Module 4
Relativity of Time, Simultaneity and Length
The Inverse Transformation Equations
As written, the Lorentz transformation equations transform position and time from frame S to frame S’. It is handy to write
down the “inverse” transform equations that transform from S’ to S. We can derive these inverse equations by solving for the
unprimed variables in the transformation equations. It turns out that this is equivalent to simply swapping primed and
unprimed variables in the transformation equations and replacing v with –v. Let’s write down both sets of equations.
Transformation Equations
Inverse Transformation Equations
x'   ( x  vt)
y'  y
z'  z
x   ( x'vt' )
(4.1)
t '   (t  vx / c 2 )
y  y'
(4.2)
z  z'
t   (t 'vx' / c 2 )
In vector notation, these look like


'  L
(4.3)


  L* '
(4.4)
where * means complex conjugate. To find the complex conjugate of a matrix, you simply take the complex conjugate of
each of its elements. You should verify that Eq. (4.4) does indeed give you the equations in (4.2).
Module 4
Relativity of Time, Simultaneity & Length
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Relativity of Time
Let’s first investigate how measured time intervals compare for the two observers. Since the time of an event measured by
Prime depends not only on the time measured by Unprime but also on the x-component of the position of the event
measured by Unprime and the relative speed of the two observers, we expect that time intervals between two events are
going to be different. Indeed, this is the case. Generally, t’  t.
Let’s investigate a special case of this inequality. Suppose that two events, such as a light turning on and a light going off,
occur at the same position as seen by Prime. Figure 4-1(a) shows this situation. Perhaps the light is from a light bulb that is
stationary in frame S’. For Prime, the first event (light turning on) occurs at x’1 at time t’1. (We can ignore the y’ and z’
components of the position since these do not affect the measured times.) The second event (light going off) occurs at x’2 at
time t’2. The time interval between the two events is t’ = t’2 - t’1. Since the two events occur at the same position, x’2 =
x’1.
What does Unprime measure for these events? Unprime sees the light bulb moving at speed v to the right as shown in
Figure 4-1(b). The light comes on at x1 at time t1 and goes off at x2 at time t2. The time interval is t = t2 - t1. Note that for
Unprime, x2  x1.
y’
S’
on
t’1
O’
y’
y
x’1
y
off
t’2
x’2
(a)
x’
v
on
t1
O
S’
O’
Module 4
x’
S
S
off
t2
Relativity of Time, Simultaneity & Length
v
x2 x
O
Fig. 4-1
x
x1
(b)
2
Let’s use the inverse transform equations, Eqs. (4.2), to transform t to the S’ frame. We do this by transforming t2 and t1 as
follows:
t  t2  t1  (t '2 vx'2 / c 2 )  (t '1 vx'1 / c 2 )
Terms are rearranged to yield
t   (t '2 t '1 )  v( x'2  x'1 ) / c 2
Recall that x’2 = x’1 so the second term in the previous equation vanishes and we are left with
t  t '
So Unprime will measure a longer time interval than Prime between the same two events. We call the time interval measured
by Prime the proper time interval because the two events occurred at the same position. Any other observer moving relative to
Prime measures a nonproper time interval because the two events are observed to occur at different positions. Using tp for
the proper interval and tnp for a nonproper interval, we write the previous equation as
tnp  t p
(4.5)
Eq. (4.5) states what is called the principle of time dilation. It states that the proper time interval between two events is the
shortest interval that is measured. It is measured by an observer that sees the two events occur at the same position.
Just keep in mind that time dilation is a special case of the general fact that time intervals are not the same for observers in
relative motion. We can compare the time intervals measured by any two observers, even if both measure nonproper time
intervals. We would have to know the difference in position for the two events in one frame to figure out the time interval in the
other frame.
Module 4
Relativity of Time, Simultaneity & Length
3
Relativity of Simultaneity
The relative of simultaneity is a direct consequence of the fact that time intervals are different for observers in relative
motion. While the phrase sounds impressive, it simply means that two observers in relative motion cannot measure the
same two events to be simultaneous. For example, if Prime measures two events to be simultaneous, then t’ = 0. These
two events will not be seen to be simultaneous for Unprime since t  t’. (You may notice that we can make t = 0 when
t’ = 0 if we have x’2 -x’1 = 0. But this cannot be since then we would have Prime seeing two events occurring at the same
time and at the same position. That is physically impossible.)
