Special Relativity, Lecture 4: The postulates of Special Relativity - 3rd Year Student Lecture

Added: 2026-05-05

[00:00:07][music] >> I guess we can start by reviewing the postulates that we got to in the first three lectures.

[00:00:20]So, good morning, everyone.

[00:00:23]Um So, what we have seen in the first three lectures um in the first two we derived this list of postulates um that Galileo and Newton were happy with.

[00:00:39]And then we saw that some of those postulates didn't agree with Maxwell's theory of electromagnetism and the Michelson-Morley experiment. So, we arose some of those and we added a new postulate.

[00:00:53]And then we ended up with four postulates, which are the postulates of a special relativity.

[00:01:00]Einstein was a very philosophical guy.

[00:01:03]He didn't want to accept anything that he didn't need to accept. And this was the minimum set that he understood that he needs to accept. And based on these, we will derive all the theory of a special relativity and then you will see that a new concept for space-time, a new conception of a space-time, arises from these four postulates.

[00:01:28]So, the first one says that there is a preferred class of frames called inertial frames where the law of inertia holds, meaning free particles move in a straight lines.

[00:01:41]And an inertial frame plus a choice of zero for time gives you an inertial coordinate system, ICS for short.

[00:01:52]Then the third postulate tell us that the laws of physics hold in any inertial frame. All the laws of physics, mechanical laws, electromagnetism, etc. And they have the same form in all in any inertial coordinate systems.

[00:02:13]And finally, the last postulate says that in all inertial frames, the speed of light in vacuum is a constant independent of the direction of propagation and the speed of the source.

[00:02:27]Okay?

[00:02:28]So, this last postulate in particular is crazy.

[00:02:33]And if you think a little bit about it, it tell us that either this is true or the Galilean transformation or addition of velocities is true. Okay? Both things cannot be true.

[00:02:47]All right?

[00:02:49]So, we will today we will show how to start from these four postulates and how to show how to construct the special theory of relativity. Okay?

[00:03:01]That's what we will do today.

[00:03:03]And we will start with one plus one dimensions.

[00:03:09]Meaning you have, let's say, space one special direction direction plus the time direction. Okay? Because all the features that are important, they appear already if you have one a special direction plus time and later in the course we will move to three plus one to three special directions.

[00:03:37]Now we see um we will start today by deriving the Lorentz transformations.

[00:03:50]And the question we want to answer now is how are inertial coordinate systems related?

[00:04:07]Okay?

[00:04:09]So, we have seen that they are I mean we I told you that they are not related by a Galilean transformation. So, the question, the burning question is how are they related? Okay?

[00:04:20]And that's the question we will answer today.

[00:04:24]So, we will assume uh the manipulations to do this are really simple. They are like two by two matrices, basically.

[00:04:35]However, the consequences are astonishing. Okay? So, we will have a lot of trivial statements that will lead to an astonishing consequence. And because of that, we will see this in two different ways, one today, another in the less in the next lecture.

[00:04:54]So, the problem we will consider So, consider two inertial coordinate systems called O and O prime Okay?

[00:05:14]And such that O prime moves at speed V Sorry.

[00:05:28]with respect to O.

[00:05:31]So, the picture that we have is that the system O let's say is this system here.

[00:05:41]This is This is O.

[00:05:44]This is X and this is T with respect to O.

[00:05:49]And then with respect to O the system that O prime uses is moving at the speed V in the in the X direction. Okay? So, this is this is O prime.

[00:06:08]And this is this is O. All right? So, this is what O will see.

[00:06:13]O This will be a static and this the system that O prime is using with respect to O will move at a velocity V.

[00:06:21]Is that okay?

[00:06:23]Yeah?

[00:06:24]Beautiful.

[00:06:25]So when they pass each other, so there is a time at which the access that O uses and the access that O prime uses Right? They overlap. So, O and O prime overlap. So, imagine I am O, I am standing here and O prime is some of you moving at the speed V with respect to me. When we are at the same place Okay? We decide to set our clocks.

