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0:00
Hi and welcome to A5 Galilean and Special Relativity. This is a higher level topic only. Now you can go down a rabbit hole on special relativity. Lots of students absolutely love this topic and you can spend hours on the interesting concepts of proper time, length contraction and so on. But my job, my job is to get you through the exam. Before we head into this pretty difficult topic I want to help you avoid a mistake that many students make and a
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lot of people think special relativity is hard and actually yes it is. Conceptually it is quite difficult to understand but I really don't want you to overthink this and get into a panic because what's going to happen is when we head to step two which is activate the application and we start looking at past paper questions you'll find out that past paper questions all follow the same theme and actually special relativity is not as difficult as you
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think it would be to get good marks in the exams. We're going to start this tutorial with defining a reference frame and understanding the concept of an inertial reference frame. The definition of a reference frame is that it's a coordinate system that is used to describe the position of an object relative to a point of reference. So a point of reference is here that's 0,0 and we have an object here and we can describe that position by using the grid
1:24
system relative to this point of reference. So the grid system is -3x to y. It's important that we have this grid system and this reference frame because it allows us to explain where things lie relative to each other. We have to take this one step further and define an inertial frame of reference. It is defined as a coordinate system where Newton's laws are valid and is either stationary or traveling at a constant velocity with no external
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forces acting on it. I've seen past paper questions that say, 'State why this is an inertial frame of reference' and I don't really think the examiners are looking for you to try about this massive sentence. I think they're just looking for you to say it's not accelerating but it is useful to understand and memorize this whole definition of an inertial frame of reference. So without further ado let's head to your revision note template and write down the definition
2:23
of a reference frame and an inertial reference frame in your notes. Then complete question one in your knowledge questions. We're now going to look at the concept of Galilean relativity and how we can relate one object moving in comparison to another. Galilean relativity is based on the principle that the laws of physics and specifically Newton's laws all can be applied in inertial frames of reference. So if you have an inertial frame of reference that's either
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traveling at a constant velocity or is at rest you can use F equals ma, all of Newton's laws, inside that frame of reference. And this is going to become really useful when we look at two frames of reference traveling in relation to each other. When two frames of reference travel in relation to each other we use something called Galilean transformations. We have two frames of reference here. Frame S is stationary. Frame S dash is moving with velocity V relative to frame S and it
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will be useful for us to calculate where things are in each frame of reference. And to do that we use this first Galilean transformation which is X dash is equal to X minus VT. Is that a formula you just need to remember? Well yeah it's also in the data booklet but it also kind of makes sense right? Because what you have here VT is a distance, speed times time. So VT gives you the distance traveled by this frame in a certain amount of time. So that frame has a position X dash
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relative to X and it can be calculated using this which is basically a subtraction of distances. It's really important to note that in Newtonian physics the time in both frames of reference is exactly the same. It's important for us to know that because when we go to special relativity time is not a constant. So in Newtonian physics time is a constant, it's absolute, it's fixed but in special relativity it's not. If we're looking at relative velocity
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and Galilean transformations it's very very simple. Just imagine yourself looking at this train. The train is moving at 100 meters per second and this person is walking back the way at 1 meters per second. We watching this train would see this person moving at 99 meters per second and that's what this simple additive or subtractive formula is. If the guy was moving this way at 1 meters per second then U dash would be 101 meters per second. Don't overthink it,
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remember what I said at the start. So head to your revision note template, write down the Galilean equations for X dash, T dash and U dash in your revision notes. You now know how to find the position, the time and the speed or the velocity of an object in frame S dash in Galilean relativity. Use that information to complete question 2 in knowledge questions. I'd like to now introduce the concept of special relativity and special relativity is based on two key postulates. These
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postulates must be memorized and postulate number one is the laws of physics are the same in all inertial reference frames and postulate number two is that the speed of light in a vacuum is the same for all observers regardless of their relative motion. Let's not overthink what the consequences these postulates are right now. All I want you to do at the moment is memorize these two postulates because they can gain you valuable marks in the exam. Now
