Special Relativity

Lectured by Prof. Arjun Berera

February 20, 2014

The Nature of Light

For a long time, physicists wondered about the nature of light. In particular, they wanted toknow if light was a wave or a particle.

It was not until the dawn of the 20th century and the advent of Quantum Mechanics before itwas understood that light, like matter, had a dual nature; it can behave both as a wave and aparticle - more on this in other lectures.

In the 19th century, Thomas Young conducted some experiments that led to a pervading beliefthat light was a wave. This thinking led to Maxwell developing his famous equations describingthe propagation of light and other electromagnetic radiation.

If light was a wave, then, physicists argued that there had to be some medium through whichthis wave could propagate - like sound does in air. They imagined that this medium, whichthey called the ether, permeated the whole of space and then it was this that light travelledthrough at a fixed velocity.

Michelson-Morley Experiment - 1

At the end of the 19th century, Michelson and Morley conducted an experiment to testwhether or not the ether was really there.

Because the Earth moves around the sun, it must mean that the Earths motion relative to thisether changes during the year.

v"

S"

D"

The experiment uses a piece of equipment called an interferometer. A diagram of thisequipment is shown. It consists of a light source (S) which fires a light beam towards thecentre.

At the centre, the beam is split into two beams (red and blue) which travel at right angles toeach other towards mirrors located a distance L from the centre.

Each beam hits the mirror and bounces back to the centre, at which point the two beamsrecombine and head towards the detector (D).

Michelson-Morley Experiment - 2

Now, suppose this interferometer is moving at some velocity v through the ether in thedirection of the arrow. The light beam going parallel to this motion (red), then, takes sometime T||1 to reach the mirror. Light travels at a speed c through the ether, so in this time ittravels a distance cT||1. Looking at the diagram, we see that the distance it has travelled isalso the length L plus the additional amount the mirror travelled in this time, vT||1. Thus, thedistance travelled = cT||1 = L+ vT||1. You can rearrange this equation to find T||1 = L(cv) .

You can look at this another way - if the interferometer is moving with velocity v with respectto the ether, it is the same as the ether moving with velocity v in the opposite direction withrespect to the interferometer.

So as the light moves in the right-hand direction, it is moving in the opposite direction to theflow of the ether. Thus the velocity of light is slowed to c v . It is similar to if you wereswimming upstream in a river - as you move forward, the river flow (or in this case, the etherflow) carries you back.

Michelson-Morley Experiment - 3 Now lets think about the journey back from the mirror to the centre. The light is now moving

in the direction of the ether flow so its speed is enhanced rather than slowed and thereforeT||2 = Lc+v . This gives a total time for the journey to and fro of T||1 + T||2 = T|| =

2Lcc2v2 .

12 vT

L

12 cT

For the journey across the ether (blue), you can apply Pythagoras theorem to the triangleshown in the diagram (in this case, the light doesnt travel with or against the ether flow so itsspeed does not change) to find that the time to go up and back from the mirror isT? = 2Lp

c2v2.

Thus, the dierence between the times is about T|| T? Lc ( vc )2. This is a positivenumber, so the beam going perpendicular to the ether flow will arrive back at the centre soonerthan the beam that goes parallel. Because of this, the two light beams going these twodierent directions when they combine again will have slightly shifted waves, which can thenbe detected.

When the experiment was done, however, it showed that there was no such shift in the lightwaves!

Towards Special Relativity - A Solution to the Null Result

After this unexpected result, physicists were confused and were scrambling to find anexplanation. Many physicists still wanted to believe the ether was there and various ideas wereproposed. For example, some thought that the Earth dragged the ether along as it movedthrough it, so there would be no relative motion. However, most of these ideas proved to beunsuccessful.

One idea, proposed independently by Fitzgerald and Lorentz, had it that in the direction ofmotion an object gets contracted. Lorentz even produced an exact formula for this contractionthat could precisely explain the result of the Michelson-Morley experiment.

This idea, however, led to a lot of complicated questions about the nature of forces - forexample, do all forces contract in that direction or just the electromagnetic force? Thesequestions had no easy answer.

In 1905, Albert Einstein came along with his now famous Theory of Special Relativity.

Towards Special Relativity - The Two Postulates

Einsteins theory rested on the truth of two postulates:

1) The laws of Physics are the same in all reference frames moving with a constantvelocity with respect to each other.

