DOCSLIB.ORG

Effects of Rotating Frame on a Vector Boson Oscillator

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Effects of Rotating Frame on a Vector Boson Oscillator Sakarya University Journal of Science SAUJS e-ISSN 2147-835X | Period Bimonthly | Founded: 1997 | Publisher Sakarya University | http://www.saujs.sakarya.edu.tr/en/ Title: Effects of Rotating Frame on a Vector Boson Oscillator Authors: Abdullah GUVENDİ Recieved: 2021-04-07 17:07:23 Accepted: 2021-05-07 20:36:07 Article Type: Research Article Volume: 25 Issue: 3 Month: June Year: 2021 Pages: 834-840 How to cite Abdullah GUVENDİ; (2021), Effects of Rotating Frame on a Vector Boson Oscillator. Sakarya University Journal of Science, 25(3), 834-840, DOI: https://doi.org/10.16984/saufenbilder.911340 Access link http://www.saujs.sakarya.edu.tr/en/pub/issue/62736/911340 New submission to SAUJS http://dergipark.org.tr/en/journal/1115/submission/step/manuscript/new Sakarya University Journal of Science 25(3), 834-840, 2021 Effects of Rotating Frame on a Vector Boson Oscillator Abdullah GUVENDI*1 Abstract We analyze the effects of the spacetime topology and angular velocity of rotating frame on the dynamics of a relativistic vector boson oscillator ( ). To determine these effects on the energy of the we solve the corresponding vector boson equation in the rotating frame of 2+1 dimensional cosmic string-induced spacetime background. We obtain an exact energy spectrum, which depends on the angular velocity of the rotating frame and angular deficit parameter of the background. We show that the effects of angular deficit parameter on each energy level of the cannot be same and the angular velocity of the rotating frame couples with the spin of the . Furthermore, we have obtained that the angular velocity of rotating frame breaks the symmetry of the positive-negative energy states. Keywords: Non-inertial effects, quantum fields in curved spaces, topological defects. the symmetric spinor can be constructed by 1. INTRODUCTION kronocker product of two dirac spinors [1,4-6]. However, it is important to underline that this The relativistic wave equations for spin-1/2, spin- symmetric spinor does not include the spin-0 1 and spin-3/2 particles can be derived via sector in 2+1 dimensions [7]. The vector boson canonical quantization of the classical spinning equation ( ) corresponds to the spin-1 part of particle with zitterbewegung. These equations can the equation in 2+1 dimensions [1,4-7] and be obtained as possible quantum states from the its massless form can also be derived via the same mentioned classical quantization procedure [1]. quantization procedure [4]. It has been declared After demonstrating that there is a unifying that massless and Maxwell equations are principle for this class of relativistic wave equivalent [4], and moreover it has been clearly equations [1], it has been shown that the well- shown that the state functions for massless form known duffin-kemmer-proca equations and the of the have probability interpretation [4]. A many-body dirac equation (fully-covariant) [1-3] few applications of the can be found in the can be derived from similar action. Spin-1 sector refs. [5-8]. In this work, we introduce the vector of the duffin-kemmer-petiau ( ) equation has boson oscillator ( ) and investigate its been derived as excited state via the mentioned dynamics in the rotating frame of the 2+1 procedure provided that the spinor is a symmetric dimensional cosmic string-generated spacetime. spinor of rank-two [1,4,5]. Under this condition, *Corresponding author: [email protected] 1Kutahya Health Sciences University, Faculty of Engineering and Natural Sciences, Department of Basic Sciences, 43100, Kutahya. E-Mail: [email protected] ORCID: https://orcid.org/ 0000-0003-0564-9899 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator Cosmic strings are predicted to be linear where the particles can be placed [20-22]. It has topological defects that probably occurred in the been shown that the phase shift in neutron early stages of the universe [9]. After Cosmic interferometry is due to the coupling of angular strings were introduced in dimensional velocity of the earth with the angular momentum spacetime [10], the obtained solution was of the neutron [26]. The effects of rotating frame naturally expanded to dimensions where on the molecules were analized [27,28] and there is a dynamical symmetry [11]. The the obtained results have indicated that rapidly spacetime backgrounds of cosmic strings include rotating molecules can acquire a topological a conical singularity on the intersection point of phase [27,28]. the string with the space-like subspace [9-11]. Even though these spacetime backgrounds are In this contribution, we investigate the influences locally flat [9-11] out of the intersection point, of rotating frame on the relativistic dynamics of a they are not flat when viewed globally [9-11]. in the -dimensional cosmic string Because of this feature, the cosmic strings can be spacetime. In order to determine the effects of