DOCSLIB.ORG

Parameterizing the Proper-Acceleration Three-Vector Phil Fraundorf, Matt Wentzel-Long

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Parameterizing the Proper-Acceleration Three-Vector Phil Fraundorf, Matt Wentzel-Long Parameterizing the proper-acceleration three-vector Phil Fraundorf, Matt Wentzel-Long To cite this version: Phil Fraundorf, Matt Wentzel-Long. Parameterizing the proper-acceleration three-vector. 2019. hal- 02087094 HAL Id: hal-02087094 https://hal.archives-ouvertes.fr/hal-02087094 Preprint submitted on 1 Apr 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Parameterizing the proper-acceleration three-vector P. Fraundorf∗ Physics & Astronomy/Center for Nanoscience, U. Missouri-StL (63121), St. Louis, MO, USA† Matt Wentzel-Long Physics & Astronomy, U. Missouri-StL (63121), St. Louis, MO, USA (Dated: April 1, 2019) Abstract The implicit idea of global time adds dissonance to the bridge between Newtonian dy- namics and high-speeds/curved-spacetimes. However Newtonian use of 3-vectors connects naturally to a “traveler-point dynamics” that starts with “minimally frame-variant” proper- time/velocity/acceleration. This allows one to express the introphysics equations of constant accel- eration as a limiting case of characteristic-time equations which work well at both high speeds and in curved spacetimes, provided that we (as usual) approximate geometric forces (like gravity and centrifugal which are invisible to cell-phone accelerometers) as proper forces. This approach can help make engineering pedagogy “spacetime smart”, and be useful in relativistic physics engines used for cinema, video games, and interstellar travel simulations. 1 CONTENTS I.Introduction 2 II.Metric invariants and their solutions 4 III.Work-energy and Newton’s laws 6 IV.Uses in simulation and in pedagogy 8 V.Conclusions 11 Acknowledgments 11 References 12 A.Supplementary Material 12 1.Intro to motion studies 12 a.start with a map 13 b.traveler-point kinematics 13 c.map-based dynamics 14 2.Symbols 17 3.Numerical examples 18 4.Sample acceleration problem 18 I. INTRODUCTION There are many ways to describe accelerated motion in spacetime, because one can choose between using: time on map-clocks or traveler-clocks; • velocity as distances on map1,2 or traveler1 yardsticks per unit time on map1 or traveler2 • clocks; acceleration as second derivatives of distance on map1 or traveler yardsticks with respect • to time on map1 or traveler3,4 clocks. 2 invari frame ants proper or ration | cele Σξ| ac = | lf mα se | light or elf speed c s s d e r i t n o n f i t γ f δ e o δ m c i e / r γ e p r g e t ro er y p e l v a / m δ y m t n δ i - r δ t - i ime δ t k p t r 3 e c / i e f e l δ a = Σ o τ l d e τ l a γ l / a e c γ c e τ δ n t t c t a P c = v δ t o e a a i g e x i E d l m m i t ( l r x e a e i e p n m e δ e δ s p s o a k r g p c r p m m c i - ^ a o c w a l m f 2 r t ^ δ l a e u w p 2 ) i a c c ) t o t r a n n o s p e r s m o δ/δt v m t δ c δ x o y o t r i d c i o n l a e t v e δ/δt m a t a = δ s / c δ v p o n e o o r r i o d t u i a r n r s a e t l e e a c a c b e o m o - k e k t e a e n p i e d r r o c o FIG. 1. Scalar and 3-vector parameters for characterizing motion using a single metric: Practical use