1
PHY3101
January 13, 2007
Relativity
Notes on homework #2
Some Taylor expansions: I had intended to go over these Taylor expansions in class
on Friday. They are useful for the homework.
√
1
1
1
1
1 − x = 1 − x + O(x2 ), √
= 1 + x + O(x2 ),
= 1 + x + O(x2 )
1−x
2
2
1−x
The mathematical notation O(x2 ) represents something that is smaller than some number
times x2 , when x is very small. Then, x2 will usually be comparable to the error in these
approximations. In this class, when x is very small, you are allowed to just use
√
√
1
1
1/(1 − x) ≈ 1 + x
1−x≈1− x
1/ 1 − x ≈ 1 + x.
2
2
−1
−2
−3
−4
−5
Try them out on your calculator for x = 10 10 10 10 10 and see how the approximation improves while x decreases. See if the error is comparable to x2 .
Details of Time Dilation
Definition: A thought experiment is an experiment that we can imagine doing, but
which would not be possible under reasonable circumstances. Thought experiments
allow us to test our understanding and to sharpen our physical intuition. Formulating a
thought experiment is particularly useful when the laws of physics, as understood, appear
superficially to be paradoxical, self-contradictory or inconsistent.
Definition: An Event is a particular place at a particular time. Something might happen
at that place and at that time, such as the snap of my fingers, or a firecracker going off.
Different frames of reference might give different coordinates for the location and time of an
event.
A thought experiment: A horizontal mirror is a height h above a source-detector of
light.
h
∗∗
#1 #2
Alice starts a pulse of light from the emitter at event #1. The light then bounces
off the mirror and is reflected back to the detector where it is received at event #2. Alice
measures the time interval to be
∆t0 = t2 − t1 = dist/speed = 2h/c
(1)
2
between event #1(emission) and event #2 (detection).
Definition Proper time interval: Notice that in Alice’s frame of reference the events #1
and #2 occurred at the same location in space but at different times. This is important:
If two events occur at the same location, then the time interval measured between these
two events in that frame of reference is special and called the proper time interval. The
smallest time interval between these two events is always the proper time interval. In any
other frame of reference, the events would occur at different locations and the time interval
would be longer than the proper time interval.
Alice’s experimental apparatus is actually on a train going through a station with a speed
v. In the station Bob sees the source-detector move a distance L before the light is detected.
d
d
#1∗
h
∗#2
L
Let Bob be in the primed frame of reference. Notice that in Bob’s frame of reference event #1 and event #2 do not occur at the same location in space—and this is not
at all surprising because the train is moving through Bob’s station. Assume that Bob
measures a change in time ∆t0 between the events #1 and #2. Then the distance between
the events, in Bob’s frame, is just
L = v∆t0 .
Bob measures the length of the light’s path and obtains
p
p
distance = 2 h2 + L2 /4 = 2 h2 + v 2 ∆t02 /4,
which is longer than the distance 2h that Alice measured. From the principle of relativity
Bob knows that the speed of light is c, so he measures
p
2 h2 + v 2 ∆t02 /4
0
∆t = dist/speed =
.
c
Now, Bob sits back and does some algebra: squaring both side gives
∆t02 = 4
(h2 + v 2 ∆t02 /4)
,
c2
and solving for ∆t02 yields
³
v2 ´
4h2
02
∆t 1 − 2 = 2
c
c
4h2 /c2
∆t2
or
∆t02 =
=
(1 − v 2 /c2 )
(1 − v 2 /c2 )
∆t
or
∆t0 = p
1 − v 2 /c2
(Use eqn. 1 on the previous page.)
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So in Bob’s frame of reference the light traveled further than it did in Alice’s frame of
reference. But, the speed of light must be the same in the two frames of reference. The
unmistakeable conclusion is that the time interval ∆t0 that Bob measures between events
#1 and #2 is longer than the time interval that Alice measures.
The time interval between two event, as measured in the frame of reference where the
events occur at the same place, is called the proper time ∆t0 between the events. Any other
observer who is moving with respect to this particular frame will measure a longer time
interval
∆t0
∆t0 = p
.
1 − v 2 /c2
Because Alice’s time interval is shorter than Bob’s time interval, Bob concludes that
Alice’s watch runs slower than his own watch. This effect of special relativity is called time
dilation.
This situation is actually symmetrical: Bob could do a similar experiment in the station.
And Alice could go by it in the train with a velocity v. But, from Alice’s frame of reference,
Bob would appear to have a velocity −v. The analysis of the time intervals is just that same
as above, except that the two events would occur at the same location only in Bob’s frame
of reference. and he would measure
p the proper time interval ∆t0 . Alice would measure a
longer time interval ∆t = δt0 / 1 − v 2 /c2 , and she would conclude that Bob’s clock was
more running slower than her own watch.
Time dilation might seem paradoxical, but here is a statement that is true: When Bob
carefully observes Alice’s watch, it will appear (to him) to run slower than his own watch.
Here is another statement that is true: When Alice carefully observes Bob’s watch, it will
appear to her to run slower than her own watch.
This conclusion is surprising and seems to be a logical paradox and to contradict our
everyday experiences involving time, trains and watches. But, it has been tested experimentally and describes the way Nature behaves.