Welcome to Java TA Special Relativity Well for an entire term in Ph 1a, you studied mechanics based on the laws of Newton. Now, I am here to demonstrate to you that the laws that are the basis of Newtonian Mechanics are wrong! At speeds approaching, or at the speed of light, the laws of Newtonian Mechanics no longer hold. This boggled science for many years until Albert Einstein introduced Special Relativity. There are two theories of relativity discovered by Einstein, one is called the Special Theory of Relativity and the other the General Theory of Relativity. The 'special-ness" of Special Relativity is that it is restricted to objects moving with a uniform velocity in a straight line. General relativity includes such phenomena as gravity. This lab will discuss Special Relativity. So what is relativity? Click Next. @N Relativity can be stated like this: "The motions of bodies included in a given space (reference frame) are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line." -Newton (Text in parentheses added) This idea of relativity has been around for a long time, the statement above is from Newton's Principia. We observe this principle everyday - if I'm sitting in my car, everything inside: the seats, the steering wheel, the doors and windows all look the same when I'm sitting in the driveway and when I'm on the highway. If I'm driving at a constant speed of 100 mph, and I throw something into the backseat, like a physics book, it won't go flying through the car (i.e., the car won't 'move' beneath the book), it acts the same as if I wasn't moving. This is accomplished mathematically using the Galilean transformations. If an observer at the side of the road sees me tossing the book into my backseat, and I give it a velocity v, then the observer would measure the book moving with my car's velocity minus its velocity, 100 - v. Read over the Newton quote again. What this says, and what is important to Einstein's relativity, is that an observer cannot tell whether he or she is moving or at rest, simply by observing bodies in his or her own frame of reference. By observing the steering wheel or the book moving, I cannot tell if my car is in the garage or on Route 66. And actually, Route 66 is in my reference frame too, so I cannot determine if I'm moving, simply by watching the highway go by either! How do I know that I'm the one moving, and not the highway? Well, we know that we're moving and the highway isn't because we approximate the earth to possess its own inertial frame, that is, we pretend that the earth is moving with a uniform velocity and in a straight line. From this inertial frame we can determine the speeds of all moving bodies relative to that frame using the Galilean transformations. These transformations worked very well for moat observations, except when we observe the very fast. Click Next. @N The Galilean transformation worked for over 200 years until science began conducting experiments on the speed of light. First, science discovered that light in vacuum moves very fast, about 186,000 miles per second. So then they thought- Let's say we build a car that goes 100,000 miles a second, relative to the ground. When this car turns on its headlights, how fast does the light from the headlights move relative to the earth? Applying the Galilean transformation, we find that the light travels a screaming 286,000 miles per second! Unfortunately this experiment could not be conducted, so physicists built other experiments using the newly discovered Maxwell's equations. These equations describe the phenomena of light, electricity and magnetism in just a few concise equations. These experiments found every time that the speed of light was 186,000 miles per second in a vacuum. Because Maxwell's equations were so new, it was assumed that Maxwell had made a mistake, and lots of zany corrections were made to his equations so that they would obey Newton's mechanics and predict all observations made thus far. However these corrections predicted other phenomena that did not exist experimentally. One correction was the idea of an 'ether', which we will discuss very soon. Click Next. @N Then along came Albert Einstein, who had a different approach to the problem. First he assumed that Newton's relativity was correct (that an object cannot determine its speed by observing things in its own reference frame). Second, he postulated that the speed of light in a vacuum did not change, but is a constant! He then asserted that it was not the Maxwell's equations that needed a correction, but it was the laws of mechanics that were incorrect! Let's figure out what this correction is, click Next. @N I will show you what problem Einstein had to solve. I said before that at the time before special relativity, physicists were trying to make Maxwell's equations fit into classical mechanics. One solution was to create ether, a colorless, tasteless, weightless and most importantly motionless substance that permeated all space. This ether solves the problem of having the earth being an inertial frame, and it also provided a medium for light to propagate through. The ether would then predict that each body had an absolute velocity - velocity with respect to the ether. (Which postulate of Einstein's does that violate?) The Michelson-Morley experiment in 1887 set out to discover the absolute speed of the earth, and using the power of Java, we'll try to do that too. Click Next, and we'll go through the basis of the experiment. @N The experiment was set up like this: Every sailor knows that it takes longer to sail against a headwind (and then return with a tailwind) than to sail across it. (Please look at the derivation at the left, you may want to come back to it after reading the next couple pages.) Because the ether didn't move and light propagated in it, light would