of the red parameters may require synchronized clocks, e.g. to measure map time at different positions, while the indigo “traveler-point” parameter set is synchrony-free, and has minimal frame- variance. In (3+1)D one can also resolve components with respect to the directon of either: (i) the ”heading” or velocity1,4, (ii) the “traveler’s map-acceleration”4 or frame-variant force, (iii) the coordinate-acceleration5, or (iv) the proper-acceleration3. None of these choices are wel- come in the Lorentz-transform world of co-moving free-float-frames6, since extended frame- works of synchronized clocks cannot accelerate lest they lose internal sync. Hence it’s not surprising that 3-vector treatments of acceleration at any speed are not generally standard- ized. Only one of these descriptions works robustly (cf. Fig. 1) for 3-vectors in curved space- time, namely that using “traveler-point” variables, namely frame-invariant proper time τ, synchrony-free2 proper velocity7 ~w δ~x/δτ, and magnitude-invariant proper acceleration ~α. ≡ This strategy mirrors the “one-map two-clock” form of the metric-equation itself, whose zeroeth through second derivative versions (with respect to traveler-time) define frame in- variants in terms of displacements on bookkeeper or map yardsticks and times elapsed on map clocks. Work with components referenced to the direction of the constant proper- 3 TABLE I. Constant proper acceleration ~α in flat spacetime: Here τ is traveler-time from “turnaround” when proper velocity ~w = ~w ~α, initial aging-factor γ = 1 + (w /c)2), and o ⊥ o o characteristic time is τ 1 (γ + 1)c/α. p o ≡ 2 o q 4-vector V invariant scalar timelike component V t/c 3-vector spacelike component V~ proper ~r = 2 1 (wo/c) ατo 2 τ t = 2 γ +1 τ + c τo sinh τ 1 τ 2 τ coordinate time o o (τ + τo sinh ) ~wo + τ (cosh 1)~α 2 τo o τo − τ h i t ~wo = 0 t = τo sinh τ h i 2 τ h i X , X~ τ → o ~wo = 0 ~r = τ cosh 1 ~α { } → o τo − w c t τ h i g µ ν h 1 i 2 √ µνX X ≪ → ≃ w c ~r ~woτ + 2 ~ατ ≪ →d~r ≃ 2 dτ ~w = lightspeed dt γ = 1 (wo/c) + ατo 2 cosh τ ≡ dτ 2 γo+1 c τo 1 τ τ velocity ≡ (1 + cosh ) ~wo + τo sinh ~α τ h i 2 τo τo c ~wo = 0 γ = cosh t ~ τo h i τ h i U , U → ~wo = 0 ~w = τo sinh ~α { } 2 τo √gµνU µU ν w c γ 1+ 1 hw i → 2 c h i ≪ → ≃ w c ~w ~wo + ~ατ proper ≪ δ2→~x ≃δ ~w 2 2 = = δ t δγ α 2 τ δτ δτ 2 = = ( ) τo sinh δτ δτ c τo 1 τ τ accelerationacceleration (sinh /τo) ~wo + cosh ~α δγ 1 h τ i 2 τo τo ~wo = 0 = sinh t ~ δτ τo τo h i δ ~w τh i A , A α → ~wo = 0 = cosh ~α { } δγ α 2h i → δτ τo w c δτ ( c ) τ √ gµνAµAν ≪ → ≃ w c δ ~w ~αh i − ≪ → δτ ≃ TABLE II. Traveler-point dynamics in flat spacetime: Conserved quantities energy E = γmc2 and momentum ~p = m ~w, where differential-aging factor γ δt/δτ, proper velocity ~w δ~r/δτ γ~v, ≡ ≡ ≡ coordinate acceleration ~a δ~v/δt. ≡ relation (1+1)D (3+1)D w c γ 1 ≪ → ≃ momentum ~p ~p = m ~w = mγ~v ~p = m ~w = mγ~v ~p m~v ≃ energy E E = γmc2 E = γmc2 E mc2 + 1 mv2 ≃ 2 felt (ξ~) map-based ↔ ξ~ = f~ + γf~ ~ ~ || ~w ⊥ ~w ~ ~ (f~) f = ξ f ξ ~ ~ ~ ≃ f = ξ|| ~w + ξ⊥ ~w/γ force conversions map-based Σf~ = δ~p/δt force:momentum Σf~ = δ~p/δt = m~α Σf~ = δ~p/δt m~a Σξ~ = m~α ≃ felt force:acceleration action-reaction f~ = f~ f~ = f~ f~ = f~ AB − BA AB − BA AB − BA work-energy δE =Σf~ δ~x δE =Σξ~ δ~x δE Σf~ δ~x · · ≃ · 4 acceleration looks particularly promising. We describe a parameterization of these flat-space equations of motion in terms of a characteristic “traveler-time from turnaround” c τ γo−1 , (1) o ≡ α 2 q after which the exponential or “e-folding” effects of constant proper-acceleration’s hyperbolic trig-functions start to “kick in”. Here c is lightspeed, α is the proper-acceleration magni- tude, and γo is the differential-aging (Lorentz) factor “at turnaround”, when the velocity- component parallel to that acceleration reverses sign. The result is a set of scalar (time, energy, power) and 3-vector (position, momentum, force) equations that reduce seamlessly to the familiar low-speed equations. Their application, initially in flat-spacetime simula- tions, is also discussed because (in addition to the hyperbolic trig functions) there are some important differences when operating in this metric-first8 or one-map two-clock9 world. II. METRIC INVARIANTS AND THEIR SOLUTIONS We start with the flat-space interval or metric equation in Cartesian form, as a spacetime Pythagoras’ theorem with “time-like” hypoteneuse: (cδτ)2 =(cδt)2 (δx)2 (δy)2 (δz)2 (2) − − − where as usual t represents scalar map-time and 3-vector ~x represents map-position, while frame-invariant proper time (elapsed on the clocks of our accelerated object) is represented by τ, and c is the space/time conversion factor e.g.
Load more

Recommended publications
  • Arxiv:1401.0181V1 [Gr-Qc] 31 Dec 2013 † Isadpyia Infiac Fteenwcodntsaediscussed
    Painlev´e-Gullstrand-type coordinates for the five-dimensional Myers-Perry black hole Tehani K. Finch† NASA Goddard Space Flight Center Greenbelt MD 20771 ABSTRACT The Painlev´e-Gullstrand coordinates provide a convenient framework for pre- senting the Schwarzschild geometry because of their flat constant-time hyper- surfaces, and the fact that they are free of coordinate singularities outside r=0. Generalizations of Painlev´e-Gullstrand coordinates suitable for the Kerr geome- try have been presented by Doran and Nat´ario. These coordinate systems feature a time coordinate identical to the proper time of zero-angular-momentum ob- servers that are dropped from infinity. Here, the methods of Doran and Nat´ario arXiv:1401.0181v1 [gr-qc] 31 Dec 2013 are extended to the five-dimensional rotating black hole found by Myers and Perry. The result is a new formulation of the Myers-Perry metric. The proper- ties and physical significance of these new coordinates are discussed. † tehani.k.finch (at) nasa.gov 1 Introduction By using the Birkhoff theorem, the Schwarzschild geometry has been shown to be the unique vacuum spherically symmetric solution of the four-dimensional Einstein equations. The Kerr geometry, on the other hand, has been shown only to be the unique stationary, rotating vacuum black hole solution of the four-dimensional Einstein equations. No distri- bution of matter is currently known to produce a Kerr exterior. Thus the Kerr geometry does not necessarily correspond to the spacetime outside a rotating star or planet [1]. This is an indication of the complications encountered upon trying to extend results found for the Schwarzschild spacetime to the Kerr spacetime.