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we know the two postulates of special relativity. Let's take those two postulates and see how they feed into two frames of reference that are traveling relative to each other close to the speed of light. Gosh, what a mouthful. We're starting to look at special relativity and this diagram should be very familiar to you. We looked at it in Galilean relativity and we said time and space are considered to be absolute and independent and transformation equations
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are linear but with special relativity it's different because this is not true. When frame S dash is traveling close to the speed of light so this is getting sort of close to the speed of light in relation to frame S, time is not a constant. In fact what happens is that time becomes a part of a four dimensional space-time continuum and we can't use these linear equations. We've got to come up with some nonlinear equations to find the position, to define
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the speed, to find the time in this frame of reference. We need different transformation equations and these different transformation equations are called Lorentz transformation equations and they're based on a factor called the Lorentz factor. Let's have a look at them. The position equation, so to find this position in a frame S dash that's traveling at a speed close to the speed of light relative to frame S you need to use this equation here and gamma is the
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Lorentz factor. It's very similar to the Galilean equation because do you remember that the Galilean equation was this but what's happened instead is we put in this Lorentz factor to account for the fact that frame S is moving close to the speed of light. If you want to find the time in frame S which is moving close to the speed of light, in the olden days we had this equation with Galilean transformation. That can't happen in special relativity because
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time is not absolute so to find time with special relativity we use the Lorentz transformation which includes this Lorentz factor. Let's have a look at the Lorentz factor in a little bit more detail. It's just a number. The formula for it is 1 over the square root of 1 minus V squared over C squared and it's going to give you a value of 1 and above. At 1 when the Lorentz factor is 1 the object is traveling very slow speeds compared to the speed of light so even
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all the way up to about 0.6 to the speed of light the Lorentz factor doesn't really change very much which means up to those speeds actually you could use Galilean transformations. But you'll notice that after 0.6 and specifically after 0.9, so speeds very close to the speed of light, the Lorentz factor increases very very significantly which means as you approach the speed of light Galilean transformations are no longer applicable and this Lorentz factor
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is going to make your transformation equations look very different indeed. Now we understand the size of the Lorentz factor and how it only really applies to speeds close to the speed of light. I just want to nip back two slides. You'll notice that we found equations for x dash, the position, and the time in a frame of reference that's moving at relativistic speeds compared to another frame of reference. But we haven't yet found an equation to find the speed of
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an object in this situation. We do have an equation for relativistic frames of reference to find the speed and this equation shows that velocities do not add the same way as in classical mechanics because the speed of light is a limiting factor here. Let's look at this we've got u. u is the speed of this rocket traveling relative to a stationary observer so imagine I'm here and I see that rocket traveling at point 8c. I also see this rocket here traveling at point
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5c. How fast does this rocket look to that rocket? That's the question we want to answer and this is the equation that we want to use to solve it. So let's use that as an example. u dash is the velocity of the other spaceship relative to the other and we have to pop in values here so for u we'd have to put in point 8c for v we have to put in point 5c we bring it all the way through and u dash is equal to point 6c. So the velocity of the spaceships relative to
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each other is point 6c and that's not a linear transformation that doesn't you can't do that easily in your head. Alright so it's important when you're looking at two frames of reference are traveling at speeds close to the speed of light then you need to use this equation to determine the speed of the frames reference relative to each other. Head to your revision note template, write down the ReLens equations for x dash, t dash and u dash in your revision notes. Draw
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and annotate the ReLens factor graph in your notes because that comes up quite a lot and complete questions 3 to 5 in your knowledge questions. This learning objective is a bit of a mouthful with a quite complicated looking equation so let's have a look at more detail we're looking at something called the invariant quantity and we'll head back to Galilean relativity. If you think about the normal world as we would call it Galilean relativity and you had a
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calculator and you measure the length of the calculator as five units. Whenever you put that calculator in that frame of reference it still has a length of five units right the length of the object doesn't change no matter what location you put it in length is an invariant quantity doesn't change and it's the same for all observers. That's not the case for special relativity. For special relativity things change the length changes the time changes different
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things change so we have to come up with a new invariant quantity that doesn't change and it's the same for all observers and it's not length like it is in Galilean it's something else and it's called the space-time interval it's given the symbol Delta S space-time interval. You can derive this you don't need to but what you need to know is for special relativity the combination of distance and time is invariant from one coordinate system to another and that's
