Chapter 28

Relativity

At the end of the 19th century, this was the picture of light:

• light is an ElectroMagnetic Wave (with wavelengths in the hundred of nanometers range) which, as predicted by Maxwell's equations, propagates, obviously, at the speed of light.
• waves are perturbation of a medium, traveling through it at a speed which is characteristic of the medium itself. Consequence : the wave velocity, as measured by an observer, will depend on the relative velocity between medium and observer. Even though this is different from the case of a moving particle, it is still true that the observed velocity of a wave should depend upon the motion of the frame of reference.
• if light is a wave, and a wave is a perturbation of a medium traveling through it, what is the wave-carrying medium in the case of light? In lack of a better answer, physicist were thinking in terms of a hypothetical ether (or aether), some very tenuous ("ethereal") substance permeating all space, from here to the farthest stars.

Towards the end of the century (in 1887) a wrinkle appeared in this otherwise nice picture. An experiment devised by Michelson and Morley, to detect variations in the speed of light in a direction parallel or perpendicular to the motion of the earth through the ether, failed to show any difference.

Even though puzzling, the result of M&M experiment was not, at the time, recognized in all of its very far reaching importance, but it was felt that some reasonable explanations would still be found, within the rules of Classical Physics. But this is when Einstein enters the scene....

In his thinking, Einstein was not so much affected by the results of M&M experiment, but by a much more fundamental issue:

as we have seen, the speed of propagation of EM waves came out of Maxwell's equations as an absolute quantity, $c = 1/\sqrt{\epsilon_{0}\mu_{0}}$, unrelated to any specific frame of reference. One had the choice of either accepting this very basic fact, or throw away the very successful (and mathematically extremely elegant) theory of ElectroMagnetism, as described by Maxwell's equations. Einstein knew which way to choose, performing what was at the same time a very simple but incredibly bold step. Here is his reasoning:

1.

Nature does not favour any given frame of reference, all frames moving with respect to each other with uniform velocity are indistinguishable

2.

Maxwell's equations produce an absolute value for the speed of EM waves, not a relative one with respect to some frame of reference. Consequently

the speed of light must be the same in any uniformly moving frame of reference

In other words, light (in vacuum) will always be observed to move at the speed c ($\sim 3\times 10^{8} m/s$), regardless of the speed of the light source or of the observer.

This is it, this is all that is necessary to build the incredibly far-reaching theory of Special Relativity. But of course this is just the starting point, the cornerstone around which you have to build in order to get the complete picture. The problem with taking a bold step, is that then you must be able to follow through. Einstein had "the right stuff" to be able to do it.

So, if you want to hold on to the idea that the speed of light is the same in any frame of reference, something else has to give... And what gave was the absolute concept of time. In fact, what reason do you have to believe that time is an absolute quantity ? Einstein would have told you that, even if this is what common sense tells us, there is no reason for it to be true. And if we allow this possibility, then we can start building the New Physics.

In Einstein's picture, we do not live in an environment with 3 (relative) space dimensions and 1 (absolute) time dimension, but our universe evolves in a 4-dimensional space, where time has to be treated on an(almost) equal footing with the three spatial dimensions. Time intervals should then not be treated as absolute quantities, but instead as something whose value can depend on the motion of the observer. Once we are willing to follow this route, it is fairly straightforward to get an expression relating the time measured by a clock at rest with that given by a moving clock, under the assumption that the speed of light is constant in any frame of reference. Using the standard example of the light clock, as discussed in the book, one gets the fundamental result that if an observer compares the time t₀ measured by a clock at rest with respect to himself with the time t given by a clock moving with velocity v, he observes that the moving clock is ticking more slowly, according to

$t = t_{0}/\sqrt{1-v^{2}/c^{2}} = t_{0}\gamma$

i.e. one "tic" on the moving clock occurs only every $\gamma$ tics of the stationary clock, with $\gamma = 1/\sqrt{1-v^{2}/c^{2}} \geq 1$. The bottom line is then that time elapses more slowly for an object in (fast) motion. And, we are nowadays convinced, this does not just apply to "light clocks" and the like, but it is a real fact, applying to all physical (including biological) phenomena.

Question : the station-master sees a fast train going by, and, in agreement with Einstein's results, he realizes that the passengers' clocks are ticking more slowly than his. How will the station-master's clock appear to run to the train passengers?

