Special Relativity, Lecture 1: Space, time and relativity before Einstein - 3rd Year Student Lecture

Okay. So, good morning everyone. Uh my name is Fernando and I will be your lecturer for this uh special relativity which is a part B course um in the direction of theoretical physics. Um basically now I will start um this course by explaining you what the title means what the special relativity means. So the first important point is that a special relativity is not special. Okay. Now when you want to make someone feel nice you tell them that they are special and that's a very nice thing. for a physical theory is not a nice thing. Okay, because for a physical theory it would mean that the theory applies only to very special situations and this is not the case with the special relativity. Okay, so a special relativity is a theory that describes the real world all of it. All right. The second important point um about this. So the way the reason why we use the word special sometimes is as opposed to general relativity where in general relativity we consider gravity and in a special relativity we do not. Okay. So that's the only reason why we use the word special. The second important point is that the word relative doesn't mean does not mean everything is relative but actually quite the opposite but a special relativity will tell us what is relative and what is absolute. Okay. when you describe a physical theory and actually understanding these sort of questions. So what is relative and what is absolute will change completely will transform our understanding of space and time. And actually special relativity has had a bigger impact in our understanding of the physical world more than any other theory of theoretical physics. I could say maybe quantum mechanics is as big and indeed a special and general relativity and quantum mechanics are the two basic pillars of theoretical physics of modern theoretical physics. Okay, it is quite um sad slashimpressive slash exciting that we don't know yet how to put these two things together. Okay, but okay, maybe you do this next year or in your future. But basically this special relativity is one of the two pillars of um theoretical physics and the course [clears throat] will be organized as follows. Today in the rest of today we will remind some aspects of Newtonian mechanics and these are basically things that you have already seen on the first uh two years here at Oxford but we will revise some of the conception preconceptions and assumptions that are actually quite important and you assume that they are true and people has assumed over you know hundreds and hundreds of years and then in the second part of the course or the second section we will see that we need to go from this point of view of Galileo and Newton to a new point of view due to Einstein because we will see we will um uh we kind of described experimental evidence that will tell us that some of our assumptions when we describe Newtonian mechanics are actually to be revised.

Okay, in a very fundamental way and I am talking about really fundamental um assumptions then we will uh start describing special relativity or the special theory of relativity proper and we will start with 1 + one dimensions and I want you to start getting used to this notation here. The first one is a time direction so it's what time it is etc. The second one is a space. Okay. So we assume that there is only one special direction and uh one time direction and then we will see some of the nice implications of a special relativity in astrophysics, astronomy and cosmology. Okay. Then we will have the tools to uh to have a look at fourdimensional and here we mean one special uh time direction and three space directions. Uh space time And then we will see relativistic kinematics and collisions. Now a problem that you are very used to in Newtonian mechanics is if you go to play snooker here in the UK uh you learn how to use the loss of Newton etc to predict what happens with little balls but in a special relativity the word is a bit different so we will see what happens with little balls in special relativity right or rather particles at particle ical accelerators. So here we will see some particle physics something quite cool actually and then we will end up with something called relativistic electronamics. All right. uh if you have any questions at any points in the course in the lectures please feel free to stop me ask me if something is not clear okay and I will also stay outside after the lectures for as long as you want okay so please feel free to stop me let's make this a very informal discussion and do you have any questions up to now so far fantastic beautiful so let's start then revising some of the things that we have learned in Newtonian mechanics. And what will be important for us is to revise the concept of relativity in Newtonian mechanics. The first uh thing that you learn in Newtonian mechanics is that in order to describe the motion of a particle, you need to choose a reference frame. Okay? So I can have a description of a particle. You may have another description of the same particle and our descriptions may be different but they are equally good. Okay? if we are reasonable choosing our systems and we will define what I mean by that. So the first definition that I introduce is the definition of reference frame in Newtonian mechanics. A reference frame And sorry sometimes I abbreviate words like this. This is due to two reasons. First I am very slow writing and second

