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0:10
Hello, I'm Norman Wildberger. In this video, we're going to carry on with our development of special relativity in a somewhat novel way. We're framing this theory not in terms of the speed of light and Einstein's famous postulates but rather in a more elementary and more mathematical setting. So what we're going to do today is talk about three rather remarkable consequences of Einstein's theory:
0:42
namely, length contraction, time dilation, and velocity addition. We'll see that these things, in fact, are necessary consequences of basic linear algebra that arise from the setting that we've established, just having to do with bats moving in a one-dimensional setting, communicating and observing by sending out sound signals and waiting for those sound signals to return.
1:11
Okay, so the basic setup is that we have the space, which is just one-dimensional, the x-axis, and we're adopting a rather Newtonian point of view about this so that we have a notion of time that's marching along uniformly. This is our time axis here, and we have our observer Bat A, which is stationary at this point here and is represented by that timeline there.
1:40
His coordinates, which are obtained by sending out sound signals and then observing when they come back, are the same as our coordinates. Now we have a new observer, Bat B, which is moving at some uniform rate along this axis and also has a clock, which is internally generated, and this arrow here represents one unit of time in B's reference frame.
2:09
Okay, we saw that if B plays the same game in terms of sending out sound signals and then waiting for them to come back and deducing what the position and the time coordinates are for various events, then we get a sort of alternate framework, an alternate coordinate system from B's point of view. So the crucial ingredients, as far as linear algebra goes, are actually very, very simple and go back to our very first video in this course. We really have two bases here:
2:43
A's basis, consisting of the unit vector E_x in this direction and the unit vector E_t in this direction, and they are, in our coordinates, the vectors (1,0) and (0,1).
2:57
Now we have another pair of basis vectors, say the B basis, consisting of E_x' and E_t', and these are these vectors here, S and R, R and S. We saw last time that we could normalize things by looking at an affine condition, a purely affine condition involving areas, so that
3:23
this vector and this vector have the property that S^2 - R^2 = 1. That ended up amounting to the fact that this vector here, which is the vector on B's timeline that represents a tick of one in B's direction, lies on this hyperbola with the equation x^2 - t^2 = -1. So what we're really going to do now is see that special relativity really follows from just the basic
3:55
exploration of how things change from A's point of view to B's point of view.
4:03
This is a perfect example of the usefulness of linear algebraic ideas and the change of basis discussion that we've had previously in this course. In particular, you might want to review Wild Lin Alge 26, where we talk about change of bases and change of basis matrices. So we'll apply that kind of reasoning here, and we're going to extract, as a consequence, these three remarkable aspects of special relativity purely from this mathematical point of view.
4:37
Alright, so let's have a look at these two bases: E_x and E_t, and E_x' and E_t'. Now, the fact that these are S and R, and R and S, really means that E_x' is in terms of A's basis vectors S times E_x plus R times E_t. That's just what we mean when we have this coordinate vector S, R representing E_x'. Similarly, E_t', being R, S, means that it's R times E_x plus S times E_t because E_x and E_t are the standard basis vectors from our point of view,
5:14
which are giving us the coordinate system that we're using to label these vectors.
5:22
Now, this relationship between the two bases can be expressed or rewritten using
5:28
the idea of matrices, which I introduced as a bit of a novelty. It's allowing us to be a little bit more flexible about the meaning of expressions that we write down.
5:41
So we can rewrite this system in the sort of vector form or matrix form E_x', E_t' = S, R, R, S, this 2x2 matrix, times the vector E_x, E_t. But we note that the entries here and here and the entries here and here are themselves vectors and not numbers.
6:04
So the first line here is just a rewriting of this equation; this is really a vector equation, so the E_x', and the E_x and E_t are vectors, and here are numbers.
6:16
Alright, so this is a change of basis matrix, and in terms of coordinates, well, both A and B can use their bases to write any given vector in terms of coordinates, and we've agreed that A's coordinates for a vector will be, say, (x, y) in square brackets, while B's coordinates for that same vector will be (x', y') in pointed brackets. That's just a convention that we're going to use just here to distinguish A's coordinates from B's coordinates.
6:54
Alright, so if we have an arbitrary vector W, which is expressed either this way in A's coordinates or this way in B's coordinates, then this one here means that it's (x, y) times E_x, E_t. That's what this means: it means it's x times E_x plus y times E_t,
7:12
and this expression means that it's x' times E_x' plus y' times E_t'.
7:19
Here are the two basis vectors for B, so again we're using this matrix form where this looks like a vector, but the entries themselves are actually the vectors in our various bases.
