 In this lecture, we will learn the transition in thinking that led from Galilean relativity to the special theory of relativity in 1905. We will learn the postulates of special relativity, which are the basis of the mathematics of the framework, and we will look at some of the consequences of those postulates even before we delve into the mathematical framework itself. In class, we looked at the lessons of the Michelson-Morley experiment, which can be summarized as follows. First, light travels at a fixed and constant speed in any medium, regardless of the relative velocity of the light source and the light observer. This is unlike any other phenomenon described in mechanics, and it implies that Newton's mechanics is actually the incomplete theory of nature. No medium is actually required for that light to propagate. Unlike a mechanical oscillatory phenomenon, a wave, to exist, light requires no medium to be distorted, it is not mechanical in origin, and this implies that Maxwell's equations are complete, or at least sufficiently complete to understand light. These lessons, however, would not be fully absorbed until about 1905, when Albert Einstein one of the most famous physicists in history published the definitive papers explaining how to reconcile mechanics, electricity and magnetism, and the results of the Michelson-Morley experiment. Now interestingly, the mathematics that Einstein would come to rely on for encoding the relationship between space and time measurements in one frame and space and time measurements in another frame, were actually laid down much earlier by Hendrik Lorenz in a famous paper on the compression of bodies in the ether. The mathematics that would later become a replacement for the Galilean relativity equations would actually be kind of sketched out, but for a completely different purpose than they would ultimately be used for. Lorenz was considering the effects of the ether on bodies that are moving through it. Now these bodies are held together by chemical bonds, they're made of atoms and those atoms are chemically bonded to each other, but chemical bonds are just electromagnetism in action. And so based on this, he arrived at a few hypotheses. Should the ether exist? First, that mechanical bodies would compress along the direction of motion in the ether, and this has a precise mathematical description for the process. And second, in transforming observations from the ether frame to other frames of reference, he would conceive of an alteration of time that also had a very firm mathematical description. Now Lorenz conceived of this during a period when the ether was still very much believed to exist. The results of the Michelson-Werley experiment were not fully digested during this period. The ether's existence would ultimately be disproven, or at least shown not to be necessary to explain anything that was then known about nature. In the decades that would follow this work, but the mathematics laid down by Lorenz during this period would still prove extremely useful, and today we know this as the Lorenz transformation, the replacement of the Galilean transformation from frame to frame. We'll come back to that in a later lecture. Let's talk about Albert Einstein and his miracle year of 1905. So Albert Einstein in 1905 was a young PhD physicist who was laboring doing physics as sort of side work in what was otherwise supposed to be his regular work at the Swiss Patent Office in Bern, Switzerland. He had this job because he was unable to secure, for instance, a faculty job after completing his PhD, and in part this was because Einstein really couldn't get any recommendations out of any of the professors that had supervised his education because he had so irritated them with his behavior during what we would consider graduate school, including skipping out entirely on classes, in particular, for instance, math classes for mathematics he didn't consider to be physically useful, and also for challenging his professors, challenging their authority, thinking of them as idiots and so forth. Now, Einstein was a very bright young man, but he was also a bit arrogant and temperamental, and this didn't do him any favors when he was trying to get a job. Now, ultimately, it was the thinking that culminated at the end of his PhD work and then into the years leading up to 1905 that would lead to a change in the way that the community of physicists thought about the supremacy of the assumptions made in Newton's mechanics versus what the laws of electromagnetism, that is Maxwell's equations, had to say about light and space and time. And in 1905, he published the work that had resulted from his PhD research in a series of about four papers, and this was his so-called miracle year. This is a highly productive year for a young and relatively unknown physicist in this day. In doing so, he reframed assumptions about space and time and what is and what is not invariant to all observers and all frames of reference recall that in the Newtonian and Galilean view of space and time, time is experienced the same way by all observers, regardless of their relative states of motion. Time would be referred to then as invariant, but what Einstein proposed challenged thinking about what was and what was not invariant in space and time. Now, in short, here's what Einstein did. He accepted the conclusion of the Michelson-Morley experiments that light has a fixed speed regardless of the motion of the source relative to the observer of the light from the source. This then implied that there's no ether as well. Using a simple thought experiment, like the one that we did in the Foundation's lecture, involving car headlights and the ether, he explained also why time is not absolute even in Newton's mechanics. Time itself is not an invariant concept, and he did a quick thought experiment that showed that it wasn't even true under Newton's way of thinking. And so he was free to abandon time as the constant thing in transformations from one frame to another frame. Instead, he chose to preserve overall the forms of the laws of physics and the speed of light, which the Michelson-Morley experiment implied was constant regardless of