In the previous two sections, we saw that our perception of length and time change when we observe objects in frames of reference that are moving with respect to us.  Length and time are two of the fundamental quantities required to describe the universe.  The third is mass, and as you might guess, our perception of mass changes when we deal with moving frames of reference.  Quite apart from the direct consequences of this, it also leads to possibly the most famous equation in physics, E = mc².

Before we discuss this equation, we must first clarify what we mean by the "mass" of an object.  Many people (including some scientists who should know better) use the terms "mass" and "weight" interchangeably, as though they are the same thing.  They are not.  The mass of an object is a measure of the inertia (resistance to motion) of that object.  The weight of an object is a measure of the gravitational force that acts on the object, in a gravitational field.

As a consequence, the weight of an object varies, depending on the strength of the gravitational field at the point at which weight is measured.  The mass of that object is a constant property of the object, its weight is not.  Thus, an object of mass 1 kg has that mass whether it is on earth or in space; the weight of the object varies, and in outer space the weight can be zero (weightlessness).

Though we will not go into it in too much detail, the mass of an object that is moving is observed by a stationary observer to be greater than when the mass is stationary.  The situation is illustrated in the following figure.

The situation is completely analogous to the time dilation and length contraction effects we looked at previously, and in the notation we have used previously, the mass of the moving object (M_A) is given in terms of the mass of the stationary object (M_B) by

where v is the velocity with which the object is moving, and c is the velocity of light.

For those who are interested, the origin of the mass increase is the requirement that momentum should be conserved in all frames of reference, as required by the first postulate of relativity.

OK, so let's take a look at where the mass-energy equation, E = mc², comes from.  Perhaps the easiest way to derive the expression is to consider an object of mass M₀, and calculate the kinetic energy (motion energy) of that object at speed v, noting that the mass of the object increases as the speed v increases.  I don't really want to go into too many details, as it gets a bit mathematical, but I will try to outline the procedure for the calculation.

To physicists, the easiest way to calculate kinetic energy from first principles is to compute the "work" that is required to move the object from being stationary up to speed v.  The first question, of course, is what do we mean here by the term "work"?  Mathematically, the term "work" is the product of the force applied to an object and the distance that the object moves during application of the force.  A simple example will illustrate why this definition is a good one for measuring the energy supplied to an object.

Consider trying to push your car by hand.  You apply a force to the car, and it moves.  If you increase the force you apply, then you supply more energy and it moves faster.  You also increase the energy supplied according to how far you push the car - the further you push it, the more energy you supply and the faster it goes.  Thus, varying the force and the distance pushed are ways to increase the energy supplied (i.e. increase the kinetic energy of the car).

The mathematical details of computing kinetic energy on this way are not difficult, though a little hairy in the maths, and if you apply this procedure to an object and allow for the increase in mass (to a stationary observer) as the speed increases, then you arrive at the following expression for the kinetic energy of the object:

where M (in the first tem on the right hand side) is the expression for the mass of the object when it is travelling at speed v:

This expression can be rewritten in the following way:

This is extremely revealing, as it tells us that the total energy of the body (the term on the left of the above equation) consists of two components.  The first is simply the kinetic energy K of the body, but the second is a term that attributes energy to the mass of the object, and which means that the object possesses energy, even when the speed (and hence the kinetic energy K) is zero!

This is the famous equation that we are seeking, the one that defines the energy of an object that arises from its mass.  Here it is, written out explicitly:

This equation states that an object has a certain amount of energy associated with its mass, and in fact it implies that mass and energy are interchangeable.  In a moment we will discuss one means by which this is manifested, but first let us get a feel for the magnitude of this "rest-mass energy", as it is often called.  The rest-mass energy of an object of mass 1 kg can be evaluated by noting that the speed of light is 300,000,000 metres per second.  Therefore, the rest mass energy is

90,000,000,000,000,000 joules,

which is a very big number!  (Note the the standard unit of energy is the joule - to put this in perspective, it requires 4,200 joules to heat one litre of water by one degree Celsius).  In practice, we can only tap into a very small portion of this energy, as it is tied up in all of the atoms and molecules of the object.

However, we can make use of this energy in a very practical way, and one which is very closely related to the work that I do in my career.  One means of generating electricity is to build nuclear power stations that are based on a process known as "nuclear fission".  In nuclear fission, heavy, unstable atomic nuclei split into smaller fragments.  However, if you add up the masses of the fragments and compare the total with the mass of the original nucleus, then you find that the total mass of the fragments is slightly smaller that that of the original nucleus - some mass has got lost somewhere.

What has happened is that some of the mass has been converted into energy, according to our expression for rest-mass energy above.  This energy is the basis for the nuclear power generation, as it can be used to heat water and generate steam, which is then used to drive generators that create electricity.  The energy that has been created from the fission process manifests itself in the kinetic energy of the fission fragments - if you like, when the big nucleus fissions, the fragments "fly off" at high speed with all that released energy.

The following diagram illustrates the process of nuclear fission.

So there you have it, a practical application of E = mc².  The opposite process - conversion of energy to mass - can also occur, and a nuclear process known as "pair production" is an example.  In this process, a gamma ray photon (which has energy but zero mass) with sufficient energy can be converted into an electron and positron (both of which have mass).

If you didn't follow the discussion in this section, the key message is as follows:

A moving object appears to have a greater mass than one that is stationary.  In addition, this consideration can be used to demonstrate that all objects have a rest-mass energy associated with them.  Under the right conditions, mass and energy can be converted from one to the other according to E = mc².

On to the Twin Paradox