responsible for several physical events in the both uniformly rotating frame and spacetime universe [12-15]. The spacetime background of topology on the system in question we solve the the cosmic string is characterized by parameter, corresponding . We obtain an exact energy which is angular deficit parameter depending on spectrum depending on the parameters of the linear mass density ( ) of the cosmic string spacetime background. [12-15]. The gravitational [9-14] and electrostatic effects [12,15] of these topological defects on the 2. GENERALIZED FORM OF THE dynamics of physical systems depend on the parameter . The influences of topological defects on the dynamics of quantum mechanical In this part of the manuscript, we introduce the systems have been widely studied [15-17]. In one generalized form of the and arrive at a set of of such works, it has been shown that the spin- coupled equations for a in the rotating frame symmetric [18] quantum state of a positronium of 2+1 dimensional cosmic string-generated system can be used to detect the topological spacetime background.The generalized form of defects [15]. The effects of topological defects on the can be written as the following [1-8], the spectral lines of quantum systems are not same for each spectral line of the quantum systems [19]. Ψ(x)=0, This means that the effects of topological defects , are distinguishable from the other effects such as doppler effect and redshift [15,19]. (1) On the other hand, the investigations about the In eq. (1), are the generalized dirac matrices, effects of rotating frames on the quantum systems are the spinorial affine connections for spin-1/2 have provided very interesting results and field, m is the rest mass of vector boson, c is the accordingly such studies have attracted great speed of light, is usual planck constant, the attention for a long time [20-22] since these symbol indicates kronocker product [28], x is investigations have discovered very important the spacetime position vector, and stand for effects [23-29]. The sagnac effect [23], mashhoon the and dimensional identity matrices, effect [24], persistent currents in the quantum ring and the coupling between the angular momentum respectively, and is the symmetric spinor of of the quantum systems with the velocity of rank-two. The dimensional spacetime uniformly rotating frame [25] can be considered geometry of the uniformly rotating cosmic string- among these effects. One of the interesting results generated background is represented through the of these investigations is that the non-inertial following line element [20-22], effects stemming from the rotating frames restrict the physical region of the spacetime backgrounds Sakarya University Journal of Science 25(3), 834-840, 2021 835 Abdullah GUVENDİ Effects of Rotating Frame on a Vector Boson Oscillator - - . (2) eq. (2) the space-dependent dirac matrices are obtained, In eq. (2), is the angular deficit parameter, depending on the linear mass = , = , = + . (5) density of the cosmic string and newtonian gravitational constant ) [15]. The azimuthal The choice of space-independent dirac matrices is angle range is in this spacetime not unique and different choices can be preferred, background. Here, we deal with 0 case, of course. According to signature of eq. (2) the since the case describes an anti-conical space-independent dirac matrices can be chosen spacetime background with negative curvature as = , = , = [29-31], in which [20-22]. Therefore, the case does not make , and are the pauli spin matrices. Also, sense in the general relativity context [20-22]. The we can obtain the spinorial connections for spin- eq. (2) describes a flat spacetime background, in 1/2 fields via the following expression [15], terms of polar coordinates, when and [15]. In eq. (2), we should notice that the radial coordinate must satisfy , since (휆,σ,μ,ν,a,b=0,1,2.), (6) this spacetime geometry can be defined for values of the inside the range: 0 . This where are the christoffel symbols obtained condition guarantees that the vector boson through =1/2 ( ) and considered is placed inside of the light-cone and are the spin operators determined by imposes a spatial constraint. That is, the wave [ ]. Now, one can obtain the function vanishes at = [20-22]. At this spinorial connections (non-vanishing) as the point, we should underline that the in eq. (2) is following, and is the angular velocity (not frequency) of the uniformly rotating frame. Therefore, the = , = . (7) rises to a hard-wall confining potential [20-22]. Now, one can obtain the tetrads [15], The coupling, describing the interaction of magnetic moment (anomalous) with a linearly varying electric field [3,16,17,21], can be introduced through the following non-minimal ( ). (3) substitution, In eq. (3), is the flat spacetime metric tensor + ( )r, and hence the indices indicate the local reference frame. The tetrads, which satisfy the ( ) = diag (1,-1,-1,1), (8) following orthonormality conditions, where is the oscillator frequency (coupling , , (4) strength) [3].
