11.2 Relativity of Time
Physics Tool box
 Proper time
t s is the time interval separating two events as seen by an
observer for whom the events occur at the same position.
 Time dilation is the slowing down of time in a system, as seen by an
observer in motion relative to the system.
 The expression
tm 
ts
v2
1 2
c
represents time dilation for all moving objects.
 Time is not absolute: both simultaneous and time duration events that are
simultaneous to one observer may not be simultaneous to another; the time
interval between two events as measured by one observer may differ from
that measured by another.
 Proper length Ls is the length of the object, as measured by an observer at
rest relative to the object.
 Length contraction occurs only in the direction of motion and is expressed as
v2
Lm Ls 1  2 .
c
 The magnitude p of the relativistic momentum increases as the speed
increases according to the relationship
p
mv
v2
1 2
c
.
 The rest mass m of an object is its mass in the inertial frame in which the
object is at rest and is the only mass that can be uniquely defined.
 It is impossible for an object of nonzero rest mass to be accelerated to the
speed of light..
Time Dilation
We have always wondered about time, does time proceed at the same rate for
everybody? Can time go backwards? It turns out that one of the most cherished
physical concepts is relative, that is, relative to the observer. There is no such thing as
absolute time.
Let’s look at an hypothetical situation.
We will have two observers measuring time intervals for the same sequence of events.
Assume a spaceship contains two parallel mirrors (one on top and one on bottom inside
the spaceship) and a method of transmitting a pulse of light from the bottom mirror to
the top mirror. Inside the bottom mirror the astronaut has placed a clock that records a
“tick” at the instant the pulse leaves the mirror and a “tock” when the pulse returns.
For the astronaut riding in the spaceship, stationary with respect to the clock, the pulse
goes up and down whether the spaceship is at rest relative to Earth or moving with a
constant high velocity relative to Earth.
The interval of the tick and tock is written as
2ts , we use the subscript s to remind us
that the astronaut, stationary relative to the clock, is taking the measurement. The
interval required for light to travel from the bottom mirror to the top is 
t s and the
distance between the two mirrors is
ct2 .
Note: Most advanced textbooks use another notation to indicate a frame that is moving
and a frame that is stationary (remember these are relative terms, as the
moving frame could correctly state they are stationary and the frame we called
stationary is the frame actually moving). The stationary frame has no prime
notation, for example S, while a moving frame would have a prime notation such
as S 
. all measured properties have a subscript zero as a reminder that the
apparatus is at rest, with zero velocity to the measurer (in what every frame the
measurer is in). That is a clock measured in a moving frame by an observer
would measure t0
Now if the spaceship is moving with a speed v relative to another observer on Earth.
From that observer’s viewpoint the pulse takes a longer interval
t m for the light to
travel from the bottom mirror to the top mirror. We write the subscript m as a reminder
that in this frame, the mirrors are moving.
Let’s look at where this extra time come from.
In the same time that the pulse moves from the bottom mirror to the top mirror, the
spaceship moves a distance of v 
t m relative to the observer on Earth. Now Einstein’s
second postulate states that light has the same speed c for both observers, therefore
using the distance triangle, it follows that
ctm vtm cts 
2
2
Now isolate
2
t m to see how it compares to t s .
ctm vtm cts 
2
2
2
c2 
tm v2 
tm c2 
ts 
2
2
2
2
v2
2
2