The relativity of simultaneity is just another statement of the relativity of time. It flies in the face of Newtonian relativity
where time is absolute. In Newtonian relativity, if two events are simultaneous for one observer, they are simultaneous for
all observers. Special relativity gives a very different model for the universe.
Relativity of Length
Before we can compare the measured lengths of an object made by our two observes, we have to establish how we correctly
make a measurement of length. Say we have to determine the length of a box that is at rest. What is the algorithm that we
use to do so? The following one will work.
1. Measure position of one end of box.
2. Measure position of other end of box
3. Subtract two positions to get length.
This algorithm works fine as long as the box remains at rest. But what if we had to determine the length if the box is
moving? If Step 1 is done first, then Step 2, we run into a problem. When we do Step 3, we will not have the length of the
box, but rather just some position difference.
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Relativity of Time, Simultaneity & Length
4
How do we change the algorithm so that it will work if the box is moving or at rest? The change is minor. We simply have
to make the measurements of the end positions of the box at the same time. Then, when we subtract the two positions, we
will definitely have the length of the box. So the algorithm to make a valid length measurement goes like this:
1. Measure positions of ends of box at same time.
2. Subtract two positions to get length.
We must use this new algorithm to determine the length of a moving object. However, we could use either algorithm if the
object is at rest. Because of this choice for objects at rest, we cannot know for sure which algorithm was used if some one
reports a length of a stationary object. This may not seem important but not knowing the choice of algorithm is key in
determining the relativity of length
Suppose that an object, say a box, is at rest in frame S’ as shown in Figure 4-2(a). Prime measures a length of l’ = x’2 - x’1
where x’1 and x’2 are the positions of the end of the box. These positions are respectively measured at times t’1 and t’2. In
frame S Unprime sees a moving box as shown in Figure 4-2(b). Unprime measures a length of l = x2 - x1 where x1 and x2
are the end positions. These positions are respectively measured at times t1 and t2.
y’
S’
t’1
y
t’2
S
t1
t2
v
O’
x’2
x’1
x’
O
(a)
x2
x1
x
(b)
Fig. 4-2
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Relativity of Time, Simultaneity & Length
5
We want to compare the two lengths, l and l’. The question is which way should we transform to get the result. Should we
take l and transform it to l’ or vice versa? The answer is found in our previous discussion of valid length. We know for sure
that Unprime measured the end positions at the same time, so t1 = t2 and t = 0. However, we don’t know if Prime measured
the end positions at the same time. Prime could have done so but we are not guaranteed this. As a result, we don’t know the
value of t’. It could be zero but not necessarily. Thus, we must transform from l’ to l. Using the transform equations (4.1)
we have
l '  x'2  x'1  ( x2  vt2 )  ( x1  vt1 )
Terms are rearranged to yield
l '  ( x2  x1 )  v(t2  t1 )
Since we know that t2 = t1 the second term in the previous equation vanishes and we are left with
l '  l
So Unprime will measure a shorter length than Prime for the same object. We call the length measured by Prime the proper
length because the object is at rest with respect to Prime. Any other observer moving relative to Prime measures a nonproper
length because the object is seen to be moving. Using lp for the proper length and lnp for a nonproper length, we write the
previous equation as
lnp  l p / 
(4.6)
Eq. (4.6) states what is called the principle of length contraction. It states that the proper length of an object is the longest
length that is measured. It is measured by an observer that sees the object at rest.
It is interesting to point out that if we assume t’ = 0 in the previous derivation, we can transform from l to l’ and come up with
l = l’ which contradicts our result of l’ = l. But remember that we cannot assume t’ = 0. Therefore, there is no contradiction.
The weird thing is that t’ could be zero. But “could be” is not “is” and the apparent contradiction vanishes.