As puzzling as it might be, we reach the conclusion that there is no absolute time; when a person A is moving carrying his own clock, the time he measures is correct for himself (and is referred to as proper time), but any other observer B in relative motion with respect to A will measure, on his own clock, a different time duration for the same event. Understandably, at the time Relativity was first introduced (in 1905), scientists had difficulties in accepting such revolutionary concepts, even though they had to admire the mathematical elegance and self-consistency of the whole theory. And for many decades, various people attempted to find holes in the theory, but invariably with no success. One of the most famous arguments is the so-called twin paradox :

two twin brothers say goodbye to each other, as one stays on earth and the other embarks on a super fast spaceship, traveling at some fraction of the speed of light for many years. To the brother on earth it appears that his twin, ages more slowly then himself. But on the other side, from the astronaut's point of view, it is the brother on earth that is moving, and therefore it is the earth based clock that is ticking more slowly. But of course both possibilities cannot be simultaneously true, when the traveler returns on earth, he must be either younger or older than his brother, but not both....Hence, it was stated, relativity leads to a conceptual impossibility.....

The resolution of the dilemma is based on the fact that, in this example, it is possible to state absolutely which of the two was moving: the astronaut, in order to leave earth, reach a high speed, go somewhere, turn around and land back onto earth, must undergo several steps of acceleration. And given that it is possible to assess which twin was in fact accelerating, then it is also possible to establish who was moving and who wasn't... In conclusion, the astronaut will come back to earth and find himself in the future

But notice that, even though you can in principle travel into the future, not even Relativity allows you to travel into the past. This is just as well, since this would give raise to some well known logical inconsistencies (i.e. you could go back in time and kill your grand-mother before your mother was born.....).

Another relevant factor that impeded the universal acceptance of the theory of relativity was that, for a very long time, there was no way to verify it experimentally. Nowadays, observation of time dilation is common place in any laboratory dealing with elementary particles, that can be accelerated to a good fraction of the speed of light. In these cases, the "lifetime" of fast moving, unstable sub-atomic particles is observed to stretch in perfect agreement with the rules of relativity.

Probably such a phenomenon was demonstrated for the first time in 1941 by measuring the flux of cosmic-ray generated muons at a higher (Echo Lake) vs. lower (Denver) altitude: knowing the lifetime of the muon "at rest", it was possible to infer which fraction of the muons present at a higher altitude could still be present at a lower altitude, and the observed result was in perfect agreement with the time dilation predicted by relativity.

The relativity phenomena we are studying here belong to Einstein's theory of Special Relativity, that was presented in an almost complete form in 1905, but that deals exclusively with zero-acceleration motions. Within a decade, Einstein extended his results to any type of motion in his theory of General Relativity. You might have heard that the first experimental verification of the theory of relativity occurred during the 1919 solar eclipse, when it was verified that light is affected by gravity. The bending of light by gravity is a prediction not of Special but of General Relativity

Hint for solving time-dilation related problems : the basic expression to apply is

$t_{v} = \gamma t_{proper}$

in any problem at hand, the crucial step is to decide which one is (or which observer measures) the proper time. And the answer is always : the proper time is the one read from a clock at rest with respect to the observer.

Once we believe in time dilation, we can understand a related concept of relativity, i.e. space contraction. Here is one way to look at it :

A superfast spaceship (v=.98c, $\gamma=5$) leaves earth on a journey to a star which is 5 light years away. As measured from a clock on earth, the journey will take 5/.98 = 5.1 years. But, 5.1 years of earth measured time correspond to only $5.1/\gamma = 1.02$ years of time on the spaceship. From the spacemen point of view, they see the distant star approaching them at a speed of 0.98c, and they see the star reaching them after 1.02 years. If they hadn't studied relativity, they would make fun of the earth based astronomer, since they would conclude that the original distance estimates were all wrong, and the actual distance to the star was just 1 light year, since d = vt = v/c c t = .98 c 1.02 = 1 light year.

