Special Relativity. Frames of Reference. Introduction

Transcription

1 Special Relativity Introduction In 1905 Albert Einstein introduced his theory of special relativity. With this theory Einstein sought to make the laws of motion consistent with James Clerk Maxwell's ( ) laws of electromagnetism. Those laws predicted that light in vacuum traveled at a speed c (about 300,000 km/s) that was independent of the motion of the observer of the light and of the light source. Special relativity is based on the idea that all motion is relative - that is, one can only judge the state of motion of an object by comparing it with some other reference point or object. For example, cars move relative to the surface of the earth. Special relativity shows that this apparently simple and obvious assertion has far-reaching consequences, especially when considering very fast moving objects. Einstein based his theory of special relativity on two basic postulates, one of which is that all motion is relative, and then proceeded to determine the consequences of those postulates. Frames of Reference If you ask most people to explain what the term "motion" means, they will probably say something about movement, transport, travel, and other similar words that convey the idea of movement from one point to another. However, most people would omit to mention that all motion is RELATIVE. That is, motion can only be judged by comparing the position of the moving object with some 1

2 other reference point or object. The following figure illustrates this. Examples of relative motion in this figure: 1. Car moves relative to the road 2. Person in car moves relative to car 3. Car moves relative to house 4. Person in house moves relative to house One can go further - the person in the car moves relative to the road, the car moves relative to the house and the person in the car moves relative to the person in the house. In the language of physics, the reference against which the motion of an object is measured is called a "frame of reference". Thus, in the figure above for example, the road is a frame of reference against which the motion of the car can be measured. The house could also be a frame of reference against which the motion of the car is measured. There is a special type of frame of reference, which is referred to as an "inertial" frame of reference. An inertial frame of reference is one in which Newton's first law of motion holds. What is Newton's first law of motion? 2

3 Basically, it states that: An object remains either stationary or in its state of constant motion if there are no forces acting on that object. The point of all this is that Einstein asserted that there is no "preferred" inertial frame of reference against which motion should be measured. In other words, all inertial frames of reference are equivalent. It does not matter which inertial frame of reference is used to judge the motion of an object. In the above example, either the road or the house are acceptable frames of reference against which to judge the motion of the car. The "relative" part of relativity Suppose you are the person who is taking this photo and the other guy seems to be drifting toward you. Are you moving toward him or is he moving toward you? 3

4 The answer is yes to both questions. From each person's point of view, he is stationary, and the other guy is moving. Relativity states that everyone's frame of reference is valid. The observations that you make from your frame of reference is just as valid as those made by some one else on a different reference frame. Postulates of Special Relativity With this in mind Einstein came up with his first postulate of special relativity, which generalizes our discussion to ALL of the laws of physics: Postulate 1: All the laws of physics are the same in all inertial frames of reference. All uniform motion is relative. In the context of our discussion above, we can give a simple example of this principle in action. Consider the person standing in the house and the person sitting in the car moving at constant speed. This postulate states that both people must observe the same laws of physics for all phenomena. Now suppose that you are in a train that is moving at a constant speed of 100 km/hr. As long as the track is straight, you can pour beverages, juggle, play cards or doing anything else without taking into account the fact that you are moving. The postulate of relative motion takes this to its logical conclusion and states that if the motion is truly uniform, and you are sheltered from effects of the matter outside the train, there is no experiment that can be performed that will tell you whether your train is in motion or standing still. The best you can do is measure its relative motion compared to some other object. 4

5 The second postulate is: Postulate 2: The speed of light (in a vacuum) is the same in all inertial frames of reference. The second postulate says that the speed of light is always observed to be the same however we, or the source, might be moving. It is a universal invariant. The consequence of Einstein's two postulates are radical: time and space become intertwined in surprising ways. Events that may be simultaneous for one observer can occur at different times for another. This leads to length contraction and time dilation, the slowing down of time in a moving frame. Every observer has her own personal time, caller proper time. That is the time measured by a clock at the observer's location. Two observers, initially the same age as given by their proper times, could have different ages when they met again after traveling along different space-time paths. Given these two statements, Einstein showed how definitions of momentum and energy must be refined and how quantities such as length and time must change from one observer to another in order to get consistent results for physical quantities such as particle half-life. To decide whether his postulates are a correct theory of nature, physicists test whether the predictions of Einstein's theory match observations. Indeed many such tests have been made -- and the answers Einstein gave are right every time! 5