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called the space-time interval and you can find it using this equation here. So let's get into some of the meatier parts of special relativity and we're going to start by defining what is meant by proper time interval and proper length then we'll look at solving problems with time dilation and then we'll look at the concept of the relativity of simultaneity. Don't worry if this means nothing to you at the moment it will very soon I promise. Every observation in physics is
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made relative to a frame of reference and this can be thought of as the perspective from which measurements are made. The proper time interval is the time between two events as measured in a frame where the events occur at the same spatial position or happen in the same place. Imagine two events a light turning on and then turning off. If you're standing right where the light is and measure the time between these events with a stopwatch then that's the
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proper time interval. Again you can see from the picture here if you're measuring a race you measure when the race starts you measure when the race finishes you stand at the same point and you're using a stopwatch then that's the proper time interval. The proper length is the length of an object as measured in its rest frame. If you measure the length of a stationary train car while you're standing next to it then you're measuring the proper length. If you're
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measuring the length of your moustache with a very odd-looking verniacaliper then you're measuring its proper length. It's the true length of the object before any relativistic events come into play. Now time dilation is one of the most fascinating aspects of special relativity. It tells us that time can stretch depending on relative velocities and there's an equation that captures this idea. Okay you have seen this this little figure before this is the Lorentz
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factor and what this equation basically says is that if you are moving very quickly at speeds close to the speed of light relative to a stationary observer then your watch will move slower or time will move slower for you than it does relative to the observer. And we can use this example here so we've got this chap here he's standing on the earth he's the observer who's stationary he's in a stationary frame of reference and he measures the time this person on the
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rocket takes to go around the world. Okay the guy who's stationary measures that time as nine hours let's say but the guy who's going round the world on the rocket at speeds close to the speed of light only measures that whole time as one hour is taken. That's because he's moving really fast and his time appears to move slower and we can find out how much slower this time appears to be by using this time dilation equation here. The proper time delta t naught is a time
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felt by the observer in that frame of reference. Lambda is the Lorentz factor and this delta t here is the dilated time so it's a time experienced by the person who's in the moving frame of reference. Another aspect of special relativity is that objects appear shorter in their direction of motion when observed from a stationary frame. So just as with time dilation the Lorentz factor plays a crucial role in this and as the object speed increases its length
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appears to contract or shrink in the direction of motion when viewed by a stationary observer. So just as we had before with time dilation we have an equation to link the proper length and the appeared length using the Lorentz factor but because it's a contraction this time we divide by the Lorentz factor because the Lorentz factor is always greater than one and we want our contracted length to be smaller than the proper length. It's likely that you'll
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have to do equations using time dilation and length contraction in your exam and they really are very simple please don't overthink them. I have some examples after this short part of the tutorial for you to practice on and you'll see whilst this seems abstract and odd the questions themselves in the exams are pretty simple. I'd like to finish off this short section of the tutorial by looking at simultaneous events. Now simultaneous events are those physical
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events that may happen at different locations but happen at exactly the same time. According to special relativity simultaneous events are not absolute that means that two events which are simultaneous with respect to one observer are not necessarily simultaneous in respect to a different observer and a different frame of reference. For some other observer one event could happen before or after the other and the best way to look at this
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is to look at Einstein's thought experiment to think this through. So what we have here in these diagrams is we have a train moving with velocity v in a given direction and on the train in the center of that carriage there's a light bulb and that light bulb lights up and of course light will travel in all directions from this light bulb but we were interested in it traveling towards front of the train and towards the bank of the train. Let's look at the observer
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on the train for the moment. When that light bulb lights up the speed of light is a constant and that means the light is going to reach A at the same time that light is going to reach B. These two events event A and event B will be simultaneous for the observer on the train. Let's now look at the observer outside the train on the platform. The train is moving at velocity v and if the light bulb is switched on light is emitted in the forward and backward
