Alexandria University Faculty of Engineering Computers And

8/14/2019 Alexandria University Faculty of Engineering Computers And

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Alexandria University

Faculty of Engineering

Computers and Systems Department

Theory of RelativityTreatise on the Applications of the Special Theory of

Relativity



Theory of Relativity

2

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Introduction

The most interesting aspects of Special Relativity are its 'paradoxes.' We put

'paradoxes' in quotes because Special Relativity is, in fact, an entirely self-consistent theory

that contains no true paradoxes (that is, no paradoxes that cannot be resolved with a little

careful thought). The fun part is thinking through the apparent paradoxes to find where the




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logical error that leads to the inconsistency lies. We have already encountered several of theclassic paradoxes of Special Relativity, and in this topic we will uncover several more.

The first section examines the relativistic Doppler Effect, which is an interesting and

useful extension of the Doppler Effect in classical mechanics. It is especially important

because relativistic speeds need to be considered for one of the main instances, in which the

Doppler Effect becomes important, the red-shift of light reaching us from far galaxies. Thesecond section deals with the most famous 'paradox' of relativity: the so-called Twin Paradox.

The third section extends what we have learned about energy, momentum and 4-vectors tomore interesting problems involving the decay of and collision between particles.

The aim of this treatise is to demonstrate that Special Relativity, as unfamiliar and

unintuitive as it may seem, does have important applications in areas of physics ranging from

the very small (sub-microscopic particle interactions) to the very large (motion of stars and

galaxies, cosmology). Moreover, the relationship between the electric and magnetic forces

and fields is bound up with Special Relativity; the interaction ultimately produces oscillations

of the electromagnetic field that is light itself. The results of Special Relativity are not as

abstract as they at first appear to be!

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The classical Doppler Effect can be observed with any type of waves. When the

source of the waves if moving towards the observer it causes the waves to bunch up, resulting

in an apparently higher frequency. Similarly, if the source is receding from the observer, the

waves are spread out and the frequency appears smaller. The effect of time dilation between a

moving source and an observer complicates this situation (frequency is inverse time); the

relativistic adjustment to the Doppler Effect is called the relativistic Doppler Effect.





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The observed frequency is just f = 1/

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Note that if the source is moving away from the observer,v/c is negative and thus f < f . For

the source approaching the observer . This result is qualitatively the same as for the

normal (non-relativistic) Doppler Effect.

Transverse Doppler E ect

Consider the x - y plane with an observer at rest at the or igin. A straight train track traverses

the line y = y 0 . A train with a laser mounted on it emits light with frequency f . Consider :

Figure I : The T ra

sverse Doppl er effect .

There are two interesting questions posed by the diagram: What is the frequency with which

the light hits the observer just as the train is at the position of closest approach to the or igin

(at point(0, y 0) --illustrated in i) )? And what is the frequency of the light emitted justas the

train passes the point of closest approach(0, y 0) , as seen by the observer (illustrated in ii)?

R ecall that we must consider the time taken for the light to reach the observer (otherwise the

distinction between the two questions above is meaningless). In the f irst case, even though

the train is already at (0, y 0) , the observer will be seeing at an ear lier time (the light takes

time to reach her), thus the photons will be observed arr iving at an angle, as shown. In the

second case, the photons have come to the observer directly along the y -axis; of course the

train will already be past the y -axis by the time this light reaches the observer.



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2010 In the f irst case, let us consider things from the frame of the train. An observer on the

train, O' sees the observer at the or iginO moving past to the lef t with speed v . The light in

question hits O just as he crosses the y' -axis in O' . Time dilation tells us that O's clock ticks

slowly such that t ' = t . To say the opposite ( t = t ' ) is not true of the time at

which O sees the light arr ive. This is because for t = t ' to hold we need x' = 0 ; this is

true of the emission of the light, but sinceO is moving in the frame of the train,O does not receive ad jacent light pulses in the same place, hence x' 0 . Thus it is true that the

frequency of the light is lower is the frame of O than in the frame of O' , but because of the

relative motion of the source and observer O observes the frequency as being higher, as we

shall see. If we want to analyze the situation from the point of view of O , we have to take

longitudinal effects into account; by usingO' we have avoided this complication. In the frame

of the train, then, the observer at the or igin gets hit by a 'peak' every t ' = 1/ f ' seconds (Here

we make the assumption that the train is close to the y' -axis and thus that the distance

between the train and the source is constant at y 0 for the time it takes the light to reach the

observer--in this way we eliminate any longitudinal effects). The observer at rest then gets hit

by a 'peak' every T seconds, where:

Thus, like the longitudinal Doppler Effect, the frequency observed at the or igin (for someone

at rest) is greater than the emitted frequency.

In the second case we can work in the frame of O without complication.O sees the clock

of O' run slowly (since O' is moving relative toO), and thus t = t ' . Here the observed

frequency is:

In this case the observed frequency (for the observer at rest at the or igin) is less than the

emitted frequency by a factor

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The so-called 'Twin Paradox' is one of the most famous problems in all of science.

For tunately for relativity it is not a paradox at all. As has been mentioned, Special and

General R elativity are both self-consistent within themselves and within physics. We will




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state the twin paradox here and then describe some of the ways in which the paradox can be

resolved.

The usual statement of the paradox is that one twin (call her A) remains at rest on the

earth relative to another twin who flies from the earth to a distant star at a high velocity

(compared to c). Call the flying twin B. B reaches the star and turns around and returns to

earth. The twin on earth (A) will see B's clock running slowly due to time dilation. So if the

twins compare ages back on earth, twin B should be younger. However, from B's point of

view (in her reference frame) A is moving away at high speed as B moves towards the distant

star and later A is moving towards B at high speed as B moves back towards the earth.