6 Special relativity includes only the special case (hence the name) where the motion is uniform. The motion it explains is only if you re traveling in a straight line at a constant speed. As soon as you accelerate or curve or do anything that changes the nature of the motion in any way special relativity ceases to apply. That s where Einstein s general theory of relativity comes in, because it can explain the general case of any sort of motion. The genius of Einstein s discoveries is that he looked at the experiments and assumed the findings were true. This was the exact opposite of what other physicists seemed to be doing. Instead of assuming the theory was correct and that the experiments failed, he assumed that the experiments were correct and the theory had failed. Einstein s conclusion was this: Time and space are not absolutes. They are not invariable, nor do they always retain the same values. The only way the speed of light can possibly remain constant is if, as an observer travels at different speeds, that observer s own perception of time and distance must change as well. Skipping past some of the complex mathematics, Einstein concluded that as a person moves faster; their own perception of time slows down accordingly (though normally to such a small degree that it is hardly even measurable, unless one is moving very quickly). Furthermore, the faster one travels, the smaller they would appear to an outside observer contracted in the direction of motion. If this was the case, then according to Special Relativity, as one s perception of time and distance changed according to their speed, light could always be measured at the same speed. 6

7 Why is it called a theory of RELATIVITY? Because time and length are no longer absolutes. You ve got your digital watch on your wrist and a meter ruler on your desk. These seem like absolutes: a second and a centimeter for you must be the same as they are for me. But they re not. There are many quantities in our lives that are relative to our frames. The direction of Columbus, Ohio from here depends on where "here" is. How fast the flight attendant's cart is moving depends if you are on the ground or inside the airplane. As an example, say you are traveling in a car going one direction at 25 m/s, and another car is heading the opposite direction at 15 m/s. From your vantage point, the other car would seem to be coming towards you at 40 m/s (simple addition: ). In other words, the other car is moving 40 m/s relative to you. Let us move this analogy to higher speeds. You are in a spaceship moving 2.0 x 10 8 m/s. Another spaceship is moving the opposite direction at 1.5 x 10 8 m/s. Do you say that the other spaceship is moving 3.5 x 10 8 m/s relative to you? However, light moves at 3.0 x 10 8 m/s. Does that mean that the other spaceship is moving faster than the speed of light relative to you? In fact, Einstein says that the other spaceship does not move at 3.5 x 10 8 m/s; it would be impossible for it to do so. Einstein calculates that the other spaceship will appear to travel about 2.6 x 10 8 m/s towards you. The simple addition does not work anymore. This is Einstein s theory of special relativity. 7

8 So what has gone wrong? The easiest conclusion to draw is that the speed of light postulate is incorrect. However, Einstein realized that one cannot give up this postulate without giving up either Maxwell's wave theory, or the postulate of uniform motion, and he was willing to give up neither. He therefore made a brilliant intuitive leap and concluded that our common sense is wrong. Einstein's faith in the speed of light postulate turned out to be well founded. Its strange consequences have since been verified experimentally. Unifying space and time Einstein s theory of special relativity created a fundamental link between space and time. The universe can be viewed as having three space dimensions up/down, left/right, forward/backward and one time dimension. This 4-dimensional space is referred to as the space-time continuum. If you move fast enough through space, the observations that you make about space and time differ somewhat from the observations of other people, who are moving at different speeds. You can picture this for yourself by understanding the thought experiment depicted in this figure. Imagine that you re on a spaceship and holding a laser so it shoots a beam of light directly up, striking a mirror you ve placed on the ceiling. The light beam then comes back down and strikes a detector. 8

9 (Top) You see a beam of light go up, bounce off the mirror, and come straight down. (Bottom) Amber sees the beam travel along a diagonal path. However, the spaceship is traveling at a constant speed of half the speed of light (0.5c, as physicists would write it). According to Einstein, this makes no difference to you you can t even tell that you re moving. However, if astronaut Amber were spying on you, as in the bottom of the figure, it would be a different story. Amber would see your beam of light travel upward along a diagonal path, strike the mirror, and then travel downward along a diagonal path before striking the 9

10 detector. In other words, you and Amber would see different paths for the light and, more importantly, those paths aren t even the same length. This means that the time the beam takes to go from the laser to the mirror to the detector must also be different for you and Amber so that you both agree on the speed of light. This phenomenon is known as time dilation, where the time on a ship moving very quickly appears to pass slower than on Earth. As strange as it seems, this example (and many others) demonstrates that in Einstein s theory of relativity, space and time are intimately linked together. If you apply Lorentz transformation equations, they work out so that the speed of light is perfectly consistent for both observers. This strange behavior of space and time is only evident when you re traveling close to the speed of light, so no one had ever observed it before. Experiments carried out since Einstein s discovery has confirmed that it s true. Time and space are perceived differently, in precisely the way Einstein described, for objects moving near the speed of light. 10