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direction at the same time. However while the light is traveling the train has also moved towards the forward direction and that means the back end of the train has moved towards the light source. The front end of the train is then moving away from light source. Given that the speed of light is constant that means that the light going backwards will reach the end of the train first and the light that's going forwards will take longer to reach the
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front of the train. So event A will happen before event B. Events A and B are not simultaneous. So let's be really clear the same two events are simultaneous for one observer, the observer on the train, but they're not simultaneous for the second observer, the observer on the platform and according to special relativity simultaneous events are not absolute. That means that two events which are simultaneous with respect to one observer are not necessarily simultaneous for other observers in
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different frames of reference and this really highlights the intricate interplay between space and time in relativity and we're going to look at that in more detail when we come on to space-time diagrams. I'd like you to get your revision notes and write down these following things. I'd like you to write down what the invariant quantity for special relativity is, write down the definitions of proper time interval and proper length and write down the
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equations for time dilation and length contraction. Then complete questions 6 to 10 in the A5 knowledge questions. We move on to learning objective 13 and 14 and this is where we start to look at space-time diagrams. Space-time diagrams are just like a distance-time graph but they help us to understand the concepts of time dilation and relativity. It kind of looks like a distance-time graph okay but it's inverted. If you think about a distance-time graph we have distance on
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this axis and time on the other axis on this x-axis but what you can see here is we have distance on the x-axis and ct, time on the y-axis. ct is basically a scaling factor for time so it's a scaling factor that allows us to look at relativity for time. Now because the units of speed times time is equal to distance the units of the ct are meters and the units of this axis here which is meters as well so they both have the same units these
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axes but we kind of can compare it to an inverted distance-time graph which means that if we look at gradient of any of the lines on this space-time diagram we can start to think about velocity and how we can find out velocities of frames of reference and objects that are traveling at relativistic speeds. Sometimes you will see the term Minkowski diagram. That's the term given to a space-time diagram that looks only in the x-axis in one dimension and of course
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there are three dimensions in space there's x y and z. You can get space-time diagrams that look at all the different dimensions we are not going to do that we're going to focus on Minkowski diagrams which only look at the x dimension. So what would I like you to get out of this slide? Okay there's two axes the y-axis is ct has units of meters and we can kind of compare it to a time axis the x-axis is distance in the x direction it also has units of
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meters and this diagram can be compared to an inverted distance-time graph. Let's start looking at what information a space-time diagram can give us. When somebody observes a stationary object its trajectory its path on a space-time diagram or a Minkowski diagram must be such that the position doesn't change but of course the time does. So just like a distance-time graph, distance is on horizontal axis, vertical axis is just like the time axis but with a scaling
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factor of c if we consider a stationary object then the position of this object with time doesn't change. So for a stationary object we're looking at a vertical trajectory or a vertical world line on the space-time diagram. A world line is a representation of the space-time trajectory. Let's now look at photons and the speed of light. Do you remember I told you to think of a space-time diagram like an inverted distance-time graph? Well if you were
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looking at a distance-time graph you would know that the gradient of a distance-time graph is equal to velocity. Well let's look at what the gradient of a space-time diagram looks like. We know that the gradient is the change in y over the change in x. So the change in y over the change in x. Change in y is c delta t over the change in x. It's just delta x. Let's take the c out and what that means is we have c delta t over delta x. Well delta t over delta x is the same as 1
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over velocity. So the gradient of a space-time diagram is equal to c 1 over velocity which is also equal to c 1 over the gradient. Let's now consider photons. We've gone from one extreme where we're looking at stationary objects to the other extreme. We're looking at particles that travel at the speed of light. So our gradient is actually limited on a space-time diagram to c equals 1 over c which is 1. Gradient of 1 is at 45 degrees on this angle here. So if you see
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this line here, this world line here at 45 degrees, that is the world line of a photon. It's the world line of the speed of light and nothing can be on this side of that world line because that implies that something is traveling greater than the speed of light and that's not possible. So when you're looking at world lines, you're always looking at world lines within this 45 degree angle here and the world line of course of a photon is at 45 degrees and a gradient of 1.
