energy model, so that such as high redshift Type Ia supernovae (SNe Ia), matter we can determine in which stage of the universe wormholes power spectra, observational Hubble parameter data (OHD), can exist by using the condition ω<−1. As a concrete cosmic microwave background radiation (CMBR), etc, have instance, we study the Ricci dark energy (RDE) traversable strongly suggested that the universe is undergoing an acceler- wormholes constrained by astrophysical observations. Par- ated phase at present [1–5]. To explain the accelerated mech- ticularly, we find from Fig. 5 of this work, when the effective anism, cosmologists have proposed a new negative pressure equation of state parameter ωX < −1(orz < 0.109), i.e., the fluid named dark energy. The simplest candidate of dark null energy condition (NEC) is violated clearly, the worm- energy is the so-called cosmological constant, i.e., the - holes will exist (open). Subsequently, six specific solutions cold-dark-matter (CDM) model, which has proved to be of statically and spherically symmetric traversable wormhole very successful in describing many aspects of the observed supported by the RDE fluids are obtained. Except for the universe. For instance, the spectrum of anisotropies of the case of a constant redshift function, where the solution is not CMBR, the large scale structure (LSS) of matter distribution only asymptotically flat but also traversable, the five remain- at linear level, and the expansion phenomena are very well ing solutions are all non-asymptotically flat, therefore, the described by the standard cosmological model. However, exotic matter from the RDE fluids is spatially distributed in this model has faced two fatal problems, namely, the “fine- the vicinity of the throat. Furthermore, we analyze the phys- tuning” problem and the “coincidence” problem [6]. The for- ical characteristics and properties of the RDE traversable mer indicates that theoretical estimates for the vacuum den- wormholes. It is worth noting that, using the astrophysical sity are many orders of magnitude larger than its observed observations, we obtain the constraints on the parameters of value, i.e., the famous 120-orders-of-magnitude discrepancy the RDE model, explore the types of exotic RDE fluids in that makes the vacuum explanation suspicious, while the lat- different stages of the universe, limit the number of available ter implies that why the dark energy and dark matter are at the models for wormhole research, reduce theoretically the num- same order today since their energy densities are so different ber of the wormholes corresponding to different parameters during the evolution of the universe. In addition, a positive for the RDE model, and provide a clearer picture for worm- cosmological constant is inconsistent with perturbed string hole investigations from the new perspective of observational theory [7]. Therefore, a realistic interpretation of dark energy cosmology. should not be simply in terms of the cosmological constant (interpreting it as quantum vacuum). In recent years, to alle- viate or even solve these two problems, cosmologists have proposed a variety of dark energy models, partly as follows: • Exotic equation of state: a linear equation of state [8], van der Waals equation of state [9,10], Chaplygin gas [11–13], a e-mail: [email protected] generalized Chaplygin gas [14,15], modified Chaplygin gas b e-mail: [email protected] [16,17], superfluid Chaplygin gas [18–20], inhomogeneous 123 484 Page 2 of 13 Eur. Phys. J. C (2016) 76:484 equation of state [21], barotropic fluid model [22], Cardassian Wormholes could be defined as handles or tunnels in the model [23–26]. spacetime topology linking widely separated regions of our • Viscosity: bulk viscosity in the isotropic space, bulk and universe or of different universes altogether [81]. The most shear viscosity in the anisotropic space [27–32]. It is worth fundamental ingredient to form a wormhole is violating the μ ν noting that the perfect fluid that occurs in many papers is just Null Energy Condition (NEC), i.e., Tμνk k > 0, and con- an approximation of the medium of the universe. Nowadays, sequently all of other energy conditions, where Tμν is the all the observations indicate that the medium of the universe stress-energy tensor and kμ any future directed null vector. is not an idealized fluid and the viscosity is investigated in In general, the wormholes in the literature can be divided into the evolution of the universe. three classes: • The holographic principle: holographic dark energy [33– • Ordinary wormholes – this class just satisfies the viola- 36], Ricci dark energy [37–40], agegraphic dark energy [41], tion of the NEC and is usually non-asymptotically flat, sin- tachyon model [42,43]. gular and consequently non-traversable. • Dynamical scalar fields: quintessence (or cosmon) [44– • Traversable wormholes – in light of ordinary wormholes, 52], ghost condensates [53,54], phantom [55] and quintom one can obtain the traversable wormholes by an appropriate [56], the model potential ranging from power laws to expo- choice of redshift function or shape function. Subsequently, nentials and, to some extent, the quintom is an interesting one can analyze conveniently the traversability conditions of combination of quintessence and phantom. the wormholes and the stabilities. • Modified gravity: f(R) gravity [57–59], braneworld • Thin shell wormholes – one can theoretically construct models [60–63], Gauss–Bonnet models [64–67], Chern– a geodesically complete