## E me2 the most famous result of all

So far our discussion on relativity seems to have brought us no nearer to the ultimate relation, E = mc2, for which Einstein is best known. The reason for this is that we have been dealing exclusively with kinematics, the study of motion without reference to force. We will now turn our attention to dynamics, the study which encompasses force, energy and momentum. In this context we are interested in how these dynamical variables appear in different frames of reference; in particular, frames which travel relative to one another at near the speed of light.

In a collision between two particles the forces of action and reaction are equal and give rise to a rate of change of momentum, but how is this rate of change affected by time dilation when speeds near the speed of light are involved? We start with a fascinating thought experiment of a collision of snooker balls from different relativistic time frames.

### 16.2.1 Bringing energy into the picture

Returning to the relationship E = mc2. The principles of relativity deal with symmetry of inertial frames and with the constancy of the speed of light. It is not at all obvious how these lead to the equivalence of mass and energy. That mass is a form of energy becomes particularly relevant in the world of the atomic nucleus.

To find the connection between mass and energy we must first look at momentum and energy from different frames of reference. Let us consider momentum first. The following is an imaginary experiment.

16.2.2 Conservation of momentum — a thought experiment with snooker balls

There are two snooker players, one in a railway station, and the other in a passing train. Each strikes a cue ball as in the diagram below. The balls are identical and each player strikes his ball with an identical speed in his own reference frame. The experiment is carefully arranged so that the two balls collide at an open window, just as the train passes the station, then bounce back as in the diagram. It is a head-on collision, so the cue balls bounce straight back along their original paths.  Figure 16.2 An absurd kind of snooker. The diagram combines the two points of view: inside the train as seen in frame S', and outside the train as seen in frame S.

There is complete symmetry in the situation. Each player is equally entitled to his point of view. Each considers himself to be stationary in his own frame of reference, and if the cue ball bounces straight back to one player, exactly the same thing happens to the other cue ball for the other player. Incidentally, the train is a 'super-train' travelling at a speed comparable to the speed of light.

To put this discussion into perspective, one of the fastest trains in the world is the TGV (Train a grande vitesse); it set a world record speed of 515 km per hour in France in 1990. This is still a far cry from our super-train, which, in order to show rel-ativistic effects, would have to travel at somewhere near half the speed of light, or 150,000 km per second, about 10,000 times faster than the TGV on its record-breaking run.

This argument is well described by the German word Gedankenexperiment', an experiment performed in one's imagination. The result also comes from the resources of the mind.

The collision is perfectly elastic. The two balls collide 'head-on' and each goes back exactly along its path in the frame of

To observer A it appears that his own ball returns straight back with reversed velocity and momentum, whereas the other ball has struck only a glancing blow and continues on with the train, travelling from left to right. Observer B in the train comes to the opposite conclusion. As far as he is concerned, his ball comes straight back to him, and the other ball glances off, travelling from his left to his right. The situation is symmetrical whether we are using classical or relativistic laws. Let us assume that at an earlier stage each observer had the opportunity to examine the two snooker balls, side by side in his own laboratory. They were satisfied that the two balls were identical. In particular, their masses were equal.

### 16.2.3 Interacting with another time frame

Our thought experiment involves a super-train travelling near the speed of light. There is a collision between two masses from different time frames. This is something Newton had never envisaged! The laws of dynamics involve time as a basic parameter and we must analyse, in our minds, what new considerations come into play in such a unique situation!

Each observer will notice something peculiar. Each uses his own clock, which measures proper time. Time in the other frame appears to run more slowly (remember, the astronaut's cigar lasts longer, and his movements appear slower). Here the other snooker player appears more lethargic, his movements seem slower as viewed from outside his rest frame, and he appears to strike the ball more slowly.

Figure 16.3 illustrates an apparent inconsistency in the dynamics of the collision. For each observer the perpendicular component of the velocity of the other player's ball is smaller.