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We've looked at a stationary object. We've looked at the fastest particle, a photon. Let's look at just your bog standard fast moving object. When somebody observes an object moving at a constant speed, the world line is sloped. Okay, it's sloped like this and it's sloped positively when the motion is in a positive direction. It's sloped negatively when the motion is in a negative direction. We only allow, like we said before, let's go to this 45 degree
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angle here, we're only allowed lines that are steeper than the 45 degree angle. Anything less steep than a 45 degree angle is not possible. Let's think about events now on this space-time diagram and by events I just mean two different things happening. Let's say event 1 is happening at position x1 and it's a light bulb being switched on. Event 2 is a position x2 on the x-axis and it's a second light bulb being switched on. You can see the position of these light bulbs
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are shown clearly on the x1 and x2 axes. You can see that the time that each of these light bulbs are put on are shown clearly as CT1 and CT2 on the CT axis. My question to you is can event 1 communicate with event 2 before it happens and therefore is it possible for event 1 to cause event 2? That seems like an odd question I suppose but what I'm saying is could the light bulb 1 send a signal to light bulb 2 to turn it on? And we can answer that question by
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checking that the world line between these two events is steeper than the speed of light. If you connect the two events and connect the world line and if you see that the gradient of that is steeper than the gradient of something being communicated by a photon then yes light bulb 1 could cause light bulb 2 to turn on. So it is possible they are causally connected. Let's have a look at more world lines. We looked at the world line back when we had a stationary
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object and we saw that the world line was just straight up and down but we can see we have lots of different colored world lines here and we'll start with the yellow one. Okay nice and easy we've seen that already. The yellow one is when body 1 moves in a positive direction with a constant speed and we can see it's quite a steep line it's steeper than 45 degrees so we know that it's possible within the physical world. Body 2 which is this purple line has
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the same speed as body 1 and we know that because they're parallel. It's got the same speed as body 1 but body 2 is moving a fixed distance away from body 1 and that fixed distance is constant all the way through its journey because they're both moving at the same constant speed. So they started off their journey a fixed distance away from each other and because they both move with the same constant speed that fixed distance is the same. The red dotted line we've seen
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this before this is when the speed of the object is equal to the speed of light and anything that's less steep than that would be impossible. What we haven't seen before is this squiggly line. This squiggly line here illustrates a body that's swinging back and forth between left and right and it's accelerating. A curve indicates an acceleration and finally this line here this blue line tells us a body flies at almost the speed of light. How do we know that?
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Because we can see to begin with the gradient is almost 45 degrees and slows down until it eventually stops. How do we know that? Well because at the end the gradient is almost infinite means stationary so it goes from the speed of light all the way to stationary and it decelerates down to that. These world lines are something you should be familiar with although the IB specification does specify that it's only looking at constant velocities so
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you could probably forget this information but it's good to know. Let's try and expand this idea of space-time diagrams to include two people. So before when we were looking at space-time diagrams we were looking at one frame of reference and these world lines were all in this one frame of reference. It'd be good now to try and incorporate a second frame of reference because that's what simultaneous events is all about in special relativity and
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so on. So observer 1 is stationary and observer 1 has the frame of reference that we're really comfortable with already. CT on the y-axis and X on the x-axis. Observer 2 is moving very fast relative to observer 1 and he also has a frame of reference we're calling S'. He also has a representation on the space-time diagram of X' and CT'. You can see that these axes are at an angle to the original axis and this angle later on will tell us how fast observer
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2 is moving and relative to observer 1. This angle here between CT and CT' should be exactly the same as the angle between X and X'. That's how we represent two frames of reference in one space-time diagram. I want to take this a little bit further. So frame S', well it's got the CT' and the X', looks different when you start putting the grids in. Can you see here the the grid is kind of squished up but it still measures a right angle geometry in their
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frame of reference. It just doesn't look like that when you're trying to put it onto paper. So we'll again we'll look at these grid lines in a bit more detail as we look at simultaneous events and so on. I mentioned before that we can find the speed of observer 2 relative to the stationary observer 1 by using the angle here and how we do that is we find the speed of this observer or the speed of frame S' by using the angle between the vertical and this world line of CT'
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and using this relationship here tan theta equals V over C. V is the speed of observer 2 moving in relation to observer 1, C is the speed of light and tan, the tan function you should be very aware of. So once you know this angle here you can find out the speed of the observer 2 in relation to observer 1 by using this equation and it's pretty simple for you to do that. Do you remember that we talked about the relativity of simultaneous events and we had two well the diagram of two trains
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and there was a light bulb in the middle and we said this this guy here experiences these two events A and B to happen at the same time but the guy who's watching from outside experiences A and B to happen at a different time and we decided that events are not simultaneous necessarily because it depends on the frame of reference of the observer. Let's try and relate that to a space-time diagram and I want to think of observer 2, the person who's moving.