traversable wormhole with a shell Simons gravity [68], Einstein–Aether gravity [69], cosmo- placed in the junction surface by using the so-called cut- logical models from scalar–tensor theories of gravity [70– and-paste technique. This class has attracted much attention 76]. since the exotic matter required for the existence of spacetime In the above, a part of various models on this topic are configuration is only located at the shell, and it avoids very mentioned, since there are too many. Hereafter, we plan to naturally the occurrence of any horizon. focus our attention on the geometric contributions, namely, Wormholes like other extreme astrophysical objects also the so-called Ricci dark energy model based on the holo- have a long theoretical formation history. The earliest graphic principle. Although a complete quantum gravity the- remarkable contribution we are aware of is the 1935 introduc- ory (QGT) has not been developed, we could still explore tion of the object now referred to as an Einstein–Rosen bridge partly the nature of the dark energy by using the holographic [82]. Twenty years later, Wheeler first introduced the famous principle which acts as an important result of present QGT (or idea of “spacetime foam” and coined the term “wormhole” string theory) for gravity phenomena. Thus, the holographic [83]. The actual revival of this field is based on the 1988 paper dark energy model (HDE) constructed in light of the holo- by Morris and Thorne [81], in which they analyzed in detail graphic principle can bring us a new perspective on the under- the construction of the wormhole, energy conditions, time lying theory of dark energy. Recently, Gao et al. [40]pro- machines, stability problem, and traversabilities of the worm- posed a new HDE model called the Ricci dark energy model holes. In succession, Visser and Possion introduce the famous (RDE), in which the future event horizon area is replaced by “thin shell wormholes” by conjecturing that all “exotic mat- the inverse of the Ricci scalar curvature. They have shown ter” is confined to a thin shell between universes [84–88]. this model does not only avoid the causality problem and After that, there were a lot of papers to investigate the above is phenomenologically viable, but also naturally solves the three classes of wormholes and related properties. coincidence problem. In the past few years, in light of the important discovery In this study, we intend to investigate the astrophysical that our universe is undergoing a phase of accelerated expan- scale properties of the RDE during the evolution of the uni- sion, an increasing and significant interest in the subject of verse through assuming the dark energy fluid is permeated wormholes has arisen in connection with the globally cos- everywhere. In particular, as in our previous work [77–79], mological scale discovery. Due to the violation of NEC in the existence of wormholes is always an important problem both cases (astrophysics wormholes and cosmic dark energy in physics both at micro and macro scales. There is no doubt for simple terms), an unexpected and subtle overlap between that wormholes together with black holes, pulsars (physi- the two seemingly separated subjects occurs. To be precise, cal neutron stars), and white dwarfs [80], etc., constitute the one can usually parameterize the dark energy behaviors by most attractive, extreme, strange, and puzzling astrophysical an equation of state of the form ω = p/ρ, where p is the objects that may provide a new window for physical discov- spatially homogeneous pressure and ρ the energy density ery.
Recommended publications
  • 2008. Pruning Some Branches from 'Branching Spacetimes'
    CHAPTER 10 Pruning Some Branches from “Branching Spacetimes” John Earman* Abstract Discussions of branching time and branching spacetime have become com- mon in the philosophical literature. If properly understood, these concep- tions can be harmless. But they are sometimes used in the service of debat- able and even downright pernicious doctrines. The purpose of this chapter is to identify the pernicious branching and prune it back. 1. INTRODUCTION Talk of “branching time” and “branching spacetime” is wide spread in the philo- sophical literature. Such expressions, if properly understood, can be innocuous. But they are sometimes used in the service of debatable and even downright per- nicious doctrines. The purpose of this paper is to identify the pernicious branching and prune it back. Section 2 distinguishes three types of spacetime branching: individual branch- ing, ensemble branching, and Belnap branching. Individual branching, as the name indicates, involves a branching structure in individual spacetime models. It is argued that such branching is neither necessary nor sufficient for indeterminism, which is explicated in terms of the branching in the ensemble of spacetime mod- els satisfying the laws of physics. Belnap branching refers to the sort of branching used by the Belnap school of branching spacetimes. An attempt is made to sit- uate this sort of branching with respect to ensemble branching and individual branching. Section 3 is a sustained critique of various ways of trying to imple- ment individual branching for relativistic spacetimes. Conclusions are given in Section 4. * Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, USA The Ontology of Spacetime II © Elsevier BV ISSN 1871-1774, DOI: 10.1016/S1871-1774(08)00010-7 All rights reserved 187 188 Pruning Some Branches from “Branching Spacetimes” 2.