Load more

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]
  • 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]
  • 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]
  • Topology and Singularities in Cosmological Spacetimes Obeying the Null Energy Condition
    Topology and Singularities in Cosmological Spacetimes Obeying the Null Energy Condition Gregory J. Galloway & Eric Ling Communications in Mathematical Physics ISSN 0010-3616 Volume 360 Number 2 Commun. Math. Phys. (2018) 360:611-617 DOI 10.1007/s00220-017-3020-9 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag GmbH Germany. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Commun. Math. Phys. 360, 611–617 (2018) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-017-3020-9 Mathematical Physics Topology and Singularities in Cosmological Spacetimes Obeying the Null Energy Condition Gregory J. Galloway1,EricLing1 Department of Mathematics, University of Miami, Coral Gables, FL, USA. E-mail: [email protected] Received: 4 June 2017 / Accepted: 17 September 2017 Published online: 9 November 2017 – © Springer-Verlag GmbH Germany 2017 We dedicate this paper to the memory of Ted Frankel Abstract: We consider globally hyperbolic spacetimes with compact Cauchy surfaces in a setting compatible with the presence of a positive cosmological constant.
    [Show full text]
  • Astrophysical Wormholes
    universe Article Astrophysical Wormholes Cosimo Bambi 1,* and Dejan Stojkovic 2 1 Center for Field Theory and Particle Physics and Department of Physics, Fudan University, Shanghai 200438, China 2 Department of Physics, State University of New York (SUNY) at Buffalo, Buffalo, NY 14260-1500, USA; [email protected] * Correspondence: [email protected] Abstract: Wormholes are hypothetical topologically-non-trivial structures of spacetime. From the theoretical point of view, the possibility of their existence is challenging but cannot be ruled out. This article is a compact and non-exhaustive review of past and current efforts to search for astro- physical wormholes in the Universe. Keywords: wormholes; black holes; spacetime topology; gravity 1. Introduction Wormholes are hypothetical spacetime structures with non-trivial topologies capable of connecting either two distant regions of the same universe or two different universes, as illustrated in Figure1. The entrances of a wormhole are called the “mouths” of the wormhole and the spacetime region connecting the mouths is called the “throat” of the wormhole. The simplest wormhole configuration has two mouths connected by a throat, but more complex structures are also possible [1]. Strictly speaking, wormholes are not a prediction of general relativity or of other theories of gravity. They are spacetime structures Citation: Bambi, C.; Stojkovic, D. that can potentially exist in curved spacetimes, so they exist in a very wide class of gravity Astrophysical Wormholes. Universe models. The formation mechanisms and stability of these spacetime structures depend on 2021, 7, 136. https://doi.org/ 10.3390/universe7050136 the specific gravity theory and often present problems, so the existence of wormholes in the Universe is challenging.
    [Show full text]
  • 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).