[00:07:02]So, when O O prime are at the same place they set their clocks to zero.

[00:07:38]Okay?

[00:07:41]So, T O says, "Okay, from now on, this is T equals zero."

[00:07:46]And T prime is also equal to zero. Okay? And the event where they coincide we will give it a name.

[00:07:56]This is called event E.

[00:08:01]So, this by definition is the time at which O and O prime are at the same place. Okay? After this, after time zero, O prime moves with respect to O.

[00:08:13]Okay? Just imagine my This is O prime. I am O. This is just moving like this at a constant speed. Okay? Of course, if we describe the things from the point of view of O prime, O prime will be a standing from the point of view of my notebook and I am the one that is moving in the opposite direction. Okay? So, we can describe both from the point of view of O or the point of view of O prime. Is that okay?

[00:08:39]Beautiful. So So, um things events O will use coordinates X and T.

[00:09:00]And O prime will use coordinates X prime and T prime.

[00:09:13]And our aim for today is to find X prime as a function of X and T.

[00:09:24]And Y prime Sorry.

[00:09:27]And T prime as another function of X and T. Okay? So, we want to say if I am measuring something, right? When something happen, etc. like the motion of a particle and I give to that particle a coordinates X and T what will the X prime T prime coordinates be with respect to O prime?

[00:09:52]Okay? How do we relate our measurements of the same thing going on. All right?

[00:09:59]Uh beautiful.

[00:10:01]I noticed that because they have set their clocks to zero uh when they are at the same place all what we are saying is that their origins coincide.

[00:10:16]The origins coincide.

[00:10:22]So, in particular uh zero has to map to zero in both coordinates.

[00:10:32]So, uh g of 0 0 is 0 and f of 0 0 is also 0. This just saying that the origins 0 0 of um O and O' coincide. All right?

[00:10:45]Beautiful. So, let's try to to work this out.

[00:10:51]And let's use the postulates, okay? So, good. So, we have the first postulate.

[00:11:01]The first postulate tell us that the law of inertia holds, okay?

[00:11:08]Yes. Sorry, I was just going to ask So, in a classical sense, what are f and g?

[00:11:13]Hm? Uh if we if we were to look at it from a classical perspective. They are just functions. So, they they they will be it's just some function of x and t that we want to find.

[00:11:22]They are linear functions actually, but they Yeah. It could be x + t. It could be x ^ 2 - t ^ 3.

[00:11:31]We are not saying much.

[00:11:32]But but if we model this from a classical perspective, is it the case that t' actually equal to t and x' is equal to So, in Galilean transformation uh x is equal to x' is x + vt.

[00:11:45]For instance, right? So, so and and and t' is equal to t. Yeah, that that that's a very good question. So, if you have Galilean transformations for instance, in Galilean transformations. Yeah, thank you for the question. So, if we have like completely classical sense um what time is it? Do you know the time?

[00:12:07]Perfect. Now, answer that question again. What time is it?

[00:12:12]17 minutes past 11. Yeah, sorry. Now, answer the question again.

[00:12:16]Beautiful. You see, your answer to the question didn't depend on whether I am moving or not.

[00:12:22]Okay? So, in Galilean mechanics, all systems whether they are moving or not, they have the same notion of time.

[00:12:32]The time is the time.

[00:12:34]Right? And that's it. We all agree. So, for instance, in Galilean transformation, t' is equal to t. We all agree with the time. We all agree with the time.

[00:12:44]And because we are saying that by definition um because this is our problem O' moves at the speed v with respect to O then in a Galilean transformation x' is equal to x - vt. So, for the case of Galilean transformation, the function f is this function and the function g is this function. Okay? We will try to find now the relativity analog of this.

[00:13:17]Thank you for the question. Beautiful.

[00:13:19]Um good. So, the first postulate the first postulate tell us that the law of inertia holds on both systems.

