SPECIAL RELATIVITY

-Postulates of Special Relativity -Relativity of time –> time dilation -Relativity of length –> length

contraction

© 2005 J. F. Becker

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

An event takes place at a certain time and place as measured in a frame of reference,

or coordinate system (x, y, z, t)

INERTIAL FRAME OF REFERENCE: Newton’s law of inertia is valid in the frame of reference. The acceleration of a body is zero when measured in the coordinates.

C 2005 J. F. Becker

Two inertial frames of reference moving with constant relative velocity v

vtC 2005 J. F. Becker

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

POSTULATES (ASSUMPTIONS)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 each other.

C 2005 J. F. Becker

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

Both the person on the truck and the observer on the earth measure the speed of light to be c, regardless of the speed of

the truck.

The speed of light postulate

C 2005 J. F. Becker

We will see some mind-boggling effects of the theory of special relativity, such as:

MOVING CLOCKS TICK SLOW!(Time dilation)

MOVING RULERS APPEAR CONTRACTED! (Length contraction)

But first, let’s consider the concept of “simultaneity”

(two events happening at the same time)

C 2005 J. F. Becker

SIMULTANEITY IS RELATIVE!Whether two events are simultaneous

depends on the frame of reference. Two lightning bolts strike the railroad car and

ground at each end.u

u

u

u

C 2005 J. F. Becker

TIME DILATIONThe proper time interval to between two

events is the time interval measured by an observer who is at rest relative to the

events and views them as occurring at the same place.

An observer who is in motion with respect to the events and who views the events as occurring at different places measures a

dilated (expanded) time interval t.The dilated time interval is greater than the proper time interval (t > to) according to the time-dilation equation: t = to / (1 –

u2/c2) ½

t > to and t indicates a slow clock!

Time dilation: t = to / (1 – u2/c2) ½

A spacecraft speeds past Earth at a constant speed of u = 0.92c and the

astronauts measure the time between ticks of the spacecraft clock to be 1.0 sec. (to =

1.0 sec.).

What time interval do observers on Earth measure? t = ?

t = to / [1 – (0.92)2]1/2 = to / 0.392 =2.6 to

t = 2.6 sec.

MOVING CLOCKS TICK SLOW!C 2005 J. F. Becker

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

A light clock: One “tick” is the time interval it takes for the light pulse to travel the round trip distance / speed of light or (2 d = c x t)

time interval = 2 d / c

For stationary clock:to = (2 d) / c

For moving clock:t = (distance) / c

d

C 2005 J. F. Becker

Time dilation

u

to t

u t

u t/2

L = [ d2 + (u t/2) 2 ] ½

The path observed by Stanley is longer than d.

d L L

C 2005 J. F. Becker

(a) Mavis measures (proper) to (= 2 d /c). (b) Stanley measures t.

The path observed by Stanley is longer:

2 L = 2 [ (d2 + (u t / 2)2 ] ½

So c t = 2 L = 2 [ (c to / 2)2 + (u t / 2)2 ] ½

t)2 = (2 L/c)2 = (2/c)2 [ (c to/2)2 + (u t/2)2]

t)2 = [ (to)2 + (u t/c)2]

t)2 - (u t/c)2 = [ (to) 2 ]

t)2 { 1 - (u /c)2 } = [ (to) 2 ]

Time dilation: t = to / {1 – u2/c2} ½

Experimental verification of time dilation – Muons are particles observed on earth

after being created in the upper atmosphere when cosmic rays from the

Sun collide with atoms in our atmosphere. The muon quickly decays into an electron and a neutrino particle.These muons travel toward earth with

speed u = 0.998 c

(Lifetime of a muon at rest = 2.2 (10)-6 s.)

(a) How long does one of these muons live according to an observer on earth?

(b) How far does a muon travel before it disintegrates?

C 2005 J. F. Becker

Verification of time dilation –

How long does one of these muons live according to an observer on earth?

The two events are the generation and disintegration of the muon. When the

muon is at rest the events take place at the same place so the lifetime is the

proper time interval to = 2.2 (10)-6 s.The muon moves at u = 0.998 c so Earthlings measure a dilated time

interval t.

t = to / (1 – u2/c2) ½

t = 35 (10)-6 s.C 2005 J. F. Becker

Verification of time dilation –

How far does a muon travel before it disintegrates?

If we mistakenly neglected relativistic effects the calculated distance would be d = ut =0.998 [300(10)6 m/s] 2.2(10)-6s

=659 m (distance too short; muons never reach earth!)

An Earthling measures a distance of d = u t =0.998 [300(10)6 m/s] 35(10)-6s

d = u t = 10,500 m

(the distance to top of atmosphere)C 2005 J. F. Becker

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

Length contraction – As measured by an Earthling the Earth-to-star distance is Lo, the proper length as measured by an observer at rest with respect to the ends of the ruler, and the time to make the trip is t. An Astronaut measures the distance to be L & the time to. Voyage to a star!

Lo L

u

u u

to

t

C 2005 J. F. Becker

Length contraction: u = 0.95 c = distance/timeThe relative velocity is: u = L / to = Lo / t

L = (Lo / t) to

And substitutingt = to / (1 – u2/c2) ½

we get L = (Lo / t) t (1 – u2/c2) ½

L = Lo (1 – u2/c2) ½

Lo = the proper length between two points as measured by an observer at rest with respect to the two points, i.e., a stationary ruler.

MOVING RULERS APPEAR CONTRACTED!C 2005 J. F. Becker

Length contraction from the muon’s frame: Recall the muons speeding toward earth at u = 0.998 c. In the muon’s reference frame it lives to = 2.2 (10)-6 s and it measures the distance to earth as L. But, as measured by Earthlings the distance is Lo = proper length = 10,500 m. Since moving rulers are contracted the muon measures L = Lo (1 – u2/c2) ½

L = 10,500 m. (1 – u2/c2) ½ = 632 m., a short enough distance to earth to cover in 2.2 s.

Lo = the proper length between two points as measured by an observer at rest with respect to a ruler (a stationary ruler).

C 2005 J. F. Becker

Time dilation:

MOVING CLOCKS TICK SLOW!

Length contraction:

MOVING RULERS APPEAR CONTRACTED!

C 2005 J. F. Becker

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.

Graph shows how the factor

(1 – u2/c2) ½

increases as the relative speed approaches c.

c ~ 300 (10)6 km/s

c ~ 670 (10)6 mi/hr

u

EQUIVALENCE OF MASS AND ENERGY

Total energy of an object: E = mc2 / (1 – u2/c2) ½

Rest energy of an object: Eo = mc2 / (1 - 0) ½

Eo = mc2

C 2005 J. F. Becker

REVIEW

C 2005 J. F. Becker

Length contraction – light pulse reflected off mirror at end of ruler

(a) Astronaut measures (proper) to. (b) Earthling measures t.

The path observed by Earthling is longer:s = [ D2 + L2 ] ½ = [ (D2 + (u t/2) 2 ] ½

L = u t/2

C 2001 Wiley, Physics Cutnell & Johnson 5th Ed.