# Special Relativity, Lecture 5: What do we mean by space and time? - 3rd Year Student Lecture

## Метаданные

- **Канал:** Oxford Mathematics
- **YouTube:** https://www.youtube.com/watch?v=-Te1fUF6hvc
- **Дата:** 11.05.2026
- **Длительность:** 53:30
- **Просмотры:** 3,647

## Описание

In the fifth of six lectures we are showing from Fernando Alday's 'Special Relativity' third year course, we ask ourselves, after Einstein, what do we mean by space and time, length and duration? We then re-derive the Lorentz transformations and explore their crazy consequences. 

You can watch other lectures in the course here: https://www.youtube.com/playlist?list=PL4d5ZtfQonW13GWAAaLi0YoSvK76coptp

You can also watch many other student lectures via our main Student Lectures playlist (also check out specific student lectures playlists): https://www.youtube.com/playlist?list=PL4d5ZtfQonW0A4VHeiY0gSkX1QEraaacE

All first and second year lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.

## Содержание

### [0:00](https://www.youtube.com/watch?v=-Te1fUF6hvc) Segment 1 (00:00 - 05:00)

— Good morning um everyone um or the few survivors uh so far. It's uh So, in the last lecture we have um written down the postulates of a special relativity and we have seen that either one uh says that the speed of light is the same for all inertial observers or we have these nice Galilean transformations between different uh frames of reference. So, that we threw away Galilean transformations because we thought the speed of light uh is constant. That's what we liked. And then we derived Lorentz transformations betw- um from that from those principles. And basically, the Lorentz transformations are the transformations between two uh inertial frames that are moving at a relative speed of B and we work them in one uh special dimension plus time. Uh today we are going to review to obtain this in a different way and I think it's a physically much more beautiful way that was the original way that Albert Einstein used. Okay? So, Albert Einstein asked himself "What do we mean by a space and time? " And in particular, what do we mean by length and duration? This, of course, is an incredibly deep question and the fact that you can sit down and try to answer this question is uh is quite remarkable and someone like only Einstein could do this. Okay? And a more precise way to put this question okay? Uh was the following. "How can an inertial observer let's say O inertial observer set up a coordinate system? " And basically in uh Newtonian and Galilean mechanics, we are assuming that an observer has like an infinite rod and with this infinite rod, you can measure coordinates. But that is not true. That is actually not realistic to assume that you can do something like this. So, we need to say what is the question and what are the tools in order to do this. Okay? And the tools that we can assume that we have the first tool is a clock. Okay? On their person meaning an observer can measure time but only where the observer is sitting. And the second tool is a beam of light. Okay? So, with these two tools we need to understand I am sitting here, right? Standing here. I have a clock, a beam of light. I need to understand how I set up a coordinate system. Okay? And that is the question that Einstein uh wanted to answer. So, first let's um let's plot a few things. So, let's say that here is the observer. Right? So is the this is the origin of my system. And I set the clock at zero and then I let the clock run with me so that all the events with coordinates T0 I can actually measure very well. Okay? So, because I have the clock with me so if the special di- dimension is zero, it means it is here. But because I have a clock, I can measure time on me very well. Okay? So, if I have a clock I can