    [Show full text]
  • 3 Spacetime 4 Unitarity Patent Application Publication Mar
    US 2012.0075682A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2012/0075682 A1 Amoros0 (43) Pub. Date: Mar. 29, 2012 (54) SPACETIME ENERGY RESONATOR: A Publication Classification TRANSISTOR OF COMPLEX DRAC (51) Int. Cl. POLARIZED VACUUM TOPOLOGY GO3H 5/00 (2006.01) B82Y IO/OO (2011.01) (76) Inventor: Richard Louis Amoroso, Oakland, (52) U.S. Cl. ............................................. 359/1977/933 CA (US) (57) ABSTRACT Method and apparatus is described for a general purpose (21) Appl. No.: 12/928,592 transistor of the higher dimensional (HD) spacetime back cloth consistent with symmetry parameters of a new cosmo logical paradigm with a unique M-Theoretic background (22) Filed: Dec. 15, 2010 making correspondence to a complex scale-invariant covari ant Dirac polarized aether where energy pathways follow Related U.S. Application Data spacetime geodesics; and Switching is achieved by manipu lating resonant modes of a Calabi-Yau mirror symmetry hier (60) Provisional application No. 61/284,384, filed on Dec. archy inherent in continuous-State brane topology of the 17, 2009, provisional application No. 61/284,896, structural-phenomenology of least cosmological units tiling filed on Dec. 28, 2009, provisional application No. the raster of spacetime as it emerges from the infinite potential 61/336,522, filed on Jan. 22, 2010. of the unified field. CONCEPTUALIZED NOETIC INTERFEROMETRY Sagnac effect quadrupole modulated RF pulse R2 1 electrons 2 nucleons 3 spacetime 4 unitarity Patent Application Publication Mar. 29, 2012 Sheet 1 of 17 US 2012/0075682 A1 Ring Laser Array Spacetime(S-T)Vacuum Symmetry Hierarchy FIGURE 1 Patent Application Publication Mar.
    [Show full text]
  • BLACK HOLES: the OTHER SIDE of INFINITY General Information
    BLACK HOLES: THE OTHER SIDE OF INFINITY General Information Deep in the middle of our Milky Way galaxy lies an object made famous by science fiction—a supermassive black hole. Scientists have long speculated about the existence of black holes. German astronomer Karl Schwarzschild theorized that black holes form when massive stars collapse. The resulting gravity from this collapse would be so strong that the matter would become more and more dense. The gravity would eventually become so strong that nothing, not even radiation moving at the speed of light, could escape. Schwarzschild’s theories were predicted by Einstein and then borne out mathematically in 1939 by American astrophysicists Robert Oppenheimer and Hartland Snyder. WHAT EXACTLY IS A BLACK HOLE? First, it’s not really a hole! A black hole is an extremely massive concentration of matter, created when the largest stars collapse at the end of their lives. Astronomers theorize that a point with infinite density—called a singularity—lies at the center of black holes. SO WHY IS IT CALLED A HOLE? Albert Einstein’s 1915 General Theory of Relativity deals largely with the effects of gravity, and in essence predicts the existence of black holes and singularities. Einstein hypothesized that gravity is a direct result of mass distorting space. He argued that space behaves like an invisible fabric with an elastic quality. Celestial bodies interact with this “fabric” of space-time, appearing to create depressions termed “gravity wells” and drawing nearby objects into orbit around them. Based on this principle, the more massive a body is in space, the deeper the gravity well it will create.
