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Solutions Manual Special Relativity a Heuristic Approach

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Solutions Manual Special Relativity a Heuristic Approach Solutions Manual For Special Relativity A Heuristic Approach Sadri Hassani Contents List of Symbols 1 1 Qualitative Relativity 5 2 Relativity of Time and Space 9 3 Lorentz Transformation 19 4 Spacetime Geometry 49 5 Spacetime Momentum 75 6 Relativity in Four Dimensions 85 7 Relativistic Photography 99 8 Relativistic Interactions 127 9 Interstellar Travel 139 10 A Painless Introduction to Tensors 151 11 Relativistic Electrodynamics 157 12 Early Universe 167 A Maxwell's Equations 177 B Derivation of 4D Lorentz transformation 181 C Relativistic Photography Formulas 185 List of Symbols, Phrases, and Acronyms a^ unit vector in the direction of ~a anti-particle a particle whose mass and spin are exactly the same as its corresponding particle, but the sign of all its \charges" are opposite. If a particle is represented by the letter p, then it is customary to denote its anti-particle byp ¯. If a particle is represented by the letter q− (or q+), then it is customary to denote its anti-particle by q+ (or q−). as arcsecond; an arcsecond is an angle 1=3600 of a degree. baryon a hadron whose spin is an odd multiple of ~=2. Baryons are composed of three quarks. Examples of baryons are protons and neutrons. β~ fractional velocity of one observer relative to another, β~ = ~v=c boson a particle whose spin is an integer multiple of ~. All gauge particles are bosons as are all mesons, as well as the Higgs particle. causally connected referring to two two events. If an observer or a light signal can be present at two events, those events are said to be causally connected. causally disconnected referring to two two events. If an observer or a light signal cannot be present at two events, those events are said to be causally disconnected. CBR Cosmic Background Radiation CM center of mass CS coordinate system e^x; e^y; e^z unit vectors along the three Cartesian axes. EM electromagnetic or electromagnetism, one of the four fundamental forces of nature. equilibrium temperature temperature of the universe at which matter and radiation densities are equal eV electron volt, unit of energy equal to 1:6 × 10−19 J 2 List of Symbols fermion a particle whose spin is an odd multiple of ~=2. Fermions obey Pauli's exclusion principle: no two identical fermions can occupy a single quantum state. Electrons, protons, and neutrons are fermions, so are all leptons and quarks, as well as all baryons. γ the Lorentz factor, γ = 1=p1 − β2 = 1=p1 − (v=c)2 gauge bosons According to the modern theory of forces, fundamental particles interact via the exchange of gauge bosons. Excluding gravity, whose microscopic behavior is not well understood, there are 12 gauge bosons whose exchange explains all the interactions: Z0, W ± and γ (photon) are responsible for electroweak interaction, while 8 gluons are responsible for strong interaction. gluons the particles responsible for strong interactions: two or more quarks participate in strong interaction by exchanging gluons. There are four gluons, which with their antiparticles comprise the eight gluons whose exchange binds quarks together. GTR general theory of relativity; the relativistic theory of gravity. Gyr gigayear, equal to 109 years hadron a particle capable of participating in strong nuclear interactions. Examples of hadrons are protons, neutrons and pions. All hadrons are made up of quarks and/or anti-quarks. half life the time interval in which one half of the initial decaying particles survive. LAV Law of Addition of Velocities lepton a particle that participates only in electromagnetic and weak nuclear interactions, but not in strong nuclear interactions. Leptons are elementary particles in the sense that they are not made up of anything more elementary. There are three electri- cally charged leptons: electron, muon, and tauon. Each charged lepton has its own neutrino. So, altogether there are six leptons. LHC Large Hadron Collider light cone (at an event E) The set of all events that are causally connected to E. light hour the distance that light travels in one hour, ≈ 1:08 × 1012 m light minute the distance that light travels in one minute, ≈ 1:8 × 1010 m light second the distance that light travels in one second, ≈ 3 × 108 m lightlike referring to two events, when c∆t = ∆x or (∆s)2 = 0. luminally connected referring to two two events. If a light signal can be present at two events, those events are said to be luminally connected. ly light year; one light year