### [5:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=300s) Segment 2 (05:00 - 10:00)

measure the coordinates of all events with coordinates T at the point zero. Is that okay? Beautiful. Now, let's imagine that we do we want to do uh something a bit more difficult. And basically we have X. Let me make this big. And we have some event A happening here. And I want to assign coordinates to that event. I want to understand how the observer would assign coordinates to that event. So, what I do is first of all the observer is here. So, we throw a ray of light and we throw it at some time T0 so that it hits the event A. Okay? And we imagine and here is in principle so in our experiment um we can do any imaginable thing that agrees with the laws of physics. It doesn't mean that there has to be someone here, etc. But we can imagine that there is a mirror at A. Right? And then this mirror reflects light all right? Reflects light and then light comes back to me at T1. All right? And T0 and T1 we can measure because I have a clock. Okay? So, T0 is the time I need to send the light such that the light hits the event A. And T1 is the time at my special origin, I receive the light back. Is that okay? And then if we have this the time coordinate for the event A let's call it TA is very easy to calculate because I would like to be here, right? So, it is nothing but the middle point between T0 and T1. Remember that if this is CT and this is X, these lines are 45°. Although they don't look from my picture, but they are 45°. So, uh so this is light. And this is light coming back. So, that TA is T0 plus T1 over two. Any questions? So, that's the T the time T I would assign to an event if I do this um operation. You get the two because of the two T0 T1? I get the two because it's the middle point between T0 and T1. So, it is the average. It's just the average between T0 and T1. Yeah. And now for XA what is the special coordinate? Uh let's say XA. That's very easy, too, because I know that the lights the rate of the ray of light the rays of light, sorry. They move at the speed C. Okay? So, the distance has to be the time which is T1 minus T0 over two times C. So, that's the special coordinate. Okay? Because we know that the that that's the amount that light uh moves in this in the time uh from here to here. Is that okay? Yeah. C is the amount of light from Well, C is the speed of light. Yeah. Right? And imagine that I tell you that you move 10 km/h right? And it took you 1 hour to reach that place, then you know that place is 10 km far. I I'm just saying that. — It's nothing that's um Okay? But with the speed of light, which is the only speed we can use. Okay? So, if we do this and this is called uh the radar method with this radar method we deduce that the coordinates that the event A

### [10:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=600s) Segment 3 (10:00 - 15:00)

has coordinates TA and XA which are equal to T0 plus T1 over 2 and C times T1 minus T0 over 2. Okay? So, all I am saying is that if I have a clock and a beam of light which I can throw two events, just by measuring how long light takes to go back and reflect, I can assign both a time and a X coordinates to all events in the universe. Is that okay? Beautiful. So, that is kind of the radar definition of light of sorry, of what do we mean by length and duration. Okay? So far, so good. Let's try to do more interesting. Yes. Um are we still saying that like something is happening at point A Yes. and then it's coming back at T0 and T1 which is That that's exactly right. So, the event itself, you could think of the event as the time and a space where light reflects. That's kind of what defines the event. And you could imagine that the observer, at least in principle, is throwing light all the time, right? And just measuring what reflects, which is exactly what a radar does. Right? So, that's the way like a radar, like if you see an object, the way that you see an object is not that you throw light to the object. You are throwing light all the time. And if there is an object, something will reflect to you. That's the way radars work. Okay? And that's why this is called the radar method. So, basically, you throw light, the light reflects, comes back at you, and just by measuring with your clock T0 and T1, you can define this. Okay? Beautiful. So, now um we will consider something much cooler than this. We will derive Lorentz transformations from the radar method. Okay? This is what Einstein did in his original paper. But let's go slowly. So, now we will consider two inertial observers. And the two observers, let's call them O and O prime, and as always O prime travels at speed V with respect to O. So, this will be true for the rest of the lecture, whenever I say that we have two systems, O and O prime. Then um O uses a system XT. O prime uses a system X prime T prime. Both use this radar method to decide what the coordinates of events are. And the question is how are X and T and X prime T prime related. Okay? And we will kind of derive this rather derive this but with just by throwing rays of light, etc. All right? Uh good. So, the picture that we do now is the following. I want to make it big because this is important. So, imagine that here is O. Okay? An event E we define the common time uh the time at which O and O prime meet. And this is O prime here. So, O prime moves at a constant speed with respect to O so that in the frame of O prime just follows a straight line, right? As we have seen before. And then uh now we imagine the following. So, imagine a time T