    [Show full text]
  • EXOTIC SPACETIME TOPOLOGY As an Alternative to Dark Matter and Energy Greg Kuri1, James Overduin1 and Richard Conn Henry2 1Towson University, Towson, Maryland
    EXOTIC SPACETIME TOPOLOGY As an Alternative to Dark Matter and Energy Greg Kuri1, James Overduin1 and Richard Conn Henry2 1Towson University, Towson, Maryland Exotic Spacetime Topology Case Study: Gravitational Lensing from Topology studies Exotic Smooth Structure the properties of space In three-dimensional Euclidean space, mathematicians and geometry that are have been able to break down complicated manifolds into preserved under contin- simpler, Euclidean pieces (called “handles”) using something uous deformations (eg. called “handlebody decomposition.” Physicist Christopher stretching or crumpling). Duston has used an analogous mechanism (based on work by Two such deformations Taubes [4]) to obtain metrics on an exotic four-manifold [7]. are homeomorphisms and When this four manifold is decomposed into blocks using diffeomorphisms. Home- “Casson handling,” we can give each block its own metric. omorphisms are inverti- (푛) Given a metric 푔 on each block 푊푛, the metric on the ble transformations that whole manifold can be determined using Fourier-Laplace do not involve cutting or Fig. 2: An artist’s interpretation of exotic (“Z-transform”) techniques, 푔 = σ∞ 푧−푛푔(푛), and takes gluing. Diffeomorphisms 푧 −∞ four-dimensional space [2]. The center is the form of what is known as an “end-periodic manifold” are differentiable homeo- the Euclidean representation of four- (Fig. 3). This mechanism gives us a way to describe curved morphisms, where we Fig. 4: G. Kuri in Towson University’s planetarium, suggesting a con- space, while the tendrils represent its space (and therefore gravity) in the alternate mathematical can perform calculus. exotic nature as it approaches infinity. nection between spacetime topology and cosmic background radiation.
    [Show full text]
  • Exotic Smoothness and Astrophysics∗
    Vol. 40 (2009) ACTA PHYSICA POLONICA B No 11 EXOTIC SMOOTHNESS AND ASTROPHYSICS∗ Jan Sładkowski Institute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland (Received October 19, 2009) The problem of possible astrophysical consequences of the existence of exotic differential structures on manifolds is discussed. It is argued that corrections to the curvature of the form of a source like terms should be expected in the Einstein equations if they are written in the “wrong” differ- ential structure. Examples of topologically trivial spaces on which exotic differential structures act as a source of gravitational force even in the ab- sence of matter are given. Propagation of light in the presence of such phenomena is also discussed. A brief review of exotic smoothness is added for completeness. PACS numbers: 98.80.Jk, 95.30.Sf 1. Introduction In 1854 B. Riemann, the father of contemporary differential geometry, suggested that the geometry of space may be more than just a mathematical tool defining a stage for physical phenomena, and may in fact have profound physical meaning in its own right [1]. But is it reasonable to contemplate to what extent the choice of mathematical model for spacetime has important physical significance? With the advent of general relativity physicists be- gan to think of the spacetime as a differential manifold. Since then various assumptions about the spacetime topology and geometry have been put for- ward. But why should the choice of differential structure of the spacetime manifold matter? Most of topological spaces used for modelling spacetime have natural differential structures and the question of their non-uniqueness seemed to be extravagant.