your state of motion. This then led to the foundation of two postulates that allowed him to then define all the mathematics that would follow. The first postulate is what I hinted at a moment ago. The forms of the laws of physics, that is, f equals ma, for instance, or Maxwell's equations, will be the same for all observers regardless of their state of relative motion, that is their frame of reference. The second postulate is that the speed of light is the same for all observers regardless of their frame of reference, their state of relative motion. Now, let's begin by breaking down the concepts that we need to dig into so that we can really understand where all of this is headed. We need to take these postulates and parse them into some phrases and words, define those things, and then go forward from there. This will allow us to build back up to a more complete understanding of the math that we'll eventually need in order to understand relativity and relative motion going forward in modern physics. First off, there's the word event. You might think you know what this is, but in physics it is given a very precise definition so that we can always try to define the concept mathematically so that everyone can agree on what an event is and what an event is not. Another phrase that's deceptive and may seem to have a common definition for you, but where we have to be careful about this in physics is the phrase frame of reference. We need to define it. It comes up a lot in our discussions and because descriptions of events can depend on the frame of reference in which the observation is made, we have to carefully define this concept. Simultaneity is another word, probably the one that causes the most consternation among people who are making the transition from introductory mechanics and electricity and magnetism into modern physics. Because simultaneity has probably been implied in a lot of things in the past, but we have to put it on some firm footing conceptually here so that we can use it and explore it going forward. It turns out that the concept of simultaneity is actually essential to many things you take for granted all the time. You've just never been forced to think about it before. This concept turns out to be a subset of the discussion of events and it's going to play a very important role, so we're going to have to define this. And then finally, you might think you're comfortable with this idea, but the phrase speed of light would benefit from some context and some description. We should really try to understand the number that is behind this phrase. It's a ridiculously large number compared to most things on the day-to-day human scale. But it actually turns out that this speed is only impressive on the scale of things that are roughly the size of planets. I'll even allow solar systems and maybe smaller, depending on how you define a solar system. But it turns out the speed of light is not as fast as we would like it to be and certainly on the scale of things like the entirety of the universe, it is pathetically slow. So let's get started in the next few slides trying to define each of these things very carefully. First of all, let's talk about the concept of an event. An event is, quite simply, anything with a location in space and time. So let's practice this concept. I will show you an event and I want you to try to describe it with words and numbers. Go ahead and pause the video while you're doing this when prompted. See if you can come up with a short sentence that describes the event using the definition that an event is anything with a location in space and time. This is excellent practice for defining events in any new situation that you will encounter as an exercise in setting up a problem for eventual solving. I've given you a one-dimensional axis, so an x-axis. And let's say that the numbers here have units of meters. That'll make it easy for us to very precisely describe an event. I've also given you a timer. The timer is capable of ticking out about 12 seconds, and the units on each of these tick marks, 1, 2, 3, and so forth, are seconds. Okay, so you have a spatial reference and you have a time reference. Given that information, let's go ahead and proceed with looking at an event and attempting to describe it. I want you to describe the event depicted above on the x-axis. Go ahead and pause the video. Come up with a short sentence that uses the definition of an event to describe it, and then resume the video when you're ready to compare to my answer. You should have come up with something like the following. The dot is at position x equals 0 meters at time t equals 0 seconds. That's an example of describing an event in physics. The dot is at a spatial location that is defined at a time that is also defined, x and t, space and time locations. If you didn't feel comfortable doing this, now that you've seen me go through it once, let's try another event. You try to describe it and let's see what you come up with. Describe the event depicted above now on the x-axis. Go ahead and pause the video. Write down a short sentence that uses the definition of an event to describe this event, and then resume the video when you're ready to see what I came up with. So what I decided to do was to describe this as follows. The dot is at position x equals 2 meters at time t equals 2 seconds. Make sure to check your space reference and your time reference when presented with an event so that you correctly mark in, say, x and t or x, y, z and t. The coordinates of an event. An event is something that has well-defined coordinates in space and time, a location in space and time. Let's now talk about a frame of reference. A frame of reference is any object or system, all of whose parts move at the same velocity with respect to an agreed-upon reference point in space. That's quite a mouthful. Let's go ahead and illustrate this with an example. I want you to consider the three objects shown below, labeled black, a black dot, blue, a blue dot, and red, a red dot. Now, one of them, the black dot, is agreed upon by the others, the red and blue dots, as the common reference point for all measurements. Now, as I've depicted