According to B, then, time should run slowly for A on both legs of the trip; thus A should be

younger than B! It is not possible that both twins can be right-the twins can compare clocks

back on earth and either A's must show more time than B's or vice-versa (or perhaps they are

the same). Who is right? Which twin is younger?

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The reasoning from A's frame is correct: twin B is younger. The simplest way toexplain this is to say that in order for twin B to leave the earth and travel to a distance star she

must accelerate to speed v . Then when she reaches the star she must slow down and

eventually turn around and accelerate in the other direction. Finally, when B reaches the earth

again she must decelerate from v to land once more on the earth. Since B's route involves

acceleration, her frame cannot be considered an inertial reference frame and thus none of the

reasoning applied above (such as time dilation) can be applied. To deal with the situation in

B's frame we must enter into a much more complicated analysis involving accelerating

frames of reference; this is the subject of General Relativity. It turns out that while the B is

moving with speed v A's clock does run comparatively slow, but when B is accelerating the

A's clocks run faster to such an extent that the overall elapsed time is measured as beingshorter in B's frame. Thus the reasoning in A's frame is correct and B is younger.

However, we can also resolve the paradox without resorting to General Relativity.

Consider B's path to the distant star lined with many lamps. The lamps flash on and off

simultaneously in twin A's frame. Let the time measured between successive flashes of the

lamps in A's frame be t A . What is the time between flashes in B's frame? As we learned the

flashes cannot occur simultaneously in B's frame; in fact B measures the flashes ahead of him

to occur earlier than the flashes behind him (B is moving towards those lamps ahead of him).

Since B is always moving towards the flashes which happen earlier the time between flashes

is less in B's frame.

In B's frame the distance between flash-events is zero (B is at rest) so x B = 0,

thus gives:



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2010 Thus the time between f lashes is less in B's frame than in A's frame. N is the total

number of f lashes that B sees dur ing her entire journey. Both twins must agree on the number

of f lashes seen dur ing the journey. Thus the total time of the journey in A's frame isT A = N

t A , and the total time in B's frame is . Thus:

Thus the total journey time is less in B's frame and hence she is the younger twin.

All this is f ine. But what about in B's frame? Why can't we employ the same analysis

of A moving past f lashing lamps to show that in fact A is younger? The simple answer is that

the concept of 'B's frame' is ambiguous; B in fact is in two different frames depending on her

direction of travel. This can be seen on the Minkowsk i diagram in :

Figure %: M ink ow sk i d iagram o f t he t win parado x.

Here is lines of simultaneity in B's frame are sloped one way for the outward journey

and the other way for the tr i p back ; this leaves a gap in the middle where A observes no

f lashes, leading to an overall measurement of more time in A's frame. If the distant star is

distance d from the ear th in A's frame and the f lashes occur at intervals t B in B's frame, then

they occur at intervals t B/ = t A in A's frame, as per the usual time dilation effect (this isthe same for inward and outward journeys). Again let the twins agree that there are N total

f lashes dur ing the journey. The total time is B's frame is then T B = N t B and for

A,T A = N ( t B/) + where is the time where A observes no f lashes (see the Minkowsk i

diagram). In B's frame the distance between the ear th and the star is (half the total

journey time times the speed) which is also equal to d / due to the usual length contraction.

Thus:



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+ t

What is ? We can see from that the slopes of the lines are the time in which A

observes no f lashes is Thus:

Compar ing T A and T B we see T B = T A / which is the same result we arr ived at above. A

measures more time and B is younger.

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The result from the theory of relativity that Mass and Energy are differentmanifestations of the same physical entity and that it is possible to convert mass into energy

finds an application in the processes of nuclear fission and nuclear fusion.

In the process of nuclear fission, a large unstable nucleus such as that of Uranium-235

decays into two smaller nuclei. Interestingly, the sum of masses of these two nuclei is smaller

than the mass of the original larger nucleus. This "mass defect" is responsible for the releaseof a large amount of energy in this process of nuclear fission. The difference in mass, when

multiplied with the square of the speed of light in vacuum (that is, c2), gives the amount of

energy released in the process by the famous equation - E = mc2

Where, E is the energy released in the process,

m is the mass difference,

c is the speed of light in vacuum.

In the process of nuclear fusion, two small nuclei combine to form a larger nucleus

again with the release of a large amount of energy. The sum of masses of the original nucleiis greater than the mass of the resulting nucleus and this "mass defect" is responsible for the

release of energy in nuclear fusion. Again the same equation of mass-energy equivalence may be applied to find out the energy released.

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One of the most obvious uses of relativity is by the nuclear reactor which is used in

nuclear power plants to generate much of the world's electrical power by employing




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to convert mass into usable energy. The atom bomb uses this same concept toinstantaneously create massive amounts of energy from a very small amount of matter.

It is very useful to physicists. Relativity nicely explains some odd questions in

physics, such as how the detection of muons on Earth is possible, since (as we observe it)

muons must travel great distances. After all, they only have a lifetime of about 2.2

microseconds and should decay after about 700 meters. Relativity says that because of itshigh velocity the muon's time slows down as observed by us so that to the muon the distance

it must travel is significantly shorter.

Relativity is also useful to cosmologists to explain how our universe came to be, and

what the ultimate fate of our universe will be.

It's possible that relativity will become a more important part of our lives astechnology advances. Physicists debate whether we will ever travel through time. If so,

relativity might be intrinsic in the development of the home time machine. Also, travel todistant galaxies would be greatly aided by the theory. You can get to the nearest star in a

decent amount of time (for you) but time on Earth will have gone by much quicker, and all

your Earth friends will have long since passed by the time you get back.

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Alexandria University Faculty of Engineering Computers And