    [Show full text]
  • Closed Timelike Curves, Singularities and Causality: a Survey from Gödel to Chronological Protection
    Closed Timelike Curves, Singularities and Causality: A Survey from Gödel to Chronological Protection Jean-Pierre Luminet Aix-Marseille Université, CNRS, Laboratoire d’Astrophysique de Marseille , France; Centre de Physique Théorique de Marseille (France) Observatoire de Paris, LUTH (France) [email protected] Abstract: I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to save the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role. Keywords: time travel; closed timelike curves; singularities; wormholes; Gödel’s universe; chronological protection; causal dynamical triangulations 1. Introduction In 1949, the mathematician and logician Kurt Gödel, who had previously demonstrated the incompleteness theorems that broke ground in logic, mathematics, and philosophy, became interested in the theory of general relativity of Albert Einstein, of which he became a close colleague at the Institute for Advanced Study at Princeton.
    [Show full text]
  • Light Rays, Singularities, and All That
    Light Rays, Singularities, and All That Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics at the Institute for Advanced Study, and also at the 2020 New Zealand Mathematical Research Institute summer school in Nelson, New Zealand. Contents 1 Introduction 3 2 Causal Paths 4 3 Globally Hyperbolic Spacetimes 11 3.1 Definition . 11 3.2 Some Properties of Globally Hyperbolic Spacetimes . 15 3.3 More On Compactness . 18 3.4 Cauchy Horizons . 21 3.5 Causality Conditions . 23 3.6 Maximal Extensions . 24 4 Geodesics and Focal Points 25 4.1 The Riemannian Case . 25 4.2 Lorentz Signature Analog . 28 4.3 Raychaudhuri’s Equation . 31 4.4 Hawking’s Big Bang Singularity Theorem . 35 5 Null Geodesics and Penrose’s Theorem 37 5.1 Promptness . 37 5.2 Promptness And Focal Points . 40 5.3 More On The Boundary Of The Future . 46 1 5.4 The Null Raychaudhuri Equation . 47 5.5 Trapped Surfaces . 52 5.6 Penrose’s Theorem . 54 6 Black Holes 58 6.1 Cosmic Censorship . 58 6.2 The Black Hole Region . 60 6.3 The Horizon And Its Generators . 63 7 Some Additional Topics 66 7.1 Topological Censorship . 67 7.2 The Averaged Null Energy Condition .
    [Show full text]
  • Chapter 9: the 'Emergence' of Spacetime in String Theory
    Chapter 9: The `emergence' of spacetime in string theory Nick Huggett and Christian W¨uthrich∗ May 21, 2020 Contents 1 Deriving general relativity 2 2 Whence spacetime? 9 3 Whence where? 12 3.1 The worldsheet interpretation . 13 3.2 T-duality and scattering . 14 3.3 Scattering and local topology . 18 4 Whence the metric? 20 4.1 `Background independence' . 21 4.2 Is there a Minkowski background? . 24 4.3 Why split the full metric? . 27 4.4 T-duality . 29 5 Quantum field theoretic considerations 29 5.1 The graviton concept . 30 5.2 Graviton coherent states . 32 5.3 GR from QFT . 34 ∗This is a chapter of the planned monograph Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity, co-authored by Nick Huggett and Christian W¨uthrich and under contract with Oxford University Press. More information at www.beyondspacetime.net. The primary author of this chapter is Nick Huggett ([email protected]). This work was sup- ported financially by the ACLS and the John Templeton Foundation (the views expressed are those of the authors not necessarily those of the sponsors). We want to thank Tushar Menon and James Read for exceptionally careful comments on a draft this chapter. We are also grateful to Niels Linnemann for some helpful feedback. 1 6 Conclusions 35 This chapter builds on the results of the previous two to investigate the extent to which spacetime might be said to `emerge' in perturbative string the- ory. Our starting point is the string theoretic derivation of general relativity explained in depth in the previous chapter, and reviewed in x1 below (so that the philosophical conclusions of this chapter can be understood by those who are less concerned with formal detail, and so skip the previous one).
