Transcript of a2-l10 ========== _0:12_: I said that we had finished chapter 8 but it chapter 6, But _0:16_: it's got to me that it would be very useful for me to show a _0:20_: sort of worked example there of using things like the _0:24_: relativistic Doppler shift just to so you can see the the the _0:29_: having been done once, we'll then moved roughly on to chapter _0:33_: seven. I think we won't get all the way through Chapter 7 today, _0:38_: but I we will encroach, I think on Lecture 11. _0:42_: But I I I have high hopes that we'll get into Chapter _0:53_: next time. _0:56_: Any questions about organisation, material and all _0:59_: of that like that? _1:02_: OK, _1:05_: what I want to do, there's an example I want to work through _1:09_: is the case of Imagine _1:11_: are _1:14_: a relativistic ambulance is going past you _1:17_: and you you see or it's going to pass at A at A at a _1:22_: a lot of speed and you see it's blue light. You see it's blue _1:26_: light as it's passing you. _1:30_: What frequency is the light? _1:33_: OK, _1:39_: so the way we work that out is using the centre that the the _1:43_: recipe that I approximation of the recipe that I mentioned for _1:46_: in Chapter 5 talking with the restaurant _1:50_: what we do is we talk about _1:52_: the _1:54_: 2 frames. _1:56_: So let's have a _1:58_: primed _1:60_: white frame frame that's in the frame of the _2:03_: I'm gonna _2:06_: and that's moving _2:08_: at some some speed V _2:11_: and the light _2:13_: in that frame being emitted _2:20_: at some angle _2:22_: Teacher prime. _2:24_: So this is the frame _2:32_: with the ambulance. _2:33_: OK _2:35_: and we want. The question is what is the and the? The light _2:39_: has frequency F primed _2:49_: OK. _2:51_: The question is what frequency do we see that light in offering _2:56_: and our. So what is _3:08_: F in _3:12_: frame _3:16_: in which _3:18_: the _3:20_: and be you Lance _3:25_: moving at _3:28_: speech? _3:29_: And that would of course be neater if I were handing this _3:31_: in. _3:34_: At that point, Once you've set the question up _3:37_: and being clear and explicit about what the frames are, what _3:42_: we have been given, Theta what we been given, we will give _3:46_: ourselves an F prime and prime. Once we've set up correctly, the _3:50_: answer _3:51_: almost pops out. It's almost automatic _3:57_: is that _3:58_: the expression for the relativistic Doppler shift F _4:01_: equals framed. _4:06_: Umm, _4:09_: Camera _4:10_: or gamma F prime 1 + v Cos Theta prime. _4:17_: So the question is this. For a question in an exercise or a _4:21_: question in a in a text or an exam, the question would then be _4:30_: if if the light were emitted perpendicular but to to the _4:33_: ambulance, what frequency would you see it as _4:39_: and what that is, is that's telling you what Theta prime to _4:42_: use? What what what angle do you use _4:44_: If if I see the light is emitted perpendicular to that that the _4:48_: light you see the flash you see that you're looking at the the _4:52_: frequency of it emitted perpendicular to the ambulance. _4:55_: What is that telling me? _4:58_: Anyone. _4:60_: What _5:01_: bit of data that we know we need is that telling me _5:08_: is telling me F _5:12_: is it not _5:14_: it will it will be upshifted, yes. But the light that that _5:18_: that that's emitted perfectly perpendicular to the ambulance _5:22_: is the ambulance driver just shines out the window. _5:27_: Is is telling us? Is it telling us Theta or Theta primed _5:31_: the depraved? Exactly. _5:33_: So it's telling us Theta framed, not Theta. _5:37_: So the light _5:42_: committed perpendicularly _5:46_: to ambulance _5:51_: at _5:54_: or -π do whatever _5:58_: future _5:60_: depriving by two. In other words, cost to prime _6:04_: is 0 _6:06_: and so that F equals gamma F prime. _6:13_: In other words, even though this light is emitted _6:18_: perpendicular to the direction of motion _6:21_: to where classically there would be no Doppler shift, _6:25_: it's _6:27_: blue shifted. _6:29_: OK, _6:30_: that now that might seem slightly surprising. _6:33_: Why is that slightly surprising _6:36_: that that that fits in with you? You know that Doppler shift blue _6:39_: blue shift things, but why is that slightly surprising that F _6:43_: equals gamma times F prime? _6:49_: It's surprising because _6:51_: the train violation _6:54_: the _6:56_: I'm really moving to relativistic speed. _6:59_: Therefore you'd expect time to slow down frequency to to to to _7:03_: to be lower. So you'd expect F _7:06_: to be lower than F prime. _7:09_: Any idea why that why That seems to be the wrong way around? _7:17_: Let's ask a different quest. _7:19_: What would I? How about what is the light of the frequency