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Lecture 13 Notes:
07 / 20
Invariance of the speed of light
The MichelsonMorley experiment, among other experiments, showed that the speed of
light in vacuum is a universal constant, as predicted by Maxwell's equations.
It is the
same for all observers, and for light coming from any direction.
This is at odds with our
everyday experience with speeds, where the speed of any object relative to an observer
depends on the motion of the observer.
Albert Einstein started with the assumption that Maxwell's equations, along with all
other fundamental laws of physics, in fact hold for all observers, and so the speed of
light really is independent from the observer.
The assumption that the laws of physics
are the same for all observers, together with a finite and observerindependent speed of
light, is known as the
principle of relativity.
The principle of relativity requires us to abandon the notion of absolute time.
In
particular, with the principle of relativity, two events that are simultaneous according to
one observer are not simultaneous according to another.
To see this, consider the
following example:
A train of length
L
is moving to the right with speed
v
.
An observer, let's call him Bob,
stands on top of the train, right in the middle, and sends a light pulse to both ends.
Since
the distance to both ends of the trains is the same, the pulses arrive at the same time.
If
we call the signal arriving at the back of the train event A, and the signal arriving at the
front of the train event B, then A and B are simultaneous according to Bob.
However, now consider that Alice is standing near the tracks, in the same spot as Bob
when he sends the light pulses.
By the principle of relativity, she also sees both light
pulses moving at speed
c
.
But now the back of the train moves towards the light pulse,
while the front of the train moves away from it, so the light reaches the back of the train
first.
Thus A happens before B, according to Alice.
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Consider the following thought experiment.
Bob is on the train.
He constructs a clock
by bouncing a pulse of light off a mirror, and measuring the amount of time it takes the
light to get back.
If the mirror is a distance
d
from the light source, then the light moves
a distance 2
d
at speed
c
, and Bob will see the light come back a time
∆
t
0
= 2d / c
later.
Now Alice watches this experiment from the ground near the train tracks.
According to
her, the apparatus is moving to the right, so the light takes a path shown below:
The light has to travel a greater distance, but by the principle of relativity, it still moves
at speed
c
.
Therefore, the time it takes the pulse to go to the mirror and back is longer
according to Alice than according to Bob.
Since Bob could use this clock to time any
kind of a physical process on the train (such as how fast a regular clock runs, or his
heartbeat, or the rate at which radioactive particles decay), all processes on the train
seem to run slower to Alice.
This is known as
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 Spring '07
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