Lecture 4
Mass and Energy

To give Relativity a complete, self-consistent structure we must add a few more "stones". From what we have learnt so far, a body cannot reach the speed of light. It is legitimate to ask what prevents it from being accelerated beyond c. Let us recall what Newton had to say :

A constant force will produce a constant acceleration , i.e.

F = m a

In a gedanken experiment, and within Newtonian Physics, we could think of applying a constant force to a body, and so keep accelerating it past the speed of light. This is obviously inconsistent with Relativity.

Parenthesis : The Physics of Star Trek (by Lawrence Krauss)

In Star Trek, the Enterprise can suddenly be accelerated to near c. Is this realistic ? The human body can only sustain accelerations of a few times the acceleration of gravity (i.e. a few "g's"). How long would it take to accelerate a spaceship and its occupants, from 0 to c, with a constant acceleration of, say, 10 g's?

Instant acceleration is therefore non-realistic, after the first episodes the Star Trek script writers introduced "inertial dampers" to shield the crew from lethal accelerations.

But let's go back to our problem. Remembering the meaning of acceleration :

$a = \Delta v/\Delta t$

Newton's law could be writtten as

$F = m \Delta v/\Delta t$

In reality, as Newton knew himself, the correct expression is

$F = \Delta(mv)/\Delta t$

For most practical cases the mass is constant so we do not have to worry about its variation, even though exceptions can be found even in Newtonian Physics: a standard problem given in Introductory Physics courses is to study the motion of a train wagon full of sand, but with a hole on the floor from which sand continuously escapes....

But what do we mean by mass ? In Classical Physics we can think of it in two respects:

1.

the mass of a body is something related to its quantity of matter , i.e. something connected to the number of atoms (as well as the mass of each atom...)

2.

at the same time the mass is also a measure of the body's inertia, i.e. its reluctance to having its velocity changed.

In Newtonian Physics these two pictures are interchangeable, but Einstein broke this correspondance : in Relativity, the inertia of an object is not a constant quantity, characteristic of each object, but it is a function of velocity :

$m = m_{0}/\sqrt{1-v^{2}/c^{2}}$

where m₀ is the rest mass, i.e. the mass of the object as measured at rest (and related therefore to its "quantity of matter"). We can then immediately see how velocities are going to be constraned below c : the more I try to accelerate an object, the more it opposes my attempts, so that, in order to accelerate it to c I would need to act upon it with an infinite Force (or for an infinite time) and neither possibility is very practical...

Putting it all together :

if we use the standard notation $\gamma = 1/\sqrt{1-v^{2}/c^{2}},\ 1\leq\gamma\leq\infty$, then we can write :

$t' = \gamma t ,\hspace{0.5in} l' = l/\gamma ,\hspace{0.5in} m = m_{0}\gamma$

Question :

Can anything move at the speed of light ?

The answer is that masssless objects can (and do) move at the speed of light. We can then distinguish between matter and radiation.

The relation between mass and velocity is verified matter-of-factly in any Particle Accelerator, where sub-atomic particles going round and round a ring are given repeated boosts of energy. After a while, the velocity of the particle hardly increases, even though the periodic kicks do result in a gradual increase of the particle's energy. There must then be a relation between the increase in mass and the increase in energy, and of course we all know the answer :

E = m c²

which, from what we have seen, could be written as

$E = m_{0}\gamma c^2$

This expression tells us two things :

1.

the increase in (kinetic) energy of an accelerating body is not related to an increase in velocity, but to an increase in mass

2.

even a body at rest possesses a (potentially enormous) amount of energy, given by E = m₀c² .

The relation between mass and energy is even more general than you might have thought. It turns out that every time a system acquires a certain amount of energy of whichever type (kinetic, gravitational, thermal, chemical, etc.), its mass increases, according to Einstein's formula. Obviously for all everyday instances, the mass differentials are completely below any measurable level. Exceptions are nuclear phenomena, where mass losses become non negligible (i.e. of the order of percent) and, moreover, masses can be measured without the need of a scale...

Deferring to a later chapter the discussion of nuclear energy (from fission or from fusion), we can for now amuse ourselves answering the following question:

is there some way to recover all (or at least most) of the energy contained in the rest mass ?

The answer is yes, there are at least two processes, even though neither of them is (yet?) very practical.

1.

MATTER-ANTIMATTER ANNIHILATION

Sub-atomic Physics research has shown that every type of particle (proton, neutron, electron, neutrino, muon, etc.) has its anti-particle partner. Anti-particle are identical to their twin particles, except that all the "charges" (e.g. electric charge) are of the opposite sign. Moreover, a particle- antiparticle pair has an intriguing property : when they meet, they can annihilate all of their mass into a blast of energy.

Unluckily, even though we know how to "create" antimatter, our procedures are extremely inefficient. And, at best, we would get out of it as much energy as we have to put in. A neat solution would be if we were to find an "antimatter deposit" somewhere in the universe, but,as of today, the chances appear to be exceedingly slim.

2.

BLACK HOLE POWER STATION

Contrary to the original belief, i.e. that nothing can come out of it, according to the current understanding, if we throw mass into a black hole, half of it is spat out as radiative energy. Unluckily, it is even less likely that such process will ever be practical.