14. Introduction to the Four-Vector
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Fundamentals of Physics I14 / 24
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BehälterbauMessungSpantSynthesizerTagFACTS-AnlageClosed Loop IdentificationSchiffsklassifikationBikristallNetztransformatorUhrRegelstabTrenntechnikFuß <Maßeinheit>ErdefunkstelleIPAD <Kernspektroskopie>KlangeffektSatz <Drucktechnik>BauxitbergbauKette <Zugmittel>Kompendium <Photographie>RaumfahrtzentrumAbstandsmessungKraft-Wärme-KopplungStandardzelleVideotechnikAbsoluter NullpunktPassfederHobelPatrone <Munition>PersonenzuglokomotiveRootsgebläseSichtverbindungWalken <Textilveredelung>DigitalelektronikKopfstützePanzerHimmelskörperTonbandgerätTreibriemenBegrenzerschaltungFahrgeschwindigkeitVorlesung/Konferenz
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TrenntechnikElektronisches BauelementErsatzteilNegativ <Photographie>SchubvektorsteuerungAnalogsignalTransversalwelleSpantGroßkampfschiffRotationszustandRootsgebläseRegelstabNanotechnologieVideotechnikBiegenNetztransformatorZylinderkopfWarmumformenFuß <Maßeinheit>GewichtsstückKlangeffektBetazerfallTiefdruckgebietWelle <Maschinenbau>FahrradständerZugangsnetzIntervallNachtÜbertragungsverhaltenMinuteBasis <Elektrotechnik>PolierenRaumfahrtzentrumKompendium <Photographie>Anzeige <Technik>MonatWeltraumMark <Maßeinheit>NeutronenaktivierungWeißQuanteninterferometerFeldeffekttransistorMessungPatrone <Munition>ProfilwalzenLithium-Ionen-AkkumulatorBarkStardust <Flugzeug>Vorlesung/Konferenz
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GasturbineGroßkampfschiffMultiplizitätDruckkraftSchubvektorsteuerungTeilchenNanotechnologieGleichstromUhrRootsgebläseFACTS-AnlageElektronisches BauelementNewtonsche FlüssigkeitKopfstützeReziprokes GitterMechanikerinMasse <Physik>BegrenzerschaltungFahrgeschwindigkeitBettBeschleunigungStücklisteBergmannIntervallSchalttransistorPhototechnikLeistungssteuerungQuanteninterferometerMaßstab <Messtechnik>Akustische OberflächenwelleMonatKraft-Wärme-KopplungMechanikSamtBildqualitätKlangeffektKartonFeldeffekttransistorHobelPassfederTagMessschieberKalenderjahrVorlesung/Konferenz
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FahrgeschwindigkeitDigitales FernsehenLeistenPick-Up <Kraftfahrzeug>DruckkraftTeilchenBiegenMasse <Physik>Elektronisches BauelementSchubvektorsteuerungNetztransformatorTagGasturbineSpantLichtgeschwindigkeitErsatzteilMechanikSerientor-Sampling-LeitungChirpAvro ArrowProzessleittechnikUhrFACTS-AnlageAnalogsignalSensorEisenbahnbetriebWasserdampfKlangeffektBeschleunigungHerbstBegrenzerschaltungDrehmasseDrechselnLichtKopfstützeVorlesung/Konferenz
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WasserstoffatomTeilchenMasse <Physik>Patrone <Munition>KernreaktionClosed Loop IdentificationKopfstützeGebläseErsatzteilIsostatisches HeißpressenBauxitbergbauErdefunkstelleImpulsübertragungRestkernFahrgeschwindigkeitSatzspiegelZylinderblockFernordnungMinuteIrrlichtMechanikBiegenUnwuchtReaktionsprinzipLeistenFeldeffekttransistorSpaltproduktNeutronQuanteninterferometerRaketeKlangeffektLeistungssteuerungPendelbahnErdölgewinnungBombeWasserdampfWasserkraftwerkRollsteigSteckverbinderWarmumformenEisProfilwalzenZylinderkopfSchaft <Werkzeug>Spiel <Technik>Uran-238Kraft-Wärme-KopplungGasturbineSchmelzsicherungVorlesung/Konferenz
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Computeranimation
Transkript: Englisch(automatisch erzeugt)
00:01
So, let me remind you that everything I did so far in class came from analyzing the Lorentz transformations. And you guys should be really on top of those two marvelous equations, because all the stuff we are doing is a consequence of that. I won't go over how we
00:21
derived them because I've done it more than once, but I remind you that if you've got an event that occurs at xt for one person, and to a person moving to the right at velocity u, the same event will have coordinates x prime is x minus ut over the square root,
00:46
and t prime is t minus ux over c squared divided by the square root. This is it. This is the key. From this, by taking differences of two events, we can get similar equations for
01:02
coordinate differences. In other words, if two events are separated in space by Δx for one person and Δx prime for another person, and likewise in time, then you get similar formulas for differences. So, differences are related the
01:23
same way that coordinates themselves are. I will write it anyway because I will use it sometimes one way and sometimes the other way. Even this one you can think of as a formula for a difference, except one of the coordinates is the origin.
01:40
This is in general, if you've got two events, not necessarily at the origin, then the spatial separation and time separation are connected in this fashion. So, I put this to work. I got a lot of consequences from that. You remember that? For example, I said take a clock that you
02:01
are carrying with you. Or let's take a clock that I am carrying with me and let's see how it looks to you. Let's say it goes tick and it goes tick one more time. So, for me, the time difference between the two ticks will be, let's say, one second.
02:21
It's a one-second clock. The space difference is zero because the clock is not going anywhere for me. So, for you, Δt prime is one second divided by the square root, which is more than one second. So, you will say, hey, in the time your clock ticked one second, I really claim two seconds have passed. Therefore, you will say my clocks are running slow.