### [15:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=900s) Segment 4 (15:00 - 20:00)

here at time T in O in the clock of O sends a beam of light to O prime. Okay? So that remember uh rates of light always move at 45 degrees. Okay? So that the beam of light let's use another color, let's say red moves at 45 degrees and it will reach O prime at some point which is that point over there. Okay? And this we call this uh B. Okay? So, this is the event where the ray of light uh thrown by T by O, sorry, reaches O prime. All right? Now uh suppose in this picture that [snorts] the light gets or arrives to O prime at time T prime measured by O prime. Okay? So when O prime sees this light arriving to them they look at their clock and their clock says T prime. Okay? Uh so this is T prime but in the system of O prime. So we define now we define uh and let me perhaps let me work here. K a constant by saying [snorts] that T prime is proportional to T and K is just a proportionality constant. Okay? Because this is a straight line and this is also point corresponds to T equals T prime equals 0 then the relation between T and T prime has to be a linear relation like this. Is that okay? That's just basic geometry but what is yeah. Let's say that you increase T by twice T prime would also increase by twice. That's it. So, they are they are proportional. Beautiful. Uh This TK has a name and is called a Bondi K factor and is a function of the relative velocity. Okay? V, which is V. Beautiful. So, now what do we do? Now, what O prime does as soon as they receive the beam of light they throw it back. Okay? So, they throw the beam back and the beam gets back to O at some time that we call T prime prime. All right? It's basic geometry but it's a lot of little steps so that if anything is not clear, please do ask me. Okay? Now um because the same consideration that applied to this problem would also apply to the problem of O prime sending a beam of light to O we deduce that actually T prime is KT prime. Okay? Because the problem is identical. So, at time T we send a light a right, a beam of light, which is received at time T prime, and this observer at time T prime send a beam of light back that is

### [20:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=1200s) Segment 5 (20:00 - 25:00)

received at time T prime prime. Okay? So, that the relation between T, T prime, and T prime is that T prime has to be KT, and KT prime. Yeah? Yes. Why do we have to have the same K? Right, because it things are proportional, and we have assumed that this K is a function of the relative speed. So, could it not be like a like a in the same way Lorentz would like we initially said it was lambda V and lambda minus V. Could it not be the same something like that? Um right, so it could So, in principle, it could. So, we are making the assumption, you are correct. assumption that it is it will be a function only of the relative V of the uh absolute value, let's say. Yeah? So, because and I think it has to do with isotropy of a space. There are different ways, but you are correct. So, here we are assuming, and it's a bit simpler. I would have to spend 15 minutes to explain why this is the case. So, it's a bit simpler to assume. Um yeah, but you're completely correct. Yeah, you could imagine situations where gamma V and gamma minus V are different. Although that doesn't happen in our universe. Uh yeah. Uh beautiful. Then yes. then yes. In the original example with the radar Yes. are we assuming that between the time for it to hit the light to travel to A and back that our observer was stationary with respect to like the event A? Right, so that Yes, that's a very good point. So, the event A is not an observer. just something at a fixed space and a fixed time with respect to O. So, it has fixed uh coordinates X and T. So, it's just one point in time and in a space. So, indeed, it is a stationary in a very clear way, simply because the coordinates are fixed both for X and T. Yeah, so that that's a good point, of course. Uh and now, actually, we will see a little bit what happens with this. But the event B uh B, sorry, um uh the same. It's some event with coordinates, but we are going to Yeah. Uh any other question? Beautiful. So, now No notice, by the way, that T prime uh equals KT prime is nothing but K squared T. Okay? Because T prime was KT. Yeah? So we deduce something very cool. Uh from this and perhaps let me uh copy here. We see that we can now use the radar method right? Because we have T, we have T prime prime, so now we can use this method, these two formulas, to say what the coordinates of B are in the reference system of O. Okay? So TB and XB So, TB is just the average of T and T prime prime, but T prime is K squared T, right? So, we simply get 1/2 K squared plus 1 T and XB is 1/2 of C K squared minus 1 T. Okay? So, these are the coordinates from the point of view of O of the event B. Okay? But now we have something very cool. Because we know that B is in the world line of O prime, right? And a bit related to your question, we know that O prime is moving with respect to O at a speed V. Okay? So, it has to be the case the case that XB divided TB is equal to V. Okay? This would be true for any point in this line.