move faster or slower depending on how the light was oriented to the direction of motion. An ether wind, like the wind past your car on a calm day, was predicted by the ether theory. This ether wind could be used to measure a body's absolute motion. Click Next and we'll try to measure a body's 'absolute motion'. @N Michelson and Morley built an apparatus, called an interferometer, that would measure even the slightest phase difference between light beams traveling equal lengths* perpendicular to one another. If one arm of the apparatus was in the direction of the earth's absolute motion, then this beam would be slowed by the factor gamma, and a different interference pattern would be measured. I've set up this experiment for you right now. In your applet window I have two light pulses travelling perpendicular to one another. Press start to send them off towards the reflecting mirror. With this set-up, which statement best describes the motion of these particles: (Notice that currently you are in the Object's frame, just as Michelson and Morley were.) A) The photon parallel to the motion of the apparatus is slowed. B) The motion of the photons are unaffected by the motion of the apparatus. C) The photon parallel to the motion of the apparatus is sped up. Choose an answer and press Next. @B Answer: B) Michelson & Morley tried this experiment, and of course in every orientation possible, they found no difference in the two beams of light. Einstein AND Newton could have told them that, because a measurement of absolute motion violates the classical law of relativity. But what about this experiment: Instead of watching the photons from their rest frame, what would happen if we stood from afar, and watched them move by? Lets now do the same experiment, except from an 'observer' frame (the frame from which .66c was measured at the top of the applet). You'll notice that things are very different in this situation. All of this aside, would we witness a phase shift between the two photons? A) Yes, the photon parallel to the motion has less distance to travel, so it will return sooner. B) Yes, the motion does not effect the anti-parallel photon, so it will return sooner. C) No, the photons return at the same time as before. Please choose your answer and click Next. @C Answer: C) This experiment gives us the same results as if we were in the object's rest frame, which means it is not a very good experiment because obviously something is going on! It turns out that the fact that no interference is found is an event that is frame independent. Just as if I sneeze someone moving fast relative to me cannot 'not see me sneeze' because my sneezing is an event. We'll see later on things that are frame dependent. Here is another experiment for you: Lets say we send up a rocket into space, at a speedy .7c, and in this rocket we send an astronaut to set up an experiment with two perpendicular rods, in which one rod is parallel to the rocket's motion, and one is perpendicular. Notice in your browser window two rods of equal length(as measured in their rest frame) that are perpendicular to one another. Right now we are in the observer's frame (set at the bottom left of the applet). Press start to send the astronaut and the rods into space, so we can watch it from earth. What is going on here exactly? A) Everything is REALLY the same as it was when the rods weren't moving, except the high speed makes the rods parallel to the motion LOOK smaller. (An optical illusion) B) Everything at rest in the rod's rest frame is contracted in the direction of the rod's motion, according to the observer. C) The earthlings have noticed that everything in their rest frame has been contracted in the direction of the rod's motion, explaining why the measurement was shorter. @B Answer: B) While the rocket jets away from earth, to the astronaut everything is as Newton's relativity would predict: nothing unusual is happening. If there is then they would know that they were moving. However, the observer notices that, oddly the horizontal rods have contracted. Why did they contract? Because if they hadn't, going back to the Michelson-Morley experiment, if they hadn't then one photon would have returned first by traveling faster than 186,000 miles per second (Making the event dependent). This contraction simply removes that gamma speed-up predicted by Michelson-Morley (and experienced by the sailor), by contracting the distance traveled. You could argue that this is simply an illusion, if you asked the astronaut whether he was contracted in any way he would answer no. However this is not an optical illusion, it is real, or as real as Special Relativity gets. It is not an optical illusion, but perhaps an illusion playing on the limits of our perception. (I can't 'see' 4-D Space-Time, can you? :) ) Now here's another question - while the rocket moves farther from the earth, the astronaut happens to looks back at earth. What does HE see? A) The earth's frame is longer, by a factor of gamma, in the direction of its motion relative to the astronaut. B) The earth looks normal, because it's not moving. C) The earth's frame is contracted, by a factor gamma, in the direction of its motion relative to the astronaut. Choose a letter and Click Next. @C Answer: C) The earth's frame is contracted. The same contraction is applied to the earth observed by the astronaut, and is contracted by the same factor gamma. When the astronaut looks at the earth, the earth is moving away from him at negative his speed relative to his rest frame, and because the gamma factor is determined by the square of the velocity, negative and positive velocities have the same effect. Well then how can BOTH frames be contracting? Its not that both frames are contracting - if you asked the astronaut or someone on earth they would tell