7:35
Okay, so what we can do is we can take this expression here,
7:39
and we can replace it right there. So imagine just replacing this thing with R, S, R, R, S, times E_x, E_t, and that shows us that (x, y) will be (x', y') S, R, R, S.
7:57
So, as a reminder of the things we talked about in Wild Lin Alge 26, the same change of basis matrix, which tells us how to express B's basis vectors in terms of A's basis vectors,
8:14
that same matrix can be used to express A's coordinates in terms of B's coordinates.
8:24
So here we have the coordinate vectors of a vector W expressed as rows, and here the matrix is multiplying on the right, and here we have these columns of vectors, and the matrix here is multiplying on the left. The same matrix is playing slightly different roles, acting either on the basis vectors themselves or on the coordinates of a general vector.
8:55
Alright, so please just make sure that you understand this little derivation here that shows the equivalence of these things. These are sort of dual expressions of the change of basis matrix acting. So, two very remarkable consequences of Einstein's point of view were that simultaneity is a relative notion. The idea that two things happen at the same time is actually an observer-dependent
9:27
idea. This causes some conflict with our intuition because we have this strong intuition about two things happening at the same time because we can see them happening at the same time.
9:42
Really, what's involved there is the fact that the speed of light, in practice, is so fast that it's essentially infinite. We see no gap whatsoever if we're looking at some distant thing across the plane or even on some distant mountain, and something happens there, and something happens here. We can see both events essentially at the same time,
10:05
but that's a little bit of an illusion or a confusion that we are led to by our biology because of the dramatically high speed of light. We're just not able to understand, physically or intuitively, that light is actually taking a finite, short but finite, amount of time to get from one place to another. With sound, this is much less spectacular, and it's sort of much easier, I think, to understand. If we think in terms of sound, as we are doing here now, which
10:37
is an advantage in framing this discussion using sound rather than light, we will see that the same equations and phenomena are happening here in this purely Newtonian situation, just with sound.
10:52
Alright, so the idea is that simultaneity is relative, and we can see this already. This is built in; it's an immediate consequence of our coordinate systems.
11:02
Okay, so two events, E and F, are simultaneous with respect to A. What does it mean? Well, it means that the time of event E equals the time of event F. So in A's frame, time is this axis here, and all the events that have time 1 are on that line. All the events that have time 2 are on this line. So if you take that point there, A reckons that the time of that event is time equals 2. That's the same event as represented by that point there,
11:37
and I remind you that that point represents an event that happens actually at that position.
11:44
Something happens at that position at time t equals 2.
11:49
On the other hand, two events are simultaneous with respect to B when B's t-coordinate, namely t', agrees on E and F. Now, what does that mean in terms of our vectors? Well, here is the time axis for B, and here is the x-axis for
12:12
B, calling this the x'-axis and the t'-axis to distinguish it from the x-axis and the t-axis of A. Now, as far as B is concerned, all of the points lying on this line have t' coordinate 0.
12:29
So this coordinate is the point (1,0) as far as B is concerned, this is the point (2,0). That means B thinks that these events are all simultaneous; they're happening at time t equals zero for her.
12:45
So we see that, in particular, simultaneity is not the same for A and B. These two events are at the same time for B, but they're obviously not at the same time for A. The time of this one would be there; the time of this one would be there. In fact, all the events which are simultaneous for B would lie on a line parallel to this one. Okay, so that would be the line t' = 1 for B, t' = 2 for B, t' = 3 for B, etc. A very different notion of what simultaneity means.
13:25
And co-location is also relative. The idea of where an event happened depends on your frame of reference.
13:40
Events E and F are co-located with respect to A—to introduce that terminology, it's not that widely used, but I think it's reasonable. We don't have such a good word for happening at the same location, so co-location is perhaps a good word. So, events E and F are co-located with respect to A when their x-coordinates are the same, but they're co-located with respect to B precisely when the x'-coordinates of the two events are the same. So x' of E equals x' of F.
14:12
So all the events on this line here are deemed to have occurred at location x = 2 for A,
14:24
and all these events are happening at x = 3. While for B, it's different; all these events happened at position x' = 0. All these events happened at positions x' = 1, x' = 2, and so on.
14:47
So this is easier if you think in terms of outer space, right? If you go back to that lovely movie "Gravity," you have a couple of people uniformly drifting in space. It's very hard to say where "here" is because I might be drifting, and you're drifting, and there's no marker to place anywhere to say, "This point in time, I can place something here, but how do I know that that thing is not moving?"