sometimes I don't know the spelling in English very well. So I just it's very nice. I don't need how to I don't need to spell mechanics. I just put mech dot and that's it. So in Newtonian mechanics, a reference frame is a choice of origin plus a set of perpendicular. And often we choose right-handed or cartesian axis. In other words, you choose an origin O. Okay. And then right-handed means that you need to choose this like X. This is Y and this is set. All right. So this is let's say something like this. Okay. And uh sometimes we denote such a reference frame with the letter O. And this O may mean origin. But actually what we mean by it is observer. Okay. So we imagine that there is an observer O attached to the origin of their reference frame and all distances all locations are measured with respect to O. Okay. So O is the person that measures that tells me distances and distances. Is that okay? Yeah. Good. So once we have chosen such a reference frame and only once we have chosen this then given such a frame then we have the position of a particle is then given by a vector R which is X of T. y of t set of t. Okay. And if you have such a position then you can define the velocity of a part of the particle which is the vector with components x y dot z dot and the acceleration of a particle which is as follows. And here uh dot means derivative with respect to time. Okay, we will also use prime but that will mean something else in a special relativity. So derivative with respect to time is just uh this t. Okay. So far so good and we haven't said anything too drastic. Okay. So here comes the first important uh important thing. So let me write it here. So the law of inertia of Newton's first law inertia says that there exist a class of frames relative to which the motion of a particle or free particle. And a free particle is a particle uh to which no forces are exerted. Right? It's just a particle uh with no forces is in a straight line and

at a constant speed. Beautiful. So I would be very uh glad if you all accept this fact. Okay. Now sometimes people talks about a zeroth law of Newton and the zeroth laws of Newton says that such a thing as a free particle exist. A free particle doesn't exist in the real universe because we it would have to be far away infinitely far away from every interaction. But we can imagine that ideally if a free particle existed then there are there is this class of systems which are called uh I will give them a name now to which that particle moves on a straight line at a constant speed. Okay. So this is an assumption built in our understanding of um of Newtonian mechanics. And the definition, our definition, another definition is that a frame in which the law of inertial H follows uh holds. is called an inertial frame. Okay. So this is an important class of frames and whenever possible and always in a special relativity we will imagine that we will choose inertial frames. Okay. Uh, beautiful. Then another definition and sorry about that. um we call an inertial coordinate system. sometimes ICS as an inertial frame plus a zero for time. Okay? So you choose when to start counting time. If you make such a choice together with an inertial coordinate system, this is what we call sorry together with an inertial frame. That's what we call an inertial coordinate system. Okay, a choice for t equals z could be when Jesus was born and a lot of people likes to do that. We could choose something else and that's also fine. Okay, all of this uh should be fine. Any questions so far? Very good. So, so far easy, right? It's uh yeah, the cool thing about this course is that today I am going to tell you a lot of things we all agree with and tomorrow we will see they are not true. Some of them okay but so far it's you know people agreed with this for hundreds of years. Beautiful. So in an inertial frame we also have the second law of Newton which says that the force exert exerted to a particle H if it is different from zero then we'll produce an acceleration on a particle. Okay. So we already introduce the acceleration. This is force and the important point is that there is an m in this equation. Okay. And this m is what we call the inertial mass of the particle. Okay. And an assumption uh which we do in Newtonian mechanics is that this quantity the inertial mass of a particle is actually a property

of the particle independent on the inertial frame or the motion of the particle. Is that okay? So we imagine let's say fundamental particles like uh snooker balls. They have a given mass and this mass doesn't depend on whether you are a steel with respect to the table or you are walking table. We will measure or we will say that the particles have the same mass. Is that okay? Yeah. Beautiful. Again, there are philosophical discussions. Some people could say that this equation is empty because it just tells me what the definition of the force is. So, this equation makes sense only if you have an independent way of determining what the force is and then you can use this equation. Okay. To compute the acceleration. Beautiful. Any questions so far? Yeah. Do you agree with everything I have said so far? Beautiful. Very nice. So, um, easy so far. Now consider we have two inertial frames. Okay. Let me call them O and O prime. And let's say for simplicity that they agree on their zero of time. Okay? So they are both using the birth of Jesus to measure time or Donald Trump or whoever you want to use and then they say that T is equal to T prime. Okay. Now they look at a particle at a given particle and I will say the location of the particle with respect to my system is x of t y of t set of t with respect to o. But the other system will describe the same motion as x prime t yprime t and set prime t. Okay. Yeah. Very nice. So the first uh task which is uh not super hard will be to understand how these two things are related. Okay. For two inertial frames. Do you prefer if I write here? I guess there is more. Sorry, this is like democracy but it's so let me work it out here. So in general without making any assumptions for these uh two particles for a given particle sorry these two descriptions are related as follows. So XY Z is equal to a given 3x3 matrix x prime yp prime z prime plus t. Okay. And here h and t uh this is a column vector that represents translations. So indeed imagine that all has a system like this. and O prime let's say you say something like this then this T parameterizes this translation between O and O prime okay while H is a 3x3 proper Orthogonal matrix