2) The speed of light, c, is the same in all reference frames moving with a constantvelocity with respect to each other.

The second of these looks quite simple, but is actually very unintuitive. We will now seewhy.

The Second Postulate

The second postulate states that all observers will agree on the speed of light no matter whatthe relative velocity between them is, so long as it is constant. For example, if you and a friendboth start at rest and your friend switches on a torch, you will both obviously measure thesame value for the speed of light, c. Now, if your friend starts to move in the direction of thelight beam coming from the torch and you stand still, you will both still agree on the exactsame value for the speed of light, c.

This is not what you would expect. Replace the beams of light with tennis balls, for example.When you are both at rest, when your friend throws the ball in front of them you will measuresome velocity v for the ball that your friend will agree with. If your friend now moves andthrows the ball in their direction of motion, they will tell you that the velocity of the ball wasagain v . You would disagree, however. You would say that the velocity of the ball was v plusthe velocity at which your friend was moving. Both of you are correct in your own referenceframes, but disagree on the actual number. This doesnt happen with light and that is whatEinstein realised.

Time and Special Relativity - 1

The postulates of Special Relativity lead to several consequences with are unintuitive. For onething, they completely altered our notion of time. We will see this now in a simple example.

Let us consider a clock composed of two mirrors separated by a distance d and between them alight pulse bounces. The light pulse going from the bottom to the top and then back to thebottom can be regarded as one unit of time, or one tick of this light clock.

Let us place two identical light clocks, one in what we will call the lab frame, which is at rest,and one in a rocket frame, which is moving at velocity v with respect to the lab. With respectto the lab frame, the clock on the rocket travels a longer path, the diagonal path shown in thediagram.

Since the speed of light is the same in both frames according to Special Relativity, the labframe will conclude that a light pulse in the rocket clock completes a tick in a time tCR = 2Dcthat is longer than the time it takes the lab clock to tick which takes some time tCL = 2dc .Therefore, the lab frame concludes that the moving clock in the rocket is ticking at a slowerrate.

Time and Special Relativity - 2

12 vtCR =

Dvc

d

D

If the clock in the lab frame measures some interval of time t, then the time interval the labframe will see the rocket clock to have, t0, will be reduced by the ratio of dD .

We can compute D using the pythagorean expression. D is the hypotenuse of the right angledtriangle in the diagram. The vertical side is the length of the light clock d . The horizontal sideis the distance the rocket has travelled in time

tCR2 , which is the time it takes for the light to go

from the bottom to the top. So, D2 = d2 + ( vc )2D2. Rearranging, this means D = dq

1( vc )2.

This means t0 is less than t by a factor, which is related to what physicists call the gammafactor, 1 =

q1 ( vc )2. This is how much slower the rocket clock appears to be going with

respect to an observer at rest with the lab clock.

Length Contraction - 1

The length of objects are also aected in Special Relativity. Looking again at the lab androcket systems, suppose now we have a rod which measures length L00 in the rocket framewhere it sits at rest. Here the zero subscript means proper length, which is the terminologyfor the length of an object measured in the frame where it is at rest.

Let us measure the length of the rod in the lab frame by noting the times t1 and t2 when thefront and back of the rod respectively go past point P. Then the length of the rod measured inthe lab frame is L = v(t2 t1). We will call t the time interval t2 t1.

The times t1 and t2 in the lab have corresponding times t01 and t02 measured in the rocket, withL00 = v(t

02 t01). Similarly we will call the time interval t02 t01 = t0.

Length Contraction - 2

With respect to the rod, the lab clock is now moving, so its time is slowed ast = t0

q1 ( vc )2

So, in the lab frame, L = vt = vt0q

1 ( vc )2 = L00q

1 ( vc )2

Thus the length of the moving rod has contracted when measured in the lab frame. Note thatthis is the same expression for length contraction that Lorentz had initially derived, but theinterpretation of it in Special Relativity is very dierent from what Lorentz had proposed. Infact Lorentz also had initially derived the expression for time slowing down we showed earlier,but again his interpretation was very dierent from that in Special Relativity.