02:41
And it's easy to see from the same equations that if you replace the primed and unprimed ones, simply change u to minus u, nothing changes. The clock that you think is at rest with respect to you will look slow to me. Then, I showed you how when you want to measure the length of a stick which is
03:01
traveling past you, you get your assistants who are all lined up on the x-axis to measure the two ends of the rod, or any particular time they like, and take the spatial difference. That's the meaning of the length of a moving rod. You've got to measure both ends at the same time. So, you make sure the two measurements are done at the
03:22
same time, so Δt is zero. The distance between them is what I call the length of the rod. You are moving with the rod, and as long as you're moving with the rod, the rod has got two points which are separated by the length the rod, and one thing occurs at one end, and the other thing occurs at the other end, the spatial difference between
03:41
them will be simply the length of the rod in your frame of reference. So, L will be L times, if you bring the square root to the other side, L will be L times the square root, whereas Δt prime would be some Δt in the rest frame, divided by the square root. Okay, another important result I
04:09
wanted to show you is that it's not absolute. So, I started with an unfortunate example of twins born in Los Angeles and New York, and you guys were pointing out that no normal woman could do that,
04:25
except the one from your mama jokes. Your mama jokes, your mama's so big she can have a kid in New York. So, I don't want to deal with such cosmological objects right now. So, I'll say, pick a better example,
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two things happen, something here, something there, separated by distance. People cannot agree that they were simultaneous. Now, you've got to say, it's very surprising, people cannot agree on length difference between two events. For example, New York and Los Angeles are
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about 3,000 miles apart. We think you can move in a train, you can move in a plane, you all have to agree on the separation, but you don't. So, you've got to be careful. That's just a consequence of now the new relativity. In the old days, before I did relativity, I was taking just the good old x and y coordinates, and I said you can take a point, you can assign to it some coordinates x and y,
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then somebody comes along with a rotated axis, and no one seems to be troubled by the fact that the same point is now given a new set of coordinates by this new person. Or, take a pair of points on
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this line. To that person in the rotated system, these events have the same y prime coordinate, because you see this is the y prime equal to zero axis. They occur at the same y prime, but for me, they're not the same y coordinate, because that has that y coordinate, and that's a different y coordinate. So, we are fully used to the
06:01
fact that having the same x coordinate for two events is not an absolute statement. It depends on the frame of reference. Having the same y coordinate is also not absolute. But somehow, in the case of time, we used to think that time differences are absolute and spatial differences are absolute, and what we're learning is, no, they are just like the x
06:22
and the y. Okay, so I'm going to do one more thing with the Lorentz transformation, which is pretty interesting. Let's take this equation for time difference. Δt prime is Δt minus uΔx over c squared,
06:42
divided by this. So, let's take two events. First something happens, then something else happens. And Δt is the separation in time between them, t2 minus t1. And let's say t2 minus t1 is positive, so that t2,
07:00
the second event, occurs after the first event, according to me. How about according to you? Well, Δt prime doesn't have to have the same sign as Δt, because you can subtract this number, uΔx over c squared, and Δx can be whatever you like. Therefore, you can find that
07:22
Δt prime could be negative for you. Now, you've got to understand that's a big deal. I say this happened first and that happened later, and you say, no, it happened the opposite way. Now, this can lead to serious logical contradictions, especially if event one is the
07:45
cause of event two. This is a standard example both in special and general relativity that people talk about. So, event one is somebody's grandmother is born, and event two,
08:01
that person is born. We say the birth of the kid takes place long after the birth of grandmother, according to me. What if in some other point of view, the kid is born but the grandmother is not yet born, and somebody goes and assassinates the grandmother? In fact, this is called killing the grandmother.
08:22
These words are taken right out of textbooks. Why this kind of violence is inflicted on grandmothers? I don't know, but it's a standard example. All logical contradictions involving time travel have to do with coming back and doing something. So, here's an example. So, a grandchild is born, grandmother is not yet born, and something is done to
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prevent her from being born, or two seconds after she's born, she's hit by the mafia. How do you explain the grandchild? Where did the grandchild come from? Because the cause has been eliminated. The cause has been eliminated. Or, event one I fire a bullet, event two somebody is hit, and you go to a frame of
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reference in which that person has been hit and I haven't fired the bullet. So, you come and you finish me off. So, now we've got this person dying with no apparent reason, because the cause has been eliminated. That simply cannot happen. Einstein recognized that there is a limit to how far he can push this community, and he conceded that if A can
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be the cause of B, you better not find an observer for whom they occur in reverse order, because if the cause occurs after the effect, then there is some time left for somebody to prevent the cause itself from happening, and we have an effect with no perceivable cause.
09:42
So, we want to make sure that Δt' cannot be reversed whenever the first event could be the cause of the second event. So, we go and ask this equation. If Δt is positive, when will Δt' be negative? Well, it's very simple algebra. You want this term to be that term.
10:02
So, that'll happen if uΔx, let me put a c here, another c here, is greater than Δt. If that happens, things are going to happen backwards, right? So, let's modify that a little bit by putting this c
10:22
over here, and rewrite it finally as u over c is bigger than cΔt over Δx. Let's understand what this equation means. This equation means is that if
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there were two events separated in time by Δt and in space by Δx, I can find an observer at a certain speed u, so that u divided by c, if it exceeds this number, for that observer, the time order of the events will be reversed.
11:02
Yep, it's not yet clear velocity is greater than c. It depends on what's happening in the top and bottom. So, we have to ask, what is cΔt? Remember, Δt is the time between two events.
