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 the classic 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. The second 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 to more 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!
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.
Theory of Relativity 2010
Longitudinal Doppler Effect : When the source is moving directly towards or away from the observer the frequency observed is:
Transverse Doppler Effect : When the source is moving past the observer, displaced in the transverse direction the observed frequency is at the moment the source is at closest approach to the observer and
When the observer sees the source at closest approach. Longitudinal Doppler Effect :- There are two distinct results for the Doppler Effect is Special Relativity. The longitudinal Doppler Effect considers the simpler case of a source moving directly towards you or away from you along a straight line. The transverse Doppler Effect, on the other hand, considers what is observed when the observer is displaced in a direction perpendicular to the direction of the motion. We will take the simpler case first. In this section we must be careful to distinguish, as we have not done elsewhere, between the time an event occurs in the observer's frame and the time when the observer sees it occur; that is, we have to calculate the time it takes for the light to travel from the event to the eye of the observer. Consider a source (say a laser beam mounted on a train) coming directly towards you. The light of the laser it its own frame is f' and the train is travelling with speed v . The overall effect of the longitudinal Doppler shift is due to both the time dilation that occurs between the frames and the normal Doppler effect due to the motion of the source. For a source moving towards you, its motion compresses the wavelength of the light, increasing the frequency that is observed. If the frequency is f' in the source's frame, then the time between the emission of 'peaks' in the light waves is t' = 1/f¶. Due to time dilation the time between emission in the observers frame is then: t = t¶. One peak travels a distance c t = c t' before the next peak is emitted. Similarly, in this time between peaks the source travels v t =v t' . Hence the distance between peaks in the observer's frame is c t ± v t = (c - v) t' , where the minus sign arises because the second peak 'catches up' with the first due to the motion of the source, decreasing the distance between peaks. This holds for all adjacent peaks. The time T between the arrival of peaks at the observer's eye is the distance between peaks divided by their speed, c, thus:
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 origin. 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 Tra sverse Doppler 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 origin (at point(0, y 0) --illustrated in i) )? And what is the frequency of the light emitted just as the train passes the point of closest approach (0,y 0) , as seen by the observer (illustrated in ii)? Recall that we must consider the time taken for the light to reach the observer (otherwise the distinction between the two questions above is meaningl ss). In the first case, even though e the train is already at (0, y 0) , the observer will be seeing at an earlier time (the light takes time to reach her), thus the photons will be observed arriving 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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In the first case, let us consider things from the frame of the train. An observer on the train, O' sees the observer at the origin O moving past to the left 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 arrive. This is because for t = t' to hold we need x' = 0 ; this is true of the emission of the light, but since O is moving in the frame of the train, O does not receive adjacent 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 using O' we have avoided this complication. In the frame of the train, then, the observer at the origin gets hit by a 'peak' every t' = 1/f' seconds (Here we make the assumption that the train is close to they' -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 origin (for someone at rest) is greater than the emitted frequency. In the second case we can work in the frame ofO without complication. O sees the clock of O' run slowly (since O' is moving relative to O), and thus t = t' . Here the observed frequency is:
In this case the observed frequency (for the observer at rest at the origin) is less than the emitted frequency by a factor
6WDW QW The so-called 'Twin Paradox' is one of the most famous problems in all of science. Fortunately for relativity it is not a paradox at all. As has been mentioned, Special and General Relativity 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? 5HVROXWLRQ The reasoning from A's frame is correct: twin B is younger. The simplest way to explain 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 being shorter 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. thus In B's frame the distance between flash-events is zero (B is at rest) so x B = 0, gives:
Thus the time between flashes is less in B's frame than in A's frame. N is the total number of flashes that B sees during her entire journey. Both twins must agree on the number of flashes seen during 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 fine. But what about in B's frame? Why can't we employ the same analysis of A moving past flashing 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 d ifferent frames depending on her direction of travel. This can be seen on the Minkowski diagram in :
Figure %: Minkowski diagram of the twin paradox.
Here is lines of simultaneity in B's frame are sloped one way for the outward journey and the other way for the trip back; this leaves a gap in the middle where A observes no flashes, leading to an overall measurement of more time in A's frame. If the distant star is distance d from the earth in A's frame and the flashes 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 is the same for inward and outward journeys). Again let the twins agree that there are N total flashes during 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 flashes (see the Minkowski diagram). In B's frame the distance between the earth 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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What is ? We can see from that the slopes of the lines are observes no flashes is Thus:
Comparing T A and T B we see T B = T A / measures more time and B is younger.
The result from the theory of relativity that Mass and Energy are different manifestations 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 release of 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 nuclei is 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.
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 to instantaneously 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 its high 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 as technology 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 to distant 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.