    [Show full text]
  • Stable Wormholes in the Background of an Exponential F (R) Gravity
    universe Article Stable Wormholes in the Background of an Exponential f (R) Gravity Ghulam Mustafa 1, Ibrar Hussain 2,* and M. Farasat Shamir 3 1 Department of Mathematics, Shanghai University, Shanghai 200444, China; [email protected] 2 School of Electrical Engineering and Computer Science, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan 3 National University of Computer and Emerging Sciences, Lahore Campus, Punjab 54000, Pakistan; [email protected] * Correspondence: [email protected] Received: 21 January 2020; Accepted: 3 March 2020; Published: 26 March 2020 Abstract: The current paper is devoted to investigating wormhole solutions with an exponential gravity model in the background of f (R) theory. Spherically symmetric static spacetime geometry is chosen to explore wormhole solutions with anisotropic fluid source. The behavior of the traceless matter is studied by employing a particular equation of state to describe the important properties of the shape-function of the wormhole geometry. Furthermore, the energy conditions and stability analysis are done for two specific shape-functions. It is seen that the energy condition are to be violated for both of the shape-functions chosen here. It is concluded that our results are stable and realistic. Keywords: wormholes; stability; f (R) gravity; energy conditions 1. Introduction The discussion on wormhole geometry is a very hot subject among the investigators of the different modified theories of gravity. The concept of wormhole was first expressed by Flamm [1] in 1916. After 20 years, Einstein and Rosen [2] calculated the wormhole geometry in a specific background. In fact, it was the second attempt to realize the basic structure of wormholes.
    [Show full text]
  • Weyl Metrics and Wormholes
    Prepared for submission to JCAP Weyl metrics and wormholes Gary W. Gibbons,a;b Mikhail S. Volkovb;c aDAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK bLaboratoire de Math´ematiqueset Physique Th´eorique,LMPT CNRS { UMR 7350, Universit´ede Tours, Parc de Grandmont, 37200 Tours, France cDepartment of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya street 18, 420008 Kazan, Russia E-mail: [email protected], [email protected] Abstract. We study solutions obtained via applying dualities and complexifications to the vacuum Weyl metrics generated by massive rods and by point masses. Rescal- ing them and extending to complex parameter values yields axially symmetric vacuum solutions containing singularities along circles that can be viewed as singular mat- ter sources. These solutions have wormhole topology with several asymptotic regions interconnected by throats and their sources can be viewed as thin rings of negative tension encircling the throats. For a particular value of the ring tension the geometry becomes exactly flat although the topology remains non-trivial, so that the rings liter- ally produce holes in flat space. To create a single ring wormhole of one metre radius one needs a negative energy equivalent to the mass of Jupiter. Further duality trans- formations dress the rings with the scalar field, either conventional or phantom. This gives rise to large classes of static, axially symmetric solutions, presumably including all previously known solutions for a gravity-coupled massless scalar field, as for exam- ple the spherically symmetric Bronnikov-Ellis wormholes with phantom scalar. The multi-wormholes contain infinite struts everywhere at the symmetry axes, apart from arXiv:1701.05533v3 [hep-th] 25 May 2017 solutions with locally flat geometry.
    [Show full text]
  • Quantum Fluctuations and Thermodynamic Processes in The
    Quantum fluctuations and thermodynamic processes in the presence of closed timelike curves by Tsunefumi Tanaka A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Montana State University © Copyright by Tsunefumi Tanaka (1997) Abstract: A closed timelike curve (CTC) is a closed loop in spacetime whose tangent vector is everywhere timelike. A spacetime which contains CTC’s will allow time travel. One of these spacetimes is Grant space. It can be constructed from Minkowski space by imposing periodic boundary conditions in spatial directions and making the boundaries move toward each other. If Hawking’s chronology protection conjecture is correct, there must be a physical mechanism preventing the formation of CTC’s. Currently the most promising candidate for the chronology protection mechanism is the back reaction of the metric to quantum vacuum fluctuations. In this thesis the quantum fluctuations for a massive scalar field, a self-interacting field, and for a field at nonzero temperature are calculated in Grant space. The stress-energy tensor is found to remain finite everywhere in Grant space for the massive scalar field with sufficiently large field mass. Otherwise it diverges on chronology horizons like the stress-energy tensor for a massless scalar field. If CTC’s exist they will have profound effects on physical processes. Causality can be protected even in the presence of CTC’s if the self-consistency condition is imposed on all processes. Simple classical thermodynamic processes of a box filled with ideal gas in the presence of CTC’s are studied. If a system of boxes is closed, its state does not change as it travels through a region of spacetime with CTC’s.