11:22
cΔt is what? It's the distance a light pulse can travel in that time. The denominator is a spatial separation between those two events. So, event 1 is here and event 2 is there. cΔt is how far a light signal can travel in the time separating these two events. If cΔt looks like that,
11:44
then there's enough time for a light signal to go from here to here. In that case, cΔt will be bigger than Δx, but then the velocity that you want of your frame will be bigger than 1, and that's not possible in
12:00
units of speed of light. The only time you will get a sensible value for the second frame, namely with u over c less than 1, would be if cΔt is less than Δx. That means here's the first event and here's the second event, a light signal could have only gone from there to there in the time between the
12:21
events. So, it is saying that if two events are such that there was not enough time for a light signal to leave the first event and arrive in time for the second event, then we can actually find an event here with a sensible velocity for whom the order of events is reversed. Therefore, what did we learn
12:45
from this? We learned from this that it should not be, if there is enough time for a light signal to go from one event to the other event, then we will not play around with the order of events. But if there is no time even for a light signal to go,
13:01
the system in fact allows you to see them in reverse order. But we have seen that events which are causally connected, their order cannot be reversed or should not be reversed. So, what this is saying is it should be impossible for an event here to affect a second event by using a signal faster
13:22
than light. In other words, we are going to say that if there wasn't enough time for a light signal to go from this event to that event, then this event could not have been the cause of that event. We will demand that. Therefore, what the theory for it to make logical sense is,
13:41
no signal should travel faster than the speed of light. Because if you had a signal that can travel faster than the speed of light, then the first event could have been the cause of the second event, because you send the signal and the signal may blow something up. But there is not enough time for light to get there in the same time. But if you allowed such things, then you will find a frame of reference with a velocity perfectly sensible in
14:02
which they occur backwards and cause and effect would have been reversed. So, the answer to the whole thing, in case you're still struggling with this issue, is the following. What the theory of relativity demands is that it should be impossible for events to influence other events with signals traveling faster than
14:22
light. Once you accept that, there is no logical contradiction in the theory, because the only time events are reversed is when there was no time for a light signal to go from here to there. So, that means there was no way this could have been the cause of that. For example, when I fire a bullet, the cause is my firing of the bullet, the effect is whatever happens to the
14:42
recipient, and the whole thing takes place at the speed equal to the speed of the bullet. In that case, cΔt is definitely bigger than Δx, because in the time it took the bullet to go from here to there, the light pulse could have also gone there and beyond. In that case, you'll never find an observer at a speed less than that of
15:01
the light for whom the events will appear backwards. So, the special theory of relativity is another way to ensure that everything is consistent, is to demand that no signal travels faster than the speed of light. No energy, no signal can travel that fast. But this leads to something
15:24
very interesting. Our view of space-time is now modified, you see. In the old days, I drew a dot. Here is space and here is time, and let's say 0,0 is where I am right now. I am at the origin, my clock says 0,
15:41
I'm here right now. Any dot I pick here is in my future, because t is positive for the events. They have not happened yet. Any dot I draw here is an event in my past because it occurred at an earlier time. So, in the Newtonian view of the world, there is something
16:02
called right now, this horizontal line, and everything above it is called later, or future, everything below it is called past. But in special relativity, after you take into account the relativistic thing, let's draw here an axis c times t. So, whenever you have
16:23
another event so that ct is bigger than x, there is enough time for a light signal to go from here to here. That's the meaning of saying ct is bigger than x. This is the line ct equals x. This is the line ct bigger than x. This event is in the future
16:43
of this event according to me, and also according to anybody else. In other words, Δt is positive for me, but if you go back to those equations, since cΔt is bigger than x, you will never find anybody who said this event occurred earlier than this event.
17:02
Another way to understand it is if I want to cause that event, suppose there is some explosion going off there, I send a signal that goes and makes that happen. That signal has to travel slower than light to get there. Therefore, it could have caused the event, and therefore there's no screwing around with the order of that event. That is in my future,
17:21
according to me and according to all observers. If you take an event here, so this is called, this whole event north of this 45 degree line is called the absolute future. By absolute, I mean future of me not only according to me,
17:43
but according to all observers. Every observer, this will occur later than this. It won't be later by the same number of seconds, but it will be later. So the Δt between this event and this event is positive for me. It will be positive for odd people. They will all agree this occurred after this, therefore the effect appeared
18:02
before the cause. I'm sorry, the cause appeared before the effect. It is not really necessary that this event be caused by this event. It's only necessary that it could have been caused by this event, because not every event in the universe is caused by some other event. It can just simply happen.
18:23
We have heard that. Stuff happens. So, stuff happens here with no obvious cause. As long as this is in the future, according to me, by this definition, it's in the future according to all people. So, this is called the future because you can affect it. For example,
18:40
if I heard something terrible is going to happen at this point, I can send my guys to go over there and do something. They have to travel at a speed that is less than the speed of light, so they can affect it. These events are called the absolute past. What that means is any event here could have been the cause of what's happening to me right now, because from that event,
19:02
a signal can be sent to arrive where I am right now at a speed less than the speed of light. This line, which is called the light cone, is a borderline case where you can communicate from here to here using a light signal. So, that's also considered as future. But what about an event
19:21
here? Suppose this distance is 2 seconds times the velocity of light. That has not happened yet. Suppose I go there and open my envelope and it says something terrible is going to happen at
19:43
this point. There is nothing I can do. There's nothing I can do. In the Newtonian days, there's something you can do. Tell someone else to really hurry up and get there and do something. Well, now you cannot do that because for that someone to leave you with the instructions to do something
20:00
about this requires that person to travel faster than the light, and that's not allowed. So, even though things can happen here in the future and you have knowledge of the fact that someone's planning to do something there, you cannot get there. Okay, that's a very important thing, even for people who are
20:23
going to law school. I know all of you guys are not going to physics, but suppose you're going to law school. You've got the DNA defense, right? My client's DNA doesn't match. Here's another defense. If your client was accused of doing something here, and was last seen here, you can argue that my client
20:42
was outside the light cone. That's called outside the light cone defense and it's absolutely watertight, better than anything. Because if that event is outside the light cone of your client, the client cannot be held responsible because your client will have to send a signal faster than light and that law is unbroken. So, what about the status of an event like this?
21:07
The status of this event is, I can actually find other observers moving at speed less than light, for whom this event occurs before this event. So, order of events can be reversed, but it will not lead
21:21
to logical contradictions because we know the two events are not causally connected. So, space and time, which used to be divided into upper half plane and lower half plane, the future and past, and a tiny sliver called present, now divided into three regions. The absolute future that you can affect, the absolute past whose events can affect you.
21:44
And this doesn't have a name. If you want, you can call it the relative future according to you, but I'll find somebody to whom that occurs before this one. Okay, that's also a consequence of the Lorentz transformation. So, you can see that the equations are very deceptively simple.
22:02
In fact, they're a lot easier looking than some equations with angular momentum, right? But the consequences are really stupendous. They all follow from that equation. And of course, if you invent a theory like this, you've got to make sure there are no contradictions, right? Suppose you find a theory, you're very happy with everything, velocity of light is
22:20
coming out right, then somebody points out to you that the order of events can be reversed. Then you've got to agree you'll be in a panic because how can I reverse the order of events? Then the theory is so beautiful internally. It says you can reverse the order of events if they could not have been causally connected, where the causally connected means a signal traveling at a speed less than
22:42
light could not have gone from the first to the second event. Then the theory does not allow them to be, if a signal could have gone, it allows them to be never reversed, and if a signal could not have gone, it does say you can find people for whom events occur in the reverse order. All right, so this is the
23:02
second thing, and it's called the light cone for the following reason. I've only shown you an x-coordinate, but if you have a y-coordinate coming out of the blackboard, the surface will look like a cone. That's why we call it the light cone. So, sitting at a point in four-dimensional space-time, there's a cone, and all points in that forward light cone are events you can affect.