11 Unifying mass and energy The most famous work of Einstein s life also dates from 1905, when he applied the ideas of his relativity paper to come up with the equation E=mc 2 that represents the relationship between mass (m) and energy (E). In this equation, the symbols stand for E: energy equivalent of mass (in Joules), m: mass of object (in kg), c: speed of light (= 3 x 10 8 m/s) This can lead to enormous amounts of energy; for example, a 1 kg book would have an energy equivalent of 9 x J, which is enough energy to supply the electricity needs of an average city of 750,000 people for well over one year. Compare this to the kinetic energy of the book if it is thrown at 4 m/s, which would be 8 J. Why don't we normally experience this mass energy? The answer lies in the fact that mass energy is somewhat like potential energy, in that it is only apparent or useful to us if it is converted into another form of energy, like kinetic energy. For example, a car perched on the roof of a 20 story building has a large gravitational potential energy, but that energy is not useful to us until it is converted into, for example, kinetic energy by letting the car drop to the ground and seeing the result. 11

12 Similarly, in normal circumstances, mass energy remains as mass energy, and as such we do not see directly its effects. One does do so, however, in the operation of nuclear power plants and the explosion of nuclear bombs, where some mass energy is converted into other forms of energy. Until the time of Einstein, mass and energy were two separate things. In the special theory of relativity Einstein demonstrated that neither mass nor energy were conserved separately, but that they could be traded one for the other and only the total "mass-energy" was conserved. The relationship between the mass and the energy is contained in what is probably the most famous equation in science, E = m c 2 The implications of this were not realized for many years. For example, the production of energy in nuclear reactions (i.e. fission and fusion) was shown to be the conversion of a small amount of atomic mass into energy. This led to the development of nuclear power and weapons. As an object is accelerated close to the speed of light, relativistic effects begin to dominate. In particular, adding more energy to an object will not make it go faster since the speed of light is the limit. The energy has to go somewhere, so it is added to the mass of the object, as observed from the rest frame. Thus, we say that the observed mass of the object goes up with increased velocity. So a spaceship would appear to gain the mass of a city, then a planet, than a star, as its velocity increased. 12

13 There is another profound implication of the above considerations that Einstein was the first to realize. We know that when you increase the velocity of an object, you are giving it kinetic energy. We also know that kinetic energy can be converted into other forms of energy, such as potential energy and heat energy and vice versa. If increasing the velocity of an object increases its inertial mass, then in some sense we are converting kinetic energy into mass. Einstein reasoned that mass and energy must somehow be different manifestations of the same thing. In particular if we can convert kinetic energy into mass, we should also be able to convert mass into energy. The special theory of relativity in fact gives an explicit expression for how mass and energy is related. E = mc 2 Special relativity and E=mc 2 led to the most powerful unification of physical concepts since the time of Newton. The previously separate ideas of space, time, energy and mass were linked by special relativity, although without a clear understanding of how they were linked. In a nutshell, Einstein found that as an object approached the speed of light, c, the mass of the object increased. The object goes faster, but it also gets heavier. If it were actually able to move at c, the object s mass and energy would both be infinite. A heavier object is harder to speed up, so it s impossible to ever actually get the particle up to a speed of c. 13

14 Until Einstein, the concepts of mass and energy were viewed as completely separate. He proved that the principles of conservation of mass and conservation of energy are part of the same larger, unified principle, and conservation of mass-energy. Matter can be turned into energy and energy can be turned into matter because a fundamental connection exists between the two types of substance. The Speed of Light Our everyday common sense tells us that if while standing on a moving ship and we hit a golf ball off the front, the speed of the ship adds to the speed of the golf ball. If we were to shine a flashlight in the same direction as we had 14

15 hit the ball we would find that the speed of the ship does not add to the speed of the light beam. Its speed would be the same as if the ship were not moving at all. The best minds of the 19th century were stumped by this fact because it totally violated a very basic and seemingly self evident notion of how things are supposed to work. Why time slows down So the speed of light is unaffected by the speed of its source. Let's see what falls out of this... Imagine having two horizontal mirrors facing each other and that one mirror is spaced above the other by the distance d. Also imagine that there is a pulse of light that bounces vertically between the two mirrors as shown in the left part of the drawing below. The time it takes for the pulse of light to do a round trip (from the top mirror to the bottom mirror and back to the top) is twice the distance d divided by the speed of light. This device would actually make a great clock since the pulse can be relied on to always travel at the same speed while doing its round trips and thus always counting out consistent time intervals. 15

