DOCSLIB.ORG

5 Four Vectors

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
5 Four Vectors Physics 139 Relativity Relativity Notes 2002 G. F. SMOOT Oce 398 Le Conte DepartmentofPhysics, University of California, Berkeley, USA 94720 Notes to b e found at http://aether.lbl.gov/www/classes/p139/homework/homework.html 5 Four Vectors A natural extension of the Minkowski geometrical interpretation of Sp ecial Relativity is the concept of four dimensional vectors. One could also arrive at the concept by lo oking at the transformation prop erties of vectors and noticing they do not transform as vectors unless another comp onent is added. We de ne a four-dimensional vector or four-vector for short as a collection of four comp onents that transforms according to the Lorentz transformation. The vector magnitude is invariant under the Lorentz transform. 5.1 Co ordinate Transformations in 3+1-D Space One can consider co ordinate transformations manyways: If x ;x ;x ;x = x; y ; z ; ict, 1 2 3 4 then ordinary rotations in x x plane around x 1 2 3 0 x = x cos + x sin cos sin 1 2 1 0 x = x sin + x cos sin cos 1 2 2 But in x x plane: 1 4 0 x = x cos + x sin cos sin 1 4 1 0 x = x sin + x cos sin cos 1 4 4 where the angle is de ned by q p 2 2 2 cos =1= 1v =c =1= 1+tan = tan v=c q p = sin = i 2 2 2 1+tan 1 v =c tan = iv =c = i : And thus one has the trignometic identity: 2 2 2 2 cos + sin = 1 =1 1 0 x = [x +icti ] = [x ct] 1 1 1 0 x = [x i x ] 4 1 4 0 ict = [ict i x ] 1 0 ct = [ct x ] 1 0 t = [t x =c] 1 So the extension to 3+1-D includes Lorentz transformations, if angles are imaginary. Really,we are considering the set of all 4 4 orthogonal transformations matrices in which one angle may b e pure imaginary. In general all angles may b e complex, combining real rotations in 2-space with imaginary rotations relativeto t. An alternate way of writing this is 0 x = xcosh ctsinh 0 ct = xsinh + ctcosh 1 where = cosh . 0 x = xcosi+ictsini 0 ict = xsini+ ictcosi and 1 = i = icosh ; tan = i = iv =c Still another notation is with x = ict 4 0 x = x + i x 1 4 1 0 x = x i x 4 1 4 The transformation matrix is then 0 1 0 0 i B C 0 1 0 0 B C B C @ A 0 0 1 0 i 0 0 Still yet another notation is with x = ct 0 0 x = x i x 0 1 0 0 x = x + i x 1 0 1 The transformation matrix is then 0 1 2 3 0 1 0 0 0 B C 1 0 0 B C B C @ A 2 0 0 1 0 3 0 0 0 1 2 5.1.1 Generalized Lorentz Transformation For spatial co ordinates the Lorentz transform ts the linear form 4 X 0 x = x 1 =1 sub ject to the condition that the prop er length X X 2 2 0 2 2 cd = ds = x = x =ct j~xj 2 is an invariant. This condition requires that the co ecients form an orthogonal matrix: X = X = X = 3 where the Kronecker delta is de ned by = = = 1 when = and 0 otherwise. The invariance group can b e enlarged to b e the Poincare0 group by the addition of translations: 4 X 0 x = x + a 4 =1 The full group includes: translations, 3-D space rotations, and the Lorentz b o osts. 5.2 The Inner Pro duct of 3+1-D Vectors The de nition of the inner pro duct dot pro duct must b e mo di ed in 3+1 dimensions. ~ ~ A B = A B + A B + A B + A B 1 1 2 2 3 3 4 4 if x = ict. But with our usual convention 4 ~ ~ A B = A B A B A B A B 0 0 1 1 2 2 3 3 or with the opp osite signature metric one has ~ ~ A B = A B + A B + A B + A B 0 0 1 1 2 2 3 3 ~ ~ A B = A B + A B + A B A B 1 1 2 2 3 3 4 4 if x = ct which is often the convention for the opp osite sign convention. It is an 4 exercise to show that the inner pro duct is unchanged under a Lorentz transformation. Can b e done simply by substitution. This can b e extended to the general class of Lorentz transformations. 