23:21
All points in the backward light cone are events that can come and get you right now. And the rest of the thing, outside the two cones, you cannot do anything about it, even though you know something's going to happen. And they cannot do anything to you. You open an envelope and it says, somebody here is planning to do something to you. You don't have to worry, because that person cannot get
23:43
to you in the time available, because the fastest signal is the light signal and that won't get there, and neither will anything else. All right, now we're going to go to something which is theoretically or mathematically very pretty. So, for those of you who like
24:03
mathematical elegance, this is certainly the best example I can offer you, because it is simple and also rather profound. But it's completely by analogy. The analogy is the following. We have seen that when we rotate our axes,
24:20
as I've shown there, that the x prime is not the same as x, but related by this formula. And y prime is also not the same as y, but related by this formula. Nothing is sacred. Coordinates of a point are not sacred.
24:42
They are just dependent on who is looking at them from what orientation. But even in this world, we noticed that x prime squared plus y prime squared is the same as x squared plus y squared. Namely, the length of the point connecting the origin to
25:01
you, or the distance from the origin is unaffected by rotations. This is called an invariant. This doesn't depend on who is looking at it. Everyone will agree on this, but they won't agree on x, and they won't agree on y, but they will agree on this. So, it's reasonable to ask, in the relativistic case,
25:22
where people cannot agree on time coordinate or space coordinate, maybe the square of the time and the square of the space will be the same for two people. One can ask that question. So, we'll look at that. We'll find out that that's not the case. T squared plus x squared is not invariant.
25:42
But even before you do that, you should shudder at the prospect of writing something like this. Right? What's wrong with this? Good. So, everyone's on top of the unit's game. So, you cannot add t squared x squared. There's no way this can be anything. So, we know how we have to fix
26:00
that. We've got to have either both coordinates in space-time measured with lengths or with time. The standard trick is to do the following. Let's introduce an object with two components. I'm going to call it capital X. The first component of that is going to be called x one. It's going to be called x
26:20
zero. The second is going to be called x one. x zero, I mean, x one is just our familiar x. x zero is going to be essentially time, but multiplied by c. Why do I switch from t and x to zero and one? It is just that if you want to go back to more coordinates,
26:43
if you want to bring back y and z, then I'll just mention for future use that in four dimensions, you really have x zero, x one, x two and x three. That is a shorthand for x zero and what you and I used to call the position r of the particle.
27:05
It's just that in one dimension, the vector r becomes the number x, and I want to call the number as x one because it's the first of the three coordinates. And if you're doing superstrings, you're going to have ten coordinates. One will be x naught and the other nine will be spatial coordinates.
27:20
So, we like the numerical index rather than letters in the alphabet because you run out of letters, but you don't run out of these subscripts. So, I'm writing it purposely in many ways because if you ever go read something or you're involved in a lab project or the group is studying something, you want to read the paper, people will refer to coordinates in space-time in many ways.
27:41
Some will call it, some people like to write the x the following way, x one, x two, x three, x four, where x four is ct. That's why you call that the fourth dimension. Well, it's still the fourth dimension, but you can either put it at the end of the first familiar three or at the beginning. So, it's very common for
28:01
people to use either notation. I'm just going to call it x zero. So, x zero is just time, okay? Multiply it by c so that it has units of length, and multiplying by c doesn't do anything. Everybody agrees on what c is. So, we will all multiply our time coordinates by c. All right, so let's ask the following question.
28:21
What does the Lorentz transformation look like when I write it in terms of these numbers? So, first take x prime is x minus ut divided by the famous square root, but t is not what I want. ct is what I want.
28:42
So, I fudge it as follows. I write it as x minus u over c times ct divided by 1 minus u squared over c squared. This I'm going to write as x is now called x one, and I'm going to introduce a
29:01
new symbol, β here, and this guy is called x zero divided by 1 minus β squared, but β is universal convention for u over c. So, if all velocities are measured as a fraction of the velocity of light, then β is a number between zero and one. So, the final way I want to
29:23
write the Lorentz transformation for a coordinate is that x one prime is x one minus β times x zero divided by square root. So, you have to get used to this β. I mean, you're following nicely what I was saying in terms of velocity u,
29:42
but you have to get used to this β. Now, we won't use it a whole lot, but I put it here so you can recognize it when people refer to it in the literature, if you go read something. What's the transformation law for time? Remember, T prime was T minus ux over c squared divided by root 1 minus
30:07
β squared. β squared is clearly what's sitting downstairs. u over c is β, so what do you think we should do? We should multiply both sides by c, because if you multiply
30:20
both sides by c, put a c there, put a c there, and get rid of this one, then you find x naught prime is x naught minus β times x one over square root of one minus β squared. So, let me write that here. Now, you see the relationship between the coordinates is nice and symmetric. If you write in terms of
30:50
x and t, the coordinates' transformation law is not quite symmetric, and that's because one has units of length and one has units of time. But if you rescale your time to turn it into a length, then the transformation laws
31:02
are very simple and symmetric between the two coordinates in space-time. There are also the other coordinates, the y direction, x two prime is x two, and in the z direction, x three prime is x three. In other words, the two coordinates' lengths perpendicular to the motion are
31:20
something you can always agree on. One can talk and talk about why that is true. Basically, what we're saying is, if you're zooming along and you drew a line that says y equal to one, and originally we agreed on it, we'll have to agree on it even when I start moving relative to you. Because you've got your own line, and there's no reason your
31:40
line should be above my line, should be above your line or below your line. If the line's originally symmetric, then we agreed. There's no reason why my line bends above yours or your line's below mine, because by symmetry, there's no reason why one should be higher than the other. And the reason this is required is that the two lines which last forever can be compared anywhere you like,
32:03
unlike rods which are running away from each other and cannot be compared. The line here, which says y equal to three, has got to be y equal to three for both of us. So, the transverse coordinates are not modified. So, that's why I don't talk
32:20
about them too much. All the interesting action is between any one coordinate along which motion is taking place and time. So now, let's ask ourselves, maybe x' prime squared minus plus x' prime squared is the same as this squared plus that squared. Well, let me just tell you that I'm not going to try
32:41
something that I know will not work. In other words, you might think x' prime squared plus x' prime squared is equal to x' squared plus x' squared. Well, that just doesn't work. That doesn't work because one over the square root is not a cos θ and β over the square root is not a
33:01
sine θ. You cannot make them cos and sine of something. If you could, that'll work, but it doesn't work. But I'll tell you what does work. Let's take x' prime squared minus x' prime squared, the square of the time coordinate minus the square
33:20
of the space coordinate with a minus. So, let's try to do this in our head by squaring this and subtracting it from the square of this one. Downstairs, I think we all agree, you get 1 minus β squared because the square root is x' squared. How about upstairs?