them here, blue and red have an associated velocity vector, shown here, and as depicted, red and blue are in the same frame of reference because they have the same velocities. Let's check that. If I roughly eyeball the length of this vector, it seems to be pretty similar to the length of this vector. So from this I could conclude that very likely blue and red have the same speed with respect to black. But velocity is not just speed, it's not just the magnitude, it's also the direction. And here I see that the directions align, they point parallel to each other, and so I conclude from this that they have the same velocities. Therefore, although blue and red are both moving, they are moving in the same way with the same velocity. They have the same state of motion, and therefore they are in the same frame of reference. Now take a look at this one. I've changed something here. Does this change alter the conclusion about blue and red? Do the red dot and the blue dot share the same or different frames of reference? Pause the video here, look carefully at the image, and then resume the video when you're ready to hear the answer. The answer is that they do not. Although their speeds are the same, the lengths of those two arrows look pretty much identical. The direction of the motion of the red dot relative to the blue dot, and all measured with respect to the black dot, has changed. This means that they have different velocities, and different velocities means different states of motion, and therefore different frames of reference have now emerged here. The blue frame of reference is no longer the same as the red frame of reference. Now I want you to consider the objects in this picture, blue, red, and now a purple dot. All of their velocities are measured with respect to the black dot as the reference point. That hasn't changed. I want you to practice a little bit more, and I want you to think about how many unique frames of reference you can identify in the above picture. Go ahead and pause the video here. I'm not going to provide the answer here, because I really want you to try to step out on a limb on this one. But feel free to talk to me as the instructor outside of class or in class if you're not confident in how to determine the answer to this question. Now let's visit the concept of simultaneity. Simultaneity is a subset of events in which two events or more are said to be simultaneous, that is to possess of this quality simultaneity, if they are observed to occur at the same moment in time. This seemingly straightforward definition of the concept should not fool you. You have to think really hard about whether events are actually simultaneous, and if there are multiple observers in different frames of reference, for whom are those events simultaneous? Finally, let's look at the speed of light. And let me be clear about the speed of light. It is the number of meters that light can travel once it's been emitted by some kind of source in a certain amount of time. That's just the old definition of speed. But light is special. It's special because the Michelson-Morley experiment tells us that no matter the state of motion of the observer or the emitter of the light, all parties will agree that when they measure the speed of that phenomenon in any frame of reference, it always comes out to be the same number. 2.998 times 10 to the 8 meters per second, at least an empty space. Now the history of the speed of light is interesting and can be cherry-picked through to take a look at what people try to do to measure the speed of this phenomenon, because it is ridiculously fast. Now Galileo Galilei famously claims to have attempted to measure the speed of light. By uncovering a lantern, having an assistant on a distant hill, who in response to seeing the light from Galileo's lantern, then uncovers one of their own, and then Galileo, upon seeing the assistant's lantern light, records the time for the round trip, taking into account human reaction time. It turns out, of course, that light moves way too fast for this to work with 17th-century technology, even if Galileo used the most precise clocks of his day, which he had invented, water clocks. There's no way that, even given 40, 50, or 60 miles of distance between him and his assistant, that that technology would have been sufficient, especially with really slow human reaction times to, in fact, measure the speed of light. So this was kind of a lost cause, but a clever technique nonetheless, and one which can successfully be used to measure the speed of sound. Another important person in the story of the measurement of the speed of light is Ul Romer. Now he would go on to use the period, that is, the time it takes to complete one cycle of Jupiter's moon Aya, which had been discovered by Galileo using the telescope. And by looking at its cycle of eclipses by Jupiter, to then make the first reasonable determination that light travels in finite time. He did this in about 1676. Revisiting his data in a modern context suggests he shouldn't have been as accurate as he was in measuring the speed of light, but he actually got fairly close to the currently accepted value, certainly impressive for its time, impressively close to the currently accepted value of the speed of light. But one could definitively conclude from his work that light does not travel instantaneously from place to place, rather it takes a finite amount of time to cross space, even if it does so very quickly. Now by the time of Albert Einstein's publications, the speed of light had been established by multiple experimental methods to be within about 50 km per second of the precision of today's methods. And that is remarkable for such a large number representing such an incredibly high speed. So let's then take a look at the modern speed of light and the number that is the currently accepted calibrated value of this speed today. And I say that because the definition of things like the meter are based on the distance that light travels in a certain amount of time. So based on the current definition of the meter and the second, the speed of light is defined to be exactly 299,792,458 meters per second, or about 2.998 