16 Suppose our "light clock" were traveling sideways at a very high (but constant) speed. Now the pulse would follow the "saw tooth" path shown on the right side of the drawing. The light must travel a greater distance now to make a round trip. Since its speed is the same as before (remember, the speed of light is not changed by the speed of its source), it will take longer to make a round trip. So our "light clock" takes longer to count out its intervals. Another way of saying this is that the clock "ticks" more slowly. Note that if the speed of light were not constant, the horizontal speed of the clock would have added to the speed of the pulse. Then, the light clock would not have slowed down since the pulse's greater speed would have compensated for the longer distance of the "saw tooth" path. This result has great significance. 16

17 Now, back to the light clock... It's very feasible to hook up the light clock to an LED display that reads out seconds, minutes, and hours. Now let s say you choose a frame of reference that's "stationary." That way you get a clock that's keeping time "properly" for you don't want a slow running clock. So your light clock is behaving like the one on the left side of the drawing above. But someone comes along (at an extremely fast speed) to spoil your day. From his point of view, you're the guy that's moving, not him. From his equally valid point of view, your light clock is behaving like the one on the right side of the drawing above. And of course, to him your clock is counting out its intervals very slowly and your LED is counting out those seconds, minutes, and hours way too slow. Since you're stationary relative to your clock, it's working just fine as far as you are concerned. Now you try to tell him this and you speak at a normal rate (about 4 words/second by your clock). To him you are speaking 4 words/second relative to your very very slow clock. You sound like an old fashion phonograph record that is set to a very slow speed. And of course since he is moving relative to you, his speech sounds equally slow. This effect is called time dilation. Time dilation is present even at the slow speeds that we are used to. But it's so tiny that our senses can't detect it. 60 miles/hour or even 3000 miles/hour is too slow compared to the speed of light at 186,000 miles/second or 669,600,000 miles/hour. At 60 miles/hour the triangles in the drawing above would have such a tiny base that they wouldn't look any different from the vertical photon path 17

18 of the stationary clock. So the stationary clock and the 60 miles/hour clock "practically" keep the same time. Time dilation becomes most apparent when one of the objects is moving at nearly the speed of light, but it manifests at even slower speeds. Here are just a few ways we know time dilation actually takes place: Clocks in airplanes click at different rates from clocks on the ground. Putting a clock on a mountain (thus elevating it, but keeping it stationary relative to the ground-based clock) results in slightly different rates. The Global Positioning System (GPS) has to adjust for time dilation. Ground-based devices have to communicate with satellites. To work, they have to be programmed to compensate for the time differences based on their speeds and gravitational influences. Certain unstable particles exist for a very brief period of time before decaying, but scientists can observe them as lasting longer, because they are moving so fast that 18

19 time dilation means the time that the particles "experience" before decaying is different from the time experienced in the at-rest laboratory that is doing the observations. 19

20 Predictions of Special Relativity Measurement of Time, Length, and Mass depend on the velocity of the observer. As velocity increases: o Lengths shrink (contract) o Masses increase o Clocks slow down 20

21 Time dilation leads to the famous Twins Paradox, which is not a paradox but rather a simple fact of special relativity. Since clocks run slower in frames of reference at high velocity, then one can imagine a scenario were twins age at different rates when separated at birth due to a trip to the stars. 21

22 Twin Paradox The story is that one of a pair of twins leaves on a high speed space journey during which he travels at a large fraction of the speed of light while the other remains on the Earth. Because of time dilation, time is running more slowly in the spacecraft as seen by the earthbound twin and the traveling twin will find that the earthbound twin will be older upon return from the journey. The common question: Is this real? Would one twin really be younger? The basic question about whether time dilation is real is settled by the muon experiment. The clear implication is that the traveling twin would indeed be younger. Despite the experimental difficulties, an experiment on a commercial airline confirms the existence of a time difference between ground observers and a reference frame moving with respect to them. It is important to note that all the predictions of special relativity, length contraction, time dilation and the twin paradox, have been confirmed by direct experiments, mostly using sub-atomic particles in high energy accelerators. The effects of relativity are dramatic, but only when speeds approach the speed of light. At normal velocities, the changes to clocks and rulers are too small to be measured. Another consequence of Einstein s theory is that your mass increases as you approach the speed of light. This additional mass must come from somewhere. It turns out that this mass comes from the energy of your moving ship. 22