3 5.3 Four Velo city So wehave the p osition 4-vectorx ~ =x ;x ;x ;x and the displacement 4-vector 0 1 2 3 ~ dx =dx ;dx ;dx ;dx . What other 4-vectors are there? That is what other 4- 0 1 2 3 vectors are natural to construct? What we mean by a four-vector is a four-dimensional quantity that transforms from one inertial frame to another by the Lorentz transform which will then leave its length norm invariant. Consider generalizing the 3-vector velo cityv ;v ;v =dx=dt; dy =dt; dz =dt x y z what can wedotomake this into a 4-vector naturally? One clear problem is that we are dividing by a comp onent dt of a vector so that the ratio is clearly going to Lorentz transform in a complicated way.We need to take the derivative with resp ect to a quantity that will b e the same in all reference frames, e.g. d the di erential of the prop er time, and add a fourth comp onent to make the 4-vector. It is clear that the derivative of the 4-vector p osition ct; x; y ; z with resp ect to the prop er time will b e a 4-vector for Lorentz transformations since ct; x; y ; z transform prop erly and d is an invariant. So we can de ne the 4-velo cityas ! dx dx dy dz dct u = ; u~ = ; ; ; 5 d d d d d Note that ! 2 2 2 dx dy dy 2 2 2 2 2 2 2 2 2 c c d = c dt dx dy dz = dt dt dt dt 2 2 2 2 2 2 2 2 c v c v v v = dt = dt x y z or the time dilation formula we got b efore q d 1 dt 2 2 q = = = 1 v =c ; and 2 2 dt d 1 v =c So we can now explicitly write out the 4-velo city using the chain derivative rule: dx dx dt u = = d dt d u~ =u ;u ;u ;u = c; v ; v ; v = c; v ;v ;v 0 1 2 3 x y z x y z Thus three comp onents of the 4-velo city are the three comp onents of the 3-vector velo city times . Note also that the norm - the magnitude or vector invariant length - of the four-velo city is not only unchanged but it is the same for all physical ob jects matter plus energy. For 3+1 dimensions the norm or magnitude is found from the inner pro duct or dot pro duct which has the same signature as the metric see just ab ove so that 2 2 1 v =c 2 2 2 2 2 2 2 2 2 2 2 u~ u~ = u u u u = c v v v = c = c 0 1 2 3 x z z 2 2 1 v =c 4 Thus every physical thing, including light, moves with a 4-velo city magnitude of c and the only thing that Lorentz transformations do is change the direction of motion. A particle at rest is moving down its time axis at sp eed c. When it is b o osted to a xed velo city, it still travels through space-time at sp eed c but more slowly down the time axis as it is also moving in the spatial directions. One should also note that as the spatial sp eed three-velo city approaches c, all comp onents of the 4-velo city u are unb ounded as !1. One cannot then de ne a Lorentz transformation that moves to the rest frame. Thus all massless particles will have no rest frame. 5.3.1 LawofTransformation of a 4-Vector We can write the transformation in our standard algebraic Lorentz notation q 0 2 A = A A =1= 1 0 1 0 V 0 A = A A 1 0 1 c 0 0 A = A ; A = A 2 3 2 3 where and refer to the relativevelo city V of the frames. 5.3.2 LawofTransformation of a 4-Velo city 0 u = u u 1 0 1 where and are for the relativevelo city of the frames and not of the particle. But in the formula for the 4-velo city u~ =u ;u ;u ;u = c; v ; v ; v = c; v ;v ;v 0 1 2 3 x y z x y z The is for the particle! So we should have lab eled it and the and for the p frame transform and . Then wehave f f 0 0 v = v c f p x f p p x 0 So we can get out a formula for v x q 0 1 p f p 0 q q v V = v V v = x x x 0 1 1 p p f q 2 2 This is our old friend on the law of transformation of 1 u =c q q 0 2 2 2 2 q 1 u =c 1 V =c 2 2 1 u =c = 0 2 1+u V=c x 5 and q q 2 2 2 2 q 1 u =c 1 V =c 0 2 2 1 u =c = 2 1+u V=c x which is simply 1 1 = 0 2 1 u V=c p f x p So v V x 0 v = x 2 1 u V=c x as derived earlier by the di erential route.