33:40
First you've got to square this guy. This guy will give you x' squared plus β squared x' squared minus 2β x' x' one. From that, I should subtract the square of x' prime squared. That'll give me minus x' squared minus β squared x' squared plus twice β
34:06
x' x' one. It's just simple algebra. And these cross terms cancel out. And you notice, I got an x' squared times one minus β squared minus x' one squared times one minus
34:23
β squared, right? This algebra I trust you guys can do at home, and the result is this is going to be x' squared minus x' one squared. That's very nice. It says that even though people cannot agree on the time or space coordinate of an
34:42
event, just like saying people cannot agree on what is x and what is y, they can agree on this quadratic function you form out of the coordinates. But it's very different from ordinary rotations where you take the sum of the squares, here you've got to take the difference of the squares, and that's just the way it is.
35:00
Even though time is like another coordinate that mixes with space, it's not quite the same. In other words, if I brought back all the transverse coordinates, then you will find x' prime squared minus x' one prime squared minus x' two prime squared minus x' three prime squared is
35:21
x' squared minus x' one squared minus x' two squared minus x' three squared. This you can do because x' one prime individually is equal to x' one, so you can put them on both sides. So, this is the actual four-dimensional result for people who really want to relieve the x-axis and move freely in
35:40
space, which you're allowed to. This is the invariant. This is the object that everyone agrees on. Allow me once again to drop this. Most of the time, I'm not going to worry about transverse coordinates. So, this is the analog of the length squared of a vector and it's called a space-time interval.
36:01
The space-time interval is not the space interval, it's not the time interval, it's the space-time interval between the origin and the point xt. And this is denoted by the symbol. There's no universal formula, but let's use S squared for
36:21
space-time separation. Notice that even though it's called S squared, S squared need not be always positive definite. You can take two events for which the time coordinate is the same, or the time coordinate is zero, the space coordinate is not zero, then S squared will be negative.
36:42
Whereas the usual Pythagoras length squared, x squared plus y squared is positive definite, always positive. The space-time interval can be positive or negative or zero. It's positive if x naught can beat the sum of the squares of those guys. It's negative if they can beat
37:00
x naught and it's equal to zero if x naught squared equal to that square. So, if you want, if you go back to this diagram I drew here, this is x1, this is x naught equal to ct. These are events for which S
37:24
squared is positive. These are also events for which S squared is positive. These are events for which S squared is negative. These are events in which S squared is zero. So, whenever S squared is positive, we have a name for that.
37:42
It's called a time-like separation. Well, it's called time-like separation because the time part of the separation squared is able to beat the space part of the separation. It's got more time component than space component. Now, you can also do the
38:01
following. Let me introduce some rotation now. Let us agree that we will use capital X for a space-time, which is what's called a four-vector. A four-vector has got four components, and the components are x naught, and since I'm tired of writing x1, x2, x3,
38:22
I will call it r. So, in the special theory of relativity, what you do is you take the three components of space and add one more of time and form a vector with four components. And we will agree that the node symbol for a space-time vector is a capital X.
38:43
Now, I've been careless in my writing. Sometimes I use small x and big X, but you guys should be careful. If you're a little more careful, big X like this with no subscript will stand for this four numbers. It is a four-vector. It's a four-vector. It's a vector living in space-time. And when you go to another frame of reference, the components of the
39:01
four-vector will mix with each other. Then, we are going to define a dot product. It's going to be a funny dot product. It's going to be called x dot x. The usual dot product was some of the squares of the vector with four components. But this funny dot product will be equal to xℏ squared minus the length of the
39:25
spatial part squared. Pardon me? This is a big X. These are all small. Big X is the name for a vector with four components. It's a position vector in space-time.
39:42
This is the same as writing xℏ squared minus xℏ squared minus xℏ squared minus xℏ squared. Suppose there are two events. One occurs at the space-time coordinate x. The second event occurs at
40:01
the new point, let me call it x-bar, whose time coordinate xℏ, whose position coordinate, this is unfortunate, a vector r with a bar on it. These are like two vectors in the x-y plane. You and I, if you're going a little bit with each other, will not agree on the components of this or the components of that.
40:21
But we will agree on xℏx'. In other words, xℏx will be the same as xℏx-bar-prime. What I'm saying is if you go to a moving frame of reference, the components of the vector x goes into x-prime, whose components are
40:45
xℏ-prime and some r-prime. And likewise, the vector x-bar has new components x-bar-prime, which are xℏ-prime and r-bar-prime. Look, the story is the same
41:04
like in two dimensions. Components of vectors are arbitrary, they vary with the frame of reference, but the dot product of vectors is the same, no matter how you rotate your axis. Because the dot product is length of A times length of B times cosine of the angle between them, and none of the three things changes when you rotate your
41:22
axis. Well, this is the analog of the dot product in space-time, because this is the guy that's same for everybody. You can say, why don't you study this combination? It's so nice. It's all pluses. I like that. I don't study that because that combination is not special. If I have one value for
41:42
the combination, you will have a different value for the combination. It has nothing privileged. It's the one with the minus signs in it that has the privileged role of playing the role of the dot product. So, space-time is not Euclidean. Euclidean space is the space in which we live, and distance squared is the sum of the squares of all the coordinates. So, it's a pseudo-Euclidean
42:02
space in which, to find the invariant length, you've got to square some components and subtract from them the square of some other components. Okay, so finally, if you take a difference of
42:20
two events, and you multiply it by the vector, in other words, take the difference of two events, then the spatial coordinate according to me, the difference in space coordinate, namely, c times Δt squared minus Δx squared will be c times
42:45
Δt squared minus Δx prime squared. Or, if you like, x naught squared minus x1 squared will be x naught prime squared minus x1 prime squared.
43:03
I'm using the notations back and forth, so you get used to writing the space-time coordinates in two different ways. So, it not only works for coordinates, but for coordinate differences. You understand? Two events occur, one here now and one there later, let's say. They are separated in time by two meters.
