JF Theoretical Physics PY1T10 Special Relativity

Transcription

1 JF Theoretical Physics PY1T10 Special Relativity 12 Lectures (plus problem classes) Prof. James Lunney Room: SMIAM 1.23, Books Special Relativity French University Physics Young and Freedman - Chapter 37 Berkeley Physics Course, Vol 1, Mechanics A biography of Einstein provided by the American Institute of Physics 3-Dec-13 1

2 Introduction In 1905 Albert Einstein published 3 remarkable papers: Special theory of relativity (STR) Photoelectric effect Brownian motion STR implies that measurements in different inertial frames of reference having relative velocities give different values of length and time. Very strange, non-intuitive, contrary to normal experience where v<<c Expect to relativistic effects when v~c In the limit of low v (v<<c) STR Newtonian mechanics 3-Dec-13 Albert Einstein, Zurich ( )

3 Course outline Inertial frames and Galilean transformations Michelson-Morley experiment Einstein s postulates, relativity of simultaneity Lorentz transformation Length contraction and time dilation Relativistic Doppler effect, moving clocks Transformation of velocities Relativistic dynamics: energy, mass, momentum Pair creation, fission, fusion Collisions, Compton effect Energy-momentum invariant 3-Dec-13 3

4 Classical Mechanics Newton I: a = 0 (v =const) when F = 0 Newton II: F = ma valid for non-accelerated (inertial) reference frame Inertial ref. frame? - frame defined by Newton I Frame fixed to surface of earth not inertial gravity, acceleration due to rotation of earth (small) Frame fixed to star is inertial frame to good very good approx. Are the laws of physics the same in different inertial frames (moving with relative v)? Hypothesis of Galilean invariance: Basic laws of physics are the same in all inertial frames (central idea in STR) Formal way of saying: If I am in a train with no windows I cannot say if I am moving or not. No trouble keeping my balance, coin falls straight down. Galilean invariance accepted by 19 th century physicists, but with light we should be able to detect uniform motion. 3-Dec-13 4

5 Galilean Transformation - 1 Galilean Transformation: Transforms coordinates of an event between inertial frames moving at a speed relative to each other. eg. A boat firing a canon from its deck. 2 inertial frames: Boat, and Water. Boat moves at 15 m/s, canon fired at 100 m/s relative to the boat. How fast relative to the water? Galilean transformation: Speed = = 115 m/s relative to the water. 15 m/s 115 m/s

6 Galilean Transformation - 2 y S vt y S v event x, y, z, t (x, y, z, t ) z o z o x, x Inertial frame S moves with velocity v relative to inertial frame S, along x-axis. Origins coincide at t = 0 and t = 0. x = x + vt y = y t = t z = z dx = dx + vdt dy = dy dt = dt dx dt = dx dt + v dt dt u x = u x + v, u y = u y, u z = u z u = u + v

7 Galilean Transformation - 3 u = u + v a = du dt = du dt = a F = m a, F = m a But m = m, a = a F = F Observers in both frames agree on magnitude and direction of Force.

8 Theories of Light Early theories (Newton): stream of particles 1667 Robert Hooke: vibration wave theory By beginning of 19 cent. there was strong evidence for wave theory Polarisation Interference Diffraction Maxwell (1861): light is an electromagnetic wave Sound can be transmitted in solid, liquid or gas, but not vacuum Expect to need a medium to carry light wave ether (aether) 3-Dec-13

9 The ether Expect light to propagate at constant velocity wrt ether The velocity of light measured by an observer will depend on his motion wrt ether If we measure velocity of light in different directions we should be able to detect our motion through the ether: Same in both directions stationary Not same in both direction moving Michelson and Morley tried to measure this effect Earth moves around Sun at m s -1 Velocity of light, c = m s -1 - need to measure very small effect 3-Dec-13 9

10 Michelson Morley experiment - 1 Albert Michelson, Ohio ( ) Edward Morley, Ohio ( ) 3-Dec-13 10

11 Michelson Morley experiment - 2 Aim: to measure the velocity of our motion through some absolute space as defined by ether (or check to see if c is the same in all directions) Mirror M 2 Compensating plate C Used multiple reflections to extend optical path length Source P Half-silvered plate P For rotation the whole interferometer floated in bath of mercury Ether wind Mirror M 1 Telescope 3-Dec-13 11