[00:13:31]Meaning if O sees a free particle that moves in a straight line O' should also see a particle that moves in a straight line. Okay? So, whatever the transformations f and g are straight lines in O map to straight lines in O'.

[00:14:07]Okay?

[00:14:09]Because of that the relevant uh change of coordinates has to be linear transformations.

[00:14:22]Very good. So, meaning x' is equal to some coefficient alpha 1 x plus alpha 2 um t and y' Sorry.

[00:14:38]And t' is equal to beta 1 x plus beta 2 t. Where alpha 1, alpha 2, beta 1, beta 2 are just constants that we need to determine. Okay?

[00:14:52]So, these are constants that may depend on v. That will actually.

[00:15:12]On this veloci- relative velocity v.

[00:15:15]Beautiful.

[00:15:17]The second the second point the second point is we are saying that O' So, we deduce So, Galilean is this.

[00:15:32]Okay?

[00:15:33]So, um for So, now we have deduce in relativity we have a linear form, but it could be a bit more complicated. So, it's alpha 1 x plus alpha 2 t and t' is beta 1 x plus beta 2 t.

[00:16:05]And the second point is that we are saying that O' moves with velocity v according to O.

[00:16:23]So, what that means what that means is that the line x = vt must map to the line x' equals 0, okay? Because the system O' from their point of view O' is at x = x' = 0, okay? But from the point of view of O O' is just traveling in this uh line here because x is equal to vt, okay? So, these are the coordinates of O' basically with respect to O and with respect to O'. So, for this to be true for this uh to map to this uh it has to be the case So, this line needs to map to this. It has to be the case that x' is is equal to some constant that we will call gamma of v x - vt.

[00:17:31]While t' we cannot say much for now. Okay?

[00:17:41]So, this is just information that with respect to O O' is traveling at the speed v.

[00:17:50]So, that if you set x = vt this will give you x' = 0 and any time prime t' we don't know.

[00:18:00]Now, it comes the beautiful thing.

[00:18:03]Then we have the fourth postulate.

[00:18:07]Okay?

[00:18:10]And here is where things become super interesting.

[00:18:17]With the fourth postulate we will do the following.

[00:18:24]When O and O' are at the same point okay? When they coincide we both send a beam of light.

[00:18:38]Okay?

[00:18:40]In that direction so, light and in that direction light.

[00:18:51]And because of the fourth postulate what I will see from my point of view of O is two beams of light, one in that direction, another in another direction, both moving away from me at the speed of light c.

[00:19:10]But O' prime from their point of view even though they are moving with respect to me, they will also see exactly the same.

[00:19:20]Two beams of light moving at the speed c in that direction and in that direction. Okay?

[00:19:28]That means that means in other words in in equations that the line x = ct maps to the line x' = ct' and the line x equals minus CT for the other beam has to map to the line x prime equals minus CT prime.

[00:20:10]Is that okay?

[00:20:12]Any questions about this?

[00:20:16]Yeah?

[00:20:17]Beautiful. So, what we are saying is let's imagine that I described or describes this beam of light.

[00:20:26]Right? So, I am describing that the coordinates the this coordinates of of the of this photon uh as it moves to the right.

[00:20:42]Right? So, so that's what I described.

[00:20:45]So, from my point of view from the O point of view I say um x How can I say that? I can say for instance that the speed of light c for that photon you can compute by saying x divided by t. Okay? But from the point of view of O prime what they would do is that they would say that c is x prime divided by t prime. Okay?

[00:21:20]So, but we know that x prime is gamma v x minus vt divided beta one x plus beta two t and this is true provided x is equal to CT. So, if you take this and you plug x equals to CT you need to get the speed of light.

[00:21:54]Okay? And that gives you conditions on gamma v beta one beta two and v. Is that okay?

[00:22:01]Is that understood? Beautiful. So, the two equations you get two equations one for each um light ray is gamma v CT minus vt equals c beta one set CT plus beta two t and another equation is that minus CT minus vt is equal to minus c minus beta one CT plus beta two t. This is just a statement [clears throat] that the light beam moves at the speed v with respect to both systems.