is 9:467 × 1015 m. mean time the time interval in which 1=e of the initial decaying particles survive. List of Symbols 3 meson a hadron whose spin is an integer multiple of ~. Mesons are composed of one quark and one anti-quark. Examples of mesons are pions. MeV million electron volt, unit of energy equal to 1:6 × 10−13 J µm micrometer = 10−6 m Minkowskian distance also called \spacetime distance," (∆s)2 = (c∆t)2 − (∆x)2 is an expression involving the coordinates of two events which is independent of the coordinates used to describe those events. MM clock sometimes called \light clock" is described on page 13. Mpc Megaparsec muon an elementary particle belonging to the group of particles named \leptons," to which electron belongs as well. Muon is called a \fat electron" because it behaves very much like an electron except that it is heavier. neutrino a neutral lepton with very small mass. Neutrinos participate only in weak nuclear force. That's why they are very weakly interacting. ns nanosecond or 10−9 s Parsec A distance of about 3.26 light years. One parsec corresponds to the distance at which the mean radius of the Earth's orbit subtends an angle of one second of arc. positron the anti-particle of the electron quarks elementary particles which make up all hadrons. There are six quarks: up, down, strange, charm, bottom, top. Quarks participate in all interactions, in particular, the strong interaction. RF reference frame spacelike referring to two events, when c∆t < ∆x or (∆s)2 < 0. spacetime distance see Minkowskian distance STR special theory of relativity tauon an elementary particle belonging to the group of particles named \leptons," to which electron belongs as well. It is the heaviest lepton discovered so far. timelike referring to two events, when c∆t > ∆x or (∆s)2 > 0. CHAPTER 1 Qualitative Relativity Problems With Solutions 1.1. A rod of length L emits light from all of its points simultaneously (in its rest frame) when a remote switch is turned on. Its center is on the x-axis and is moving on the axis in a plane parallel to a very large photographic plate and infinitesimally close to it. When it reaches the middle of the plate, the switch is turned on. (a) Compare the length Ljj of the image on the photographic plate with L when the rod is along the x-axis: Ljj > L, Ljj = L, or Ljj < L? Give a reason for your answer. (b) Compare the length L? of the image on the photographic plate with L when the rod is perpendicular to the x-axis: L? > L, L? = L, or L? < L? Give a reason for your answer. Solution: (a) Length parallel to the direction of motion shrinks regardless of who sees the events of light emission simultaneously. See the discussion in Section 1.3 for the reason (as well as how to capture the length of a moving object). (b) Length perpendicular to the direction of motion does not change. Note the importance of the fact that the distance between the rod and the photographic plate is zero. 1.2. A rod is placed along the x-axis with its center at the origin. A pinhole camera C1 is located on the z-axis and takes a picture of the stationary rod. Now the rod starts moving along the x-axis parallel to itself from −∞. Camera C1 is removed and another pinhole camera C2 replaces it on the z-axis. As soon as the center of the rod reaches the origin (call it t = 0), C2 takes a picture. (a) Is the pinhole of C2 collecting the light rays from the two ends of the rod that were emitted at t = 0? 6 Qualitative Relativity (b) Is the pinhole collecting the light rays from the two ends of the rod that were emitted simultaneously, but not at t = 0? (c) If the answer to (b) is no, which end emitted its light first, the trailing end or the leading end? (d) Is it possible for the image of the rod in C2 to be longer than its image in C1? Hint: Consider the location of each end as it emits the light ray captured by C2. Solution: (a) No. It takes time for the light to reach the camera once it leaves its source. (b) No. (c) The trailing end is farther away from the camera, so it must emit the light sooner than the leading end. (d) The trailing end emits its light, the rod moves a little, then the leading end emits its light. So, the distance between the source of the light from the trailing end and that of the leading end is indeed larger than the length of the rod. Note that the image in camera C2, which is longer than the image in C1, has nothing to do with the actual length of the rod! 1.3. A rod is placed along the y-axis with its center at the origin. A pinhole camera C1 is located on the z-axis and takes a picture of the stationary rod.
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