PHYS 205 Chapter 28, Fall 2017

Chapter 28 Special Relativity 1. Events and Inertial Reference Frames 2. The Postulates of Special Relativities 3. The Relativity of Time: Time Dilation 4. The Relativity of Length: Length Contraction 5. Relativistic Momentum 6. The Equivalence of Mass Energy 7. The Relativistic Addition of Velocities Outline Physics at the turn of the century Mechanics Newton Statistical and thermal physics  laws of thermodynamics and kinetic theory of gases Electromagnetism  Maxwell Atomic Structure ? Propagation of light in space ? Radioactivity ? Photoelectric effect? Light emission from gases? Modern physics 1900 … present An event is a physical “happening” that occurs at a certain place and time. To record the event, each observer uses a reference frame that consists of a coordinate system and a clock. Each observer is at rest relative to her own reference frame. An inertial reference frame is one in which Newton’s law of inertia is valid. 1- Events and inertial reference frames In this example, the event is the space shuttle lift off. ˆ velocity of system S with respect to S' x v v i   Relation between the coordinates of the two systems ' ' ' ' p p p p p p x x vt y y z z t t             ' ' ' px px py py pz pz a a a a a a         ' ' ' px px py py pz pz v v v v v v v          Inertial frames are reference frames in which Newton’s Laws of Mechanics are valid (and vice versa). These are Galilean transformations of coordinates velocity and acceleration. All Newtonian physics applies in such frames. 2- The postulates of special relativity 1.The Relativity Postulate. The laws of physics are the same in every inertial reference frame. 2.The Speed of Light Postulate. The speed of light in a vacuum ,measured in any inertial reference frame, always has the same value of c, no matter how fast the source of light and the observer are moving relative to one another. 3- The relativity of time: time dilation Time measured by O’ (observer at rest in the vehicle) 2 distance Proper time speed p d t c     For the observer O at rest in the the stationary frame, Speed remains constant, but distance has increased distance speed t   p t    t A clock in motion runs more slowly than an identical stationary clock (muons, twins paradoxes) Indeed,     2 2 2 2 1 1 with 1 1 p p t t t v c v c               2 2 2 2 2 2 1 d d c t c v v c         2 2 2 2 4 4 c v t d           2 2 2 2 2 c t v t d                 Example 1 Time Dilation The spacecraft is moving past the earth at a constant speed of 0.92 times the speed of light. the astronaut measures the time interval between ticks of the spacecraft clock to be 1.0 s. What is the time interval that an earth observer measures?   s 6.2 .0 92 1 s 0.1 1 2 2 2        c c c v t t o The time interval measured at rest with respect to the clock is called the proper time interval. In general, the proper time interval between events is the time interval measured by an observer who is at rest relative to the events. Proper time interval ot  Proper time interval 4- The relativity of length: length contraction Lp is the proper length of an object as measured by someone at rest relative to the object L is the length measured by the observer in the spaceship Relative to Earth, t = LP / v to go from star 1 to star 2 Relative to spaceship, (due to time dilation ) p t t     2 2 P 1 v L L c   LP L    p t L v t v       Length contraction takes place only along the direction of motion No contraction Example 4 The Contraction of a Spacecraft An astronaut, using a meter stick that is at rest relative to a cylindrical spacecraft, measures the length and diameter to be 82 m and 21 m respectively. The spacecraft moves with a constant speed of 0.95c relative to the earth. What are the dimensions of the spacecraft, as measured by an observer on earth.     26 m .0 95 1 82 m 1 2 2 2      c c c v L L o Diameter stays the same. 5- Relativistic Momentum v is the speed of the particle, m is its mass as measured by an observer at rest with respect to the mass When v << c, the denominator approaches 1 and so p approaches mv (classical). 2 2 1 mv p mv v c     6- The equivalence of mass and energy The definition of kinetic energy requires modification in relativistic mechanics The kinetic energy is written: KE = mc2 – mc2 The rest energy is written: ER = mc2 The total energy 2 2 TOT 2 2 mc E = mc = 1 v c   Energy and Relativistic Momentum E2 = p2c2 + (mc2)2 When the particle is at rest, p = 0 and E = mc2 Massless particles (m = 0) have E = pc This is also used to express masses in energy units mass of an electron = 9.11 x 10-31 kg = 0.511 MeV/c2 Conversion: 1 u = 929.494 MeV/c2 Conceptual Example 9 When is a Massless Spring Not Massless? The spring is initially unstrained and assumed to be massless. Suppose that the spring is either stretched or compressed. Is the mass of the spring still zero, or has it changed? If the mass has changed, is the mass change greater for stretching or compressing? 6- The relativistic addition of velocities vmo = velocity of the motorcycle with respect stationary observer. Vbm = velocity of the ball with respect to motorcycle vbo = velocity of ball with respect to the stationary observer 2 1 bm mo bo bm mo v v v v v c   