Physics 325
Lecture 17
Special Relativity
As you no doubt know already, Newton’s laws are only an approximation to classical
mechanics.
A nearly perfect approximation for everyday velocities of objects, but an
approximation nonetheless.
The Newtonian world has a universal clock that ticks on at a constant rate independent of
the motion of an observer.
Vector components however, clearly depend on the reference
frame, and the transformation of vectors is described by a matrix that preserves dot
products.
This puts time and space on different footings since time is independent of the
observer while space has very definite transformation properties.
The concept of an
invariant dot product is expanded in special relativity in such a way that time and space
are put on equal footing.
The theory of special relativity (I will henceforth refer to it as just
relativity
, but I will
mean special, not general relativity) is based on the following two postulates:
I.
The laws of physics are the same in all inertial reference frames.
II.
The velocity of light in vacuum is a universal constant and is independent of the
relative motion between the source and the observer.
The first postulate is a holdover from Newtonian mechanics.
The second postulate is
revolutionary. In the Newtonian world all velocities add like simple vectors, so if a
person is moving towards you at a velocity
v
G
and shines a light at you that is traveling at
velocity
in her reference frame, then the light is traveling towards you at velocity
(this is called a “Galilean transformation”).
Not so says Einstein!
The light
is
traveling at velocity
in the running person’s frame, and it is also traveling toward you
at
in your reference frame.
c
G
vc
+
GG
c
G
c
G
This counterintuitive postulate seems to be disproved by our everyday experience.
There are many examples in which we observe the simple addition of velocities when
moving from one frame to another.
For instance, if you walk on a moving conveyor belt
the speed with which you pass stationary objects is certainly the sum of your walking
speed and the speed of the belt.
Relativity avoids this contradiction by redefining the
addition of velocities in such a way that they work “normally” at velocities small
compared to the speed of light, but saturate at large velocities so that the result is never
larger than the speed of light.
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View Full DocumentWe have seen that Newton’s laws are invariant under Galilean transformations between
two reference frames that move with a constant velocity
0
v
G
relative to one another:
( ) ( )
() ()
0
0
rt
rt v
t
vt
v
′
=
−
′
=
−
G
GG
G
(17.1)
Here we have assumed that space and time are completely separable so that the
t
in the
first equation above can be defined independent of the reference frame (when we invert
the equation to write
r
in terms of
G
r
′
G
, the same
t
appears.
Therefore
tt
′
=
for a Galilean transformation.
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 Spring '10
 Makins
 mechanics, Special Relativity

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