43:21
They're separated in, I'm sorry, in space by two meters in time by 11 seconds. I'm sorry, I didn't mean to write this. I meant to write Δx naught squared minus Δx1 squared equal to Δx naught prime squared minus
43:42
Δx1 prime squared. So, do you guys follow the meaning of this statement? Most things are relative, distance between events relative, time interval between events relative. But this combination somehow
44:02
is invariant. Everybody agrees on what it is. Okay. Now, we are going to understand the space-time interval. Δs squared is a name for a small or a space-time
44:21
interval, and the space-time interval formed out of differences. Now, we are going to apply this to the following problem. I'm going to give you a feeling for what space-time interval means when applied to the study of a single particle. So previously, Δx and Δt were separation between two random unrelated events or
44:42
arbitrary events. But now I want you to consider the following event. Here is a particle in space-time and it moves there. This is where it is in the beginning, that's where it is at the end. So now, I want Δx to be distance particle travels,
45:09
and I want Δt to be the time in which it did this.
45:20
So, these are two events in the life of a particle, two events lying on the trajectory of a particle. In the xt plane, you draw a line, it's a trajectory, because at every time there is a vertical motion, it's an x and that describes particle motion. And this is the Δx and this is the Δt. Let's look at the space-time interval between the two
45:41
events. Δs square will be cΔt square minus Δx square. But we're going to rewrite this as follows. I'm going to write it like this, cΔt is equal to cΔt
46:04
square times 1 minus v square over c square, where v is Δx over Δt. Because these are the two events in the life of a particle, the distance over
46:21
time is actually the velocity of the particle. I will always use v for the velocity of a particle that I'm looking at, and u for the velocity difference between my frame of reference and your frame of reference. So, u is always the speed between frames, and v is the speed of some particle I'm looking at. So, let's take the square root
46:42
of both sides and we find Δs equal to cΔt times square root of 1 minus v square over c square. So, the space-time interval between two events in the life of a particle is c times the time difference times this
47:03
factor. Now, this is supposed to be invariant and invariant. By that, I mean no matter who calculates the space-time interval, that person's going to get the same answer. So, let's calculate the space-time interval as computed by the particle itself.
47:23
What does the particle thing happen? Well, the particle says, in other words, you're riding with the particle. So, in your frame of reference, for the particle now, the time difference between the two events, let me call it some
47:42
Δτ. And the space difference, what's the space difference between the two events? Yep? Zero, because as far as the particle is concerned, I'm still here. If I'm moving from my vantage, my x-coordinates,
48:02
wherever I chose to put it in the beginning, and that's where it'll follow. So, the two events, particle sighted here and particle sighted there, have different x-coordinates for a person to whom the particle's moving. But if you're co-moving with the particle, then your x-coordinate as seen by you does not change. Therefore, Δx is zero.
48:21
Therefore, the space-time interval will be simply c times Δτ, where τ is the time measured by a clock going with the particle. So, Δτ is the time in particle's frame. In other words, if the particle had its own
48:43
clock, that's the time it will say has elapsed between the two points in its trajectory. So, it's not so hard to understand. So, I'm the particle, I'm moving, I'm looking at my watch, and I'm saying time will pass for me, right? One second and two seconds, that's the time according to
49:01
me. If you guys see me, you will disagree with me on how far I moved and how much time it took. That's your Δx and your Δt. That's only Δτ. There's no Δx for me. So, the space-time interval, when you describe particle behavior, is essentially the time elapsed according to the particle.
49:23
So, it's not hard to understand why everybody agrees on that. See, you and I don't have to agree on how much time elapsed between when the particle was here and when it was there. But if we ask, how much time elapsed according to the particle, we're all asking the same question. That's why we all get the same
49:40
answer. So, the space-time interval is called the proper time. So, proper time is another name for the time as measured by a clock carried with the particle. So, let's write it this way. The space-time interval is really c times Δτ.
50:01
And Δτ is also an invariant. Namely, everyone agrees on Δτ. And what's the relation between Δt and Δt? Δt is the time according to any old person, and Δτ is the time according to the clock. And you saw that the
50:22
space-time interval, c Δτ, was equal to Δt times square root of 1 minus v squared over c squared. So, we are going to, I'm sorry, the c cancels on both sides,
50:42
and this is the relation. So, I will be using the fact that Δτ over dt is equal to square root of 1 minus v squared over c squared, or dt over dτ is
51:00
1 over square root of 1 minus v squared over c squared. In other words, the time elapsed between two events in the life of a particle, as seen by an observer for whom it has a velocity, compared to as seen by a clock moving with a particle. So, that ratio is given by this.
51:21
As the particle speeds up, let's see if it makes sense. As v approaches c, this number approaches zero. That means you can say it took, the particle has been traveling for 30 hours, but with this number almost vanishing, the particle will say I've been traveling for a very short time. That's the way it is. The particle will always think it took less time to go from here to there,
51:41
compared to any other observer, because the clock runs fastest in its own rest rate. Okay, now this is the new variable I'm going to use to develop the next step. The next step is the following. In Newtonian mechanics, particles had a position x and maybe y and z, but let me just say x. Let me take one more.
52:03
It had x and it had y. And these were varying with time. Then I formed a vector r, which is i times x plus j times y. And the vector is mathematically defined as an
52:24
entity with two components, so that when you rotate the axis, the components go into x prime and y prime, which are related to x and y with the cos θ's and sin θ's. That's defined to be a vector. Now, if I went to you and said, okay, that's one vector, the position vector,
52:42
can you point to me another vector? Anybody know other vectors in Newtonian mechanics? Yes, sir. You don't know any other vectors in the good old mechanics days, and the only vector you've seen is position?
53:01
Very good. Yeah, that's right. So, you're right. You certainly know the answer, so you shouldn't hesitate. So, how do you get to velocity from position? You take the derivative. So, you've got to ask yourself, why does the act of taking derivative of a vector produce another vector? Well, what's the derivative? Well, the derivative is the derivative. You change the guy by some δ and you divide by the
53:23
time. Now, change in the vector is obviously a vector, because difference of two vectors is a vector. Dividing by time is like multiplying by the reciprocal of the time. That's like multiplying by a number, and I've told you multiplying a vector by a number also gives you a vector maybe longer or shorter. Therefore, δR is a vector
53:43
because it's the difference of R later minus R now. Now, dividing by δT is the same as multiplying by 10,000 or 100,000 or a million. It doesn't matter. That's also a vector, and the limit is also a vector. Therefore, when you take a derivative of a vector, you get a vector. And then, once you've got
54:03
this derivative, it becomes addictive. You can take second derivatives, and as you said, you can have acceleration. Then, you can get, you can take the acceleration and multiply a mass by a mass. Now, that is called a scalar. The mass of a particle is a vector, and the mass of a particle is no direction.