Snooker quality control laboratory

Snooker quality control laboratory  super-train

his frame his frame my frame collision as seen from my frame

Why is his change of momentum in the z-direction (grey arrows) smaller than mine?

Figure 16.3 Apparent momentum imbalance.

Each observer will infer a momentum imbalance, the change in momentum of his own ball being larger. (Each observer remembers that the two balls have the same mass.)

We must assume that the law of conservation of momentum holds at every stage and in all frames of reference. If this were not the case, one could identify a 'privileged' frame in which momentum is conserved, as opposed to other frames in which this is not the case. This would contradict the theory of relativity.

An alternative way to express the inconsistency:

According to Newton's third law, action and reaction are equal. But the active and reactive forces are each equal to the rate of change in momentum of the bodies on which they act. This rate of change is slower in the other frame because of time dilation.

### 16.2.4 Momentum from another frame of reference

The two sets of observations can be made consistent by multiplying the classical expression for the momentum of the other ball by the factor 7, which compensates exactly for the decrease of velocity due to time dilation.

As will be seen below in Equation (16.2), the juxtaposition of this factor with the mass term can be interpreted as saying that the mass of the particle in the other frame increases by the relativistic gamma factor, g = ■ 1 = (v is the speed of the train*) f?

We can define the rest mass of an object m0 as the mass as measured in its own frame of reference (rest frame). Once the object is moving relative to the observer, we of course no longer observe it in its rest frame.

For small velocities, we can use Newton's definition of momentum p = const x v, where the constant is defined as the inertial mass.

In summary, we can define rest mass m0 by

If momentum is to be conserved in all frames of reference, then it must be defined as:

Relativistic momentum formula (16.2)

### 16.2.5 A new look at the concept of mass

Mass is not an invariant quantity. Viewed from another frame of reference, the mass of a particle is always greater than its 'rest mass' m0 measured in its own frame of reference. A moving particle has greater mass.

* If we want to be very strict, we can replace v by VR , the resultant velocity of the ball in the train. Figure 16.4 Relativistic mass as a function of velocity.

Relativistic mass = ym0 = mass of an object moving with speed v relative to an observer. m0 is the mass when the object is at rest relative to the observer.

As the speed approaches c the relativistic mass approaches infinity. If you have mass, you cannot go with the speed of light, not to mention 'faster than light'.

### 16.2.6 The relativistic formula for momentum

The concept of frames of reference gives us an insight into the fundamental symmetry of the laws of nature. Once the concept has been established and the relativistic formula has been derived, we can use it in any convenient frame of reference. We can take away the speeding train and simply consider a glancing collision between a snooker ball travelling at high speed from left to right, and another ball travelling in a direction perpendicular to the first ball. (Since we happen to be inhabitants of the earth, the rest frame of the earth is normally the most convenient.)

Even though the high speed component of the first ball is left to right, i.e. in the x direction, the observer will find that the y components of the velocity of the two balls A and B are not equal in magnitude either before or after the collision. If he uses the classical formula for momentum (p = m0v) he will find an apparent momentum imbalance. The situation is rectified as before, by using the relativistic formula p = jm0v. The mass of the 'glancing' ball appears bigger because it has a large component in the x direction and therefore its resultant velocity is bigger. The fact that the main component of v is 'sideways', i.e. in the x direction, is of no consequence. Remember, the gamma factor is independent of direction. Things which are moving have a greater mass than they had when they were stationary.

Once we apply the relativistic formula for momentum to both of the balls A and B, the momentum will balance exactly in every direction. Figure 16.5 The collision as seen from the earth frame of reference.

Momentum of A =

mrvv

0U A

Momentum of B =

m0v R

16.2.7 Energy in different frames of reference

We now turn our attention to the concept of energy, which in classical Newtonian physics is related to work done.

