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introduction to
Special Relativity

In order to describe motion we need to describe how that motion is seen by a particular observer. In effect the observer is considered stationary and motion is described relative to that stationary viewpoint. This is the original sense of relativity and is commonly referred to as Galilean Relativity to acknowledge the important contributions of the scientist Galileo Galilei (1564-1642).

Galilean Relativity

Consider two trains side by side in a railway station.

2 mph

1 mph

As a passenger in the blue train, looking out the window directly at the red train, it is easy to be confused as to which of you is moving. The red train can fill your field of view. You can feel the vibration of the carriage, but it is unclear if your train is stationary, moving forwards, or moving backwards. All you can be really sure of is that, relative to you, the red train is moving forwards.

The Speed of Light

We see things using light. If objects start moving with speeds approaching the Speed of Light, we should not be surprised if strange things happen.

The Speed of Light in free space is typically denoted by c

c = 299,792,458 m/s or roughly 670 million mph.

The Meaning of Simultaneous

Everybody knows that simultaneous means happening at the same time, but achieving this is not necessarily straightforward.

For an (old style) battle where the cavalry and the foot-soldiers need to strike simultaneously, the foot-soldiers have to set off first when crossing open ground since the horses charge faster than the soldiers can run. We should conclude that if the time to get from one place to another (the transit time) is significant, we need to be quite careful with our thinking concerning simultaneous events.

Truly Simultaneous Events

Consider two flashing light sources A & B and a detector D. If we ensure that D is exactly the same distance from both A and B we can be sure that if we receive light flashes from each source at the same time, the flashes originated at exactly the same time.

A

B

D

For the sake of completeness we should ensure that the time between flashes is much longer than the transit time from the source to the detector so that we don't accidentally synchronise to one light source a whole number of cycles out of phase with the other light source!

Multiple Simultaneous Events

I have arranged a group of similar flashing light sources into a straight line. The one marked R is the same as all the rest, but I have arbitrarily decided to use that one as my reference. I adjust each source in turn, moving the detector D so that it is equidistant between the reference and the source being adjusted. In the picture above I am adjusting the source labelled A. Notice that the line of sources can be arbitrarily long. After having been calibrated, they emit light pulses at exactly the same time and therefore define a line of simultaneous events.

R

A

D

Space-Time Diagram

This is a conventional space-time diagram. By only using one space direction the diagram is easier to draw. The yellow dots represent the positions of the flashing light sources at some arbitrary zero on the time axis. The red lines are called world lines and show how the objects move in space-time. In the example above the light sources are stationary and therefore their x-values stay constant as time progresses.

x

t

0

Space-Time Diagram

A space-time diagram is most useful when dealing with high speeds. It is therefore convenient to scale the time axis such that the speed of light corresponds to a 45 line.

x

t

The red lines are the light rays from the sources that happen to go towards the time axis.

0

Space-Time Diagram

In general, the speed of an object is inversely proportional to the slope of the world line.

x

t

t

x

0

Speed Ratio

Rather than using big numbers, it is convenient to express any velocity, v, as a ratio compared to the speed of light, c.

Things only start to get interesting when

but since this is greater than 30 million mph, it is not the sort of speed you would ordinarily encounter!

Space-Time Diagram

We have already decided to represent the speed of light by a 45 line. We can then say

x

t

t

x

provided we measure time in distance units.

NOTE: Some people prefer to write ct on the time axis, rather than t, in order to achieve the same result.

0

Lorentz Factor

The Lorentz factor simplifies our calculations

At low speeds (beta) is close to zero and (gamma) is close to one.

Where does this complicated factor come from? In 1887 Michelson & Morley were trying to measure the effect of the movement of the Earth through space to a higher accuracy than earlier attempts. The null result was soon explained by FitzGerald as being due to contraction of moving objects along the direction of motion, and explained in more detail by Lorentz in 1892. It was not until 1905, however, that Einstein really explained the factor when he published the Special Theory of Relativity, a fundamental postulate of which is that light travels at the same speed, regardless of the relative (un-accelerated) motions of the source and the observer(s).

Lorentz Transformation

The Lorentz transformation tells us how to convert from one observer's viewpoint to another. Once you understand how to apply this simple rule you will be able to solve many problems relating to high speed phenomena.