[00:22:53]We are almost done.

[00:22:58]You can solve for beta one and beta two if you use these two equations and what we find then is that x prime is gamma v x minus vt and t prime is gamma v minus v over c square x plus t.

[00:23:28]Beautiful. Almost there.

[00:23:30]So, we have derived the the transformations the Lorentz transformations up to a constant gamma v that we will determine momentarily.

[00:23:46]Any questions about the manipulation so far?

[00:23:50]So, we just used the fact the origins agree the transformations have to be linear because the systems are inertial.

[00:23:59]O prime is moving with respect to O at velocity v and then both they describe these beams of light as moving at the same speed c. Okay?

[00:24:11]And that that's all all we have done.

[00:24:13]And then we derive this.

[00:24:15]Now, there is another trick a very very nice trick.

[00:24:23]So, imagine now so so this let's call it equations system of equation number one.

[00:24:31]Imagine now that you want to consider a slightly different problem the opposite problem you want to write x as a function prime let's call it of x prime t prime and t as a function of x prime t prime. What this means is now we are going to solve the inverse problem.

[00:24:57]We are going to say imagine that we have x prime and t prime so from the point of view of O prime and then from that we want to compute what x and t are.

[00:25:11]This is the same problem we have already solved but you replace v by minus v.

[00:25:18]Because with respect to O prime O is moving at the speed minus v.

[00:25:24]Okay? So, this is the same problem same problem with v replacing minus v.

[00:25:40]So, if you do this the solution we will find is that x is equal to gamma minus v x prime plus vt prime and t is gamma minus v v over c square x prime plus t prime.

[00:26:06]So, it is the same solution as solution number one but we exchange the xt by x prime t prime and v by minus v. Okay? Because from my point of view O prime is moving at the speed v but from their point of view I am the one moving at the speed minus v. Okay?

[00:26:30]So far so good. Uh any questions?

[00:26:39]Beautiful.

[00:26:40]So, now we can we can uh do the following trick. We can take equation two and replace it into equations one.

[00:26:51]Okay? So, you just take these two expressions for x and t and plug them into uh equations one and that should be a true equation for all x prime and t prime and if you do this you obtain a beautiful equation that gamma v gamma minus v times one minus v square over c square has to be one.

[00:27:26]Okay?

[00:27:28]And now there is another another point and we would like to claim we would like to claim that gamma v is equal to gamma minus v.

[00:27:43]Meaning this factor that we are looking for it depends on the relative speed not the velocity of of the two systems. Okay? But it depends only on on the speed the absolute value of v and not the sign of v.

[00:28:04]Okay?

[00:28:05]And here there are three approaches that that that you can do. Some books tell you that this is an assumption and that's it.

[00:28:17]Or you can do a change of transformation which is called parity and you take x to minus x tilde and that is a bit complicated but there is a better argument which I just saw actually in the shower.

[00:28:34]And it has to do with the isotropy of a space.

[00:28:45]And because this is like a nice lesson for you guys I thought I would explain you why this is the case. So, isotropy of a space means the following. It means that if you are in vacuum all directions imagine you are in vacuum in the universe all directions they look exactly the same.

[00:29:06]Okay? And the space space time more precisely but a space is the same in any direction that you look at. Okay?

[00:29:17]Now, when I told you that I was going to describe things in one a special dimension when we say that we will choose one a special dimension of course what we are doing is the following. We are in a space time.

[00:29:39]Right? We pick one random direction and I declare that this direction is the x axis.

[00:29:49]Okay? And then we do this computation using that this direction is the x axis.

[00:29:55]But of course, I could have used my arm is the x axis. Do you all agree I could have used this direction instead?

[00:30:04]Why not, right? You are in a space, you could have used in this direction.

[00:30:09]Or you could have used this direction and you would have obtained the same gamma v.