    [Show full text]
  • Physics 325: General Relativity Spring 2019 Problem Set 2
    Physics 325: General Relativity Spring 2019 Problem Set 2 Due: Fri 8 Feb 2019. Reading: Please skim Chapter 3 in Hartle. Much of this should be review, but probably not all of it|be sure to read Box 3.2 on Mach's principle. Then start on Chapter 6. Problems: 1. Spacetime interval. Hartle Problem 4.13. 2. Four-vectors. Hartle Problem 5.1. 3. Lorentz transformations and hyperbolic geometry. In class, we saw that a Lorentz α0 α β transformation in 2D can be written as a = L β(#)a , that is, 0 ! ! ! a0 cosh # − sinh # a0 = ; (1) a10 − sinh # cosh # a1 where a is spacetime vector. Here, the rapidity # is given by tanh # = β; cosh # = γ; sinh # = γβ; (2) where v = βc is the velocity of frame S0 relative to frame S. (a) Show that two successive Lorentz boosts of rapidity #1 and #2 are equivalent to a single α γ α Lorentz boost of rapidity #1 +#2. In other words, check that L γ(#1)L(#2) β = L β(#1 +#2), α where L β(#) is the matrix in Eq. (1). You will need the following hyperbolic trigonometry identities: cosh(#1 + #2) = cosh #1 cosh #2 + sinh #1 sinh #2; (3) sinh(#1 + #2) = sinh #1 cosh #2 + cosh #1 sinh #2: (b) From Eq. (3), deduce the formula for tanh(#1 + #2) in terms of tanh #1 and tanh #2. For the appropriate choice of #1 and #2, use this formula to derive the special relativistic velocity tranformation rule V − v V 0 = : (4) 1 − vV=c2 Physics 325, Spring 2019: Problem Set 2 p.
    [Show full text]
  • Lecture Notes 17: Proper Time, Proper Velocity, the Energy-Momentum 4-Vector, Relativistic Kinematics, Elastic/Inelastic
    UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 17 Prof. Steven Errede LECTURE NOTES 17 Proper Time and Proper Velocity As you progress along your world line {moving with “ordinary” velocity u in lab frame IRF(S)} on the ct vs. x Minkowski/space-time diagram, your watch runs slow {in your rest frame IRF(S')} in comparison to clocks on the wall in the lab frame IRF(S). The clocks on the wall in the lab frame IRF(S) tick off a time interval dt, whereas in your 2 rest frame IRF( S ) the time interval is: dt dtuu1 dt n.b. this is the exact same time dilation formula that we obtained earlier, with: 2 2 uu11uc 11 and: u uc We use uurelative speed of an object as observed in an inertial reference frame {here, u = speed of you, as observed in the lab IRF(S)}. We will henceforth use vvrelative speed between two inertial systems – e.g. IRF( S ) relative to IRF(S): Because the time interval dt occurs in your rest frame IRF( S ), we give it a special name: ddt = proper time interval (in your rest frame), and: t = proper time (in your rest frame). The name “proper” is due to a mis-translation of the French word “propre”, meaning “own”. Proper time is different than “ordinary” time, t. Proper time is a Lorentz-invariant quantity, whereas “ordinary” time t depends on the choice of IRF - i.e. “ordinary” time is not a Lorentz-invariant quantity. 222222 The Lorentz-invariant interval: dI dx dx dx dx ds c dt dx dy dz Proper time interval: d dI c2222222 ds c dt dx dy dz cdtdt22 = 0 in rest frame IRF(S) 22t Proper time: ddtttt 21 t 21 11 Because d and are Lorentz-invariant quantities: dd and: {i.e.
    [Show full text]
  • Uniform Relativistic Acceleration
    Uniform Relativistic Acceleration Benjamin Knorr June 19, 2010 Contents 1 Transformation of acceleration between two reference frames 1 2 Rindler Coordinates 4 2.1 Hyperbolic motion . .4 2.2 The uniformly accelerated reference frame - Rindler coordinates .5 3 Some applications of accelerated motion 8 3.1 Bell's spaceship . .8 3.2 Relation to the Schwarzschild metric . 11 3.3 Black hole thermodynamics . 12 1 Abstract This paper is based on a talk I gave by choice at 06/18/10 within the course Theoretical Physics II: Electrodynamics provided by PD Dr. A. Schiller at Uni- versity of Leipzig in the summer term of 2010. A basic knowledge in special relativity is necessary to be able to understand all argumentations and formulae. First I shortly will revise the transformation of velocities and accelerations. It follows some argumentation about the hyperbolic path a uniformly accelerated particle will take. After this I will introduce the Rindler coordinates. Lastly there will be some examples and (probably the most interesting part of this paper) an outlook of acceleration in GRT. The main sources I used for information are Rindler, W. Relativity, Oxford University Press, 2006, and arXiv:0906.1919v3. Chapter 1 Transformation of acceleration between two reference frames The Lorentz transformation is the basic tool when considering more than one reference frames in special relativity (SR) since it leaves the speed of light c invariant. Between two different reference frames1 it is given by x = γ(X − vT ) (1.1) v t = γ(T − X ) (1.2) c2 By the equivalence