15:16
That's essentially what's happening. It's easier for us to understand that. It's harder for us to understand that the same thing is really happening with respect to time because we're fooled by the very fast speed of light. But when we start thinking about sound and how a bat actually determines when something happened, then it's not too hard for us to look at concrete examples and work out that yes, Bat A and Bat B disagree about two events happening at the same time.
15:48
Time is relative, just as location is relative. And that's in a Newtonian world where we just have bats with the critical property that each one is just measuring locally; they don't have a whole network of clocks and rulers laid out. They're just point creatures, and they're forced to just perform their local operations. That's the key reason why this is happening.
16:19
In fact, it's not so different from the way we find ourselves on planet Earth.
16:23
Here on planet Earth, we're just a little speck in some vast cosmos. Essentially, we also, if we want to observe our universe, are forced to really make measurements from this one point. We can send out signals and wait for them to come back, but that's really only useful in the vicinity of our solar system because otherwise, we would have to wait too long, and chances are the signals would be dispersed. But we do get signals from distant stars and
16:55
distant galaxies, but still, the point is we're just really an essential point. We don't have a whole grid of delineated positions with observers with calibrated clocks, whatever that means.
17:12
We might think about that kind of context in a laboratory or in some very small-scale experiment that we perform here on planet Earth, but that's not our situation globally. What we're doing here with this model is really sort of adjusting ourselves more in the sight of what really happens when we're looking at the universe in a large-scale way. In that case, we're much like the bats, just sitting there making our little observations, trying to deduce what's going on
17:40
from a rather limited perspective, and these are necessary consequences of that point of view. Okay, so now we come to another dramatic consequence of special relativity that has mystified laymen for decades, which is time dilation. The fact that we have two observers that are going in uniform motion with respect to each other, and they each have their clocks, okay? Then each observer is going to observe that the other person's clocks seem to be running slower.
18:18
Okay, so there's this popular conception out there that this means that the clocks somehow really are running slower, but it's important to emphasize that from this mathematical point of view that I'm adopting here, it's not that they really are running slower; it's just a consequence of the measurement system that the bats are forced to use. So our two bats just have different measurement systems, but it's interesting that they both really have the same conclusion
18:47
about each other's clocks. The situation becomes very symmetrical between A and B, even though A, in fact, is stationary in our Newtonian system and B is moving along. Nevertheless, their conclusions vis-a-vis each other are exactly the same.
19:03
Alright, so here's our basic relation again relating these two basis vectors, and
19:11
here's our matrix S, R, S, and I remind you that S^2 - R^2 = 1, or in other words, this matrix has determinant 1. That's an easier way to remember what's going on. Okay, so let's have a look at the
19:29
second basis vector for B, E_t'. This row just says that this is R times E_x plus S times E_t,
19:40
and time dilation is just encoded in this equation. Time dilation is an immediate consequence of just looking at this equation and noting that there's an S here. That's all.
19:54
This S represents time dilation. How? Well, let's have a look in the diagram. Okay, so what's this saying? Here's the vector E_t', and I've blown it up a little bit so we can see what's happening a little bit more clearly. Here is E_x, and here is E_t, and I've positioned E_t' so it really is on this hyperbola with the equation x^2 - t^2 = -1. Okay, so what is this telling us? This is telling us that this vector here,
20:27
if we write it in terms of the E_x and the E_t components, that this E_x component will be R, and this E_t component will be S. In other words, the coordinate on the t-axis for A of this vector here is S. That's exactly what we're saying here, that E_t' is R, S. Okay, so what does that mean? Well, remember this is the vector that corresponds to a single tick of B's clock. So this point right here actually corresponds to an event; it's
21:10
the ticking of B's clock. There's a tick at zero, and that's the tick of one. So you say tick tock.
21:20
Now, as far as A is concerned, this represents when his clock ticks: tick tock, tick tock.
21:31
And you can see that from A's point of view, if he measures when does B's clock tick, he gets the answer that it ticks at time S. Okay, so time S is not the same as time one. This is happening at a different time, so if S is bigger than one, which we're assuming in this case because we sort of put this vector along this hyperbola S^2 - R^2 = 1,
22:06
that's a working assumption, so that means that S is actually bigger than one, and so A measures B's clock as ticking slower because it takes longer for B's clock to tick one than it does for A's clock to tick one. Okay, now similarly, when we invert this relation, and we do that by multiplying by the inverse of this matrix. Okay, I remind you the inverse of this matrix is just the same as that one except there's a minus sign in front of the RS.