23 Conclusions from special relativity Einstein was a theoretical physicist; he did not spend any time in laboratories trying out his ideas with experiments. Instead, he tried to determine using logic and mathematics alone what the consequences would be of his initial assumptions. Eventually he was able to derive mathematical equations that described the physical properties of systems in motion. Some of the conclusions he drew were the following: 1. The length of an object is a function of the speed at which it is traveling. The faster the object travels, the shorter the object becomes. 2. The mass of an object is also a function of the speed at which it is traveling. The faster the object travels, the heavier it becomes. 3. Time slows down as an object increases in speed. Think for a moment about the logical consequences of just these three points. First, none of the effects is of much practical importance until an object is traveling close to 23

24 the speed of light. If you tried to detect changes in length, mass, or time in a moving train, you'd have no success at all. It is not until one approaches speeds of about 167,000 miles (270,000 kilometers) per second (about 90 percent the speed of light) that such effects are noticeable. But what effects they are! An object traveling at the speed of light would have its length reduced to zero, its mass increased to infinity, and the passage of time reduced to zero. Any clocks attached to the object would stop. Tests of relativity One of the most remarkable things about Einstein's theories is the speed with which they were accepted by other physicists. As revolutionary as his ideas were, physicists quickly saw the logic of Einstein's arguments. Some physicists and many nonscientists, however, wanted to see experimental evidence in the real world that his ideas were correct. One proof for Einstein's theory is his equation representing the relationship of energy and mass, E = mc 2. It is upon this principle that nuclear weapons and nuclear power plants operate. But other pieces of experimental proof were eventually discovered for Einstein's theories. One of those was obtained in Einstein had predicted that light will be bent out of a straight path if it passes near to a very massive object. He said that the gravitational field of the object would have an effect on light much as it does on other objects. 24

25 An opportunity to test that prediction occurred during a solar eclipse that occurred on May 29, Astronomers waited until the Sun was completely blocked out during the eclipse, and then took a photograph of the stars behind the Sun. They found that the stars appeared to be in a somewhat different position than had been expected. The reason for the apparent displacement of the stars' position was that the light they emitted was bent slightly as it passed the Sun on its way to Earth. Significance of relativity theory Einstein's theories have had some practical applications, as demonstrated by the use of E = mc 2 in solving problems of nuclear energy. But far more important has been its effect on the way that scientists, and even some nonscientists, view the universe. His theories have changed the way we understand gravity and the universe in general. In that respect, the theories of relativity produced a revolution in physics matched only once or twice in all of previous human history. END 25

26 The Twin Paradox 1. Consider two twins, each aged 20 years. Let's call them John and Sue. 2. Suppose that Sue gets into a rocket ship, and makes an outward journey of 20 light years at 80% of the speed of light. (A light year is a unit of distance, equal to the distance that light travels in one year, which is about 6,000,000,000,000 miles) 3. Sue then turns around and comes back, also at a speed of 80% of the speed of light. Now, let's analyze the times required for the journey, and how they are perceived by John and Sue. As you might guess, the effects of time dilation are going to come into play here! Note that both John and Sue have biological clocks, and the effects of time dilation are therefore applicable to the life cycle of our twins. John's frame of reference is the surface of the earth, and Sue's frame of reference is the rocket ship. 4. To John, Sue is in a moving frame of reference, and so appears to be living more slowly that John, because of time dilation. In fact, application of the time dilation formula shows that to John, the total trip (20 light years out and 20 light years back) seems to take 50 years, and so when Sue comes back, John is aged 70 years (20 at the start + 50 year journey). 5. However, to John, Sue's clock has been running more slowly (because of time dilation), and in fact applying 26

27 the time dilation formula shows that her biological clock has advanced only 30 years - and so Sue is aged 50 years (20 at the start + 30 years during the journey). Now for the interesting bit, and where the paradox arises! 6. As we noted above, the time dilation effect is reciprocal, so that as far as Sue is concerned, John is in a frame of reference that is moving away from her, and so to Sue, John appears to age more slowly, and on her return it might be expected that John is only 50 years old and she is 70 years old, i.e. the exact opposite of what John expects!! This is the twin paradox, possibly the most discussed paradox of them all. 27