Take a point ( x, t ) on the space-time diagram for one observer and convert it to a new point ( x, t ) on the space-time diagram for another observer.

Notice how time and space become more and more inter-mixed as the speed becomes larger. At low speeds with = 0 and =1 no special co-ordinate transformation is necessary.

Conventional notation in the study of Special Relativity uses a superscript mark on one set of co-ordinates. This mark can be read as either prime or dash. Thus x can be read as x dash.

Lorentz Transformation

Since we want to compare what one observer sees with what another observer sees, we need to be able to swap between the two co-ordinate systems freely. Hence we need two sets of transformation equations.

Given one pair of Lorentz transformation equations, it is not difficult to re-arrange them to give the other pair of Lorentz transformation equations shown above. (Try it!) Notice that the only difference is in the sign of the velocity term.

The Barn & Pole Puzzle

In this historic puzzle there is a stationary 10m long barn with open ends. A runner is carrying a 20m long pole, with the pole horizontal and pointing in the direction the runner is moving. Will the whole length of the pole fit in the barn if the runner is going so fast that =2 ?

In this plan view the barn (red) is shown with dotted (open) ends. The pole (blue) is shown at the moment it is just about to enter the barn. I choose this point as the origin of both co-ordinate systems for simplicity.

Please note that I have not drawn this sketch to scale. The reason for this will become evident very soon!

The Barn & Pole Puzzle

We have 4 points to draw on our space-time diagram and 4 space-time paths (world lines) to draw from these points. This is enough to solve the problem!

We are going to draw the space-time diagram in the frame of reference of the barn. We are using dimensions of metres throughout as the problem is stated in metres. The first point is the entrance of the barn which we agreed would be at ( 0, 0 ). The second point is the exit of the barn which is therefore at ( 0, 10 ). The third point is the leading end of the pole which is again at ( 0, 0 ) because of our earlier definition. It is the fourth point which is the interesting one.

We know that the trailing end of the pole is at ( -20, 0 ), in other words we know its co-ordinates only in the frame of reference of the runner, the dash frame. This is where we use the Lorentz transformation

The Barn & Pole Puzzle

barn entrance

x

t

10

barn exit

First plot the known barn co-ordinates

0

The Barn & Pole Puzzle

barn entrance world line

x

t

10

barn exit world line

Next plot the barn entrance and exit world-lines

0

The Barn & Pole Puzzle

x

t

10

barn exit world line

Now plot the leading edge of the pole's world-line

0

Don't try too hard to make your drawing to scale. It will never be accurate enough to read values from measured lengths. Use it as a sketch and calculate all the critical intersection points.

Remember that the world line of the pole has a slope

We can therefore calculate the point at which the the world lines of the barn exit and the pole intersect as

The Barn & Pole Puzzle

x

t

10

barn exit world line

Now plot the trailing edge of the pole's world-line, parallel to the leading edge world-line since both world-lines have the same slope

0

barn entrance world line

In retrospect it seems as if we could have made the drawing simpler if we had defined the origins of the co-ordinate systems as the point when the leading edge of the pole is just about to exit the barn. Try it as an exercise.

The Barn & Pole Puzzle

Our simple sketched drawing was not accurate enough to answer the original question. However we know the key points now. The front end of the pole starts to exit the barn at time

All we now need to do is to establish where the other end of the pole will be at this same time. The trailing end of the pole starts from the point (-40, -40 ) as established earlier.

The required x value is therefore evaluated as

So the puzzle is solved. The pole fits exactly within the barn.

Lorentz Contraction

We can generalise the calculation we have just done using a rod of length L as measured in the dash frame. The start of the rod is at ( 0, 0 ) and the end of the rod is at ( -L, 0 ). We Lorentz transform these two points so that the start is at ( 0, 0 ) and the end is at (-L, -L)

x

t

0

-L

-L

-L

x

t

The rod is most conveniently measured at the 0 time value, as shown in the space-time diagram shown to the right.