22:41
So if we invert this relation and write E_t as a linear combination of these two, then we get E_t = -R E_x' + S E_t', and here the coefficient of E_t' is S, the same S that's appearing here.
23:00
These two coefficients happen to be different, but we're not interested in them right now; we're just interested in this S because this is the t-coordinate of E_t in the B reference frame.
23:13
What does that mean? So here's the vector E_t.
23:18
And this represents how E_t is written as a linear combination of E_x', which is its vector, and E_t', and this is telling us that it's a certain combination of E_x', in fact, some negative combination, plus S times E_t', plus S times this vector. Alright, so if S is, say, roughly maybe 1.6 in this diagram, then here S is 1.6. That's one, that's 1.6, and we can see that geometrically because this line here, which is parallel to B's x'-axis, all
24:02
these points here, okay, all these points here are at the same time as far as B is concerned. They're all at time t' = S, and what this is telling us is that B, if B is measuring when does A's clock tick, it ticks here at time t = 0, and the next place where it ticks is at time t = S. I remind you here are the times of B's clock: that's time zero, that's time one, that would be time two, time three, and so on. This is time t = 1.6 roughly.
24:50
So B measures A's clock to tick slower, just exactly the same as A measures B's clock to tick slower. So both seem to tick slower by a factor of S to each other.
25:05
They both send the message to each other saying, "I'm sorry, it looks like there's something wrong with your clocks. Your clock is seemingly ticking slower by a factor of S." A sends that message to B, and B will send that message to A. They both make the same observation. This is a necessary consequence of the relatively simple way of using sound to determine position and times for events. That's all it is.
25:33
So now we come to length contraction, which is another seemingly paradoxical consequence of Einstein's special relativity. Again, in the popular conception of things, people often think that this means that somehow things are actually really shrinking out there.
25:53
Well, that's not really the way you need to think about it. It's again really more a question of how we measure things. Okay, it's a consequence of how we measure things.
26:05
So we're going to make the same kind of analysis as we did for time dilation. In fact, the derivation is very similar; it's just a little bit dual to that previous one. So let me first of all just mention something though. Here's a ruler, right? If I ask you how long this ruler is, you're going to measure how long this ruler is. Alright, it's pretty easy if I hold it still like this, okay? You can take a picture, so basically you can record the position of this endpoint, you
26:38
can record the position of this endpoint, and then you make some calculation to determine its length. Alright, that's easy. What if I do like this, and I say, "Alright, what's the length of this ruler?" I should go in the line, so suppose I'm going in a line, and I say, "Please measure the length of this ruler." Well, it might be alright if I'm doing it slowly. If I'm going really fast, you might say, "Wait a minute, hold still,"
27:02
and I say, "Well, why do I need to hold still? Why don't you just find out, you know,
27:07
what about the position of this thing is and then find out the position of this thing and then take the difference?" And you'll rightly say, "Well, that's not going to work because the thing is actually moving. We have to record the positions of these two places on the ruler at the same time." Okay, that's the critical thing. To make a measurement of length, you have to measure these things at the same time. Okay, so here is our relationship between the
27:36
two basis vectors, and I've given you the inverse one now, which we talked about last time. I didn't write it down, so here's E_x and E_t in terms of E_x' and E_t', and here's that inverse matrix of the one we had in the previous one. Alright, so let's have a look at the E_x vector here. The first line just tells us it's S E_x' - R E_t', and let me rearrange that a little bit just by dividing by S, so I get 1/S E_x = E_x' - R/S E_t', or if we look at it the other way around,
28:11
we express E_x' in terms of E_x and E_t, well then we'd have to use the inverse matrix, and we would get a similar equation: 1/S E_x' = E_x + R/S times E_t.
28:25
Now, I claim that both of these equations show that both A and B actually view each other's rulers as having shrunk by a factor of 1/S. The fact that there's a 1/S here and a 1/S here means that A measures B's ruler as having shrunk by that factor 1/S,
28:45
and B measures A's ruler as having shrunk by a factor of 1/S.
28:49
Alright, so let's see why these equations give us this length contraction. Let's have a look at
28:57
A's ruler, okay, given by that mirror that A used to basically set up his original length and time signal. So here is A's world line, and the world line of his mirror is the vertical line going straight up because A and his mirror are not moving. Okay, so if you ask B, "How far apart are A and his mirror?" That's the question: "How far apart are they?" Well, he's going to say, "Alright, I'm going to measure the separation between A and A's mirror."