of _7:23_: what? The frequency of the light that I see when I look directly _7:28_: across the path of the of the embrace of motion and see those _7:32_: when I see the light come in _7:36_: directly in this direction. What? What different? How is _7:38_: that different? How is that a different question? _7:45_: Different. _7:47_: They're the same 2 frames, so it's the frame of the ambulance _7:50_: is moving in this direction with positive V and I'm in this _7:54_: frame. _7:55_: The same 2 frames, same speed. _7:58_: But how have I changed the question? _8:03_: In the first case I talked about the _8:09_: that that not F&F primed but _8:12_: Theta and Theta brain. _8:14_: So in the first case I was asking what is the the frequency _8:18_: that I see of the light that's emitted perpendicular to that to _8:22_: to its motion by the by the ambulance. _8:25_: Now I'm asking what are the light, the frequency of light _8:28_: that I see arriving perpendicular to the direction _8:31_: of motion of of the ambulance. _8:34_: And they are not the same thing _8:36_: because in the second case _8:40_: the light _8:43_: or _8:44_: they've they've _8:48_: particular _8:52_: feature equals _8:54_: π by 2. _8:56_: And if I look at similar like F prime equals gamma F _9:04_: 1 -, v Cos Theta _9:08_: if Theta equals root of the counterpart to this. Going in _9:13_: another direction, if Theta is equal to 0, is a π by two, then _9:18_: Cos Theta is equal to 0 and F primed _9:27_: framed _9:28_: is F _9:31_: if you could _9:32_: yeah F primed over _9:35_: gamma. _9:38_: And _9:39_: in the first case, _9:42_: if Theta primed is equal to 0, _9:46_: then cost teacher will be equal to well, V some number which is _9:50_: less than one. _9:52_: So the light will come in at a different direction from π by 2 _9:57_: and if looking at that we're owned. If Costa was 0 because _10:01_: I'm looking at the light I'm observing from the from this _10:06_: direction, then the counterpart of that the that light was _10:10_: emitted in a different direction by by the ambulance _10:15_: and that's why there's so the light that I'm seeing _10:19_: arriving perpendicular to me that's nicely red shifted _10:23_: because obtained election _10:26_: as opposed to the light which is emitted perpendicularly and _10:29_: making a fuss. That's not because you have to care very _10:32_: much about observing the colour changes of of ambulance lights, _10:35_: but because _10:37_: in I have asked two different questions in that example _10:42_: which are different questions. _10:44_: They were carefully worded and I would word them more carefully _10:48_: if we were saying that as a an exam answer. But in both cases _10:52_: the thing you had to work out was what does he mean? Which _10:55_: does he mean Theta or Theta? Primed I was talking about π by _10:59_: Theta prime is π by two or Theta is π by two. If I'm talking _11:02_: about light being emitted forwards or backwards is that's _11:06_: Theta equals 0 or Theta equals equals π and vice versa. So _11:09_: several of the exercises _11:11_: essentially _11:13_: questions of interpretation. How do you read the the the question _11:16_: and turn it into which you know. So that's why it's it's like _11:20_: the recipe that I talked about for doing Minkowski for doing _11:24_: our transformation things in Chapter 5. _11:27_: Step one is writing down, being clear and explicit what the _11:30_: frames are. _11:32_: Step 2 is writing down what you know, what you've been told. And _11:37_: that's that's an the non trivial step _11:40_: going from what the question says to that means it's that _11:43_: that that that, that that it's that, that is that that figure _11:46_: that picture frame would ever be told, not theatre. And toward _11:50_: the end you're turning the handle, you just plugging things _11:53_: into expressions. _11:55_: So that's a a rather handy question. The diagram at the _11:60_: top, yeah _12:02_: time is there's like a street lane and then there's the wave _12:05_: coming in. Yeah, the angle there is not shown to be π / 2. Ohh no _12:09_: no no, not that example no no. So that's more general. So in _12:12_: the, in the, in the case that we're talking about _12:15_: and that would be in the X prime, _12:19_: my prime. So there it's, it's. _12:27_: Is that is that the? _12:30_: So yes, I drew the the the general sort of general diagram _12:34_: 1st and then particularised it given the information in the in _12:39_: the question. _12:42_: OK. And again, not very pretty, not very neat, but _12:47_: good enough for no talking over _12:50_: any other questions about that. _12:53_: There are half a dozen, I think, questions in the exercises in _12:57_: part in in chapter 6 which are variants of that. _13:05_: OK, right. Then let us move on _13:08_: and _13:10_: and we can talk about _13:13_: dynamics, _13:21_: OK. _13:23_: Kinematics, I said, was about describing motion. _13:34_: Let's stick with that for more. Can immatics is about describing _13:37_: motion, dynamics is about explaining motion. So dynamics _13:40_: is where we talk about forces and momentum and and so on. The _13:43_: the stuff that you learned that you're thoroughly schooled in, _13:47_: in terms of Newtonian mechanics. _13:57_: In the in the Kinematics chapter we talked about relativistic _14:00_: velocity. _14:03_: There's all we'll go back on to this. _14:06_: We talked about rustic velocity and it's slightly unusual _14:10_: properties. _14:12_: Now _14:13_: we are very familiar with how we get we get non relativistic _14:17_: momentum from velocity, we just multiply it by a mass. _14:20_: So let's do the same. _14:22_: Let's define a thing P _14:27_: where? Where _14:29_: we just the the the momentum of an object is its mass times the _14:34_: speed of the object. So the just as a momentum I've I've lost _14:39_: before vector. There's a momentum 4 vector _14:43_: and the mass here just the massive. It's nothing _14:44_: complicated, but the matters. It's how much? _14:48_: Yeah. How much material the race will you hold in your hand. _14:51_: OK, And the math doesn't change. The mass is just a number. It's _14:55_: it. It it's a, it's a a variant _14:59_: that's too far so good. That is nice and simple _15:07_: in the kit, but we remember that in the case where the particle _15:11_: is not moving, _15:13_: it's velocity _15:15_: will be _15:18_: A4 vector pointing along the time axis. So it has it's time, _15:22_: It's time component is 1 and it's special components are 0. _15:26_: In that frame, _15:29_: the _15:32_: P will be equal to _15:34_: M _15:37_: 0, _15:39_: so P dot P _15:43_: will be equal to _15:46_: m ^2, _15:48_: which is our Lorentz invariant, so that that's that. The length _15:51_: of the momentum 4 vector is just the mass squared. The length _15:55_: squared is just the mass squared, and that's true in _15:58_: every frame. So because this is a four vector, in different _16:01_: frames this momentum will have different components. They're _16:05_: nice and simple in this, in the frame of which The thing is not _16:08_: moving _16:09_: been other frames, you know the the the tank component will be _16:13_: bigger, the spatial components will be bigger and and so on. _16:16_: But when you work out the length squared of the object, it'll be _16:19_: m ^2 again as before, _16:21_: because that's the range and variety. The the the length _16:24_: length squared of the vector is extremely dependent. _16:29_: OK, now imagine we've got some. _16:40_: Umm, _16:45_: a couple of questions I'm not going to talk about. Imagine _16:49_: we've got. _16:51_: Uh, _16:56_: a collision like this? _16:59_: So there's a _17:01_: 2.2 particles come in and two particles come out. So it's not _17:04_: it's not a very exciting collision from the point of view _17:06_: of particle theory, particle physics, but it's nice and _17:09_: simple from the point of view of our analysis of it. Two parts _17:12_: come in P1 and P2. _17:13_: Something happened and they and two parts come out, P3 and P4. _17:19_: Now we're going to make a wild guess _17:22_: and suppose _17:24_: that _17:26_: P1 plus P2 _17:28_: equals _17:30_: P 3 + P _17:33_: 4. _17:34_: We're going to suppose _17:36_: that maintenance formulation is conserved. We know that three _17:39_: momentum is conserved in non relativistic collisions _17:42_: and you're doing with that I trust _17:45_: and we're just going to suppose that the same is true for this _17:48_: thing. What we've got here, we've no evidence for that. Is _17:51_: that just a guess? Because we like doing that with momentum. _17:54_: At the moment, it's just a guess. _17:56_: OK, So what is that? Let us _17:60_: do _18:05_: that and equation _18:07_: and equality between an equation between 4 vectors. _18:10_: So it's true component by component, _18:15_: and what that means is that. So P here _18:19_: and will be. _18:22_: He won, for example, will be _18:26_: M1 _18:28_: gamma _18:33_: V if I. If I look back at the _18:37_: slightly strange notation that I mentioned in the in the other _18:42_: notes, just to show you that you have seen that before, _18:49_: it's just this _18:50_: with an aim in front of it. _18:53_: OK, _18:57_: so it's it's and written down component by component. What _19:03_: that means is that gamma 1 _19:09_: M1 _19:10_: gamma V1 plus M2, gamma V2 equals M3 gamma V3 plus M4 gamma _19:22_: 3/4. _19:23_: And all I'm doing there just rating out long hand _19:28_: the let's see the the the the X component _19:32_: of _19:33_: the expression above. _19:35_: So that's just the the the see the X component of that vector _19:40_: equation above it. _19:42_: So as you can see, _19:47_: sorry _19:51_: and _19:55_: yeah, so, so, so, so that is, it is. _20:01_: And _20:03_: I've written that wrong incorrectly. _20:06_: There should be a V _20:10_: and _20:11_: in each of these V1. _20:13_: Me too _20:14_: P3 _20:16_: before my note. The equation 7.2 B in the notes should have V's _20:25_: in _20:27_: throughout and that is just P. _20:34_: It's actually gamma _20:36_: 1P1 plus gamma 2P2 equals gamma 3P3 plus gamma 4P4. In other _20:43_: words, that is just _20:49_: the spatial components of this recover the normal conservation _20:52_: of momentum just for some extra gammas in here, which of course _20:56_: in the non relativistic case are all approximately 1. _20:60_: So this this relativistic expression conservation of _21:03_: relativistic momentum is consistent with the _21:09_: with the _21:11_: conservation of _21:13_: ordinary special momentum. _21:15_: In the limit when gamma is _21:18_: it won. You know there's a a slow speeds so. So this isn't _21:21_: seeing anything different from what you're really familiar _21:23_: with. _21:24_: OK, _21:29_: OK _21:31_: mumble mumble positive Mumbles _21:36_: and _21:41_: so much for the spatial part. _21:47_: Now if we look at the the time component of that, _21:51_: what we see _21:53_: is that the. If we look back at _21:59_: this expression here _22:01_: we see that the time component _22:04_: is _22:05_: and _22:08_: yeah I'm I'm, I'm gonna I'm gonna switch away briefly from _22:11_: the _22:13_: conservation of momentum and look just at the the the the the _22:17_: time component of our four vector. So if if the overall _22:21_: thing is _22:25_: gamma one _22:27_: the then the time component, the 0 component of this former _22:30_: mentor of this 4 vector is just gamma north, _22:36_: right? _22:37_: That's that's not surprising at all. I'm just all I'm seeing is _22:41_: is that's fairly obviously the the, the, the time component, _22:45_: the 0 component of that expression. _22:48_: But _22:50_: yeah, _22:52_: like anything to ask what's the low speed limit of that? Can we _22:56_: find an interpretation for this zeroth component of this _22:59_: momentum? _23:01_: Yes we can _23:02_: because if we _23:04_: look at the _23:07_: low V expansion of VS that using the a Taylor series _23:13_: then gamma which you could do 1 -, v ^2 to the power minus 1/2 _23:20_: is is going to be a 1 + V ^2 / 2 plus things of order _23:27_: V to the 4th. _23:28_: So that is going to be _23:32_: M plus _23:35_: half _23:36_: MV squared plus terms of order _23:40_: the 4th. _23:42_: And that is, I hope, a rather suggestive expression. _23:46_: You've got something in there which is which goes like 1/2 MV _23:49_: squared, _23:50_: which looks a lot like the kinetic energy of the particle. _23:53_: And what that hint to us _23:55_: is that _23:57_: if _23:58_: momentum is conserved, momentum is conserved, _24:02_: then each of the components are conserved. And we saw that with _24:06_: the spatial momentum there's a hinting that this there's _24:09_: indicating, indicating also that this zeroth component is _24:13_: conserved _24:14_: in collisions and something that looks like energy _24:17_: to that, prompting us to interpret this 0 component _24:22_: of the four momentum as the energy _24:25_: of the of the moving particle. So the spatial components _24:28_: correspond to the spatial momentum that we're familiar _24:31_: with. _24:32_: The youth component corresponds to the energy of the particle, _24:37_: sort of, but it's not. It's clearly not the engine familiar _24:41_: with because there's a half MV grid in there, but there's also _24:44_: this M. _24:46_: In other words, this is telling us that this thing that we that _24:49_: is a bit like the energy of the particle, _24:52_: because it's conserving collisions, _24:54_: isn't zero when the particle isn't moving _24:57_: to review is equal to 0. That reduces to just P not equals M _25:04_: and if we. _25:08_: Look at this expression here. P not. _25:10_: And we're going to write that as E. _25:13_: But we're going to jump to the conclusion there. If you go to _25:16_: Gamma M _25:18_: this remember is in the units where _25:22_: she is equal to 1. _25:25_: So we can ask what is the _25:27_: the The version of that expression in physical units for _25:30_: CC is not equal to 1, _25:32_: so we can do that conversion. We can put C back in or _25:37_: and make sure that with with the right power so that the _25:40_: dimensions work in physical units _25:44_: and we get _25:45_: gamma, MC squared _25:53_: and the zero speed limit of that _25:56_: we're gonna be equal to 1. _25:59_: The _26:01_: is _26:03_: MC squared, _26:05_: which has been called perhaps the most famous equation of the _26:07_: 20th century. _26:11_: So what we've done here so, so this seems, well, equals MC _26:14_: squared. We've got that _26:16_: as you, as I'm sure you you hope we would at some point. _26:19_: All we've done here _26:22_: is guess _26:23_: that by putting an arm in front of the relative velocity we've _26:26_: got, we've got, we've got I think which we're calling the _26:28_: formentor which is physically meaningful. _26:31_: We are reassured that it's physically meaningful because _26:35_: when we decide to say _26:37_: let's suppose that the formentor was conserved, _26:41_: then we get first of all the _26:44_: something which looks like the conservation of three momentum _26:46_: or which which reduces the conservation theory momentum in _26:49_: the rugby limit. _26:51_: And we've also got something which looks a bit like energy, _26:54_: which we said called energy. So this this P naughty is what we _26:57_: have now on calling the energy of the particle _27:01_: and the fact that we we're giving it that name is is _27:04_: plausible because we we spot that express the kinetic energy _27:08_: in there and we discover that it's non 0 _27:12_: with the things isn't moving _27:14_: and recover. This equals Gamma MC squared. _27:17_: In other words, _27:18_: as far as relativity is concerned, as far as the _27:20_: dynamics of relativity is concerned, _27:23_: the energy that you that that's important is not the kinetic _27:27_: energy, the 1st order term in in that expression, but the whole _27:31_: thing which is non 0 even when the particle is stationary. So _27:35_: there is energy in mass. _27:40_: And by the way, this so, so, and all this reassures us that we _27:44_: are right to say _27:47_: this former mentum. _27:48_: It's conserved in collisions. And that is the third example of _27:52_: a physical statement that I've made In this course. _27:56_: The first two physical statements were the 1st 2 _27:58_: axioms, _27:60_: the expression of Dalian rotor. You can't tell you. Moving the _28:03_: 2nd axiom, everything moves C. _28:05_: Those are things that could be otherwise, but I'm saying that _28:07_: in our universe they appear to be that way. _28:10_: The other things we've done the last 10 weeks have been logical _28:13_: consequences of that, _28:15_: so that you know they can't be otherwise. If you take those _28:18_: axioms as true, the other things just follow. _28:21_: This conservation of four momentum is another physical _28:23_: statement. _28:25_: You can imagine that being otherwise, and you know it would _28:28_: be mathematically wrong for it to be otherwise. But in our _28:30_: universe it appears that's just true. _28:33_: OK. And that's a statement about physics _28:36_: or astronomy over you. _28:39_: And it's important to be clear with the sanctions, those two _28:41_: things _28:46_: and more things I want to see on that section. _28:50_: So, So what that also means. _28:53_: As you can see here that this is the zeros component of the. _29:00_: This kind of empty squared is 0 component of momentum and that _29:04_: changes _29:06_: at a freedom dependent thing. _29:08_: So the _29:10_: spatial components of the four momentum and the time component _29:13_: of the four momentum are different in different frames. _29:17_: So the the the the energy, the spatial momentum of a particle _29:21_: is different different frames. That's not surprising because I _29:26_: mean that's true in _29:29_: non rustic physics too, but also the energy of a particle is _29:31_: different in different frames. _29:35_: The energy being the 0 component of all for momentum, _29:38_: but always _29:41_: so so so so so energy isn't it? I think you can stripe to _29:44_: article anymore but what you can notice is that what will still _29:49_: be true is that P dot P _29:52_: will be m ^2 _29:54_: because that that because as I've said repeatedly the _29:58_: dot product of 24244 vectors and I thought you were either doctor _30:04_: 4 vector with itself is framing variant. So the length squared _30:10_: of of the four momentum vector _30:14_: is always going to be the the mass of the particle square, _30:17_: even though it have different components and different _30:22_: umm. _30:24_: I will also just quickly write down that UM _30:29_: with _30:32_: rating equals Gamma M&P. _30:36_: Lower case _30:38_: P is equal to _30:45_: gamma M _30:47_: V _30:52_: and _30:56_: yeah the former momentum P is going to be. _31:02_: It's 10 component is the energy, _31:04_: that's just P not And the special components are what I'm _31:08_: writing out. The the the the the