Load more

Recommended publications
  • On the Status of the Geodesic Principle in Newtonian and Relativistic Physics1
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilSci Archive On the Status of the Geodesic Principle in Newtonian and Relativistic Physics1 James Owen Weatherall2 Logic and Philosophy of Science University of California, Irvine Abstract A theorem due to Bob Geroch and Pong Soo Jang [\Motion of a Body in General Relativ- ity." Journal of Mathematical Physics 16(1), (1975)] provides a sense in which the geodesic principle has the status of a theorem in General Relativity (GR). I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton- Cartan theory) [Weatherall, J. O. \The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52(3), (2011)]. Here I compare the interpretations of these two the- orems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case. 1 Introduction The geodesic principle is the central principle of General Relativity (GR) that describes the inertial motion of test particles. It states that free massive test point particles traverse timelike geodesics. There is a long-standing view, originally due to Einstein, that the geodesic principle has a special status in GR that arises because it can be understood as a theorem, rather than a postulate, of the theory. (It turns out that capturing the geodesic principle as a theorem in GR is non-trivial, but a result due to Bob Geroch and Pong Soo Jang (1975) 1Thank you to David Malament and Jeff Barrett for helpful comments on a previous version of this paper and for many stimulating conversations on this topic.
    [Show full text]
  • Refraction and Reflexion According to the Principle of General Covariance
    REFRACTION AND REFLEXION ACCORDING TO THE PRINCIPLE OF GENERAL COVARIANCE PATRICK IGLESIAS-ZEMMOUR Abstract. We show how the principle of general covariance introduced by Souriau in smoothly uniform contexts, can be extended to singular situ- ations, considering the group of diffeomorphisms preserving the singular locus. As a proof of concept, we shall see how we get again this way, the laws of reflection and refraction in geometric optics, applying an extended general covariance principle to Riemannian metrics, discontinuous along a hypersurface. Introduction In his paper “Modèle de particule à spin dans le champ électromagnétique et gravitationnel” published in 1974 [Sou74], Jean-Marie Souriau suggests a pre- cise mathematical interpretation of the principle of General Relativity. He names it the Principle of General Covariance. Considering only gravitation field1, he claimed that any material presence in the universe is characterized by a covector defined on the quotient of the set of the Pseudo-Riemannian metrics on space-time, by the group of diffeomorphisms. This principle be- ing, according to Souriau, the correct statement of the Einsteins’s principle of invariance with respect to any change of coordinates. Actually, Souriau’s general covariance principle can be regarded as the active version of Einstein invariance statement, where change of coordinates are interpreted from the active point of view as the action of the group of diffeomorphisms. Now, for reasons relative to the behavior at infinity and results requirement, the group of diffeomorphisms of space-time is reduced to the subgroup of compact supported diffeomorphisms. Date: April 6, 2019. 1991 Mathematics Subject Classification. 83C10, 78A05, 37J10.
    [Show full text]
  • Quantum Phase Space in Relativistic Theory: the Case of Charge-Invariant Observables
    Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 3, 1448–1453 Quantum Phase Space in Relativistic Theory: the Case of Charge-Invariant Observables A.A. SEMENOV †, B.I. LEV † and C.V. USENKO ‡ † Institute of Physics of NAS of Ukraine, 46 Nauky Ave., 03028 Kyiv, Ukraine E-mail: [email protected], [email protected] ‡ Physics Department, Taras Shevchenko Kyiv University, 6 Academician Glushkov Ave., 03127 Kyiv, Ukraine E-mail: [email protected] Mathematical method of quantum phase space is very useful in physical applications like quantum optics and non-relativistic quantum mechanics. However, attempts to generalize it for the relativistic case lead to some difficulties. One of problems is band structure of energy spectrum for a relativistic particle. This corresponds to an internal degree of freedom, so- called charge variable. In physical problems we often deal with such dynamical variables that do not depend on this degree of freedom. These are position, momentum, and any combination of them. Restricting our consideration to this kind of observables we propose the relativistic Weyl–Wigner–Moyal formalism that contains some surprising differences from its non-relativistic counterpart. This paper is devoted to the phase space formalism that is specific representation of quan- tum mechanics. This representation is very close to classical mechanics and its basic idea is a description of quantum observables by means of functions in phase space (symbols) instead of operators in the Hilbert space of states. The first idea about this representation has been proposed in the early days of quantum mechanics in the well-known Weyl work [1].