you that their frame is proportioned quite nicely, but they only APPEAR contracted to an observer moving at a very high speed relative to their frame. This of course begs the question of what is real. We have come to a point in physics where we have to stop thinking about things as objects and begin seeing them as incorporated concepts, because what we are observing goes beyond our usual powers of perception. What is real in this case is this - the speed of light is always c and Newton's relativity. These laws bring about these odd happenings. Click Next, it gets weirder. @N OK, here is another experiment that takes advantage of the speed of light always being c. In the object frame I have created a light clock. A light clock is made up of a photon bouncing back and forth between reflectors with a constant period. In the object frame this light clock has a period of 60 units - it takes 60 units of time for the photon to go from one reflector, to the other reflector and then back. However, when I give this clock a velocity v close to the speed of light and observe it, what happens? A) The clock's period is now longer - to the observer it takes longer for the pulse to get back to the starting reflector, which means the object's time is slowed. B) The clock's period is now longer - because of its movement this clock is no longer a functioning timepiece. C) The clock's period is now longer - to the observer the clock ticks off more units of time meaning that the object's time moves quicker. Choose a letter and Click Next. @A Answer: A) Time is slowed or dilated. Because light always moves at the speed c, for the first leg it chases the front reflector. Although the second leg is shortened because it meets with the back reflector, more time is spent on the first part. Thus time must be dilated or slowed by a factor gamma or light wouldn't travel at the speed c or someone in the object frame could determine his or her own speed. Why is it that time is slowed and the clock is still a valid clock? Imagine having a real clock and a light clock both moving at the same time. If there was any discrepancy between the two, then someone in the object's rest frame could find out how fast he or she was moving! I've set up another experiment, this time with a real clock. You'll notice that in both frames, the clock that's moving relative to the frame at rest is slowed or dilated. This seems strange; there must be some sort of paradox. Click Next. @N Let's say there is a farmer, who has a barn of length L, and a very long ladder with length 2L. One day the farmer really wants to put his big ladder away, but he cannot fit it in the barn at all. So the framer's son, who has just read the first couple slides of this tutorial, has an idea. If he ran really fast, say at about .87c, or 160,000 miles per second, then the ladder would be contracted by a factor of 2 and it would fit in the barn. However, the farmer disagrees, asserting that in the ladder's rest frame, the barn would be contracted by a factor of two, and then there would be no way of fitting the ladder inside. Let's say that the front and the back doors of the barn are open, and 'being' in the barn means the end of the ladder is inside before the front leaves out the back door. I've started this experiment for you. So who is right? A) They are both right, whether or not the ladder fits into the barn is frame dependent. B) The farmer is right, there is no way a ladder of length 2L fits in a barn of length L. C) The farmer's son is right, if he runs at .87c, gamma = 2 and the ladder fits quite nicely. Choose a letter and click Next. @A Answer: A) Both are right. They are both right because whether or not the ladder fits in the barn depends on the frame in which it is observed. This is because we cannot prescribe any specific event to the situation, so the situation depends on the frame. Look at it like this: in the Michelson-Morley experiment there COULD NOT be any interference on the interferometer because having no interference was an event, and is independent of frame. If I fail a physics test, I fail no matter how anyhow views me in my rest frame. The same thing goes for the Michelson-Morley experiment. If different interference patterns are seen from different rest frames, then something is definitely wrong with one or all of Einstein's assumptions. So what classifies something as an event? In the barn and ladder paradox, we are asking whether or not the end of the ladder was in the barn before the front left therough the back. This is a question of whether or not something came before or after something else, which is a frame dependant question. However, in the Michelson-Morley experiment we ask the question does something happen or not happen (or 'does something exist'). The reason why the order of things changes from frame to frame is because of the change of the time and space axes. You may have noticed on the space-time diagrams that when something is given a high velocity, their time and space axes are no longer perpendicular, but become 'crooked'. This means that two things simultaneous, i.e., have the same time and lie on a straight line perpendicular to the time axis, are no longer simultaneous when the time axis is moved. This demonstrates that while events are independent of frame, simultaneity and causality are not. Click Next to see how Einstein corrected classical mechanics. @N Before I said that Einstein found that it was classical mechanics that needed a correction, not Maxwell's equations or Newton's relativity. So what exactly is this correction? What needs to be done is to replace the Galilean transformations by the relativistic Lorentz transformations. These Lorentz transformations (see left) encompass all the contraction and dilation phenomena described before, and at the limit where v<