Module 4
Relativity of Time, Simultaneity & Length
6
Evidence of Time Dilation and Length Contraction
We have experimental evidence of time dilation and length contraction. These experiments usually involve the decay of
high speed particles, as only particles can reach speeds where the gamma factor is significantly greater than one. Let’s
discuss such an experiment that was performed by David Frisch and James Smith in the early 1960’s. This is a nice one to
examine because of its relative simplicity. It did not require a particle accelerator nor sophisticated detectors.
We travel to Mt. Washington in New Hampshire. Frisch and Smith examined the reduction in the muon flux between the
top of the mountain and the base of the mountain. Cosmic ray protons produce pions when they collide with oxygen and
nitrogen nuclei in the atmosphere. The pions quickly decay into muons. The muons also decay but on a much larger time
scale. At speeds much less than c, muons have a lifetime T of approximately 2.2 s. As time progresses, then, the muon
flux R (number of muons per time) obeys the exponential relationship
R(t )  Ro e  t / T .
(4.7)
With a velocity selector, Frisch and Smith were able to count muons that traveled at 0.995c. They found that the flux at the
base, 2000 meters below the peak, was about 70% of the flux at the top, i.e.
R ~ 0.7Ro .
But this doesn’t seem to make to sense. To travel from the top of the mountain to the base takes a time interval of t of
(2000 m) / (0.995c) = 6.7 s. The flux at the base should then be
R = Ro exp(-6.7 s / 2.2 s) ~ 0.05R0 .
(4.8)
Their measured flux was significantly greater.
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Relativity of Time, Simultaneity & Length
7
Time dilation provides the solution to the conundrum. We are mixing time intervals from different frames of reference in
Eq. (4.8). Imagine the scenario as shown in Figure 4-3. Frisch and Smith are in frame S, at rest with respect to the
mountain. A muon is at the origin O’ in frame S’ where frame S’ is moving at v = 0.995c with respect to S. For this relative
speed,  = 10. The muon considers itself stationary and sees the mountain rushing past at v.
The 6.7 s flight time is measured in frame S and is a nonproper time interval. The two events, the muon leaving the top of
the mountain and the muon arriving at the bottom, occur at different positions for Frisch and Smith. Now consider things
from the muon’s perspective. The time of flight for the muon is a proper time interval. As far as the muon is concerned, the
top of the mountain passes origin O’ at t’1 and the bottom of the mountain occupies the same position at time t’2. The two
events occur at the same position.
From Eq. (4.5), this proper time interval is
t p  tnp /   6.7 μs / 10  0.67 μs
The trip “down” the mountain takes much less time for the muon so there
should be more muons at the base. Using Eq. (4.7), we find
y
O
S
R = Ro exp(-0.67 s / 2.2 s) ~ 0.7R0
which agrees with the observed result!
An alternate way to view the situation is in the frame of Frisch and Smith. The
time interval down the mountain is indeed 6.7 s for them. But the muons live
longer for the scientists. The proper lifetime for the muons is 2.2 s. (No one
can keep a muon at rest so any measured lifetime is really a nonproper lifetime.
But at low speeds, nonproper and proper time intervals are essentially equal.)
However, the nonproper lifetime is dilated when the muons travel much faster.
For the muons in this experiment,
Tnp  Tp  10(2.2 μs)  22 μs
y’
O’
S’
muon
v
x
x’
Fig. 4-3
so that R = Ro exp(-6.7 s / 22 s) ~ 0.7R0 once again.
Module 4
Relativity of Time, Simultaneity & Length
8
Great! We have experimental evidence for time dilation. But what about length contraction? We can refer to the same
experiment! All we have to do is to shift our focus to lengths. The height of the mountain measured by Frisch and Smith is
a proper length since the mountain is at rest in frame S. Thus, lp = 2000 meters. The muon measures a nonproper length for
the mountain since the mountain is moving in frame S’. From Eq. (4.6)
lnp  l p /   2000 m / 10  200 m
As far as a muon is concerned, it only takes the mountain a time to pass by of (200 m) / (0.995c) or 0.67 s. We once more
arrive at R = Ro exp(-0.67 s / 2.2 s) ~ 0.7R0 . Simply put, if we buy the principle of time dilation, we also have to buy the
principle of length contraction.
Module 4
Relativity of Time, Simultaneity & Length
9