    [Show full text]
  • Spacetime Diagrams(1D in Space)
    PH300 Modern Physics SP11 Last time: • Time dilation and length contraction Today: • Spacetime • Addition of velocities • Lorentz transformations Thursday: • Relativistic momentum and energy “The only reason for time is so that HW03 due, beginning of class; HW04 assigned everything doesn’t happen at once.” 2/1 Day 6: Next week: - Albert Einstein Questions? Intro to quantum Spacetime Thursday: Exam I (in class) Addition of Velocities Relativistic Momentum & Energy Lorentz Transformations 1 2 Spacetime Diagrams (1D in space) Spacetime Diagrams (1D in space) c · t In PHYS I: v In PH300: x x x x Δx Δx v = /Δt Δt t t Recall: Lucy plays with a fire cracker in the train. (1D in space) Spacetime Diagrams Ricky watches the scene from the track. c· t In PH300: object moving with 0<v<c. ‘Worldline’ of the object L R -2 -1 0 1 2 x object moving with 0>v>-c v c·t c·t Lucy object at rest object moving with v = -c. at x=1 x=0 at time t=0 -2 -1 0 1 2 x -2 -1 0 1 2 x Ricky 1 Example: Ricky on the tracks Example: Lucy in the train ct ct Light reaches both walls at the same time. Light travels to both walls Ricky concludes: Light reaches left side first. x x L R L R Lucy concludes: Light reaches both sides at the same time In Ricky’s frame: Walls are in motion In Lucy’s frame: Walls are at rest S Frame S’ as viewed from S ... -3 -2 -1 0 1 2 3 ..
    [Show full text]
  • Relativistic Dynamics
    Chapter 4 Relativistic dynamics We have seen in the previous lectures that our relativity postulates suggest that the most efficient (lazy but smart) approach to relativistic physics is in terms of 4-vectors, and that velocities never exceed c in magnitude. In this chapter we will see how this 4-vector approach works for dynamics, i.e., for the interplay between motion and forces. A particle subject to forces will undergo non-inertial motion. According to Newton, there is a simple (3-vector) relation between force and acceleration, f~ = m~a; (4.0.1) where acceleration is the second time derivative of position, d~v d2~x ~a = = : (4.0.2) dt dt2 There is just one problem with these relations | they are wrong! Newtonian dynamics is a good approximation when velocities are very small compared to c, but outside of this regime the relation (4.0.1) is simply incorrect. In particular, these relations are inconsistent with our relativity postu- lates. To see this, it is sufficient to note that Newton's equations (4.0.1) and (4.0.2) predict that a particle subject to a constant force (and initially at rest) will acquire a velocity which can become arbitrarily large, Z t ~ d~v 0 f ~v(t) = 0 dt = t ! 1 as t ! 1 . (4.0.3) 0 dt m This flatly contradicts the prediction of special relativity (and causality) that no signal can propagate faster than c. Our task is to understand how to formulate the dynamics of non-inertial particles in a manner which is consistent with our relativity postulates (and then verify that it matches observation, including in the non-relativistic regime).