(remember: to get the value of distances in light-years, you need to express velocities in terms of units of c, and times in terms of years)

In reality, what has happend is that lengths are not absolute quantities either, but their value depends on whether they are measured for an object at rest or in motion with respect to the measurer: the 5 light years measured at rest get transformed into $5/\gamma =1$ light year when measurer and measuree are in motion with respect to each other. In other words, fast moving objects appear to shrink, according to the expression:

$L' = L\sqrt{1-v^{2}/c^{2}} = L/\gamma$

where L is the length of an object when measured at rest, i.e. its proper length and L' is its length when moving at velocity v. Since the square root term goes from 1 to 0 as v goes from 0 to c, lengths gradually shrink to zero as velocities approach c. But be aware that, in Einstein's formulae, this length contraction occurs only along the direction of motion. A square object moving in the direction of its base will shrink to a thinner and thinner rectangle of constant height.

Remembering the long living muons, whose observation proved the correctness of time dilation, length contraction can explain the same phenomenon, as seen from the muon's point of view :

muons speeding towards the ground, will see the distance between them and the ground shrunk by an extent determined by their actual velocity. It will then not be a surprise to them that they can do the trip before their lifetime has elapsed !!.

Time dilation and length contraction are tied together in the global relativity picture, and they can be shown to provide a perfectly consistent picture, when taken together with another consequence of relativity, i.e. the non-simultaneity of events in different reference frames. Two events that appear to occur at the same time in a frame at rest, will not appear to be simultaneous in a moving frame.

You might wonder whether, in this sea of relativity, there is some rock of stability. The answer is yes, and examining it will give us some more insight in the theory of relativity.

In "ordinary" Physics (and Geometry), the length of an object is the same, regardless of the frame it's measured in. By Pythagoras theorem, in 3-dimensional space lengths are given by :

l² = x²+y²+z²

regardless of the choice of reference frame (x,y,z) . We have just seen that, in Einstein's picture, lengths are not absolute anymore, but it could also be shown that, even if measured in a moving frame, the following "length" is invariant :

l² = x²+y²+z²-(ct)²

This expression highlights the fact that, in the correct relativistic description of the world, time should be considered at the same level as the space coordinates. We are not living in a 3 but in a 4-dimensional universe.

The last step to complete the picture is to understand the correct relativistic relation among velocities. A simple example will explain the need for this last step:

Suppose you are moving at 0.8 c towards some observer, and shoot at him a projectile at a speed (as measured with respect to yourself) of 0.5 c . What will be the speed of the projectile seen by the observer, according to Galileo or Newton ?

The answer (1.3c) is in obvious conflict with Einstein's relativity, and in fact in relativity the "naive" Galileian rule for velocity combination has to be abandoned in favour of a new rule that can be deduced within the framework of Einstein's theory:

if an object is moving with velocity u in a given frame F, and this frame is moving with velocity v with respect to another frame F', the velocity w of the object in F' is given by

w = (u+v)/(1+uv/c²)

If u and/or v are much less than c, then this expression reduces to the familiar w = u+v. But notice also that, in the extreme case of u=c, then w=c, i.e. an object moving at the speed of light in a given frame, will do so in any other frame.

Mass and Energy

To give Relativity a complete, self-consistent structure we must add a few more "stones" to the building. 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

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, but Einstein theory shows how these ideas are to be modified. Remembering the definition 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 encountered in Classical Physics, 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 :

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

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 constrained to be less than 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 :

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

Question :

Can anything move at the speed of light ?

The answer is that masssless objects can (and do) move at the speed of light.

The relation between mass and velocity is verified matter of factly in particle accelerators, where protons or electrons are going round and round, and at each turn they receive an extra boost 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 :

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

This expression tells us two things :

1.

the (kinetic) energy of an object can become infinitly large, even though its speed can never exceed c.

2.

a body at rest ($\gamma = 1$) possesses a (potentially enormous) amount of energy, given by E = m₀c² . m₀ is normally referred to as the rest mass

When an object is moving with velocity v, we know that it possesses a certain amount of kinetic energy (=mv²/2 in Classical Physics). In Relativity, he total energy of a moving object is $mc^{2} = m_{0}\gamma c^2$,and this must be the sum of "rest energy" plus kinetic energy. The kinetic energy will then be

$E_{kin} = E_{tot}-E_{rest} = m_{0}\gamma c^2 - m_{0}c^2 = m_{0}c^2(\gamma-1)$

For v<<c this expression reduces to the familiar one.

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-particles 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.

You should be aware anyway there presently there is at least one practical application of matter-antimatter annihilation : this is the PET (Positron Emission Tomography) Scan

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.