The Lorentz contraction formula

Remembering that

x

t

0

-L

-L

-L

x

t

The Barn & Pole Puzzle - Again

Now that we have the Lorentz contraction formula we can see that the 20m pole fits into the 10m barn because the pole is moving and the Lorentz contraction factor is . But wait a minute, as far as the runner is concerned the barn could be thought of as moving. In this case it is the barn that should shrink. It is this aspect that is the real puzzle. Let's redraw our diagram in the frame of reference of the runner. This time I am going to define the origins of both co-ordinate systems to be when the leading edge of the pole is about to leave the barn.

x

t

0

-20

world line of trailing end of pole

world line of leading end of pole

world line of barn exit

world line of barn entrance

-5

Notice that this time I haven't bothered to calculate anything. I know that the barn is Lorentz contracted to 5m so I can just draw it in on the x axis. I know the slope of the barn world lines are 1/ so again I can just draw them in.

Special Relativity

In summary, Special Relativity relates to frames of reference that move with uniform (un-accelerated) motion relative to each other and where the effect of gravity can be neglected. It is needed when the speeds involved become a significant fraction of the speed of light.

If observers are moving relative to each other with a significant speed, they can no longer be guaranteed to even agree on the order in which events occurred. They will strongly disagree on length measurements in the direction of motion.

As we have seen with the Barn-Pole puzzle, the different observers see something completely different from each other. People get very upset with these paradoxes as surely both viewpoints can't be correct. In reality the pole can't both fit inside the barn and not fit inside the barn.

No amount of calculation or drawing will ever help with this difficulty. No particular viewpoint is any more valid than any other, but the world will appear different to these different observers.

Velocity Transformation

Let the event A in the dash frame be due to a particle moving from the origin to some (arbitrary) x value of p at a speed of . Then A= ( p, p/ )

The co-ordinate frames are synchronised such that ( 0, 0 ) = ( 0, 0 ) and the dash frame is moving in the x direction at a speed of . Transform the co-ordinates of A. The ratio of x over t values then gives the velocity.

x

x

x

t

t

A

x

t

t

A

Notice that the (dimensionless) velocity never goes above 1, even if both and are 1.

Relativistic Doppler Shift

x

t

-T

-2T

-3T

t

x

Periodic light flashes in the dash frame

transform to periodic pulses in the non-dash frame, but not at x=0

( 0, -T ) = ( -T, -T )

-T

-T

-T

0

So we move the flashes back to the x=0 line at the speed of light to find the true interval T in the non-dash frame (red lines).

which can also be expressed as

Note that the sign of also changes when the sign of x changes.

Wrap Up

Whilst Special Relativity is essentially universally accepted as true nowadays, over 100 years since its inception, this was not always the case. It took many years of increasingly precise experimental work and debate amongst physicists for this theory to gain widespread acceptance. Therefore do not be surprised if this brief presentation by a non-specialist is entirely convincing to you.

The fact is that much of the deep physics of Special Relativity, General Relativity, Quantum Mechanics, Quantum Electro-Dynamics (QED), Quantum Chromo-Dynamics (QCD), and Quantum Field Theory (QFT) is beyond everyday common sense notions of how things ought to be. These are nevertheless very successful theories in terms of their prediction / calculation capabilities.

When accelerations and/or gravitational fields are involved, the subject moves from Special Relativity to General Relativity. Particle accelerators are the key places where relativistic particles are created and used.

In terms of normal experience, the use of GPS systems for in-car navigation are fairly common. What is less commonly known is that the clocks in the satellites used to provide the GPS system are affected by Special and General Relativity such that corrections need to be applied to accurately synchronise and use them. (Relativity in the Global Positioning System by Neil Ashby [2003])

Further Study

There are plenty of internet science / physics forums to discuss these ideas or ask questions of people who know much more about this subject than me.

There are a great many text books on Special Relativity available, and you would have to find one that best suits you (if you are in the market to actually buy one). On the other hand, there are plenty of free resources available on the web including a free wiki book on Special Relativity.

Hyperlinks can go dead quite quickly, but a search engine should be able to find what you need.

Wikipedia has a great many articles related to Special Relativity and the linked pages can help you to explore the depth of knowledge available.

Some general internet search topics include Special RelativityTime dilationMinkowski diagram / Minkowski spacetime

Physics for Future Presidents (free) video lecture series by Prof Richard Muller, especially lectures 21 and 22 (on Special Relativity).

To boost your general physics knowledge you could try the (free) video lecture series by Prof Walter Lewin.

Special Relativity & Electrodynamics (free) video lecture series by Prof Leonard Susskind. This is very mathematically based lecture series, part of his Theoretical Minimum series, and could reasonably be described as hard work to study.