    [Show full text]
  • Reflection Invariant and Symmetry Detection
    1 Reflection Invariant and Symmetry Detection Erbo Li and Hua Li Abstract—Symmetry detection and discrimination are of fundamental meaning in science, technology, and engineering. This paper introduces reflection invariants and defines the directional moments(DMs) to detect symmetry for shape analysis and object recognition. And it demonstrates that detection of reflection symmetry can be done in a simple way by solving a trigonometric system derived from the DMs, and discrimination of reflection symmetry can be achieved by application of the reflection invariants in 2D and 3D. Rotation symmetry can also be determined based on that. Also, if none of reflection invariants is equal to zero, then there is no symmetry. And the experiments in 2D and 3D show that all the reflection lines or planes can be deterministically found using DMs up to order six. This result can be used to simplify the efforts of symmetry detection in research areas,such as protein structure, model retrieval, reverse engineering, and machine vision etc. Index Terms—symmetry detection, shape analysis, object recognition, directional moment, moment invariant, isometry, congruent, reflection, chirality, rotation F 1 INTRODUCTION Kazhdan et al. [1] developed a continuous measure and dis- The essence of geometric symmetry is self-evident, which cussed the properties of the reflective symmetry descriptor, can be found everywhere in nature and social lives, as which was expanded to 3D by [2] and was augmented in shown in Figure 1. It is true that we are living in a spatial distribution of the objects asymmetry by [3] . For symmetric world. Pursuing the explanation of symmetry symmetry discrimination [4] defined a symmetry distance will provide better understanding to the surrounding world of shapes.
    [Show full text]
  • “Geodesic Principle” in General Relativity∗
    A Remark About the “Geodesic Principle” in General Relativity∗ Version 3.0 David B. Malament Department of Logic and Philosophy of Science 3151 Social Science Plaza University of California, Irvine Irvine, CA 92697-5100 [email protected] 1 Introduction General relativity incorporates a number of basic principles that correlate space- time structure with physical objects and processes. Among them is the Geodesic Principle: Free massive point particles traverse timelike geodesics. One can think of it as a relativistic version of Newton’s first law of motion. It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein’s ∗I am grateful to Robert Geroch for giving me the basic idea for the counterexample (proposition 3.2) that is the principal point of interest in this note. Thanks also to Harvey Brown, Erik Curiel, John Earman, David Garfinkle, John Manchak, Wayne Myrvold, John Norton, and Jim Weatherall for comments on an earlier draft. 1 ab equation (or of the conservation principle ∇aT = 0 that is, itself, a conse- quence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic prin- ciple can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed).
    [Show full text]
  • Physics 9Hb: Special Relativity & Thermal/Statistical Physics
    PHYSICS 9HB: SPECIAL RELATIVITY & THERMAL/STATISTICAL PHYSICS Tom Weideman University of California, Davis UCD: Physics 9HB – Special Relativity and Thermal/Statistical Physics This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. The LibreTexts mission is to unite students, faculty and scholars in a cooperative effort to develop an easy-to-use online platform for the construction, customization, and dissemination of OER content to reduce the burdens of unreasonable textbook costs to our students and society. The LibreTexts project is a multi-institutional collaborative venture to develop the next generation of open-access texts to improve postsecondary education at all levels of higher learning by developing an Open Access Resource environment. The project currently consists of 13 independently operating and interconnected libraries that are constantly being optimized by students, faculty, and outside experts to supplant conventional paper-based books. These free textbook alternatives are organized within a central environment that is both vertically (from advance to basic level) and horizontally (across different fields) integrated.
    [Show full text]
  • Line Element in Noncommutative Geometry
    Line element in noncommutative geometry P. Martinetti G¨ottingenUniversit¨at Wroclaw, July 2009 . ? ? - & ? !? The line element p µ ν ds = gµν dx dx is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - . ? !? The line element p µ ν ds = gµν dx dx & ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - !? The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance ? -Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry).