[00:30:14]But also, you could have used this direction, which is the opposite as the one we started with.

[00:30:21]Okay? So, all these directions they should give you the same gamma v.

[00:30:26]In particular, this direction and the opposite direction. And because of that, gamma v cannot depend on the direction of of the velocity, but it depends on on this.

[00:30:41]And this is true, okay?

[00:30:43]But this is is very cool because then we solve for gamma.

[00:30:53]All right?

[00:30:55]And is square root of what Sorry, 1 over 1 minus v squared over c squared.

[00:31:06]This is called a Lorentz factor.

[00:31:14]It appears everywhere in a special relativity. If you need to memorize one expression, memorize that guy.

[00:31:24]Um And with the Lorentz factor, we have found our our final version our final version for the Lorentz transformations. So, we will write them here.

[00:31:45]Lorentz transformations.

[00:31:58]So, x prime is gamma x minus vt.

[00:32:11]While it is it is customary to multiply t by c.

[00:32:18]So, ct prime is equal to gamma ct minus v over c x.

[00:32:32]Sometimes it is useful to have the inverse ones so that x prime x Sorry, x is gamma x prime plus vt prime.

[00:32:47]And ct is equal to gamma ct prime plus v over c x prime where gamma is what I just wrote. Square root of 1 over 1 minus v squared over c squared.

[00:33:11]Okay?

[00:33:12]So, this is the way This is the way that inertial coordinate systems are related in a special relativity.

[00:33:23]Okay?

[00:33:24]There is a lot of fascinating things about this.

[00:33:28]So, the So, first First of all one um technical thing one technical thing you can write this as matrices in terms of 2 by 2 matrices and sometimes you you will see them written like this.

[00:33:47]So, ct prime x prime is equal to gamma 1 minus v over c minus v over c 1 ctx or the inverse.

[00:34:32]And something fascinating about these transformations, they are a bit more complicated than the Galilean ones, which were something very simple. But notice notice that in the limit in which v over c goes to zero gamma goes to 1 and you do recover the Galilean transformations. So, the Galilean Sorry.

[00:35:06]So, the Galilean transformations are obtained as a limit as the speed Our speeds are smaller than the speed of light, much smaller. So, that if the speeds are smaller than the speed of light, much smaller then you can still use Galilean transformations.

[00:35:27]So, that the special relativity corrects Galilean transformations, but these corrections are important only when the speeds of objects are are kind of of the same similar to to to the speed of light. However they are different and this is what they are, okay? Any questions?

[00:35:56]The most beautiful feature of this and let me let me write down let me let me ask um Let's solve a little problem with this.

[00:36:11]Which shows how crazy these transformations are.

[00:36:14]Is that a space and time a space and time are kind of intertwined and connected.

[00:36:23]So, you cannot separate them and you can only talk about a space time as a thing, okay?

[00:36:32]And let's try to solve Let's try to solve a very cute little problem.

[00:36:47]That will make that manifest.

[00:36:53]Something that we have in in Galilean transformations is the concept of simultaneity.

[00:37:01]Okay? So, if two events happen at the same time every inertial system in Galilean or Newtonian mechanics will agree that these two things have happened at the same time.

[00:37:17]This is not true anymore now.

[00:37:20]Okay?

[00:37:21]So, let's try to see an example where two things happen at the same time for one system but happened at different times for another system. Okay?

[00:37:33]With numbers. Let's put numbers, okay?

[00:37:36]Uh beautiful. So, the the problem is the same is the following, sorry.

[00:37:44]So, imagine that in a given ICS two simultaneous events are separated by 4 * 10 to the 7 m.

[00:38:21]This is us basically the distance between two opposite points in the equator of Earth.

[00:38:28]Okay? So, on Earth, that's how far you can get as far as you can get. This this is it, right?

[00:38:34]And then we have a second ICS moving with respect to the first at speed v equals 3 * 10 to the 4 m per second. Okay? Remember that light is 3 * 10 to the 8.

