May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 1

Special Relativity• Postulates

– The laws of physics are the same in every inertial frame of reference

– The speed of light in a vacuum is the same in every frame of reference and independent of the motion of the light-source

• Reference Frame– Inertial Reference Frame

• History and Experimental Support• “Beta” and “Gamma”• Space-time events• Time Dilation / Proper Time• Length Contraction• Simultaneity• Relativistic Velocity Addition• Galilean and Lorentz transformations• Spacetime interval / Relativistic invariants• Relativistic Force and Acceleration• Relativistic Momentum & Kinetic Energy• Twin Paradox• Energy and mass conversion

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 2

Special Relativity• Reference Frame

– A reference frame is an environment in which we make measurements. It’s important features are its origin, angle, and relative velocity.

– Length and time within a reference frame are simple. The difficulty comes in transforming between reference frames.

– Inertial Reference Frame – A reference frame that is moving at constant velocity. The ONLY type of reference frame that special relativity covers.

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 3

Coordinate Transformations

u

Galilean

Lorentz

2

'

''

'

( )

( )

x utyz

xy

ut

z

t xc

γ

γ

= −==

= −

'

'''

x x utyz

yzt t

= −===

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 4

Proper Time

• Proper Time (“Tau”)– If you time two events that are at rest relative to

you, you have measured the “proper time” between the events.

– If you are measuring the time between events occurring on a rocket flying by, you are NOT measuring the proper time between those events.

– The “proper time interval” is always the shortest time interval.

( )'t γ τ∆ = ∆proper time−

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 5

Space-time Events(time interval)

1 1

2 2

0, 0

0, 1 sec

x t

x t micro

= == =

'

'1

2

'2

2

2

2

2

( )

( )

(0 (0)) 0

(0 (0)) 0

(0

'

'

'

(1 sec))

(1 sec (0))

x utu

t xcuucu micro

x

t

x

umicr

x

otc

t

γ

γ

γ

γ

γ

γ

= −

= −

= − =

= − =

= −

= −

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 6

Classical Relativistic

2

0

0

0

20

0

30

2

2 2 2

'

2 20

( )

( )/

( 1)

''1

'

'

1 '

( )

u

u

x ut

t x u cL L

t t

p m v

K m c

F m a

F m a

v uv

u

x

vc

E p c m

t

c

γ

γγγ τγ

γ

γγ

γ

β ββ

β β

⊥ ⊥

= −

= −=

∆ = ∆ = ∆

=

= −=

=

+=

+

+=

+

− =

P P

'

'

0

2

( )( )

(1 2)

'x utt

L L

tp mv

K mvF maF ma

v u

xt

v

τ

= −==

∆ = ∆=

===

= +

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 7

Beta and Gammaandβ γ

8

8

6

2 10 / 0.66

1.5 10 / 1/ 2

3 10 / 0.01

vc

v m s

v m s

v m s

β

β

β

β

≡

= × =

= × =

= × =

22

1 11

1

0.6 1.250.8 1.660.99 7.10.9999 70.7

vc

γβ

β γβ γβ γβ γ

≡ =− −

= == == == =

Gamma vs. Beta (Relativistic "Weirdness factor")

1

3

5

7

9

11

13

15

0 0.2 0.4 0.6 0.8 1

Beta (v/c)

Gam

ma

(1/s

qrt

(1-

bet

a^2)

)

May 2005 New Mexico Tech Physics -- R.Sonnenfeld

Page 8

Motion relative to a wave propagation medium

Note that boat “catches up” to bow wave forming a “shock wave”

Supersonic aircraft produce shock waves (sonic boom)