We tend to think of the mass of a body as a
constant property of the body itself. 

In some sense this is justified: if you go
to the moon, you might weigh about six times less than you weigh on
earth but your mass doesn't change (assuming you keep the same diet!). 

However, we will be concerned here with the
mass of a body as measured by an observer in motion relative to
that body. 

Any measurements made on distant, rapidly
moving bodies must involve the use of light or some other
electromagnetic radiation, as was discussed here. 

What follows is a possible method by which
observer A can make measurements from which the mass of spaceship B
can be calculated. 

We will start with A and B close together and at rest relative to
each other. 



B now starts the rocket
motor and accelerates away from A. 



B leaves the motor running for a given time
(according to his/her clock, of course). 

Now, by an amazing coincidence, just at the
very instant the B stops the motor, looking out the window,
he/she sees C. 

If that's not enough, it
is observed that B and C are stationary relative to each other
(now, there's a real coincidence!) 



A now makes a measurement of the velocity of
B (and C) relative to A and let's say this turns out to be half the
velocity of light, as shown in the diagram above. 

As soon as this
measurement has been made, B starts the motor again and leaves it
running at the same power, for exactly the same time (on
his/her clock) as during the previous period of acceleration. 



We must now have the situation shown in this
diagram (seen from C's point of view). 

Having been informed by A that the measured
velocity was 0.5c, C must conclude that the velocity of B relative
to C is also 0.5c (starting from rest, same rocket motor power, same time, therefore,
same change in velocity). 

We can now use the equation for combining
relative velocities, to transform C's observations to A's frame
of reference: 



This means that during the first period of
acceleration, the velocity of B relative to A changed by 0.5c. 

During the second
period of acceleration (starting from rest, same rocket motor power,
same time), it only changed by 0.3c. 

Since the force causing the acceleration
is exactly the same in the two cases, A must conclude that,
according to his/her measurements, the mass (inertia) of the
spaceship has increased. 

From A's point of view, it is getting harder
and harder to increase the speed of B. 



We conclude that the (inertial) mass of a
body is a relative concept. 

Observers in relative motion will
obtain different values for the mass of a body. 



The mass of a body measured by an observer
at rest relative to the body is called (not surprisingly)
the rest mass of the body. Other observers will find values
greater than the rest mass. 



As the variation of (measured) mass is
basically due to the time dilation effect, you should not be
surprised to find that, if the rest mass of a body is m_{o},
then its mass, m, as measured by an observer moving with speed v
relative to the body is given by 



abbreviated to 



Looking at this equation we see that as v
approaches c, the mass, m approaches infinity. 

As infinite mass seems unreasonable, we can
consider this as an explanation of why we can never measure a
relative velocity between two bodies (possessing rest mass) equal to
(or greater than) the velocity of light. 



Momentum 

As in Newtonian mechanics, the concept of
momentum (or "quantity of motion" as Newton called it), is also
useful in special relativity. 

In special relativity momentum is defined in
exactly the same way (the product of mass and velocity) but we must
be careful to stress that the velocity, v, is the velocity of
the body relative to the observer 



This reminds us that
momentum too must be a relative concept. 

The momentum possessed by a body
depends on the state of motion of the body relative to the observer
measuring the momentum 