12 Michelson Morley experiment Dec-13 12

13 Michelson Morley experiment - 4 See interference fringes in telescope Fringes are parallel lines if M 1 not exactly perpendicular to M 2 Suppose equipment moves at speed v in direction PM 1 wrt frame defined by ether, ie. ether wind is blowing in direction M 1 P Rotate interferometer by Then PM 2 points into ether wind Expect to see fringe shift : Why? What value δ? v m s -1, c = m s -1, therefore v/c = 10-4, λ = m 1881 Michelson used l =1.2 m giving δ = 0.04, observed δ = 0.02 (max) 1887 Michelson used l =11 m giving δ = 0.4, observed δ = 0.01 (max) Conclusion No fringe shift when apparatus is rotated We cannot detect any motion relative to ether (absolute space) 2 2 lv c 2 3-Dec-13 13

14 So what about the ether? No ether 1890 s not ready to accept this conclusion OR Ether exists, motion through it is real, but compensating effects at work In M-M expt; along ether wind;, perp. to ether wind: Lorentz-Fitzgerald contraction: If a body contracts along its direction of motion through the ether by factor (1-v 2 /c 2 ) 1/2 then t 1 = t 2 and there is no fringe shift. Not just an ad hoc idea. t 2lc v c We know that electric forces are affected by the motion of electrified bodies relative to the ether and it seems a not improbable supposition that the molecular forces are affected by the motion and that the size of the body alters consequently G F Fitzgerald, 1889 t 2lc v c 3-Dec-13 14

15 Lorentz-Fitzgerald contraction George Francis Fitzgerald: Professor and Fellow of Trinity College Dublin, Nephew of George Johnston Stoney, who coined the term electron. Fitzgerald proposed using it to describe particles found by JJ Thomson. Was one of the first to suggest one cannot surpass the speed of light. Proposed Fitzgerald Contractions, which state that bodies contract along direction of motion the faster they travel. ("The Ether and the Earth's Atmosphere" 1889) George Francis Fitzgerald, Dublin ( ) The contraction hypothesis was correct, but for the wrong reason. Contraction now called Lorentz-Fitzgerald Contraction after Fitzgerald and Hendrik Lorentz, who independently derived them. Hendrik Lorentz, Leiden ( )

16 Measurement of length and time Einstein: Analysis of motions based on an abstraction - existence of universal absolute time. We should not rely on a metaphysical notions about time. Rather, we make observations with physical devices clocks ( pendulum clock, watch with vibrating quartz crystal, rotating earth, vibrating molecule To measure velocity of a body: v = r 2 r 1 t 2 t 1 r 1, t 1 (r 2, t 2 ) Clock at r 1 reads t 1 when body arrives at r 1 Another clock at r 2 reads t 2 when body arrives there. Need to define what we mean by same time at two locations 3-Dec-13

17 Einstein s Postulates Postulate 1: All inertial frames are equivalent wrt the laws of physics Postulate 2: The speed of light in empty space always has the same value, ie. the speed of light is independent of the motion of the source or receiver P2 explains the null result of the M-M expt. M-M does not prove P2. Rather we can use P1 and P2 to make predictions which can be tested by experiment. Use Galilean transformation (GT) to describe Newtonian mechanics in different inertial frames. But P2 is not consistent with GT need to find new set of transformations. Later we will see that Newtonian mechanics does not work for very high velocity. 3-Dec-13 17

18 Synchronising clocks If we could transmit signals at infinite speed, no problem. Use large, but finite speed of light. Observation stations A and B at rest in same frame of reference. Clock at A can record time differences between events that occur in immediate vicinity of A. Similarly for B. We have: A time and B time To establish common time: By definition time for light to travel A B equals time for B A Send out light signal from A at t =0 Reflect light at B and return to A at time t = t 0 Time when light arrives at B is t = t 0 /2 We have synchronised the clocks at A and B 3-Dec-13 18