[00:39:14]Just for for reference. And this for reference is basically the speed of Earth in the solar system. Okay? With respect to the sun. So, Earth is moving at that speed. So, it's just some numbers. So, one system tell us that there are two events poof the explosion of something the birth of two people, whatever, right? They happen at exactly the same time.

[00:39:42]But another system is moving at this speed. So, what will the other system see?

[00:39:50]Let's see. Beautiful. We just have these tools that we have derived, right? So, let's call let's call the first system T prime and X prime, right?

[00:40:07]And the coordinates for the events uh let's say that in my time they happened at T prime equals zero. So, this is the first um reference system.

[00:40:23]And they happen at uh with the following coordinates. So, the time is zero and the distance is X prime zero for the first event.

[00:40:36]And for the second event is X prime plus D, okay?

[00:40:41]Where this D is 4 * 10^7 m.

[00:40:49]All right?

[00:40:51]Beautiful. Then the second system the second system will say well with respect to me if I want to find X and T I could use these transformations here, okay? I want you to use the inverse, but it's just this transformation here. All right? So, you take these transformations you plug T prime equals zero X zero prime for the first event. And you plug T prime equals zero X zero prime plus D for the second event, okay?

[00:41:27]And what the X will see, what O will see is that CT1 is gamma V over C X zero prime for the first event. But for the second event this is gamma V over C X zero prime plus V.

[00:41:55]And behold they are different.

[00:41:59]Okay?

[00:42:01]So, you can ask how different are they?

[00:42:06]So, T1 is different from T2.

[00:42:10]And you can compute the time difference, which is just um T2 minus T1.

[00:42:17]And this for large speeds uh for small speeds, sorry is gamma V over C squared D which is 0.00001 seconds.

[00:42:30]Okay?

[00:42:33]Really small really hard to measure but non-zero.

[00:42:39]Okay?

[00:42:40]So, two things that are simultaneous with respect to one system they don't have to be simultaneous, in general they are not to a second system.

[00:42:51]This is crazy, right? I mean, people were uh mind-blown by this. Even more actually, you can find examples where I will think that A happened before B but if you are moving fast enough you will think that actually B happened before A.

[00:43:08]So, even the order of things can be reversed in a special relativity. Okay?

[00:43:16]Any questions about this?

[00:43:20]How How do you arrange for the order to be reversed?

[00:43:25]Yeah, it doesn't seem like if I generalize the I mean, what do you want to make T negative? Right. So, you could play with X zero. So, you have to play you have to play with T zero and X zero.

[00:43:39]And and uh that that will be that will be more clear. Actually, it is very clear in in what I will do uh in a bit uh with diagrams. But you will have problems in the problem sheets where you show this. Sure. But you can play you have to play with all the coordinates.

[00:43:56]Okay. Yeah, you're right. Because from this it seems this goes away in the difference. And this always have a given sign. Yeah. That's right, yeah.

[00:44:07]Uh now in general you need to imagine that there is a D1 here and that the time is different, okay? So, if the time is different because they are not simultaneous then you have like zero here and something positive here. But then you need to plug this and you can see whether you can adjust things. Yeah.

[00:44:27]Beautiful.

[00:44:29]So, the last thing I will be uh telling you today is a nice thing um a nice idea to draw things in a special relativity and is the concept of space-time diagrams.

[00:44:50]And we have already seen a little bit of this. But let me tell you a few things.

[00:44:57]So, basically for in order to draw these space-time diagrams um let's say we choose O with coordinates X and often instead of T you use uh CT or CTX in the order given by these transformations.

[00:45:22]And then in an axis you draw X in one axis like this.

[00:45:35]And CT in the other axis.

[00:45:39]Okay?

[00:45:40]And then you use the same units or better to say uh the same scale.

[00:45:51]Okay?

[00:45:56]For instance if you use uh years to measure your time you need to use light years to measure distance. Or you could use seconds and light seconds. Inches light inches you need to use them in the same scale. Okay?

