456 CHAPTER 37
concepts and Equations
Term Description
Relativity and Simultaneity All fundamental laws of physics have the same form in inertial reference
frames. The speed of light in vacuum is the same in all inertial frames of ref
erence and is independent of the motion of the source. Simultaneity is not
absolute: Two events occurring simultaneously in one frame might not appear
simultaneous in a second frame.
Time Dilation The proper time ata is the time interval between two events that occur at the
same spatial point in a frame of reference. If this frame moves with a constant
velocity U relative to a second frame, the time interval at between events
observed in the second frame is longer:
at = ata = yato'
VI - u2/c2
y= 1
VI - u2/c2'
----;::
This effect is known as time =
dilation.
=.
Length Contraction The proper length 10 is the distance between
= two points at rest in a frame of
reference. If this frame moves with a constant velocity u relative to a second
frame, the distance I measured parallel to the frame's velocity in the second
frame is shorter:
This effect is known as length contraction.
Lorentz Transformations Lorentz transformations relate the coordinates and time of an event in an
inertial coordinate system S to the coordinates and time of a second inertial
frame S' moving with constant velocity u relative to the first frame. The
Lorentz transformations are
'
z = z,
For one-dimensional motion, the velocities in the two systems are related by
Vx - U
V x-
, _
1 - UVx1 c2'
V� + u
Vx
= 1 + UV�/C2'
Doppler Effect for The Doppler effect is a shift in the frequency of light from a source
Electromagnetic Waves that is moving relative to an observer. For a source moving towards an
observer at speed Lt, the frequency shift is given by
/ch o.
f= \j�f
RELATI VITY 457
Relativistic Momentum and Energy The relativistic momentum and energy are respectively given by
P
-+
-
-V
1
-+
mu
- u
2 / c
2 - I
-+
- 'Vmu
,
The total energy of a particle is the sum of its kinetic energy and its rest mass
energy Eresl = mc2, given by
The total energy can also be expressed as a function of momentum and rest
mass:
conceptual Questions
1: When does relativity become important?
A train is speeding past a platform. At what speed must the train be traveling for the proper time mea
sured on the train to differ from the time measured on the platform by 0.1 %?
Solution
The time dilation relation will solve this problem. The proper
I D E N T I F Y, S ET U P, A N D E X E C U T E
time ( �to ) on the train will transform to the time measured on the platform ( �t ) by
�to
�t = --=
VI - v2/e2
--;:. ==::
== ::;:==:::
We want to find the velocity corresponding to a 0. 1 % difference in time, so we set �t to 1 .001 �to:
�to
�t = 1 .001 �to = v'
1 - v2/e2
Solving gives
v= 1 - � (_)
1 2
1 .001
e = 0.045e.
A 0. 1 % time difference will occur when the train is moving at 0.045e, or 1 .3 X 107 m/s.
EVA L U AT E For a very small relativistic change, the train must be moving at over 13 million meters
per second, or 48 million kph. It is highly unlikely that we will see such effects on any train we will
ever travel on.
2: Proper time and length
You are traveling on a spacecraft moving at O. le past a space station. If you are holding your physics
textbook and flashing LED, who measures the proper length of the book and the proper time interval
for the flashing LED, you or an observer on the space station?
Solution
I D E N T I F Y, S E T U P, A N D E X E C U T E The proper length is measured in a frame in which the book is
at rest. Therefore, you measure the proper length of your physics textbook. The proper time is also
measured with respect to a frame at rest. Again, you measure the proper time for the flashing LED.