    [Show full text]
  • Mathematics of General Relativity - Wikipedia, the Free Encyclopedia Page 1 of 11
    Mathematics of general relativity - Wikipedia, the free encyclopedia Page 1 of 11 Mathematics of general relativity From Wikipedia, the free encyclopedia The mathematics of general relativity refers to various mathematical structures and General relativity techniques that are used in studying and formulating Albert Einstein's theory of general Introduction relativity. The main tools used in this geometrical theory of gravitation are tensor fields Mathematical formulation defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity. Resources Fundamental concepts Note: General relativity articles using tensors will use the abstract index Special relativity notation . Equivalence principle World line · Riemannian Contents geometry Phenomena 1 Why tensors? 2 Spacetime as a manifold Kepler problem · Lenses · 2.1 Local versus global structure Waves 3 Tensors in GR Frame-dragging · Geodetic 3.1 Symmetric and antisymmetric tensors effect 3.2 The metric tensor Event horizon · Singularity 3.3 Invariants Black hole 3.4 Tensor classifications Equations 4 Tensor fields in GR 5 Tensorial derivatives Linearized Gravity 5.1 Affine connections Post-Newtonian formalism 5.2 The covariant derivative Einstein field equations 5.3 The Lie derivative Friedmann equations 6 The Riemann curvature tensor ADM formalism 7 The energy-momentum tensor BSSN formalism 7.1 Energy conservation Advanced theories 8 The Einstein field equations 9 The geodesic equations Kaluza–Klein
    [Show full text]
  • (12) Patent Application Publication (10) Pub. No.: US 2006/0167784 A1 Hoffberg (43) Pub
    US 2006O167784A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2006/0167784 A1 Hoffberg (43) Pub. Date: Jul. 27, 2006 (54) GAME THEORETIC PRIORITIZATION Related U.S. Application Data SCHEME FOR MOBILEAD HOC NETWORKS PERMITTING HERARCHAL (60) Provisional application No. 60/609,070, filed on Sep. DEFERENCE 10, 2004. (76) Inventor: Steven M. Hoffberg, West Harrison, Publication Classification NY (US) (51) Int. Cl. G06O 40/00 (2006.01) Correspondence Address: (52) U.S. Cl. ................................................................ T05/37 Steven M. Hoffberg, Esq. (57) ABSTRACT MILDE & HOFFBERG, LLP Suite 460 A method for providing unequal allocation of rights among 10 Bank Street agents while operating according to fair principles, com White Plains, NY 10606 (US) prising assigning a hierarchal rank to each agent; providing a synthetic economic value to a first set of agents at the a high level of the hierarchy; allocating portions of the Syn (21) Appl. No.: 11/005,460 thetic economic value by the first set of agents to a second set of agents at respectively different hierarchal rank than the first set of agents; and conducting an auction amongst agents (22) Filed: Dec. 6, 2004 using the synthetic economic value as the currency. Time Update ("Predict") Measurement Update ("Correct") (1) Project the state ahead (1) Compute the Kalman Gain x = AX-1 + Buk K = P, HT (HP, H+R) (2) Project the error covariance ahead (2) Update estimate with P = AP A+ Q measurement I k Xk = x + K. (Ik- HS) (3) Update the error covariance P (I- KH) P. Initial estimates for X-1 and P-1 Patent Application Publication Jul.
    [Show full text]
  • Resource Constrained Adaptive Sensing
    RESOURCE CONSTRAINED ADAPTIVE SENSING by Raghuram Rangarajan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering: Systems) in The University of Michigan 2007 Doctoral Committee: Professor Alfred O. Hero III, Chair Professor Jeffrey A. Fessler Professor Susan A. Murphy Professor Demosthenis Teneketzis Assistant Professor Clayton Scott c Raghuram Rangarajan 2007 All Rights Reserved To my mom, my dad, and my brother ii ACKNOWLEDGEMENTS I would like to extend my sincere thanks and deepest gratitude to Professor Alfred Hero for his invaluable guidance, encouragement, and patience during the course of my research. Through the years, I have come to admire Professor Hero’s vast knowledge and deep insight on any scientific field, creativity in problem solving, and his exceptional time management skills. I consider myself extremely lucky to have found an advisor in Professor Alfred Hero and his attributes will definitely exert a great influence in all my future endeavors. My sincere thanks and gratitude also goes to Raviv Raich, with whom I have collaborated on many of my research topics. His invaluable inputs on my research and his ability to breakdown problems have helped me find solutions much quicker and more efficiently. It has also been an absolute pleasure interacting with him on a day-to-day basis for the last 3 years on many other topics of research and life in general. I am grateful to my committee members Professor Jeffrey Fessler, Professor Susan Murphy, Professor Demosthenis Teneketzis, and Professor Clayton Scott for their valuable input on my work and their helpful comments on my dissertation.
    [Show full text]