### [25:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=1500s) Segment 6 (25:00 - 30:00)

Because that's what defines this line from the point of view of O. Right? It's just the fact that O prime is moving at a speed V. So, any point in this line the two coordinates would have to satisfy this. In particular, this point here. All right? Yeah? Uh but then, this means — [snorts] — that this is equal You just take these two things. You consider the ratio of them. And this is C K squared minus 1 over K squared plus 1. All right? And this implies that K given by this. Can you just go over XB over TB again? Yes. So the equation So, we are assuming assuming that the O prime travels at a speed V with respect to O. Okay? So, that means that the equation of this line for any point of this line right? The equation of this line is basically that um X in this line is V times T in this line. Because this is what this line is. So, this line is X equals VT. So, that is the equation of this line. Okay, so V so T So, for any point of this line any coordinate of this line the X and the T will be You can give it any name. They will be related like this. Th- This is a line that crosses through zero and has a slope V. Yeah? So, th- This is just a line. A straight line. And you can ask what is the equation of this straight line. Right? And line is X equals VT. Yeah? Yeah? Yeah. So, like if you said, for instance, X equals CT, this would be 45 degrees in our conventions, because this is CT. Right? And this would be something moving at the speed of light. If V is a smaller than the speed of light, you just get a straight line doing like this. Yeah? Good. Is that okay? Yeah? Very nice. Uh Good. So, because of this we have this beautiful relation between K and V. Okay? And we can deduce um this. A- And going back to your question, uh so the precise statement is that K is given by this. And a more precise Now, I can answer the question more precisely. You can see that you get a quadratic equation for K, and basically, this is the solution we are choosing. You could wonder why we didn't choose the other solution. But basically, we are choosing the same solution for both problems. So Yeah, so that that's the statement. Um yeah. Uh Very nice. I- Is that okay? Yeah? A- Any questions about this? Yes. So, you said that um this is what Einstein used to do in his paper. Yes. If Lorentz already derived it, why did Einstein bother to do it anyway? S- S- Say that again. Lorentz already derived the transformation. So, okay, so the story is a bit complicated. And people kind of knew the kind of transformations that had to be true to be consistent with the Maxwell equations. But they did not didn't know what it meant for a space and time. They didn't understand that. Okay? So, they understood at the level of a set of equations, and this set of equations had a symmetry, and this symmetry have to had these transformations. Okay? So, if you look at the Maxwell equations, and you perform a Galilean transformation, that's not a symmetry of the Maxwell equations. So, Lorentz worked out what the symmetries of the Maxwell equations were. And they found he found out the Lorentz transformations. However, Einstein realized what it meant really