    [Show full text]
  • Non-Linear Monte Carlo Ray Tracing for Visualizing Warped Spacetime
    Non-linear Monte Carlo Ray Tracing for Visualizing Warped Spacetime Avirup Mandal∗, Kumar Ayush∗ and Parag Chaudhuri Indian Institute of Technology Bombay, Powai, Mumbai, India Keywords: Raytracing, Non-linear Monte Carlo, Warped Spacetime, Relativity. Abstract: General relativity describes the curvature of spacetime. Rays of light follow geodesic paths in curved space- time. Visualizing scenes containing spacetime regions with pronounced curvature requires tracing of these light ray paths. We present a Monte Carlo approach for non-linear raytracing to render scenes in curved space- time. In contrast to earlier work, we can accurately resolve ray-object interactions. This allows us to create plausible visualizations of what happens when a black hole appears in a more known environment, like a room with regular specular and diffuse surfaces. We demonstrate that our solution is correct at cosmological scales by showing how spacetime warps around a stationary Schwarzschild black hole and a non-stationary Kerr black hole. We verify that the solution is consistent with the predictions of general relativity. In the absence of any curvature in spacetime, our renderer behaves like a normal linear ray tracer. Our method has the poten- tial to create rich, physically plausible visualizations of complex phenomena that can be used for a range of purposes, from creating visual effects to making pedagogical aids to understand the behaviour of spacetime as predicted by general relativity. 1 INTRODUCTION momentum is referred to as a Schwarzschild black hole (Schwarzschild, 1916). A Kerr black hole (Kerr, General relativity changed the way in which we un- 1963) on the other hand, possesses angular momen- derstand our universe.
    [Show full text]
  • Proper Time Foliations of Lorentz Manifolds
    Proper Time Foliations of Lorentz Manifolds D. H. Delphenich† Physics Department University of the Ozarks Clarksville, AR Abstract. Some standard definitions and results concerning foliations of dimension one and codimension one are introduced. A proper time foliation of Minkowski space in defined and contrasted with the foliation defined by the time coordinate. The extent to which a Lorentz structure on a manifold defines foliations, and the issues concerning the extension of the proper time foliation of Minkowski space to a Lorentz manifold are discussed, such as proper time sections of a geodesic flow. 0. Introduction. Certainly much has been said about the geometrical consequences of Einstein’s general theory of relativity [1-4]. This is because specifying a metric (or pseudo-metric) on a manifold implies the existence of a unique zero-torsion metric connection − the Levi-Civita connection − from which all of the rest of spacetime geometry − curvature, geodesics, conjugate points, etc. − follows by construction. Less of a physically definitive nature has been said about how, or whether, spacetime topology differs from a Euclidean one (1). Certainly, the definition of a manifold weakens that assumption to being only locally true in general; however, the real issue is whether the process of raising time to the status of a dimension on a par with the other dimensions in the eyes of geometry is also true in the eyes of topology. A well-known result of general relativity [5] says that the mere existence of a maximal Cauchy hypersurface (2) Σ, as would be essential to expressing the metric of spacetime as a solution to an equation of time evolution, such as in the Hamiltonian formulation [6], implies that the topology of spacetime is simply the product Σ .
    [Show full text]
  • Spacetime: Arena Or Reality?
    Spacetime: Arena or Reality? H. I. Arcos1 and J. G. Pereira2 1 Universidad Tecnol´ogicade Pereira, A.A. 97, La Julita, Pereira, Colombia [email protected] 2 Instituto de F´ısicaTe´orica,Universidade Estadual Paulista, Rua Pamplona 145, 01405-900, S~aoPaulo SP, Brazil [email protected] 1 Introduction The concept of fundamental particle has been quite elusive along the his- tory of physics. The term fundamental is commonly used as a synonymous of structureless particles. However, this assumption is clearly contradictory. For example, it is impossible to explain spin without assuming a structure for the particle. In fact, a point particle is by definition spherically symmetric, a sym- metry violated by the presence of spin. This problem is usually circumvented by saying that spin is a purely quantum property, which cannot be explained by classical physics. This means to keep it as a mysterious property of nature. If one assumes that a fundamental particle is a point-like object, several arguments against this idea show up immediately. First, as discussed above, a point-like object seems to be inconsistent with the existence of spin. Second, if we try to reconcile general relativity with point-particles, which are singular points in a pseudo-Riemannian spacetime, unwanted features, like for example ultraviolet divergences, will appear. A natural alternative would be to assume that a fundamental particle is a string-like object, a point of view adopted by string theory [1]. Similarly, one can introduce membranes as fundamen- tal objects, or even extended objects with certain geometries. These models, however, are also plagued by problems.
    [Show full text]