Note there is something interesting about the gamma factor. You can only take the squareroot of a positive number, so 0 v2

c2 1! v c. Thus, this equation shows that an object

cannot move any faster than the speed of light. In fact, it turns out that anything with masshas to travel strictly slower than the speed of light. This would seem to violate the tennis ballexample from earlier, then - it was said that the velocity is just additive. However, in reality, itis not exactly additive and there are corrections to this. These corrections are tiny unless youstart going at a speed comparable to that of light and at this point, these corrections becomeimportant and act in such a way as to not allow an object to reach the speed of light.

Invariant Interval

In Special Relativity, if two events are measured with time separation t between them andlength separation x between them, then it turns out the quantity c2t2 x2 is the samenumber with respect to coordinates in all dierent frames moving at constant velocity withrespect to each other - lets see this in an example.

In the previous case where the length of the rod was measured, in the lab frame a time tlapsed between measuring the front and back of the rod. Both measurements were done at thesame point in space, so x = 0.

Now in the rocket frame, the time lapse was t0 between measuring the front and back of therod and there was a spatial separation x0 = L00. We also have that L

00 = vt

0.

Lets now check the value of this invariant interval expression in both frames and ask ourselveswhether the two are equal, as expected by Special Relativity. Namely, we ask doesc2t2 = c2t02 x02 = c2t02 v2t02?

We had t = t0q

1 ( vc )2 by the relation for time slowing down. Inserting this above, wefind indeed that this expression for the invariant interval is the same in the two frames.

Energy-Mass Relation - 1

Lab$

x$

t$

x$

t$ Par*cle$

Finally, we now look at the relation between energy and mass. For this let us look at themotion of a particle of mass m in two dierent frames. We will call the prime frame the one atrest with the particle and the lab frame the one where the particle is moving at velocity v .

In the particle frame, at all times t0 the particle is at position x0 = 0, since as we said theparticle sits at rest in this frame, so its spatial coordinate never changes. If we now take the

derivative dt0

dt0 = 1 anddx0dt0 = 0. Recall here that in the mathematics of calculus, the derivative

for example dx0

dt0 is simply the ratiox0(t0+t0)x0(t0)

t0 in the limit that the time interval t0 gets

smaller and smaller. It describes how one quantity changes with another quantity.

So the expression m2c2 dt0dt0

2 m2 dx0dt0

2is simply the same invariant interval we just discussed

c2t02 x02 except with both expressions divided by the same factor of t02. Also note the interval is multiplied by an overall factor of the particle mass squared m2. So

what we get for this invariant interval expression is m2c2.

Energy-Mass Relation - 2

Now lets look in the lab frame where the particle is moving. At time t in the lab the particle isat position x .

We have mc2 dtdt0 which we are simply calling pT . Note here we take the derivative with respect

to the same quantity, t0, and we have simply called this expression pT to give it some name.We also have m dx

dt0 = mu = p, which is just the momentum of the particle.

So we have for this interval p2Tc2 p2 and since this expression must equal the invariant interval

in the particle frame, it means it equals m2c2. This therefore leads to the relationpT =

pm2c4 + p2c2.

When the first term is much larger than the second in the expression in the square root, wefind that pT mc2 + p

2

2m . The second term is simply the kinetic energy of a particle of massm. So we identify pT as the energy of the particle, pT = E .

Now, we have the general relation E 2 = m2c4 + p2c2. As such, when the particle is at rest,p = 0, we have the famous relation, E = mc2. But this expression is more general and tells usthe energy of the particle also when it is moving with nonzero momentum p. It tells us energyis equal to the mass of the particle plus its energy of motion, through the p2 momentum term.

Energy-Mass Relation - 3

So this equation tells us the relationship between energy and mass. For example, in nuclearreactions such as atomic explosions and nuclear power plants, you start o with some mass andthen convert that to energy.

But this relation tells us of eects that work the other way, too. For example, in the LargeHadron Collider, we collide two protons with a lot of kinetic energy each. They smash togetherand a lot of this energy of motion goes into creating new particles - thus, to mass.

This concludes our discussion of Special Relativity and we now understand the origin of therelation between energy and mass.

References

Spacetime Physics, Edwin F. Taylor and John A. WheelerW. H. Freeman 1963.

Physics Volume I, Duane E. Roller and Ronald BlumHolden-Day 1981.

Foundations of Modern Cosmology, John F. Hawley and Katherine A. HolcombOxford University Press 2005

Space Time Matter, Herman WeylDover Books 1952.