39. Relativity
39.4 Consequences of the Special Theory
of Relativity
39.6 Relativistic Linear Momentum and the
Relativistic Form of Newton's Laws
Suggested HW problems:
Chp39: 5-12,16-21,27-31
Consequences of the
Special Theory of Relativity
a) Simultaneity of two events
b) Length Contraction
c) Time Dilatation
d) The Twins Paradox
1
Simultaneity of two events
two events are simultaneous if t1= t2 ; then, in general
the events are not
simultaneous in
another reference
frame
2
∆t ' = t2′ − t1′ = γ ( ∆
!t − V ∆x / c )
=0
=−
γV
c
2
∆x = −
γV
c2
( x2 − x1 ) ≠ 0
Example:
two lightning bolts striking the ends of a moving boxcar
occur simultaneously for a stationary observer at O
(AO=OB) but not for a moving observer at O’
Length Contraction
1
2
proper length
in the rest frame
1
2
to measure the length of the ruler in the frame in which
it moves with velocity v= βc requires the simultaneous
determination of the coordinates x2 and x1 (i.e., t2-t1=0)
(in the rest frame you may “take your time” to measure the
proper length of the ruler which is motionless)
Lp = x2′ − x1′ = γ ( x2 − x1 ) = γ L ⇒ L = L p 1 − β 2 < Lp
2
Time Dilatation
the proper time , i.e., the time shown by a clock in its
rest frame, is
τ = t2′ − t1′
x2′ = x1′
and clearly
in the “lab(oratory) frame” (in which the clock moves
with velocity v , from the Lorentz transformations
∆t = γ [( t2′ − t1′) + (v / c 2 )( x2′ − x1′)] = γτ > τ
!
"#$
=τ
=0
If v is not constant:
τ
dt ′
0
1− v / c
∆t = ∫
2
2
<τ
Time Dilatation
proper time
interval between
sending and
receiving the light
signal is
2d
τ=
c
The time measured by the observer O
∆t =
τ
1 − (v / c )
2
=
2d
c −v
2
2
>τ
3
The Twins Paradox
S
G
before the cosmic trip
after the cosmic trip
∆tS = 1 − (v / c) 2 ∆tG < ∆tG
assuming
∆tG = 50 years, and v / c = 0.95 ⇒ ∆tS = ?
The Twins Paradox: Space-Time Graph
Finish
Start
4
Problem
Relativistic Linear Momentum
p=
mv
1 − (v / c ) 2
= γ mv
• p is conserved if the sum (or impulse) of all forces
acting upon the system is zero (e.g., in collisions)
• for c→∞ (or v<<c), p assumes its classical expression
5
Relativistic Form of Newton's Laws

dp d 
mv

F=
= 
2

dt dt 1 − (v / c) 


ma
mv (v ⋅ a )
=
+
2
2 3/ 2
1 − (v / c )
1 − (v / c) 
where
a=
dv
dt
is the acceleration
Problem
6
For next Lecture:
From the text READ Sec. 39.7-9!
39. Relativity
39.7 Relativistic Energy
39.8 Equivalence of Mass and Energy
39.9 Relativity and Electromagnetism
Suggested HW problems:
Chp39:35-42,46-50
7