Lecture 20
Relativistic Effects
Chapter 26.6  26.10
Outline
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Relativity of Time
Time Dilation
Length Contraction
Relativistic Momentum and Addition of Velocities
Basis for Special Relativity
Einstein’s relativity:
• The laws of physics are the same in any inertial
frame of reference.
• The speed of light (c = 300,00 km/s) is the same for
all observers, irrespective of their relative speeds.
Consequences:
• Relativity of Time
• Time Dilation
• Length Contraction
Relativity of Time
Explanation of Time Relativity
Simultaneity in Relativity
Conclusions from the box car experiment:
• Two simultaneous events in one reference frame
can be non-simultaneous in another reference
frame, moving with respect to the first one.
• Simultaneity is not absolute, but depends on the
observer’s state of motion.
• There is no preferred inertial reference frame.
Time Dilation
Experiment
ct/2
vt/2
d
tp= 2d/c  time interval for light to reach the mirror
at the stationary reference frame (proper time)
t  this interval in the moving reference frame
Speed of light c is equal in both frames
2d
ct 2 v t 2
t = tp
 =  + d2  t = 
2
2
c(1 v2/c2)
 = 1/(1 v2/c2)
Conclusions from Time Dilation
t = tp
 is always grater than 1
The time interval between two events measured by an
observer moving with respect to the clock is longer
than that between the same two events measured by an
observer at rest with the clock.
A clock moving past an observer runs slower than an
identical clock at rest with the observer.
Proper time is the time interval between two events as
measured by an observer who sees them at same
position.
Twin Paradox
Twin paradox is a consequence of time dilation.
The age of the traveling twin is indeed smaller
than that of the non-traveling one, because the
former experiences acceleration and lives in a
non-inertial reference frame.
Only the non-traveling twin can apply Einstein’s
relativity and determine the time dilation.
Length Contraction
The proper length Lp is the length of an object
measured at rest with it.
In a moving reference frame, the length L is shorter.
L = Lp/ = Lp (1 v2/c2)
Length contraction occurs only in the direction of
motion.
Example: A right triangle with the sides of 10 m each
at the right angle. How would the angles change if the
triangle moves at 0.9 c along one of the short sides?
 = 45
Solution: l = 10 (1  0.92) = 1.9 m
tan ' = 1.9/10 = 0.19  ' = 10.8
Length Contraction
10
'

10
10
1.9
v = 0.9 c
Other Effects of Relativity
Momentum is the product of mass and velocity
p(rest) = mv
p(relativistic) = mv
Relativistic momentum obeys the law of momentum
conservation even at velocities, close to c.
Relativistic addition of velocities
(works for parallel or antiparallel paths)
vx  speed of object x
vy speed of object y
vxy  relative speed of
objects x and y
vx + vy
vxy = 
1 + (vx vy)/c2
Relativistic Addition of Velocities
Equivalence of Mass and Energy
Kinetic energy of a non-relativistic object KE=mv2/2
For a relativistic object: KE = mc2 – mc2
The term independent of velocity is the rest energy
ER = mc2
Total energy, defined as E = mc2, is the sum of the
kinetic and rest energies.
Any object has energy by virtue of its mass alone.
Mass can be transferred into energy and back.
Summary
• There are two postulates of special relativity:
1. The laws of physics are the same in any inertial
frame of reference.
2. The speed of light is the same for all observers,
irrespective of their relative speeds.
• The consequences (relativity of time, time
dilation, length contraction, velocity addition)
seem unusual for our experience, but can be
verified by experiments or observations of
distant objects.