1. USC2001 Energy Lecture 4 Special Relativity Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749

2. VELOCITY OF LIGHT Wave velocity = velocity of the wave relative to the medium – velocity of observer relative to medium. 1964 experiment at CERN (European particle-physics laboratory) showed that velocity of light emitted from neutral pions travelling at 0.99975c was the same as the velocity of light emitted by stationary neutral pions Bullet velocity = velocity of the source relative to the observer + velocity of the bullet relative to the source. 1887 experiment by Albert Michelson and Edward Morley, using the Michelson Interferometer, showed that the velocity of light is independent of the velocity of the observer (there is no ether = medium for light)

3. ALBERT EINSTEIN’S POSTULATES 1905 Albert Einstein, an employee at the patent office in Bern, Switzerland, published his Special Theory of Relativity. His theory asserted: The Speed of Light Postulate: The speed of light in vacum has the same value c in all directions and in all inertial frames (c = 299792458 m/s). The Relativity Postulate: The laws of physics are the same for observers in all inertial frames. He courageously worked out the mind boggling logical consequences of these simple assumptions.

4. MEASURING EVENTS An event is something that happens to which an observer can assign three space coordinates and one time. Examples include turning on a small lightbulb, collision of two particles, passage of a pulse of light through a specified point in space, an explosion, the coincidence of the hand of a clock with a marker on the rim of a clock. In any inertial frame space coordinates can be measured by setting up a three dimensional grid of rulers and time coordinates can be measured by a grid of clocks synchronized by transmitting a single light pulse from one clock to all the other clocks using the Speed of Light Postulate.

5. TIME SIMULTANEITY Speeding Sally Stationary Sam In Sam’s frame light emitted simultaneously from A and B will meet at his middle C, but to the left of the Sally’s middle C’ so she measures that the light left A after it left B – so simultaneity depends on the frame Stationary Sam Speeding Sally

6. RELATIVITY OF TIME mirror Speeding Sally mirror Stationary Sam Sally sends a light pulse to a mirror located distance D above her train and measures time Sam measured time must satisfy the equations

7. RELATIVISTIC MANNERS mirror Speeding Sally mirror Stationary Sam When two events occur at the same location in an inertial frame, the time interval between them, measured in that frame, is called the proper time Sam’s improper time Sally’s proper time Lorentz factor Speed parameter

8. RELATIVITY OF LENGTH The length of an object measured in the rest frame of the object is called its proper length. The length measured in any frame that is in relative motion parallel to the length is always less than the proper length. Speeding Sally Stationary Platform Sally measures proper time for platform to traverse the front end of her train Stationary Sam measures

9. DERIVATION OF THE LORENTZ TRANSFORMATION We need 4 equations to compute the 4 matrix entries Since light moving right, left has velocity +c, -c

10. DERIVATION OF THE LORENTZ TRANSFORMATION We need 2 additional equations to compute Since the primed frame is moving with velocity v

11. DERIVATION OF THE LORENTZ TRANSFORMATION The Relativity Postulate implies that

12. DERIVATION OF THE LORENTZ TRANSFORMATION

13. CONSERVATION OF MOMENTUM does not hold in moving frames, but it does hold for the modified momentum The law of conservation of classical momentum

14. EQUIVALENCE OF MASS AND ENERGY accelerated to velocity The mass and energy added to an object of rest mass is and this led Einstein’s to his famous formula

15. TUTORIAL 4 1. What time elapses on Stationary Sam’s watch when he observes Speeding Sally’s watch advance by one Minute? Does he think that her watch runs too fast or too slow. What does she think about his measurements – does she feel inclined to buy a new watch ? What does she say about Sam’s watch ? 2. An elementary particle known as a positive kaon has on the average a lifetime of 0.1237 microseconds. Compute the average distance it moves in a laboratory reference frame if its speed relative to the laboratory is 0.990c. Hint: the particles ‘internal’ clock runs at a different speed than the laboratory clock