times 10 to the eighth meters per second. A good rule of thumb, something that will aid you whether you're thinking about how long signals will take to propagate in electronics or if you're thinking about how long it will take for a light signal to propagate across some space for a communication system or something like that. A good rule of thumb is that light travels roughly one foot in one billionth of a second. That is, it goes one foot per nanosecond. It's a handy little thing to remember for engineering purposes going forward. Now let's begin to look at the consequences of the postulates of special relativity. And I say special because there's a more general theory of relativity, a more general theory of space and time that Einstein would spend another decade working out after 1905. What makes the early theory of space and time that he developed special is that it focused on what are called inertial frames of reference, those in which there are no net unbalanced forces. Now that doesn't mean that accelerations can't be present, but it is a special case of a more general theory of reference frames, space and time. Now under this special condition, an object in motion will appear to all observers in all frames to have a constant velocity, even if observers in different frames disagree on the magnitude and direction of the vector. So let's recall his postulates again in light of this special condition for the frames of reference that we're talking about here. Postulate one is that the forms of the laws of physics are the same for all observers regardless of their state of relative motion. That is, regardless of the frame of reference in which they find themselves. We've looked at the definition of the terminology frame of reference. The second postulate is that the speed of light is the same for all observers regardless of their frame of reference. All observers, no matter their relative state of motion, when they measure the propagation speed of light signals, will always find, and this is based on experimental observation, that light travels at the same speed in every frame of reference, even if that frame of reference is moving with respect to the source of the light. This is taken to be the thing that is invariant from one frame of reference to another frame of reference, not time, the speed of light. Now let's look at some of the consequences from these postulates. Starting with the first postulate. So the consequences of the first postulate are both straightforward and a little surprising. So one of the conclusions you can draw from the first postulate, assuming that it's true, is that all physical laws like Newton's laws or Maxwell's equations will all have the same observed form in all inertial reference frames. Now this is pretty helpful actually, because what it means is that regardless of our relative states of motion, the basic laws of physics that we can uncover by doing experiments, observations of the natural world, are not dependent on your current state of motion. The moon goes around the earth, so from our perspective, the moon appears to be moving. But the law of gravity has been tested on the moon by dropping objects there. We see no difference between the law of gravity on the moon and the law of gravity on the earth, despite the fact that we are definitely in relative motion to one another. This has been tested more precisely than just dropping things on the moon, but the basic conclusion is that this postulate holds, and as a consequence of that, basic laws of physics can be determined regardless of what your state of motion actually is. But this consequence has a flip side. It's impossible based on determining the laws of physics by making observations in your frame of reference to determine whether or not you are actually in motion. The analogy I like to make for this one is being a little sleepy on a train. If you've ever been on a light rail car or a real passenger train, you've been a little tired, you're sitting on the car waiting at the station for the train to leave, and another train is parked next to you. You might doze off for a moment while sitting there looking at the other train. And then you might wake up, and during the time when you were slightly unconscious, your train began to move. With ever so slight an acceleration, you started to gain some velocity. So when you wake up, you've missed the fact that there was an acceleration in your frame of reference that caused you to start moving, and you might look out the window and see the train next to you moving past you and draw the conclusion that the other train is pulling out of the station. You conclude, therefore, that you're in the rest frame with respect to the Earth, your train is standing still because you feel no forces, and the train next to you is moving. But then suddenly you reach the end of the train next to you, and you realize that your train is the one moving with respect to the ground, and that other train was sitting still the whole time. You had no way of knowing that you were actually the frame in motion with respect to the Earth, because there were no cues, and there's no experiment you could have done in that 30 seconds while you're passing the other train that would have definitively told you you were moving and the other train was not, or that the other train was moving and you were not. And that's one of the consequences of the first postulate. There's no way to measure even the most fundamental statements about nature, the laws of nature, and figure out that you are moving and not something else. So as a result of this postulate, it has to be concluded that there is no such thing as an absolute state of rest or an absolute state of motion. All motion is relative. All motion in nature is relative to a reference point. You have to pick what that reference point is, depending which one you pick may change the degree of your state of motion or the state of motion of the other frame of reference. All motion is relative as a consequence of this postulate. There is no experiment that could be done if this postulate holds forever that would tell you that you were moving and something else wasn't or vice