### [30:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=1800s) Segment 7 (30:00 - 35:00)

for space and time. That's a huge leap. Yeah. Okay? It's like, you know, you have a system of equations and you find some symmetry, that's one thing. Mathematically, you can mathematically see that it works. When you want to mess up with the structure of a space and time that people thought they knew for like 2,000 years, that's something else. Okay? And that's what Einstein was doing. And and his point, okay, we will see what his point was. Okay? So, ask me this in like 10 minutes. But basically, he was trying to answer this question, okay? — [clears throat] — Uh sorry. Yeah, this question here. Yeah. Beautiful. So, then uh very nice. So, notice um notice then that we have K um and this is very cool because now we know that T prime is equal to KT which is C plus V divided C minus V times T. Okay? This is a beautiful relation. And notice the following. So, now already we are going to go into science fiction movies and stuff that people uses in science fiction movies. Um and the stuff is the following. Now, according to O so, for O time between E and B is equal to TB. Okay? Where E is this event and B is this event here. Okay? That's just the time coordinate uh TB with respect to O. On the other hand, according to O prime right? How much time has passed between I met this nice guy called O and I receive a letter from him, right? In the form of a beam light, a beam of light. Okay? Now, for O prime uh O prime will say, well um let's say that she says that the time elapsed between E and B, sorry, that means the same as E and B, above, is actually T prime. Okay? Because T prime is we said the time that when she receives uh this uh laser beam. Which is K times T. And now we have something very cool. We have that the time measured by O divided prime is equal you just do this ratio okay? Where TB um yeah, it is given here. And what we find very nicely is that this one over one minus V squared square root of V squared minus C squared. And this is just gamma. Okay? So, O and O prime measure that the time elapsed between the same two events is different. And the difference between the two times is this factor gamma which is what appears in the Lorentz transformation. Uh we will come back to this. This is known as time dilation and it leads to a lot of paradoxes in science fiction movies that we will understand how they are solved next lecture. But uh yeah. Good. So, we are almost there. So, any questions so far? It's all basic geometry but I understand that is uh that maybe a bit complicated. So, so please do ask me if you have any questions. So, now what we want to do

### [35:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=2100s) Segment 8 (35:00 - 40:00)

we draw the same plot. So, this same plot here right? So, do practice these things because once you practice they become very very easy. Like I promise that this is like basic it is kind of high school geometry actually. It's not too complicated, but you need to practice. Uh so, this is the system O. O prime. Okay? And there is an event B and both of them O and O prime want to assign coordinates to B. Okay? Then I imagine the following. Imagine a beam of light from O to B and then reflects back like this from B to O. And now let's call this time T1 according to O this time we know what it is, right? Because it's K T1 because that's what the Bondi factor does. If you multiply by it, it just gives you the time in the next reference system. Let's call this point here let's call it T2 right? With respect to O prime but then this one will be just KT2. Okay? So, this point here will be K T2. All right? So, what um O will think, so O can use the radar method, right? These formulas here, exactly the formulas that we wrote to the left, but T0 will be T1 and this time up will be KT2. Okay? So, according to O so, coordinates of B according to O are as follows. Are TB equals 1/2 T1 plus KT2 and XB 1/2 of C K T2 minus T1 and just applying the formulas of the radar method that we derive on the left right? And this is the system of equations number one. But if you ask to O prime so, coordinates of B in O prime can do the same, but now with times KT1 and T2. Okay? So, T prime B is equal to 1/2 KT1 plus T2 and X prime B 1/2 C T2 minus K T1. Okay? Where K Sorry, where K is equal to this here. Okay? So, yeah, with this formula for K. Okay? And now we are almost there. Okay? Because all we need to use, do is use two so, the second system of equations to solve for T1 T2 and then, right? You take this you solve for T1 T2 in terms of T prime and X prime. Yeah? Uh as functions of X prime B and T prime B, and then plug these into one, into the first time. Okay? That's all we do. It's like linear. You can use Mathematica if you

### [40:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=2400s) Segment 9 (40:00 - 45:00)

don't know how to diagonalize 2 by 2 matrices, as I don't. So, like you take this, you solve for T1 T2 in terms of T prime X prime, and that you plug them here. Okay? That's all. And then, if you do this, we find that X is equal to gamma X prime plus VT prime and CT is gamma CT prime plus V over C X prime. And these is nothing but the Lorentz equations. Okay? So, this relationship between uh X and T as said by Lorentz that you should find. Okay? Imagine how good you have to be. Imagine how huge your intuition must be for you to sit down, ask what does duration mean, and work it out, and end up with the Lorentz transformations. Right? That's kind of what Einstein did. And um yeah, so the second way we derived the Lorentz transformation is exactly the transformations we found before, but doing this way that can be useful for many other things. Is that okay? Yeah? Now, this way um although mathematically is very simple, conceptually is very complicated. And it means we need to abandon some notions. Now, if you want to do theoretical physics, you need to think in this way. There is no other way, because that's the way the world works. Okay? So, there is no way around it. Now, uh do you have any questions about these? Yeah? Okay. So, let's try to uh let's try to work out two uh very nice neat examples. And basically, we will only use these Lorentz transformations. And with the Lorentz transformations, we can see Sorry, some I shouldn't have erased gamma. Uh We will work the following. So, the first one, is the concept of um of Lorentz contraction. Uh and it's the following. So, imagine imagine uh that O prime has a rod of length L. Okay? And the question And the question is what does O think the length of the rod is? Okay? So, that's a very basic question. So, what do we mean? So, in the system of O prime, so O prime has a rod and basically one end is at X prime equals zero, and the other end at X primes equals L. Okay? This is what it means to have a rod of length L. Okay? You have a rod between X equals zero and X prime, sorry, prime equals L. And we just want to use the Lorentz transformations to see what will O see. All right? Uh beautiful. So, being a bit more