    [Show full text]
  • A One-Map Two-Clock Approach to Teaching Relativity in Introductory Physics
    A one-map two-clock approach to teaching relativity in introductory physics P. Fraundorf Department of Physics & Astronomy University of Missouri-StL, St. Louis MO 63121 (December 22, 1996) observation that relativistic objects behave at high speed This paper presents some ideas which might assist teachers as though their inertial mass increases in the −→p = m−→v incorporating special relativity into an introductory physics expression, led to the definition (used in many early curriculum. One can define the proper-time/velocity pair, as 1 0 well as the coordinate-time/velocity pair, of a traveler using textbooks ) of relativistic mass m ≡ mγ. Such ef- only distances measured with respect to a single “map” frame. forts are worthwhile because they can: (A) potentially When this is done, the relativistic equations for momentum, allow the introduction of relativity concepts at an earlier energy, constant acceleration, and force take on forms strik- stage in the education process by building upon already- ingly similar to their Newtonian counterparts. Thus high- mastered classical relationships, and (B) find what is school and college students not ready for Lorentz transforms fundamentally true in both classical and relativistic ap- may solve relativistic versions of any single-frame Newtonian proaches. The concepts of transverse (m0) and longi- problems they have mastered. We further show that multi- tudinal (m00 ≡ mγ3) masses have similarly been used2 frame calculations (like the velocity-addition rule) acquire to preserve relations of the form Fx = max for forces simplicity and/or utility not found using coordinate velocity perpendicular and parallel, respectively, to the velocity alone.
    [Show full text]
  • RELATIVISTIC GRAVITY and the ORIGIN of INERTIA and INERTIAL MASS K Tsarouchas
    RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS K Tsarouchas To cite this version: K Tsarouchas. RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS. 2021. hal-01474982v5 HAL Id: hal-01474982 https://hal.archives-ouvertes.fr/hal-01474982v5 Preprint submitted on 3 Feb 2021 (v5), last revised 11 Jul 2021 (v6) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Relativistic Gravity and the Origin of Inertia and Inertial Mass K. I. Tsarouchas School of Mechanical Engineering National Technical University of Athens, Greece E-mail-1: [email protected] - E-mail-2: [email protected] Abstract If equilibrium is to be a frame-independent condition, it is necessary the gravitational force to have precisely the same transformation law as that of the Lorentz-force. Therefore, gravity should be described by a gravitomagnetic theory with equations which have the same mathematical form as those of the electromagnetic theory, with the gravitational mass as a Lorentz invariant. Using this gravitomagnetic theory, in order to ensure the relativity of all kinds of translatory motion, we accept the principle of covariance and the equivalence principle and thus we prove that, 1.
    [Show full text]
  • The Extended Relativity Theory in Clifford Spaces
    THE EXTENDED RELATIVITY THEORY IN CLIFFORD SPACES C. Castroa and M. Pav·si·cb May 21, 2004 aCenter for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta bJo·zef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia; Email: [email protected] Abstract A brief review of some of the most important features of the Extended Rela- tivity theory in Cli®ord-spaces (C-spaces) is presented whose " point" coordinates are noncommuting Cli®ord-valued quantities and which incorporate the lines, ar- eas, volumes,.... degrees of freedom associated with the collective particle, string, membrane,... dynamics of p-loops (closed p-branes) living in target D-dimensional spacetime backgrounds. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, noncommuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable di- mensions/signatures that allows to study the dynamics of all (closed) p-branes, for all values of p, on a uni¯ed footing. It resolves the ordering ambiguities in QFT and the problem of time in Cosmology. A discussion of the maximal-acceleration Rela- tivity principle in phase-spaces follows along with the study of the invariance group of symmetry transformations in phase-space that allows to show why Planck areas are invariant under acceleration-boosts transformations and which seems to suggest that a maximal-string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying n-dimensional spacetime results from taking di®erent n-dimensional slices through C-space. The conformal group emerges as a natural subgroup of the Cli®ord group and Relativity in C-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl's gauge ¯eld of dilations.