54:21
So, the product of a number and a vector is another vector, and that vector, of course, is the force. So, derivatives of vectors and multiples of vectors by scalars, namely things that don't change when you rotate your axis, are ways to generate vectors. So, what I want to do is I want to generate more vectors.
54:42
The only 4-vector I have is the position 4-vector, which is this guy capital X, whose components I write for you again are x naught, which is a code name for ct, and x1, which is a code name for x, and you can put the other components if you like.
55:05
Now, take this x to be the coordinate in space-time of an object that's moving. And I want to take the derivative of that to get myself something I could call the derivative of x, which is a vector in relativity. But the derivative cannot be
55:22
the time derivative. You guys have to understand that if I take these things, of course, in space-time the particle's moving, I can certainly tell you where it is at one time and where it is a little later, and I can take time derivative. But the time derivative of a vector in four dimensions is not a vector because time is
55:44
like any other component now. It's like taking the y derivative of x, that doesn't give you a vector. You've got to take a derivative with respect to somebody that does not transform, that does not change from one observer to the other. So, do you have any idea where I'm going with this?
56:01
Yes? Pardon me? Yes, or you can take derivative with respect to the time as measured by the clock moving with the particle. So, you can take the tau derivative. In other words, let the particle move by some time, and let it move by some amount of delta x,
56:21
namely, Δx0, Δx1, Δx2, etc. That difference will also transform like a vector. We have seen many, many times differences in coordinate transform like a vector. Divided by this guy now, which is the time according to a clock moving with the
56:43
particle. Everybody agrees on that because we're not asking how much time elapsed according to you or according to me. We are going to quarrel about that indefinitely. We are asking how much time passed according to the particle itself. Then, no matter who computes the time, you will get the same answer, and that's what you want to divide it by. So, I'm going to form a new
57:04
quantity called velocity, which is the derivative of this with respect to tau. Then, I'm going to use the derivative of that as dx dt times dt dtau.
57:24
That becomes then 1 over 1 minus v squared over c squared times dx0 dt, dx1 dt, dx2 dt, etc. So, you find the rate of
57:45
change as measured by a clock moving with a particle, and that has the virtue that what you get out of this process will also be a four vector. By that, I mean its four components will transform when you go to a moving frame, just like the four components of x.
58:02
You remember when you took x and y and you rotated the axis, x prime is x cosine theta plus y sine and so on. If you take time derivatives, you'll find the x component of the velocity in the rotated frame will be related to the x and y in the old frame by the same cosines and the same derivative of sine because the act of taking derivatives doesn't change the way the object transforms.
58:22
So, this is my new four vector. In fact, let me put the third guy to dx3 over dt. So, what I did was I took the tau derivative for which we don't have good intuition and wrote it in terms of a t derivative for which we have a
58:41
good intuition, because no matter how much Einstein tells you about space and time, you'll find that the time is different from space. So, we'll think in terms of time derivatives. But to form a four-dimensional vector, it's not enough to do that, is what I'm saying. Every term there is divided by this because that's the way of rewriting d by d tau as d by dt times this factor. So now, I am ready to define
59:05
what I'm going to call momentum in relativity. It's going to be called the four momentum. Everything is the four something, the four vector for the position. This is called the four velocity. Now, I'm going to define something called the four momentum.
59:25
The four momentum is going to be the mass of a particle when it's sitting at rest, multiplied by this velocity. So, what is it going to be? Let me write it out.
59:40
What is dx0 over dt? You guys remember x0 equal to ct. Sorry, x0 was ct and dx0 over dt will be equal to c. You're right, if that's what you were
01:00:00
I agree with you, yes. So, this is a vector, m zero c over 1 minus v square over c square. Then, the other guy, other component, let me write simply as m zero times the familiar velocity times 1 minus v square over c square.
01:00:24
If you wanted to keep the x, y, and z velocities, you can keep them as a vector, or if you just wanted to live in two dimensions, one space and one time, you can drop the arrow. So, we have manufactured now a new beast. It's got four components.
01:00:40
What is it? I modeled it after the old momentum. I took the mass and I multiplied by what is going to pass for velocity in my new world, and I got this creature with four parts. You've got to understand what the four parts mean. It's got a part that looks like an ordinary vector, and it's got a part that has no vectors in it.
01:01:02
Completely in analogy with the fact that x had a part that looked like a vector, namely the three spatial components, and a part that in the old days was called a scalar because it didn't transform. But of course, in the new world, everything gets mixed up. We've got to know who this is, and we have to know who this is. So, I want to show you what they are.
01:01:26
So, here is my new four-vector, m zero c over one minus v square over c square, and m zero v. Let me drop the other components except just in the x direction.
01:01:42
Let me not worry about that now. I wrote that because I want to be able to call it a four-vector. It's crazy to call this a four-vector. I should call it a two-vector. You guys should say, well, you've already seen two vectors. Well, you've seen two vectors in the xy plane. This is a two-vector in space-time. Or it's part of this big four-component object.
01:02:02
It's a four-vector in four-dimensional space-time. So, we have created something, and we are trying to understand what we have created. So, let me look at this guy first. This guy looks like m zero v divided by this factor. I don't know what it stands for, but I say,
01:02:23
well, relativistic physics should reduce to Newtonian physics when I look at slowly moving objects, because we know Newton was perfectly right when he studied slowly moving objects. If I go to a slowly moving object, so v over c becomes negligible, I drop that, and that becomes mv.
01:02:41
I know that's the old momentum. So, this guy is just the old momentum, properly corrected for relativistic theory. So, this quantity here deserves to be called momentum. A little p, you can put an arrow on it if you want to keep the three components, or don't put an arrow.
01:03:04
So that, of that four-vector, the three components are just momentum, but not defined in the old way. It's not m not v. It's m not divided by this. So, the momentum of a particle is very interesting in relativity. Even though nothing can go
01:03:21
faster than light, that doesn't mean the momentum has an upper limit, m zero times c, because the momentum is not mass times velocity, mass divided by this crazy factor. So, as you approach the speed of light, as v approaches c, the denominator is very close to zero, this number can become as big as you like. So, why are people building
01:03:41
bigger and bigger accelerators? If you ask them how fast is your particle moving, for everybody in Fermilab, in CERN, it's all close to the velocity of light. So, the velocity of light is 99.9999, and the other person has a few more nines. Nothing is impressive when you look at velocity, but when you look at the momentum, it makes a big difference how much of the one you have subtracted in the bottom.