spatial momentum P _31:12_: to P _31:13_: dot P _31:15_: which is equal to m ^2 _31:18_: is going to be _31:20_: e ^2. _31:21_: Nice _31:22_: P ^2 _31:28_: and the instant physical unit is it is also eastward equals _31:36_: P ^2 C ^2 + m ^2 C _31:40_: 4th. I'm not going to go through the details of that, but is it _31:43_: good to see that expression written down at some point? The _31:45_: point is that there's a a nice relationship between the mass, _31:48_: the energy and the the rest of the energy and the relative _31:51_: momentum. _31:53_: That's a key point I want to mention before moving on. But _31:56_: before I go on and talk about photons, _31:58_: are there questions about that? _32:01_: Over what question are you excellent questions? From the _32:04_: working there was this _32:06_: gamma, _32:09_: what it was, and it was to the minus 1/2. That's right. Yes. _32:13_: OK. _32:16_: So the definition of gamma _32:19_: is it's 1 / 1, one of us square root 1 -, v ^2. _32:24_: So that's one of most V ^2 to the power minus 1/2. _32:29_: Now if you're if you catch me back you have you done Taylor _32:33_: Series. I'm McLaren Series and all that stuff so you will _32:37_: remember possibly or or you may not remember but you can go back _32:41_: and and and just confirm. You seen before that 1 + X to the _32:45_: power N _32:47_: The Taylor expansion of that is 1 plus N, X + 1/2 N _32:55_: X ^2 plus _32:58_: I and so on. _33:01_: Adequate, general _33:02_: for general thing and and and doesn't have to be integer. _33:05_: I think that's also called binomial theorem, blah blah _33:08_: blah. There's got a couple of right. It's very special cases, _33:12_: have a couple, have a variety of different names, but the _33:15_: expression just above there is just that _33:18_: applied to _33:21_: the one most V ^2 to the power minus 1/2. _33:25_: I think it's raising it. _33:29_: One, one, one 1 -, -, 1/2 * V ^2 _33:34_: question, _33:36_: four term and that just never matter because it would just be _33:39_: such a small number. Yes, for vehicle small. For very small, _33:42_: yeah. So so this is is the sort of thing that you you will see _33:45_: again and again in in _33:49_: in this sort of context and and a bit of bits of physics. And _33:52_: when something is small, _33:54_: you're also often interested in the leading, so-called the _33:56_: leading order behaviour. _33:58_: So what? What is the? The the the behaviour of something when _34:03_: the _34:04_: the the the next leading order terms had you know could be _34:08_: ignored because they're small. So yeah, so this the the _34:11_: expression for the Taylor series is _34:15_: Yeah, _34:17_: it's valid for v ^2 less than one. _34:20_: Thanks _34:23_: and and the thing that that that's telling us is is fairly _34:27_: obviously tells us that that that gamma goes to one when v = _34:31_: 0. But also is telling us that it goes to one quite quickly as _34:35_: V goes to zero it goes to one that the the deviations from _34:39_: from one goes as v ^2. So when V gets small, gamma gets very _34:43_: close to 1. _34:46_: Yeah. Thank you for _34:48_: anything else. _34:49_: OK. _35:04_: When we were talking about. _35:15_: Driving the the the full velocity, you may remember I did _35:19_: it by just differentiating the displacement 4 vector delta R _35:24_: term by term, Dr nought by D Tau DR1 by tour Dr 2 by 2 and so on. _35:30_: Does that work? Ohh And the momentum of of particle was just _35:33_: the mass times that that full velocity. _35:36_: Does that work for photons? _35:39_: That particular plan doesn't work for photons because if you _35:43_: remember, photons always move to be right and that means that the _35:47_: displacement _35:49_: they will, which is is that they'll move as much through _35:53_: time. The the the the the time component of the displacement of _35:57_: a photon is always going to be the same as the spatial _36:01_: component of the photon. In other words, which means that _36:05_: the times training component squared minus the spatial _36:09_: component squared will be 0 or. Photons always move along a _36:13_: light like or null _36:15_: well _36:17_: paths, _36:19_: so the full velocity. So by that definition the full velocity of _36:24_: our _36:25_: of of a photon is always null. _36:29_: So if we were to naively talk about the momentum of a photon _36:34_: in the same way as here, we discovered that photo 0 momentum _36:39_: because it's mass times a null vector. So we don't do that. _36:44_: So what that is saying is that the the prescription we had for _36:48_: deriving a physically meaningful quantity didn't work in the in _36:51_: the extreme case where the the, the the the the the vector in _36:54_: question was null. _36:57_: Uh, what? We can instead do _36:59_: what was so that for a photon. _37:04_: I'm right P gamma. As for that that is always null. _37:09_: But if we look back at. _37:12_: This expression Here _37:15_: we see that for a massive object we simply need to have the _37:20_: energy and the momentum _37:23_: equal to yeah equal to each other. _37:27_: So for a Masters particle _37:29_: eastward _37:31_: will be equal to P ^2 _37:39_: as this article. And what that means in practise is something _37:43_: like the form. The formatum of our photon will be _37:49_: something like HF. _37:51_: Where I have you remembered a little bit quantum connect _37:56_: perhaps to be told that the former meant that the energy of _37:60_: a photon is Planck's constant frequency _38:04_: that to the. If we suppose that is the energy of this of a _38:08_: photon with this momentum, then we know that the _38:12_: we will only X axis. Then the X component must be the same value _38:16_: in order that the _38:19_: in a product of this _38:21_: 4 vector _38:22_: with itself end up being dull. _38:25_: So we can by this by this means talk sensibly about the four _38:30_: momentum of _38:33_: a massless particle moving through a light. _38:37_: Uh, _38:38_: in the way that the that we can't, we we couldn't with a _38:42_: prescription before. _38:47_: There's not a lot I could read here, but that that is I think. _38:52_: Thank you. _38:53_: We are making good progress here. That's that's _38:58_: before moving to the next section. Other things wrapping _39:01_: up things to say about before _39:06_: OK, _39:10_: So what what what other things we have does that _39:13_: Ohh yes so so so that should have a. _39:19_: Have I have I just? If that's wrong, it's wrong for a very _39:23_: long time. _39:26_: Yeah, those those those shoots, there should be V's in there _39:31_: gamma gamma V1M1V1. I think I've written the one is not one to _39:35_: write the second time whatever. So that's that's incorrect and _39:39_: as as expression seven point _39:41_: 277.2 B. _39:50_: OK, _39:53_: now let's look at the _39:57_: simplest situation where 2 particles. _40:01_: Collide and form _40:03_: a single outbound particle, perhaps that. Perhaps the _40:07_: Collider that and it bounced off into the beyonder. Perhaps it _40:11_: stayed at rest or something. _40:13_: Two particles arrive and collide _40:17_: into one particle _40:20_: and as before P1 plus P2 will be equal to P3. Just form into _40:23_: conservation, but form into conservation rather 3 minute _40:26_: conservation. _40:29_: OK, _40:32_: so let us step through this. _40:47_: In each of those cases, _40:49_: right, Pi will be _40:53_: Gamma I MI one _40:57_: VI where Gamma I is just shorthand for Gamma _41:04_: of the eye. So that's for I123. So for the three different _41:08_: particles, _41:10_: OK, _41:16_: so let's go through the numbers here _41:20_: and _41:22_: and what we're gonna do is go to _41:25_: assume that these particles are travelling only in the along the _41:30_: X axis, so that the the incoming _41:34_: in this direction One Direction and the resulting particle will _41:37_: end up also moving all in the X direction. So I'm going to miss _41:40_: out why and Z, just to make things easier to write. _41:48_: So let's suppose that M1 _41:52_: and M2 _41:53_: are 8 units of mass. I'm not gonna, I'm just not gonna worry _41:57_: about what the what the unit of the mass are. They're not very _42:01_: big. OK for for particles and the first one, _42:04_: you're travelling at speed _42:07_: 15 seventeens. _42:10_: Will this be late? _42:12_: And the second one, _42:16_: it's stationary. _42:17_: OK, so this is a particle particle sitting there. Particle _42:20_: one comes in at 5017, super light hits it, they join _42:23_: together and we're interested in what the speed of the particle _42:27_: is going along the X axis afterwards. _42:30_: OK, you've got the picture. _42:32_: I trust _42:39_: that means _42:41_: that we can start to look at the _42:48_: and _42:50_: let's see _42:59_: if I ask what is the? _43:03_: Zeros component of part of of the of the energy of particle _43:08_: one. That's going to be gamma 1 _43:13_: M1. _43:15_: If V1 _43:16_: is equal to _43:18_: 15 or 17, then gamma, 1 _43:22_: gamma or V1 will be 17 / 8, _43:30_: one of the Pythagorean triples. So why did I pick that odd _43:33_: fraction? Because 8 ^2 + 15 ^2 is 17 ^2 1 of the Pythagorean _43:37_: triples, right? Nice and easy to do. _43:40_: That means that P that's used component of particle one is 17 _43:46_: eighteen 17817 / 8 times the mass of particle one which is _43:54_: the _43:55_: X component of particle particle 1 momentum is gamma 1M1V1 _44:03_: which is that times 15 seventeens _44:09_: 2/1 _44:11_: and