    [Show full text]
  • Observer with a Constant Proper Acceleration Cannot Be Treated Within the Theory of Special Relativity and That Theory of General Relativity Is Absolutely Necessary
    Observer with a constant proper acceleration Claude Semay∗ Groupe de Physique Nucl´eaire Th´eorique, Universit´ede Mons-Hainaut, Acad´emie universitaire Wallonie-Bruxelles, Place du Parc 20, BE-7000 Mons, Belgium (Dated: February 2, 2008) Abstract Relying on the equivalence principle, a first approach of the general theory of relativity is pre- sented using the spacetime metric of an observer with a constant proper acceleration. Within this non inertial frame, the equation of motion of a freely moving object is studied and the equation of motion of a second accelerated observer with the same proper acceleration is examined. A com- parison of the metric of the accelerated observer with the metric due to a gravitational field is also performed. PACS numbers: 03.30.+p,04.20.-q arXiv:physics/0601179v1 [physics.ed-ph] 23 Jan 2006 ∗FNRS Research Associate; E-mail: [email protected] Typeset by REVTEX 1 I. INTRODUCTION The study of a motion with a constant proper acceleration is a classical exercise of special relativity that can be found in many textbooks [1, 2, 3]. With its analytical solution, it is possible to show that the limit speed of light is asymptotically reached despite the constant proper acceleration. The very prominent notion of event horizon can be introduced in a simple context and the problem of the twin paradox can also be analysed. In many articles of popularisation, it is sometimes stated that the point of view of an observer with a constant proper acceleration cannot be treated within the theory of special relativity and that theory of general relativity is absolutely necessary.
    [Show full text]
  • Gravitational Redshift, Inertia, and the Role of Charge
    Gravitational redshift, inertia, and the role of charge Johannes Fankhauser,∗ University of Oxford. (October 5, 2018) I argue that the gravitational redshift effect cannot be explained purely by way of uni- formly accelerated frames, as sometimes suggested in the literature. This is due to the fact that in terms of physical effects a uniformly accelerated frame is not exactly equivalent to a homogeneous gravitational field let alone to a gravitational field of a point mass. In other words, the equivalence principle only holds in the regime of certain approximations (even in the case of uniform acceleration). The concepts in need of clarification are spacetime curvature, inertia, and the weak equivalence principle with respect to our understanding of gravitational redshift. Furthermore, I briefly discuss gravitational redshift effects due to charge. Contents [Einstein, 1911]. His thought experiment ini- tiated the revolutionary idea that mass warps space and time. There exist some misconcep- 1 Introduction 1 tions, as evidenced in the literature, regarding the nature of the gravitational field in Einstein's 2 Gravitational redshift 2 General Theory of Relativity (GR) and how it 3 Uniformly accelerated frames and relates to redshift effects. My aim is to give the equivalence principle 3 a consistent analysis of the gravitational red- shift effect, in the hope of thereby advancing in 4 Equivalence and gravitational red- some small measure our understanding of GR. shift 6 Moreover, I will show that when charge is taken into account, gravitational redshift is subject to 5 Redshift due to charge 7 further corrections. 5.1 The weight of photons . .7 5.1.1 Einstein's thought exper- iment .