01:04:01
If you only got a difference in 19 decimal place, well, you got a huge momentum. So, particles have limited velocity in relativity, but unlimited momentum. It also means to speed up this particle is going to take more and more force as it picks up speed. So, the momentum will change. In other words, you'll be pushing it like crazy.
01:04:20
It won't pick up speed, but its momentum will be going up. It'll pick up speed, but the pickup in speed will be imperceptible, because in the 19th decimal place of v over c, it'll go from 999999 to something near the end. That's the last digit we'll change. But the momentum will change a lot. But I don't want to stop before looking at this guy.
01:04:40
So, we don't know who this is. So, we take this thing. This is the zero component of p, just like I called it x naught in the old days. I'm going to call it p naught. p naught was m zero c divided by 1 minus v square over c square. So, I don't know what to make of this guy either.
01:05:01
If I put v equal to zero, I get m zero c. That looks like the mass of a particle, but I have no idea. Okay, so it's the mass of a particle. See here, when I put v equal to zero in the bottom, because the top had a v, not v over c, something was left over that looked like something familiar,
01:05:20
namely momentum. Here, if I take v equal to zero too quickly, what's left over is nothing reminiscent of anything in Newtonian mechanics. So, you want to go in what's called the next order in v over c. You don't want to totally ignore it. You want to keep the first term. You want to keep the first non-zero term, so we write it as m zero c times 1 minus
01:05:42
v squared over c squared to the power of minus one-half. I've just written it with the thing upstairs for the minus one-half as the exponent. Then, I remind you the good old formula. 1 plus x to the n is 1 plus nx plus dot dot dot, if x is very small. So, if you use that,
01:06:03
you get m zero c times 1 plus v squared over 2c squared plus higher powers of v squared over c squared.
01:06:20
So, what are these terms coming out? Well, there's the first term, for which I have absolutely no intuition. The second term looks like one-half mv squared divided by c. So, for the first time, I see something familiar. I see the good old kinetic energy, but not quite,
01:06:42
because I have this number c here. So, I decided maybe I shouldn't look at p naught, but I should look at c times p naught. Maybe that looks something more familiar. That looks like m naught c squared plus one-half mv squared plus more and more terms, depending on higher and higher powers of v squared over c squared. But now we know who this guy
01:07:03
is. This is what we used to call the kinetic energy of a particle by virtue of its motion. As the particle picks up speed, the kinetic energy itself receives more connections, because the other terms with more and more powers of v over c are not negligible. You've got to put them back in. Basically, you have to compute
01:07:21
this object exactly. For the low velocities, this is the main term. So, this is what led Einstein and mainly him to realize that this quantity is talking about the energy of an object. And this is certainly energy I recognize. I can put it to good use, right?
01:07:41
You run windmills and so on with kinetic energy, or hydroelectric power is from churning kinetic energy of water into work. So, you figure this is also part of the energy. So, the first conclusion of Einstein was even a particle that's not moving seems to have an energy, and that's called the rest energy. If the particle is moving,
01:08:03
then go back to the derivation and you can find that. So, c times p naught is m naught c square over this thing. This is the full expression for the energy of a moving particle, not the approximate one, but if you keep the whole thing, this is what you get. So, this says when a particle moves, it's got an energy that
01:08:22
looks like some velocity dependent mass times c squared. And when the particle is at rest, it has got that energy, and that's the origin of the big formula relating energy and mass. So, it leaves open the possibility that even a particle is at rest. You think when a particle is at rest, you have squeezed everything you can get out of
01:08:41
the particle, right? What more can I do for you? You gave it all it's got and it stopped. Well, according to Einstein, there is a lot of fun left. You can do something more if there's a way to destroy this. So, in his theory, he doesn't tell you how you should do that. But he did point out that if you can get rid of the mass, because it was energy,
01:09:01
some other form of energy must take its place by the law of conservation of energy. So, that's how all the nuclear reactions work. In nuclear reactions, you can take two parts, like two hydrogen atoms, hydrogen nuclei, you can fuse them into helium, and you will find the helium's mass is less than the mass of the two parts. So, there is some energy
01:09:21
missing, some mass missing, that means some energy is missing. That's the energy released in fusion. Or you can take uranium nucleus and give it a tap and break it up into a barium and krypton and some neutrons, and you will find the particles together have a mass less than the parent. And the missing mass times c squared is the energy of the reaction that comes out in the kinetic energy of the
01:09:42
fast-moving particles. So, Einstein is wrongly called the father of the bomb, because this is as far as he went. He didn't specify how to extract energy from mass, but showed very clearly that rest energy, that a particle at rest seems to have a term that should be called energy, because this guy certainly is. Yes?
01:10:08
Student There are more corrections to kinetic energy. In other words, why is this an important concept? If two particles collide,
01:10:22
we believe energy is conserved. So, you want to add the energy before and comparative energy after. If you want them to perfectly match, if you stop here, your bookkeeping will not work, before will not equal after. You've got to keep all this infinite number of terms to get it exactly right. But if you're satisfied to one part in a million or
01:10:41
something, maybe that's as many terms as are needed. And at very low velocities, you don't even need that, because this is never changing in a collision. Every particle has a rest mass. This is what we balanced when we collided. Remember, we collided blocks and we balanced the kinetic energy? Actually, that energy is there in every block, but it's not changing.
01:11:00
It's canceling out in the before and after. So, unless the two blocks annihilated and disappeared, then you really have something interesting. But you don't have that in the Newtonian physics. Okay, so I should tell you that I've given you guys two more problems, one of which is optional. You don't have to do that, but it's a very interesting
01:11:20
problem. And I want you to think about it, because it's a problem where there's two rockets headed towards each other, both of the same length, say, one meter. And this guy thinks he's at rest, and the second rocket is zooming past him like this. When the tail of this rocket passes the tip of this, he sends out a little torpedo aimed at this one.
01:11:42
Clearly, it doesn't do any harm. But look at it from the point of view of this person. This person says, well, I'm a big rocket. I'm moving like this. Then, when my tail met the tip of this rocket, he sent out an explosive, which would certainly hit me down here. So, the question is, does this upper rocket get
01:12:03
attacked? Does it get hit by the lower one or not? From one point of view, you miss. The other point of view, when you expand this one and shrink this one, you definitely got to hit. So, you've got to ask, I mean, do you get hit or don't get hit? You cannot have two answers to one question. So, I've given the problem, I've given you a lot of hints.
01:12:20
And I think even though it's optional, if you do that problem, you've got nothing to fear, okay? I'll give you good practice on how to think clearly about any number of these problems. And I've given you enough hints on how to do that.