[00:46:16]In this way if you use the the this is scale in my coordinates in these coordinates a rate of light will always be traveling at pi over 4 or 45°.

[00:46:35]Okay? So, with these beautiful coordinates light will be traveling either like this right?

[00:46:47]Or if it is light in the opposite direction sorry. This was supposed to be 45.

[00:46:56]It's traveling like this. Okay? Where this angle at pi over 4 uh which is 45°.

[00:47:10]Okay?

[00:47:13]In addition imagine that you have this plot and there is another observer another observer with respect to me this observer is traveling also at a constant velocity, okay? And this constant velocity, let's say our origins agree let me use another color.

[00:47:38]Green. So, this is light.

[00:47:42]And this here is O prime which is another observer.

[00:47:52]Notice that we will always have and we will prove that that other observers um they can never travel at a speed higher than light. So, that their slope is always above the slope of light.

[00:48:08]Okay? So, all the particles they will the trajectories of the particles will always be inside this cone.

[00:48:17]Okay? Because this is how fast light can go and never uh not nothing can go faster than this. All right?

[00:48:25]So, um there is something else that sometimes uh we like to plot. And for instance, if you have a system O prime uh and by that means the origin, so that this line is the line X prime equals zero.

[00:48:42]In the same system you could draw uh the line T prime equals zero.

[00:48:50]Okay?

[00:48:54]Beautiful.

[00:48:55]So, whenever you have an observer at sorry, whenever you have these uh space-time diagrams uh the lines of O prime will always be straight lines that are inside this cone.

[00:49:10]And the lines T prime equals zero, etc. will be lines here. And something that is cool is that this angle and this angle is always the same.

[00:49:21]Okay?

[00:49:21]So, so that that we have this this nice uh symmetry. So, basically, all we are doing is is is a very simple representation.

[00:49:31]Is we are choosing the units right.

[00:49:34]And we are just plotting the trajectory of things uh into this diagram. But the beautiful thing is that because light um moves in these 45° angles and we know things cannot go faster than light, we will prove that. Particles, etc., they can never cross this this kind of cones, okay? And And that's something very useful. Is that okay?

[00:50:00]It's um Yeah, it just a diagram and the more you use them the more use you will get to to using them.

[00:50:08]Did you say T prime equals zero must be an exact reflection? Yes. So, basically, yes, that's right. So, if this angle, thank you, is uh let's say alpha, this angle should also be alpha.

[00:50:21]Or if this angle uh So, the point is Right. So, the point is that all observers will agree that this is light. Right?

[00:50:33]So, that this red line should always be in the middle of these two axes, in the middle of these two axes, and so on, because all observers agree that light is a kind of 45°. Is that okay?

[00:50:49]Um And this, of course, looks a squeeze like this because of the factor gamma.

[00:50:57]Uh Yeah. And the faster the observer O prime goes with respect to O, the closer this will get to to to the light from our point of view, from the point of view of O. Is that okay?

[00:51:10]Any other questions?

[00:51:14]Beautiful. So, give me one more minute and then I will let you go.

[00:51:19]The second way that we will have uh to derive the Lorentz transformations, uh of course, we will get the the same result.

[00:51:30]It is It is something that Einstein introduced and then people um kind of now we think of physics in that way, but it's something extremely deep and extremely important philosophically.

[00:51:45]How do we measure things?

[00:51:47]Okay? So, the question that Einstein asked was the following. Okay, you tell me that you will assign coordinates X and T to events and in Galilean transformation, we somehow assume that we all have this infinite ruler, right? And with this infinite ruler, we can see the location of things.

[00:52:14]But that's not true in in in a special relativity. It cannot be true. So, first Einstein devised a way to measure positions of particles, etc., and then with that way, he derived the same equations. Okay? Which I think is beautiful. So, basically, we will uh re- derive and and revise the paper by Albert Einstein in the next uh lecture.

[00:52:40]Anyway, thank you very much.