Chapter 26

Relativity

Introduction

• Newtonian Mechanics has been found not to be valid at speeds close to the speed of light

• Albert Einstein (1879 – 1955)– Published special theory of relativity in

1905 at the age of 26!

• Newtonian mechanics was found to be a special case of Einstein’s theory

Theory of Relativity

• Special theory of relativity based on two postulates:– Laws of physics are the same in all

coordinate systems either at rest or moving at velocity with respect to one another

– The speed of light has the same value, c = 3×108 m/s, regardless of the speed of the observer or of the velocity of the source of light

Galilean Relativity

• Frame of reference– Coordinate system at rest with respect

to something• For example – your frame of reference can

be the school, which is at rest with respect to the ground but moving with respect to the sun

• Galilean Relativity – laws of mechanics are the same in all inertial frames of reference

Speed of Light

• Suppose you are traveling on a boxcar that is moving with velocity v– You send a light pulse with velocity c

relative to the observer in the boxcar– Common sense would suggest that the

speed of the light pulse to an observer outside of the boxcar would be c + v

– However, Einstein’s theory says that the velocity should be the same for an observer both on the train and outside of it

Speed of Light, cont.

• One of two conclusions must be drawn:– Either the addition law for velocities is

wrong– The laws of electricity and magnetism are

not the same for all reference frames• If this is true then there is one reference frame

for which the speed of light is c but all other reference frames would have a velocity greater or less than c

Electromagnetic Theory

• Says that speeds travel through empty space with speed equal to c

• Mechanical waves require a medium to support the wave disturbances

• In the 19th century, physicists thought light waves would have to also have a medium through with to propagate – This was called luminiferous ether

Ether

• This was assumed to be everywhere, including in open space

• Because planets travel through it, it has to be massless and have no effect on the motion

• Absolute frame – frame of reference at rest with respect to the ether

• The ether had to be moving through the labs on Earth as “ether wind”

Michelson-Morley Experiment

• Albert A. Michelson (1852 – 1931)– Won the Nobel Prize in 1907

• First performed the experiment in 1881 and then was later repeated with Edward Morley (1838 – 1923)

• Experiment designed to calculate the velocity of earth with respect to the ether

Michelson Interferometer

Michelson Interferometer, cont.

• Should observe different fringes depending on the orientation of the interferometer

• Experiment resulted in negative results– No fringe shift was observed– This contradicted the ether hypothesis

• Light is an electromagnetic wave that requires no medium

Einstein’s Principle of Relativity

• Two postulates of special relativity– Principle of Relativity – All laws of

physics are the same in all inertial reference frames

– Constancy of the speed of light – the speed of light has the same value, regardless of the velocity of the observer or the source

Time Dilation

• A result of special relativity• Consider a beam of light on a

spaceship moving at a speed close to the speed of light

From the spaceship…

From the earth…

Time Dilation

Time Dilation, cont.

• On the spaceship, the time it takes to go from the source to the receiver is given by

distance traveled 2

speedp

dt

c

Time Dilation, cont.

• For someone observing from outside of the spaceship, the time is found by:

2

c t

2

v t

d

2 22

2 2

c t v td

Time Dilation, cont.

• Solving for the time, you get

• From before,

• So

2 2 2 2

2 2

1 /

d dt

c v c v c

2

p

dt

c

2 21 /

ptt

v c

Time Dilation, cont.

• From this equation, the time interval observed by a person outside of the spaceship is longer than the interval for the person on the spaceship

• Proper time – the time interval as observed by someone who sees both events at the same position

Twin Paradox

• Consider two twins. At some point, one of the twins decides to travel to a planet 20 lightyears from earth. – His speed is 0.95c– When he returns, the twin that stayed

behind has aged 42 years.– However, the twin that traveled has only

aged 13 years

Length Contraction

• Proper length, Lp, of an object is the length measured by someone at rest with respect to the object

• For an observer on earth, the distance traveled by a spaceship is Lp, and the time it takes to travel is Lp/v

• Because of time dilation, a person on the ship thinks that the time is smaller; therefore, the distance traveled is shorter

Length Contraction, cont.

• Therefore,

• This only takes place along the direction of motion

2 21 /pL v t v t v c

2 21 /pL L v c