we can therefore build up the table that's in _44:18_: in the news. I'm not going to write it out. _44:21_: So there I I worked out what _44:23_: the serious component of Article _44:29_: momentum for vector was. _44:32_: The one component X component, _44:35_: the velocity, the _44:37_: gamma factor, and _44:40_: notice that the _44:42_: mass, the length squared of that four vector is 17 ^2 - 15 ^2 _44:50_: with the youth component minus the spatial component which is 8 _44:54_: ^2. The square root of that is _44:56_: through. The mass of this particle is still 8, as you _44:60_: would expect, _45:02_: which is the same thing. Same calculation I'm going through. _45:05_: Go through the steps _45:06_: Tim Cook calculation. For the particle that's at rest _45:10_: then it's _45:13_: V is 0 so it's gamma is 1 _45:17_: so equals gamma M _45:21_: for the for particle 2 is m = H * 1 + 8. _45:27_: The the velocity of that particle is 0 so the the spatial _45:30_: component of the former momentum is 0. So the four mentum T and _45:35_: the components are 8 zero plus 20. Gamma is 1 and H ^2 - 0 is 8 _45:39_: ^2. _45:40_: Whether that is 8, so again the particles you know the mass is _45:45_: correct. _45:48_: What about the particle particle 3? We can know what code what _45:51_: the energy momentum of the particle _45:55_: are. Going particle is just by conserving momentum. _45:60_: So here the 0 component of particle 3 is 17 + 8. It's just _46:05_: the sum of the _46:08_: of 0 components of the incoming particles. 70 + 8 is 25 _46:14_: 15 plus 08/15 _46:17_: so the outgoing particle has four momentum 2515 _46:24_: and then _46:26_: getting a no to get the these from that. _46:37_: We know that _46:41_: looking at this expression here that VI is equal to P. _46:47_: This is the _46:51_: get me through this front. Yeah, the one component _46:54_: divided by zero component, _46:56_: but for each particle. _46:59_: So the velocity of this particle is 15 / 25 _47:04_: 3/5 which gives us the gamma factor. _47:08_: And to find out the length of this four of this 4 vector, _47:13_: we have 25 ^2 -, 15 ^2 which is 20 ^2 _47:21_: and we were filled out the rest of the table. _47:26_: No, _47:27_: there are a couple of points to make here in the I'll make these _47:30_: points again at the beginning of our next lecture, _47:32_: just because quite important. _47:37_: So the the two incoming particles _47:40_: have this have have _47:43_: former mentor _47:44_: which have the same length, _47:47_: so the the same _47:49_: moment energy momentum _47:52_: in both cases. _47:53_: Now that may seem surprising _47:56_: since one of them is moving in, other one isn't. _47:59_: But if but since the length of the _48:03_: and you mentioned Victor _48:05_: is Freeman variant _48:07_: issue with the the Case No matter what frames we pick, _48:11_: so it has to be the same in all frames. So in the frame with the _48:15_: particle of all particle one, it's particle 2 that's moving. _48:20_: So in in three of particle one it's particle 2 that is moving _48:23_: fast. _48:23_: So therefore it should not be surprising _48:26_: that in both cases the argumentum of the two particles _48:29_: is the same. Of two incoming particles is the same even _48:33_: though in the frame in our in our lab frame only one of them _48:36_: is moving. _48:38_: So the energy momentum is that the total length instrumental _48:42_: vector squared length is the mass of the particle. _48:47_: It doesn't reflect how fast the particles moving the direction _48:51_: of the four momentum vector _48:54_: reflect how fast the particle moving in your frame, but that's _48:57_: because the components are framed dependent. _48:60_: You change change frames, you have different components. _49:06_: The mass _49:08_: the length of this of the outgoing particle is not the sum _49:11_: of the ingoing particles _49:14_: is not 16. _49:20_: It's the because it's not because because mass or the _49:23_: length of the of the four vector is not a thing that's that's _49:27_: conserved in collisions. The length of the four vector, the _49:31_: the total amount of energy momentum of the particle has is _49:34_: framed independent, _49:36_: meaning the same in all frames, but it's not conserved, meaning _49:40_: it's it's the same through a collision. _49:43_: Whereas the components of the forum mentum are conserved, _49:48_: meaning they stay the same through a collision, but the _49:51_: note frame independent, there are different different frames. _49:54_: OK, so 20 is not eighteen 8 + 8. _49:60_: Ohh yeah, and that's a good point. So M ^2, _50:03_: the length of the four vector, is frame invariant but not _50:06_: conserved. _50:08_: The individual points are conserved but not being negative