Classical physics

How we calculate kinetic energy Reminder of classical definitions:

Work (W) = Force x distance in direction of force, W = Fd Energy (E) = Ability to do work

Kinetic energy (KE) = Ability to do work due to motion Classical expression for kinetic energy

Force = rate of change of momentum

Kinetic energy T = work done in accelerating a mass m from rest to velocity V = Force x distance in direction of force

16.2.8 High energy particle accelerators

At CERN the original accelerator was the Proton Synchrotron (PS), built in 1959. It was at that time the highest energy accelerator in the world, with 200 magnets positioned around a ring of diameter 200 m. It accelerated protons up to a maximum energy of 28 GeV.

The Super Proton Synchrotron (SPS), completed in 1976, had 1000 magnets around a ring 2.2 km in diameter, located in an underground tunnel. At the time the SPS was built it was also the largest accelerator in the world, with a maximum proton energy of 450 GeV. At the time of writing (2008) the CERN complex consists of a series of machines with increasingly high energies, injecting the beam each time into the next one, which takes over to bring the beam to an even higher energy, and so on. The flagship of the complex, scheduled for completion at the end of the year, will be the Large Hadron Collider (LHC), which will produce colliding beams of protons, each of energy 7 TeV (7000 GeV).

'^SlA / OXcTFJ GtMSwttll}

P Pbtoiis

The first step in the process, the original pre-injector of the Proton Synchrotron was a Cockroft-Walton accelerator developed when they first split the atomic nucleus in 1932.

Courtesy of An Post, Irish Post Office.

### The cost of a little extra speed

As far as speed is concerned, the work done to accelerate a particle nearer and nearer to the speed of light soon reaches a point of rapidly diminishing return. The speed of 400 GeV protons '^SlA / OXcTFJ GtMSwttll}

P Pbtoiis Courtesy of An Post, Irish Post Office.

emerging from the Super Synchrotron is 0.99999753 c, compared to the protons from the original 28 GeV machine, which travel at 0.999362 c. Building the new giant accelerator resulted in an increase in speed of just 0.006%! The Yfactor comes into its own in a very practical way in terms of euros or dollars or Swiss francs, in the cost of the accelerators!

However it is energy and not speed which is important, in the quest to investigate the innermost structure of matter or, as we shall see in the next chapter, to make new kinds of matter out of energy.

### Calculating the relativistic energy

The classical calculation is of course perfectly valid at 'normal' velocities, when the gamma factor can be taken as equal to 1. But now, as the mass increases, the force required for a given acceleration rapidly becomes greater with increasing velocity, and each increment of the work integral has to be multiplied by the value of y appropriate to that instant.

In Appendix 16.1 we insert into the integral the relativistic expression for momentum p = Ym0v. Since y is a function of v, the integration, while somewhat more difficult than in the classical case, can still be done by standard methods, and gives the total work done as

Work done on a particle = (change in its mass) x c2 By doing work we create mass

The classical concept that work done is invested into energy, be it kinetic energy, potential energy or heat energy, still holds, but has to be expanded to include mass energy.

For an object in empty space, with no force fields of any kind, there are no potential hills to climb, and only kinetic energy comes into play. Kinetic energy can be defined in a given frame of reference, as the difference between the energy of an object at rest and its energy when in motion.

= Total energy - rest energy

Einstein's famous relationship E = mc2 gives the total energy, which includes kinetic and rest energy taken together. The rest energy is one of the great new concepts introduced by the theory of relativity. Every particle of matter possesses energy, purely by virtue of its existence.

Mass is a form oof energy.

### Relativistic and classical kinetic energy

The expression 16.6 appears to bear no resemblance to the classical formula for kinetic energy. There seems to be no reference to the speed of the object. We can show however that the rela-tivistic expression can easily be reduced to the It is included V---L-* classical form for velocities small compared to the speed of light. Relativistic formula KE = mc - m0c The classical formula is an approximation to the relativistic formula when v is small compared with the speed of light.

How many joules of energy in a unit mass?