### [45:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=2700s) Segment 10 (45:00 - 50:00)

precise, we have events of coordinate where X prime of T prime is equal to zero for all T prime, and then X prime of T prime is equal to L for all T prime. These are two lines, and basically we want to see what these two lines map to the system O. Okay? So, here um So, the lines in O Easy. We use the Lorentz transformations. Okay? So, let's do it very slowly. So, CT X for the first line, let's say, for this, will be equals to gamma one V over C one CT prime zero. Okay? And this is gamma CT prime VT prime. And this is what the first end of the rod maps to in the system O. Okay? So, this is the first line. Uh The second line, or the second end, maps to the following. It maps to CT X equals gamma one V over C one Sorry, CT prime L. And this is Jesus Christ. And this is gamma CT prime plus L V over C VT prime plus L. Okay? And now And now it comes the very important point. We need to compute the distance, right? Between this point and this point. But for O prime, the length L is the distance between simultaneous events. Okay? Let's say, zero T prime and L T prime. So, you take two events, one of them leaves at one end of the road, another one leaves at the other end of the road at the same time T prime, because O prime is the one measuring the length now, and the distance between these two things is L. Okay? Now, when it comes to O O, what the important point here is simultaneous, and this is simultaneous with respect to O prime. Okay? Now, O will look for two points which are simultaneous for or with respect to O. Okay? So, the definition of length for O prime is that the two events have to be simultaneous in O prime. But for O, is that two events have to be simultaneous with respect to O. Is that clear? Yeah? That's the way you measure length. It's the distance between two simultaneous events for you. Okay? So, let's write it here.

### [50:00](https://www.youtube.com/watch?v=-Te1fUF6hvc&t=3000s) Segment 11 (50:00 - 53:00)

Let's say then that you choose So, you are O now and you choose the first end at t equal zero and x1 equal zero. So, with respect to O the coordinates of the first end are zero zero. But then you see from here that if you want the second end to be at t equal zero because it has to be simultaneous with this one then you need that this combination is zero. Okay? Because this is what t is for the second end. All right? So, this happens then this implies that ct prime is equal to minus L v over c and this implies that the x2 the x of the second road is gamma minus L v squared over c squared plus L which is nothing but L times a square root of 1 minus v squared over c squared which is L over gamma. Okay? So, if O has a road of metal or whatever material actually of length O prime sorry, of length L O will think that the same road has length L over gamma. A bit smaller. Okay? This is called Lorentz contract contraction. And the important point here is that both systems define distances between events in such a way that the events are simultaneous. But simultaneous for O and simultaneous for O prime are two slightly different things. Because simultaneous for O prime means that t prime is the same. Okay? So, While for O um while for O is a bit different because it means t has to be the same, not t prime. Is that okay? Anyway, so tomorrow we will um deal with something actually quite beautiful. We will see a very similar paradox but with time which is called uh time dilation and we will uh understand the twin paradox and we will see why it's not really a paradox, but it will be nice to understand that. Thank you.

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*Источник: https://ekstraktznaniy.ru/video/51879*