    [Show full text]
  • Radiation from an Inertial Mirror Horizon
    universe Communication Radiation from an Inertial Mirror Horizon Michael Good 1,2,* and Ernazar Abdikamalov 1,2 1 Department of Physics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan; [email protected] 2 Enegetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan 010000, Kazakhstan * Correspondence: [email protected] Received: 27 July 2020; Accepted: 19 August 2020; Published: 20 August 2020 Abstract: The purpose of this study is to investigate radiation from asymptotic zero acceleration motion where a horizon is formed and subsequently detected by an outside witness. A perfectly reflecting moving mirror is used to model such a system and compute the energy and spectrum. The trajectory is asymptotically inertial (zero proper acceleration)—ensuring negative energy flux (NEF), yet approaches light-speed with a null ray horizon at a finite advanced time. We compute the spectrum and energy analytically. Keywords: acceleration radiation; moving mirrors; QFT in curved spacetime; horizons; dynamical Casimir effect; black holes; Hawking radiation; negative energy flux 1. Introduction Recent studies have utilized the simplicity of the established moving mirror model [1–6] by applying accelerating boundary correspondences (ABC’s) to novel situations, including the Schwarzschild [7], Reissner-Nordström (RN) [8], Kerr [9], and de Sitter [10] geometries, whose mirrors have asymptotic infinite accelerations: lim a(v) = ¥, (1) v!vH where v = t + x is the advanced time light-cone coordinate, vH is the horizon and a is the proper acceleration of the moving mirror. These moving mirrors with horizons do not emit negative energy flux (NEF [11–16]). ABC’s also exist for extremal black holes, including extremal RN [17,18], extremal Kerr [9,19], and extremal Kerr–Newman [20] geometries, whose mirrors have asymptotic uniform accelerations: lim a(v) = constant.
    [Show full text]
  • Relativistic Mechanics
    Relativistic Mechanics Introduction We saw that the Galilean transformation in classical mechanics is incorrect in view of special relativity. The Lorentz transformation correctly describe how the space and time coordinates of an event transform for inertial frames of reference. We also saw that the simple velocity addition law that results from the Galilean transformation is incorrect, and velocities are seen to transform according to rules consistent with the Lorentz transformation. These results should make one think about the structure of classical mechanics itself, and whether all the laws of classical mechanics, including such seemingly immutable ones as conservation of momentum and conservation of energy, need to be reformulated. We now take a look at such quantities as mass, velocity, momentum, and energy in the context of special relativity, and build a new, relativistic mechanics based upon this framework. Relativistic Momentum and Energy First, we take a look at momentum and its conservation. Note that momentum itself is not of any special importance in classical mechanics. However, momentum conservation is a very important principle which one would certainly like to retain in special relativity. We now show that the law of conservation of momentum is inconsistent with special relativity if the classical definition of momentum is maintained.1 u1 u u3 u4 1 2 2 3 4 (S frame) m1 m2 m3 m4 Before After Classically, momentum is conserved in the collision process depicted above: m1u1 + m2u2 = m3u3 + m4u4. (3.1) We are assuming a one-dimensional collision to simplify the discussion. (Thus, we have dropped the subscript "x".) Also, we have assumed that the masses can change as a result of the collision (interaction).
    [Show full text]
  • Worldlines in the Einstein's Elevator
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 March 2021 doi:10.20944/preprints202103.0230.v1 Worldlines in the Einstein's Elevator Mathieu Rouaud Boudiguen 29310 Querrien, FRANCE, [email protected] (Preprint: March 1, 2021) Abstract: We all have in mind Einstein's famous thought experiment in the elevator where we observe the free fall of a body and then the trajectory of a light ray. Simply here, in addition to the qualitative aspect, we carry out the exact calculation. We consider a uniformly accelerated reference frame in rectilinear translation and we show that the trajectories of the particles are ellipses centered on the horizon of the events. The frame of reference is non-inertial, the space- time is flat, the metric is non-Minkowskian and the computations are performed within the framework of special relativity. The deviation, compared to the classical case, is important close to the horizon, but small in the box, and the interest is above all theoretical and pedagogical. The study helps the student to become familiar with the concepts of metric, coordinate velocity, horizon, and, to do the analogy with the black hole. Keywords: special relativity, Einstein, elevator, lift, horizon, accelerated, ellipse, circle. 1. INTRODUCTION We imagine a portion of empty space infinitely distant from all masses. We have a large box in which an observer floats in weightlessness. With the help of a hook and a rope, a constant force is exerted on the box thus animated by a rectilinear translation motion uniformly accelerated. The observer then experiences an artificial gravity. We will study in the elevator's frame the motion of light, then of a massive particle, and, finally, we will do a comparison with the black hole during a free fall from rest.
    [Show full text]