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.sgTel (65) 6874-2749

VELOCITY OF LIGHTWave 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 theobserver + 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 velocityof the observer (there is no ether = medium for light)

ALBERT EINSTEIN’S POSTULATES

1905 Albert Einstein, an employee at the patent office in Bern, Switzerland, published his

SpecialTheory 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.

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.

TIME SIMULTANEITY

Speeding Sally

Stationary Sam

v

AA BB

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

CC

A C B

CA B

Stationary Sam

Speeding Sally

RELATIVITY OF TIME

tv

mirror

DSpeedingSally

mirror

DLL

StationarySam

Sally sends a light pulse to a mirror located distanceD above her train and measures time c/D2t

0

Sam measured time must satisfy the equations t22 D)2/tv()c/2(c/L2t

2/12

0 ))c/v(1(tt

RELATIVISTIC MANNERS

tv

mirror

DSpeeding Sally

mirror

DL LStationary 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

0

t

t Sam’s improper time

c/v,21/1,tt0

Sally’s proper time

Lorentz factor

Speed parameter

RELATIVITY OF LENGTHThe 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.

L

0L

Speeding Sally

v

Stationary Platform

Sally measures proper time for platform to traverse the front end of her train

0t

0tvL

Stationary Sam measurestvL

0

0LL

DERIVATION OF THE LORENTZ TRANSFORMATION

t

x

mm

mm

t

x

2221

1211

We need 4 equations to compute the 4 matrix entries

xx

0tx

0tx

v

Since light moving right, left has velocity +c, -c

)tcx()ctx( )tm)ct(m(ctm)ct(m

22211211

221121

2

12mm,mcm

DERIVATION OF THE LORENTZ TRANSFORMATION

t

x

mm

mcmt

x

1121

212

11

We need 2 additional equations to compute

xx

v

Since the primed frame is moving with velocity v

)tvx()0x( )tm(vtmc

1121

2

2

1121c/vmm

2111 m,m

DERIVATION OF THE LORENTZ TRANSFORMATION

t

x

1c/v

v1

t

x211m

xx v

The Relativity Postulate implies that

t

x

1c/v

v1

t

x211m

DERIVATION OF THE LORENTZ TRANSFORMATION

xx v

t

x

1c/v

v1

t

x211m

1)c/v1(222

11m

t

x

c/v

v

t

x2

CONSERVATION OF MOMENTUM

does not hold in moving frames, but it does hold for the modified momentum

The law of conservation of classical momentum

txmmvp

txmtxmmvp 0

EQUIVALENCE OF MASS AND ENERGY

accelerated to velocity

The mass and energy added to an object of rest mass

2c)1(mdxE

x

0dtdp

m v is

and this led Einstein’s to his famous formula2mcE

TUTORIAL 41. 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