t

 m  2 tm ts 
c
2
2 v 
2
tm 1 2 ts 
 c 
t m 
t s
v2
1 2
c
t s is the time interval for the observer stationary relative to the sequence of
events and 
tm is the time interval for an observer moving with a speed v relative to
Where
the sequence of events.
1
Note: The expression
is so common in relativity applications that is is given
v2
1 2
c
the symbol
The equation
. So the expression tm 
t m 
ts
v2
1 2
c
shows that
t s
v2
1 2
c
becomes
t m 
ts
tm ts . That is the time interval seen by the
Earthbound observer, moving relative to the mirrors, must be greater than the
corresponding time interval seen by the observer inside the spaceship.
Another way of looking at this is if the time interval represents the beating of a heart,
as measured by the observer with the clocks (also known as proper time). Now the
Earthbound observer would say that the interval of time between the beats is longer.
A phrase that may help you remember this is “A moving clock runs slower” when
observed by a stationary observer, this is called time dilation.
You can think of time “dilated” or “expanded” on a moving object. This may initially
seem to be a bit confusing as we determined 
t s t m . But remember from the
stationary observer the local interval (eg watching 1 hour on rocket ship’s clock,
t s ) is
expanded (dilated), thus slowed down.
This holds for any “clock”, from watch, oscillating spring, heart beat, all events are
slowed down (relative to stationary observer). There is no absolute time.
v2
ts can be a real number if 1  2 is positive, that is
c
2
2
v
v
1  2 0  2 1  v 2 c 2  v c
c
c
Note:
This is the “speed limit” proposed by Einstein, No object with mass can have a speed
that is equal to or greater than that of light.
These time dilation effects have been verified more than any other experiment. Every
time somebody uses a GPS to check their position, the effects of time dilation (from
both special relativity and General relativity) have to be taken into consideration for an
accurate position to be calculated.
Proper time
There is only one frame of reference in which a clock is at rest, and there are infinitely
many frames of reference in which the clocks are moving. Therefore the time interval
measured between two events (such as ticks on a clock) that occur at the same point in
a particular frame is more a fundamental quantity, than the interval between events at
different points. We use the term proper time to describe the time interval 
t s between
to events that occur at the same point.
It is important to note that the time interval
t m in our equation involves events that
occur at different space points in the frame of reference S (non moving). Any difference
between tm and proper time ts are not caused by differences in the times required
for light to travel from those space points to an observer at rest is S. We assume that
the observer is able to correct for difference in light transit times. We say that the two
clocks have been synchronized by either (or both observers) prior to the observations.
Example
Mr Spock and his Vulcan spaceship has a speed of 0.700c with respect to Earth. He
determines, by observing an event on Earth , that 24h was the time interval between
two events. What value would they determine for this interval if their ship had a speed
of 0.950c with respect to Earth.
Solution:
First, since motion is relative, from Mr. Spock’s perspective, it is the Earth that is
moving, and since the Vulcan’s measured time was 24h for the event on Earth, 
t m . So
we need to determine how long the event would be measured on Earth,
t s (which
should take less Vulcan time).
Note:
t m 
Proper time is always the shortest time, and it is where the clocks that
measured the two events are at the same place.
ts
v2
1 2
c
v2
t s t m 1  2
c
0.700c 
2
24.0h 1 
c2
17.1h
Now, when Spock’s ship has a velocity of 0.95c and
t s 17.1h , we need to determine,
t m , the time interval in Vulcans’s frame.
t m 

ts
v2
1 2
c
17.1h
0.95c 
2
1
c2
54.8h
The Twin Paradox
This was one of Einstein’s famous thought experiments.
Consider identical twin astronauts named Bob and Sam. Bob remains on Earth while his
twin Sam takes off on a high speed trip through the galaxy. Because of time dilation,
Bob sees Sam’s heartbeat and all other life processes proceeding more slowly than his
own. Thus to Bob, Sam ages more slowly, when Sam returns to Earth he is younger
(has aged less) that Bob.
Now here is the paradox: all inertial frames are equivalent. Can’t Sam make exactly the
same arguments that Bob is in fact the younger? Then each twin measures the other to
be younger when they are back together, and that’s a paradox.
To resolve the paradox, we recognise that the twins are not identical in all respects.
While Bob remains in an approximately inertial frame at all times, Sam must accelerate
with respect to that initial frame during parts of his trip in order to leave, turn around,
and return to Earth. Bob’s reference frame is always approximately inertial; while Sam’s
is often far from inertial. Thus Bob is correct, when Sam returns, he is younger than
Bob.