which represents the rotations rotation of axis. For instance, it could be that this observer is in the United Kingdom. Australia. So they will have their axis rotated. And because it's a proper orthogonal matrix, it will satisfy that h transpose H is the same as H. HRpose and this is the identity matrix. So this one here means the identity matrix H and is proper. So the determinant of H is equal to one where this is really the number one. Okay. Are you comfortable with this? Thank you. Yeah. Because mathematicians will sometimes will not be very happy about this. Uh beautiful. Now in principle these two h and t things can be smooth functions of time. Okay. However by assumption we are asking for what is the relation between two inertial frames. Okay. So imagine we have a free particle and the system O describes the motion of this free particle. Then because the particle is free, x dot dot, y dot and set dot is equal to zero. Okay. However, because also O prime is an inertial frame, the transformation has to be such that this is true if and only if X prime dot Y prime dot and Z prime dot is equal to zero. Okay. Then you can simply take this relation here. You can assume a general transformation and you can start taking dots on both sides. All right? And you will uh see that this is true if and only if the only way this can work. So this is true for a general rectilinear trajectory if and only if H dot is equal to zero and t dot is equal to zero. Okay. So that actually means that for the transformation between two inertial frames the transformation is of this form but h is a constant and t dot is also a constant. Okay. Yeah. Questions. — Oh, sorry. Rectilinear. Yes, sorry. Um, okay. Let me rectilinear. It actually in equations it basically means this. Now rectilinear means on a straight line. So you know that a free particle because of the first law of Newton a free particle has to move on a straight line that means rectilinear motion. It's just this uh in Latin rack and a straight is the same. So rect ractus actually it also means anus but okay but it also means like a straight something straight and is uh so this is just like a straight line basically so the point is that if you have any free particle

the motion of a free particle has to be a straight line both in the system O and also prime because by assumption both of them are inertial. Okay. And that means that basically means that if you take this for any generic xt yt sett and you take dot on both sides, right? Both sides are equal provided this is true. Yeah. Yes. You had a question — is so for a free particle it has acceleration zero. Is there any other condition on a free particle which is just — that that's fantastic question. No, it's just acceleration zero. Yeah. Uh yes. However, when so there is a physics answer and a math answer in formulas, acceleration zero also implies that the particle will move on a straight line. So if you have something rotating at a constant speed, that thing doesn't have zero acceleration. Okay? So if you look at the trajectory of the moon around Earth, the moon doesn't have zero acceleration. Okay? So by acceleration, it doesn't necessarily mean that it's moving faster and faster or slower and slower. It just means that x dot dot, y dot dot, z dot is actually zero. Yeah. Yes. — Um, is there an assumption here that uh a free particle in one reference frame is free in another reference frame? — Beautiful. Yes. Fantastic. Thank you very much. And the reason for this why this assumption is true is because I am looking for the transformation between two inertial frames. And by definition, an inertial frame is a frame on which a free particle has no acceleration. Yeah. If you look for a transformation between an inertial frame and another frame which is not inertial, then this wouldn't have to be true for a free particle. So the reason why this is true for a free particle is because we are assuming both O and O prime are both inertial. Okay. — But we also assume that the property of being free is independent of your inertial frame. — That's exactly correct. Yes. And this Yes. Thank you so much for the question. And this is um our definition of inertial basically. So a frame in which the law of inertia holds that's by definition an inertial frame. Yeah that's a very important assumption. Very good. Uh beautiful. Thank you for the questions actually. I love that. Very nice. So okay. So if we uh put this h constant t dot equals constant and in addition we add the fact that now uh one of the systems may choose a different origin for time which is perfectly uh okay. then we end up let me write it here with what we call a Galilean transformation. Okay. So the definition a Galilean transformation is the general transformation. two inertial frames. Okay. And the here we will have our first glimpse of a spacetime exciting moment for you guys. So if you put X, Y and Z set together with then this you have one 0 0. This V1 B2 B3 and here is the matrix H. H11 1 2 1 3 2 21 2 1 3 1 sorry 2 3 2