    [Show full text]
  • More on Vectors Math 122 Calculus III D Joyce, Fall 2012
    More on Vectors Math 122 Calculus III D Joyce, Fall 2012 Unit vectors. A unit vector is a vector whose length is 1. If a unit vector u in the plane R2 is placed in standard position with its tail at the origin, then it's head will land on the unit circle x2 + y2 = 1. Every point on the unit circle (x; y) is of the form (cos θ; sin θ) where θ is the angle measured from the positive x-axis in the counterclockwise direction. u=(x;y)=(cos θ; sin θ) 7 '$θ q &% Thus, every unit vector in the plane is of the form u = (cos θ; sin θ). We can interpret unit vectors as being directions, and we can use them in place of angles since they carry the same information as an angle. In three dimensions, we also use unit vectors and they will still signify directions. Unit 3 vectors in R correspond to points on the sphere because if u = (u1; u2; u3) is a unit vector, 2 2 2 3 then u1 + u2 + u3 = 1. Each unit vector in R carries more information than just one angle since, if you want to name a point on a sphere, you need to give two angles, longitude and latitude. Now that we have unit vectors, we can treat every vector v as a length and a direction. The length of v is kvk, of course. And its direction is the unit vector u in the same direction which can be found by v u = : kvk The vector v can be reconstituted from its length and direction by multiplying v = kvk u.
    [Show full text]
  • Physics 200 Problem Set 7 Solution Quick Overview: Although Relativity Can Be a Little Bewildering, This Problem Set Uses Just A
    Physics 200 Problem Set 7 Solution Quick overview: Although relativity can be a little bewildering, this problem set uses just a few ideas over and over again, namely 1. Coordinates (x; t) in one frame are related to coordinates (x0; t0) in another frame by the Lorentz transformation formulas. 2. Similarly, space and time intervals (¢x; ¢t) in one frame are related to inter- vals (¢x0; ¢t0) in another frame by the same Lorentz transformation formu- las. Note that time dilation and length contraction are just special cases: it is time-dilation if ¢x = 0 and length contraction if ¢t = 0. 3. The spacetime interval (¢s)2 = (c¢t)2 ¡ (¢x)2 between two events is the same in every frame. 4. Energy and momentum are always conserved, and we can make e±cient use of this fact by writing them together in an energy-momentum vector P = (E=c; p) with the property P 2 = m2c2. In particular, if the mass is zero then P 2 = 0. 1. The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time di®erence between the events according to an observer moving at u = 0:8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0:8c. Answer: According to the formula for a Lorentz transformation, ³ u ´ 1 ¢tobserver = γ ¢tEarth-Sun ¡ ¢xEarth-Sun ; γ = p : c2 1 ¡ (u=c)2 Plugging in the numbers gives (notice that the c implicit in \light-minute" cancels the extra factor of c, which is why it's nice to measure distances in terms of the speed of light) 2 min ¡ 0:8(8:3 min) ¢tobserver = p = ¡7:7 min; 1 ¡ 0:82 which means that according to the observer, event B happened before event A! If we reverse the sign of u then 2 min + 0:8(8:3 min) ¢tobserver 2 = p = 14 min: 1 ¡ 0:82 2.
    [Show full text]
  • 8. Special Relativity
    8. Special Relativity Although Newtonian mechanics gives an excellent description of Nature, it is not uni- versally valid. When we reach extreme conditions — the very small, the very heavy or the very fast — the Newtonian Universe that we’re used to needs replacing. You could say that Newtonian mechanics encapsulates our common sense view of the world. One of the major themes of twentieth century physics is that when you look away from our everyday world, common sense is not much use. One such extreme is when particles travel very fast. The theory that replaces New- tonian mechanics is due to Einstein. It is called special relativity. The effects of special relativity become apparent only when the speeds of particles become comparable to the speed of light in the vacuum. Universally denoted as c, the speed of light is c = 299792458 ms−1 This value of c is exact. In fact, it would be more precise to say that this is the definition of what we mean by a meter: it is the distance travelled by light in 1/299792458 seconds. For the purposes of this course, we’ll be quite happy with the approximation c 3 108 ms−1. ≈ × The first thing to say is that the speed of light is fast. Really fast. The speed of sound is around 300 ms−1; escape velocity from the Earth is around 104 ms−1; the orbital speed of our solar system in the Milky Way galaxy is around 105 ms−1. As we shall soon see, nothing travels faster than c.
    [Show full text]