versa. Now let's look at the consequences of the second postulate. The speed of light is the same for all observers regardless of their frame of reference. Now the consequences of the second postulate are typically more surprising to a general audience of individuals who start really thinking about this for the first time on their own. And these conclusions tend to put most people well outside the comfort zone of typical human experience. So let's take a look at these. So all observers agree that light moves at a fixed speed. This is a singular invariant independent of states of relative motion. Now that's already a bit freaky in the sense that you could be driving in a car at 70 miles an hour and switch on your headlights and somebody on the side of the road standing still with respect to the earth measures the speed with which the light from your headlights passes them and they measure 2.998 times 10 to the 8th meters per second. Exactly, not 70 miles per hour faster, exactly the speed that it would travel in empty space if it were emitted from rest. You in your frame of reference could get out on your hood and do some very careful experiment to measure how fast light is moving when it's emitted from your head lamps and you would draw the same conclusion that the speed of the light is exactly that number from a few slides ago. Even though the person on the ground sees you and the source of your light is moving, they still measure the same speed of light you measure. That is freaky. That somehow light is immune to a state of motion of the emitting source but that is an observational fact. It may be freaky, but it's also reality and that means you need to rethink the universe at a fundamental level particularly rethink space and rethink time. And so as a consequence of this observational fact of nature the belief that humans typically hold that say time or space or both are experienced in the same way by observers in different states of motion has to be completely abandoned. If we are to hold the speed of light constant in all frames of reference you have to abandon the absolute nature of, for instance, time. Time passing will be experienced differently by observers in different frames of reference. So as a result of this postulate there's just no such thing as an absolute measure of time or an absolute measure of space. We already have to abandon the notion of an absolute frame of reference in space from the first postulate. But in the second postulate we also find that we can't hold on to this seemingly intuitive belief that time passes at the same rate for all people regardless of their state of motion. Measurements in one frame of reference regarding space and time distances need not agree with measurements in a different frame of reference. But all observers will agree that light signals travel at a specific and fixed speed independent of the relative states of motion. So really special relativity is not so much a theory of what is relative. It's a theory of what is invariant between observers in different frames of reference and it allows us to define a mathematical framework to figure out how to relate our observations. So let's take a look at the relative nature of time briefly using a variation on Einstein's thought experiment or as he called them, Gadunkin experiments. Gadunkin from the German for thoughtful or mindful. What freed Einstein to write down the postulates ultimately of what we now call special relativity was his ability to be able to abandon Newton's old idea of absolute time. That is time that passes the same way for all observers regardless of their state of motion. It was this thing that was really a key moment for Einstein of insight. A moment when he relates that the damn kind of broke moment in his mind and he was freed to draw the conclusions that ultimately led down the correct path to the correct description of nature. So let's take a look at a variation on the Gadunkin experiment that he felt liberated him from the sort of tyranny of absolute time that had been passed down through the generations as an assumption that turned out not to be true. So while riding on a streetcar in Bern, Switzerland where he worked as a patent clerk Einstein began to think more carefully about what it meant to know the time by observing the clock tower. So shown here on the right hand side of the slide is a picture of the Bern clock tower and here you can see the tram lines in the street that likely carried the streetcar on which he was riding at the time when he finally had one of his moments of insight into this question. What does it mean to know the time by observing the clock tower? Well, we're going to do a modern version of his thought experiment because analog clocks are not as common as they were in his day. Time is the measure of distance, if you want to think about it that way, between events that occur, for example, at the same spatial coordinates. So imagine not an analog clock on the face of this clock tower but rather a large blinking light and when the light is on, that marks a moment of time. It's an event. It has a location in space and a location in time. And then the light goes out and then it comes back on later in the same position in space. The gap between the two blinks is what we refer to as a duration of time and we could use that gap between these regular blinks of the light to define a standard unit of time, whatever we choose that to be, the second, for instance. Now, Einstein realized that the way you know that time is passing is you see these two events, but to see these two events, you need to receive light from the blinking light and light has to travel through space. So if you're on the streetcar and the streetcar is moving away from the clock tower, the light from the clock tower has to travel from the tower to your eyes. So you see the blink after it's actually occurred but in your frame of reference in the streetcar, it's the arrival of the light that tells you that a moment in time in event has occurred and then you wait for the next blink to occur. But by then the streetcar's moved a little further away so the light has to travel a little bit further and that takes a