The relationship E = mc2 also gives the rate of exchange from one form of energy to the other. In SI units the energy contained in one unit of mass (1 kg) equals 9 x 1016 units of energy (joules). You get a lot of energy for a tiny amount of mass. Rest mass energy must not be confused with chemical energy of matter, such as the energy obtained from burning coal. Chemical energy is smaller by a large factor. Mass energy is present whenever mass exists, but is generally securely locked into the atomic nucleus and released only under very special circumstances in reactions which transform the atomic nucleus.

### Example

What is the energy contained in a grain of sand of mass = 0.001 g (= 10-3 kg)? Assuming that it were possible to convert the grain of sand completely into energy, how far, using that energy, could you drive a car against an average resistive force of 1000 newtons?

1000

In practice, we can never change mass completely into energy. In nuclear reactions, only a small fraction of mass is turned into energy.

### 16.2.9 Nuclear structure

In the classical world it is self-evident that the mass of any structure is equal to the sum of the masses of its parts. So, for example, mass of the house = sum of the masses of bricks, mortar, wood, glass and even the coat of paint. There is no need for mortar to hold the atomic nucleus together. It is composed of protons and neutrons, which are held together by the strong nuclear force — a strange kind of mortar, which manifests itself in that but the sum of the masses of the protons and neutrons is greater than the mass of the whole nucleus. If we want to take it apart, we must supply energy to create that extra mass. The percentage mass difference is very small, but because of the very unfavourable rate of exchange of mass for energy, it is enough to make it very difficult to take the nucleus apart.

rBTiass increases if taken apart rBTiass increases if taken apart

En KJ

The nuclear house

The table below gives a listing of the masses of the constituents, and the total masses of some elements at the bottom of the periodic table:

 Name Symbol Mass (a.m.u.)1" Sum of parts Difference Neutron o n 1.008987 Proton ® p 1.008145 Deuteron o® d 2.014741 2.017132 0.002391 Triton o®o t 3.016997 3.026119 0.009122 Helium3 o®® He3 3.016977 3.025277 0.008300 Helium4 oo®® He4 4.003879 4.034264 0.030385

The accounts in the table above do not balance. The whole is not equal to the sum of its parts.

For example: The sum of the masses of the constituents of stable helium

The helium nucleus has a smaller mass than the sum of the masses of two neutrons and two protons. The deficit is 0.030385 a.m.u. = 28.3 MeV/c2, or 0.76% of the total mass.

The mass deficit, Am, expressed in energy units, is called the binding energy of the structure.

16.2.10 Nuclear fusion — nature's way of powering the sun

The sun is powered by nuclear energy which should last for another 10 billion years. The basic reaction involves making helium out of deuterons. The mass difference is converted into kinetic energy of the reaction products which appears as heat and t 1 atomic mass unit (a.m.u.) = 1.66 x 10-27 kg = 931.5 MeV/c2.

creates a temperature of the order of 109 degrees Celsius inside the core of the sun. At this temperature more deuterons are driven together, thus sustaining a thermonuclear fusion reaction.

We can calculate the energy released when two neutrons form a helium nucleus, as follows:

Energy = (Am)c2 = 5.64 x 10-30 x 9 x 1016 = 5.25 x 10-13 J

Energy = 3.3 MeV 16.2.11 Nuclear fission

A fission reaction occurs when the nucleus of a heavy element such as uranium is broken into two or more fragments. An example is a neutron-induced reaction such as

In fission the sum of masses of the parts is less than the mass of the whole, and energy is released. Moreover the fission fragments, in this case nuclear isotopes of barium and krypton, are neutron-rich and highly unstable, and almost instantaneously eject two or three neutrons. These neutrons in turn can strike other uranium nuclei, causing more fissions. U238 is a relatively stable nucleus and fission will occur only by an energetic neutron, so that the chain reaction will not be sustained. However, the isotope U235 is much less stable; if a critical mass of U235 is brought together a stray cosmic ray neutron can initiate an immediate chain reaction, such as the explosion of an atomic bomb.

= 5.84 x 10-30 kg d d neutron + U238 ^ Ba145 + Kr94