prime X prime Yprime Z prime plus some constant fourth vector. So here this is a constant vector of length four. uh this v three and h is this 3x3 matrix a constant matrix that we already introduced before. Okay. And the this uh they have a name uh this t represents translations. This V is called boosts and this H rotations. Let me explain them nice and easy. Erh so translation we have already seen what it means it's just O and O prime could have chosen systems in which O is translated with respect to O prime. Okay. But in addition, one could have a velocity with respect to the other. And this is measured by V by this V which has kind of the structure of a velocity. And then in addition, H is a rotation of the axis. For instance, if someone is upside down and the other observer is just straight up, then they would have different uh rotations. Is that okay? So if you uh write this for generic one v1 b2 b3 and rotation h and a constant vector of length four this is called a galilean transformation. Okay. And basically if you are Galileo, you are Newton and you have two observers and these two observers want to translate between one set of information and the other one, they use this Galilean transformation to go from one to the other. Okay, so this is the most general transformation that satisfies precisely these relations. So not notice that because this t is constant t dot is constant you can write it as v * t. So plus another constant let's say so this is the this is why we have uh b1 b2 b3 ant here. Is that okay? Yeah. If it is not I am very happy to do it more slowly. But uh yeah another thing you can do you can take this you can multiply here and then you can put it in this form and see that h is this h and then we have this. Yeah. Where this is V and then this is the translation. Is that okay? Yeah. Okay. Any questions? Beautiful. Have you seen Galilian transformations before? So you have seen it. You have seen many examples. So for instance, something that you see is transformation of velocities. All right? Like they tell you when they teach you how to drive, if you are driving really fast, you know that actually it is way better to smash against a car going in your same direction that against you. Okay? And the reason for that is that in one you will feel the added speed of both cars but in the other the relative speed is what matters. Now if two systems are describing uh the motion of a particle then you need to subtract the speed of the relative of speed of one in order to find the speed of with respect to the other one. Is that okay? Yeah. If it is not please do tell me is you don't feel it's like there are no silly

questions. I see some really basel faces. So I am uh shall I do this in more detail or you are happy with it? Yes. What? Don't be shy. Just more detail. Perfect. Okay. Good. Very nice. So here it comes. Uh yeah. So if you want um you can decompose this, right? like here I don't know you can call this constant C 0 C1 C2 C3 right then for instance you can see that you can multiply here from the right and you can see that it's a nice exercise that you recover exactly this form okay beautiful so we have the following Now uh the two following statements the nice thing about Galileian transformations is that they preserved Newton's laws. In other words, if the laws of Newton are true in one reference frame and if you do a Galileian transformation of that, then the loss of Newton will also be true in this other frame related to the first one by these Galileian transformations. Yes. — Over there in the matrix form, we said that the translation is constant. — Yes. But over here we're saying only it derivative is constant. — Sorry. Yeah. Inde. Yeah. Thank you very much. So let's do it. Let's do it very precisely. So we because we said that the derivative is constant. Okay. Of the translation then you can always if you have t dot equal constant this means that t is linear in time. Okay. Because linear The way you can write t which remember in this part of the board t is a column vector of dimension three. Okay. So because t dot is constant t itself is linear on time. So we write it as V a three vector times T plus a real constant thing C that we call three right where this is uh C1 C2C3 and this here is B1 B2 B3 and this B1 B2 B3 is exactly this one here B1 B2 B3 and this C123 is this one here and this is zero is the this constant here. Yeah. Thank you for the question actually. Uh fantastic. [snorts] So the first uh point is that the Galilean transformations they preserve Newton's laws. So Newton's law are true uh in one system then they will be true in another system if both of them are related by the Galilean uh transformation and the important point and this is called Galilean principle of relativity and this is the most important part of this lecture and it's the following. as far as the loss of mechanics are concern all inertial frames.