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The space-time diagram grid axes for this guy who's moving is this set of axes here and we saw on the last slide that the axes set parallel within it they're kind of squished up. So if we now consider two events in the moving frame of reference P and Q, we'll do P and Q's of A and B, if you extrapolate back a parallel line back to the time axis you can see for the observer who's moving P and Q hit the same point of the CT axis. That means for observer 2
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events P and Q are simultaneous but let's now consider P and Q for observer 1. For observer 1 you're using a different set of axes we're using these axes and what we want to do again is extrapolate back at grid lines that are parallel to their frame of reference so we extrapolate back here to the CT axis and back here to the CT axis and if we go up the CT axis in time moving forward we see that event P happens first and then event Q happens next. So despite those
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events being simultaneous for observer 2 they are not simultaneous for observer 1. I'd like to solidify this with an example so slightly different just two different events A and B and the question is do events A and B happen simultaneously? We'll start in observer 2's frame of reference looking at this CT dashed axis. If we extrapolate back to CT dashed axis event A happens there and event B happens here on the CT dashed axis. As we move up the CT dashed axis A
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happens first and B happens second so A and B are not simultaneous for observer 2. What are they like for observer 1? For observer 1 if you extrapolate back to the CT axis A's extrapolated back B's extrapolated back they both happen at the same time so events A and B are simultaneous for observer 1. Here's another example of a question you might be asked with a space-time diagram. You might have two positions on your space-time diagram A
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and D and the question might be can a pulse of light go from A to D? I'd like to start in the frame S X CT axis frame. In this frame we know that a pulse of light will travel at the speed of light which means that we can draw something called a light cone. So from point A we draw two lines at 45 degrees and what those two lines represent is the position of a pulse of light traveling the speed of light and anything in this side of it is not
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possible and anything within the cone of light is possible. We can see that in the X frame of reference this line A to D the world line from A to D is possible because it sits within the light cone that means that a pulse of light can go from A to D because it's not traveling faster than the speed of light to get there. Is it possible in the other frame of reference? Let's have a look. We're now looking at the frame of reference of X dash and CT dash and we want to draw a
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light cone in there and what you'll see is yes that A to D falls within the light cone on that frame of reference so yes it's possible for a pulse of light to go from A to D. Another thing you might be asked to do with a space-time diagram is to determine the value of the Lorentz factor and to do that you'd head to unit one on the X dash axis and the CT dash axis and extrapolate them back to the CT axis and you'll find this value here where it's extrapolated back is
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equal to the Lorentz factor. One final thing you can do with space-time diagrams is determine the time dilation and length contraction using the coordinates of these space-time diagrams. On this space-time diagram there's an event being chosen here. If you extrapolate back there and there that will give you the X dash position and the CT dash position. If you extrapolate back to the frame S, frame of reference, that will give you the X position and
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the CT value. Having these values then allows you to make calculations using time dilation and length contraction. I went through quite a lot of different examples of space-time diagrams there and it's time for you to go to the graph section of your revision note template and draw those graphs down. I really don't want you to overthink this okay. I know that some of it's been quite complicated. I promise you it's going to be okay as we head to the next section
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step two which is activate the application and we start looking at taking these concepts and putting them into exam style questions. Now if you're anything like me you get to the end of this topic and you think yeah but how do you know? Where's the proof? This all sounds like a lot of rubbish to me. Well I'm going to explain to you how we can prove special relativity and where the experimental evidence comes from. We're going to start by looking in space.