little bit longer and so in your frame of reference in the streetcar, time appears to be slowing down and this is just using a Newtonian view of the universe. I haven't even invoked the postulates of special relativity here. This is just the simple fact that light has to travel across a distance and it does so in finite time and the time intervals are stretched by moving away from the clock tower because light has to catch up to you. So even in Newton's view of the universe, time measurements cannot be absolute as a result of this. So imagine two observers that are using a blinking light to measure time. They agree that the blinking of this light is how they will define their standard time units. Now one of the observers is at rest on the ground with respect to the source of the light, maybe standing right next to the blinking light and the other is on a super train that's racing away from the light source and it's doing so at a ridiculous speed, half the speed of light. So the two observers agree to count how many blinks occur while the super train makes a journey of 2 million miles. Now I chose that because this is about how far light can travel in 10 seconds. Now on the ground, the observer at rest with respect to the blinking light counts 10 blinks during the journey, each blink being one second apart. But for the observer in motion, not all of those 10 blinks will have had time to reach the super train by the time it arrives at its agreed upon destination. It will have marked off fewer observed blinks from the light and thus an observer on the train would rightly claim that less time was required than the 10 blinks that the person on the ground saw to make the journey. Two observers disagree on how much time has passed using a common reference point. So even in Newton's view of space and time, the notion of an absolute time measurement is just not correct. Now this thought experiment is essentially based on an optical effect. You could even say it's based on an optical illusion, the transit time of light through space. But nonetheless, because it already, using a Newtonian view of the universe, disproves that there is such a thing as a notion of an absolute unit of time that passes the same way for everybody. This completely then frees a thinker from abandoning the concept of absolute time as a necessary tenet of reality. So the speed of light is the same for all observers, regardless of their frame of reference. And since space and time displacements are not experienced the same in frames with different relative states of motion, even based on this optical illusion-based thought experiment, observers at rest looking at an object in a frame of reference that's moving with respect to them will observe that that object is contracted in length along the direction of motion. Now we will firmly see that when we explore the Lorenz transformation for relating observations in one frame to observations in another frame. But already you could have concluded that since it's the speed of light that remains fixed, not time or something else, that you're going to have to give something in this process. And what you find out from all of this is that objects in motion from the perspective of people who are in the frame that's agreed upon as being the rest frame will be observed to shorten along their direction of motion. This is known as length contraction. So hang on to that phrase because it will come up over and over and over again. It refers to this phenomenon of spatial measurements from the perspective of observers at rest looking at the moving frame getting contracted in the moving frame. Now observers in motion relative to other observers will also experience a slower passage of time. It's not an optical illusion that you need to use to explain this. It's a physical change in the experience of time itself. No optics required to explain the phenomenon. It simply is a behavior of time that for objects in motion relative to other observers if they stop moving and then compare their clocks to people on the ground they'll find out that they have experienced less passage of time than their colleagues who remained in what was agreed upon to be the original rest frame. This is known as time dilation that time slows down in a moving reference frame relative to a frame that's at rest. Now it will be a lot easier to appreciate the degree of these consequences as we actually explore the postulative relativity in class and then in the next section of this class look directly at the Lorenz transformation which is the correct way to relate observations between frames of reference. So I want you to get these notions of terminology down. You don't necessarily have to agree that this is what happens right now because I have done no math to prove to you that this is possible and I've certainly shown you no experimental results to tell you that this is what happens. But for now look at the terminology what a length contraction or a time dilation is so that we can carry that terminology forward with us. So to review what we have done in this lecture we have learned about the following things. We've learned about the transition in thinking that led from Galilean relativity to the special theory of relativity in 1905. We've learned about the postulates of special relativity which are the basis of the mathematics of the framework. And further we started looking at the consequences of those postulates from the fact that it's impossible to tell from looking at the laws of physics in different reference frames that a given frame is in motion relative to any other. All motion is therefore relative and also that different observers in different frames of reference while they'll all agree that light moves at the same fixed speed regardless of their relative states of motion they will disagree on the lengths of objects and the durations of time that are passing in different frames. These consequences will carry forward into the next section of the course, a discussion of the Lorenz transformation and preview the conclusions that will draw from the correct mathematics that relates observations from one frame of reference to another frame of reference.