are on equal footing. This we take it as granted but it's an amazing concept. Okay. So people got burned for defending this concept. All right. So in particular it means that we cannot say that an inertial frame is at rest. Okay. That is better than another. All inertial frames are equally good. And if you described the loss of nature, you can use as far as the laws of mechanics are concerned, you can use any inertial frame. Is that okay? And all of them are equivalent. That's why relativity it has the word relativity on it. All right. Any questions about this? Beautiful. As we will see and let me spend one minute on this. Er no I will not but tomorrow we will see why this is highly non-trivial and actually um people discussed a lot about this all right so basically humankind first we thought that earth was at the center of the universe and was somehow preferred then we thought that the sun was the preferred thing and we could describe things with the sun at rest. Now that we see that not even that is preferred and actually all inertial frames they are all equivalent okay and that's an important point and I think uh with the years you are very young you will see how deep this is like if you keep going and thinking about this you will see how amazing this is okay so in the last 10 minutes let's discuss the structure of an important thing. Galilean spacetime. The first definition that I need to introduce is the definition of an event. Okay. An event is something very simple. is a specific point in a space. at a specific time. Okay? So it's like saying here and now. um if you choose in a given coordinate system The event sometimes denoted by E is labeled by a time t and coordinates x, y and set. Okay. Now another system another frame will denote this the same event by t prime xp prime yprime zprime sorry where uh the two of these are related by a galilean transformation. However, we would like to think of the vent itself as something independent of any coordinate system. Okay. So, if we want to give coordinates to the event, then we need to think of it, we need to choose a frame. It's very much imagine

you have a map, right? And you have a mountain. you think of the mountain independently of the coordinates that you choose to to describe points in this map. Okay? So we think of the bent as something independent of my system that o describes with these coordinates and uh o prime describes with these coordinates. Okay. And here t x y and set are something fixed. Yeah. So this could be 1 second and this 1 meter, half a meter and 2 mters. Beautiful. So now uh we have the following statements which are invarian statements. given two events. And the reason why it's important uh to understand this is that special relativity uh will tell us how space and time come together. So let's try to think in Newtonian relativian sorry mechanics. Um so this if you have two events right there are two things there are many things sorry that are invariance and for instance one is the statement A and B are simultaneous. Okay, by invariant I mean that if a inertial frame thinks that A and B are simultaneous, every other inertial frame will agree with that. Okay, in Newtonian mechanics. All right. So if two things look at this, these two things have happened at the same time. Okay, and someone on a plane would agree with that. All right. B happens time t after a. This is also a statement that will not depend on your choice of inertial frame. And also A and B are simultaneous and separated. by a distance d. Sorry, I will. Okay, so all these statements are statements that do not depend on the inertial frame that you choose. Okay, sorry. So, two more one more minute. So, we're almost uh out of time. So, let me just uh end up perhaps with a statement which is not invariant. So for instance, if you have the following statement uh A and B happen at the in the same at the same place at different times. This is not a very good statement. Okay? Because imagine that the first event is this and the second event is this. Okay? We all agree that the two claps happen at the same time at the same place. However, imagine someone on a plane, right? And that someone on a plane sees me clapping and then the second time I

clapped I was like 1 kilometer away from the first time I have clapped. So according to that inertial frame this will not be true. Okay. So this statement is not a statement which is invariant under the choice of inertial frame while these statements are. Okay. Beautiful. So the next tomorrow we will continue a little bit and revise some of these other preconceptions uh and we will start seeing why in the 22nd uh century in the 20th century people started to doubt all this and then in the third lecture how Einstein resolved all these problems and this will lead to a new picture of a spacetime. Thank you very much and see you tomorrow.