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In space we have something called cosmic particles and cosmic rays and those rays are made up of helium and hydrogen particles careering through space and what they do is they collide with particles as they hit our Earth's atmosphere. Our Earth's atmosphere is around about 10,000 meters above the surface of the Earth. When the hydrogen and helium cosmic particles collide with the particles in our atmosphere these collisions cause a shower of other
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particles and these other particles can be things like pions, kaons, positrons, electrons and muons. Let me tell you something about muons. They have a half-life of 22 microseconds. Oh before we go any further this micro here means 10 to the minus 6 but it's also the symbol we give to muons so that could be a little bit confusing but I'm just trying to relate it to the context. Okay so muons have a half-life of 2.2 microseconds before they decay into other
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particles and they travel really close to the speed of light. So let's say we have 100% of muons that are created when they're hit by a hydrogen particle. 2.2 microseconds later half of them have decayed into other things. 6.6 seconds later actually only 12% of those muons are left. In 6.6 microseconds we expect all of these muons that were created in the Earth's atmosphere by colliding with the hydrogen particles we expect them to be completely decayed and in real life
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most muons will decay between 2.2 microseconds and 6.6 microseconds. That means if we take their speed which is a factor of 0.995 of the speed of light they should only travel between 660 meters and 2,000 meters towards the Earth's surface after they've collided with these particles. But do you remember I told you that the Earth's atmosphere is 10,000 meters above the Earth's surface? Wait wait wait wait, how do we know that they should only travel between 600 and
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2,000 meters? Well distance equals speed times time and speed is 0.995 times the speed of light times the time let's just say 6.6 microseconds which gives us around about 2,000 meters. So in common sense around about 2,000 meters below the Earth's atmosphere we should have very little muons there but what we found is when we detect muons on the Earth's surface is that there's way more muons detected on Earth using particle detectors than expected and this
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is explained by special relativity. If you calculate the Lenz factor for speeds of 0.995 to the speed of light that gives you a Lorentz factor of 10. So in proper time if their half-life is 2.2 microseconds then their dilated time is going to be 22 microseconds. Oh 22 microseconds you can travel a lot further and actually some of them are still existing within 6.6 microseconds in their proper time which gives a dilated time of 66 microseconds which
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means they can be traveling up to 20,000 meters before they're all decayed. So suddenly with time dilation it explains why we are detecting way more muons on our surface than we should be without time dilation. Let's now think of special relativity in the frame of reference of the muon. The muon experiences proper time and in the muon's life it only lives for 2.2 microseconds on average but what it sees it sees the Earth moving towards it at 0.995 times the
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speed of light. They see the Earth moving towards them really fast. Because they see this Earth moving towards them really fast they experience length contraction and they experience that length contraction by the same Lorentz factor of 10. So instead of the Earth having to travel towards the muon 10,000 meters it only has to travel towards the muon a thousand meters in its frame of reference before it hits the Earth's surface. So from the muon's point of view
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it has to travel a smaller distance to get to the Earth and therefore it's more likely to hit the Earth's surface than it would be if we weren't considering special relativity. So there we have it, muons provide direct experimental evidence of special relativity because we detect so many of them on the Earth's surface that there must be something that's dilating their time in our frame of reference and contracting their length in their frame of reference to
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allow them to reach the Earth's surface. I'm going to be honest with you I do not find it easy to teach special relativity. I find the concepts a little bit mind-blowing. My objective and logical brain struggles with this idea and if you're struggling with these concepts that's okay you just got to pass the exam. You know what these concepts are now let's just help you pass the exam. Head on to step two and we'll help you do that alright?
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