29:32
Now, in order to do that, B has to measure them at the same time. It's no good measuring the positions of A and his mirror at two wildly different times, and to measure something at the same time means that B has to get the two objects on the same line parallel to this x'-axis. This x'-axis consists of those events that have the same time, time t = 0 for B. These are all simultaneous events, so the world line of the mirror meets
30:14
this x'-axis at this point here, and so as far as B is concerned, this point and this point are simultaneous events. So it's along this axis here that B will measure the separation between A
30:34
and his mirror, and he will look at this separation and say, "Ah, compared to my unit of length, that's only 1/S times my length of one." So the separation between A and his mirror is, in fact, only 1/S; it's less than one.
30:56
Alright, on the other hand, it also works the other way around. So if we look at E_x', that point lies on the world line of B's mirror, okay, which is going to be parallel to B's world line because they're both moving at the same speed together. Alright, so here is the world line of B's mirror, here's the world line of B. Okay, so if we wanted to ask A, "So how far apart are B and her mirror?" A will say, "Well,
31:24
I need to measure them at the same time. I can only measure their separation at the same time." So which events correspond to B and her mirror being at the same time?
31:38
Well, these events here. Okay, so when A, at time t = 0 according to A, B is right here, and at time t = 0, her mirror is right here. This is where the mirror is at time t = 0 according to A,
32:02
and so A just measures that and says, "Ah, that's not one; that's only 1/S." That's what this is telling us. This point here, 1/S times E_x, that's this point here, is E_x' minus some multiple of E_t'. That's that vector there, so this point here is obtained by taking this thing here and going in this direction here.
32:27
So A measures the separation of B's ruler to be 1/S,
32:34
and B measures the separation of A's ruler to be 1/S. They both are firmly convinced that each other's rulers seem to have shrunk by a factor of 1/S.
32:48
Have the rulers, in fact, shrunk by a factor of 1/S? Well, no, they visibly have not to us. We're just watching one bat stationary there and another bat moving along.
33:00
We know, well, at least we think we know, that there hasn't been any shrinkage of rulers. This is a consequence of the way the bats are forced to measure their world—the fact that they're point observers essentially, and it's sound that they have to rely on to make their measurements, and sound propagates at a finite speed, and it takes a certain amount of time to get there, and that's going to create this space-time diagram, and it's the geometry of this
33:32
diagram that leads to these simple algebraic equations, and it's just these coefficients here that are determining the length contraction factors—the apparent length contraction factors.
33:48
And I note that this is sort of the opposite of the time dilation. We saw that time was seemingly expanded by S, and here we see that the length is seemingly dilated or contracted by 1/S. So the more time is dilated, the more length is contracted by exactly an inversely proportional amount. Alright, so let's enrich our little experiment here with bats and introduce a third bat. Let's call the bat C. So now we have three bats: A, B, and C, and
34:21
we're going to adopt the same convention here: A's basis vectors here, B's basis vectors here, and now we'll let C's basis vectors call them E_x" and E_t". Let's let them be Q, P, and P, Q.
34:38
And let's let the coordinates of a general vector
34:42
with respect to C be {x", t"}. Alright, so the coordinates of a vector for A will look like those kind of square brackets, for B will be pointy brackets, and for C will be these brace brackets. And we're going to see the same kind of normalization that we have for
35:02
B. So here we had S^2 - R^2 = 1. We would similarly assume that C has normalized his coordinates so that the basic parallelogram is of area one. In other words, Q^2 - P^2 = 1.
35:19
And here we might have an example. So here's E_x and E_t, and here is E_x' and E_t',
35:27
these basis vectors, and now here is the timeline for a third bat, C,
35:32
and here is the corresponding x"-axis. So we have a basis vector E_x" and E_t". It's the same kind of relation as these two; they're symmetrical with respect to the lines of slope one or minus one.
35:52
Alright, so we know that the relationship between A's basis vectors and B's basis vectors is given by that relation there, and if we replace B with C now,
36:07
then exactly the same kind of relationship will hold between C's vectors and A's vectors.
36:16
Okay, because these are all actually in A's and our coordinate system. So E_x", E_t" will be this matrix Q, P, P, Q times E_x and E_t.
36:31
And now I want to just rewrite this a little bit by replacing the E_x, E_t here with this vector here obtained from this first equation. I can do that by re-juggling this first equation by multiplying by the inverse of this matrix, as we've seen several times now. So if we replace E_x, E_t here with the inverse of this matrix times E_x', E_t', there's the inverse right there,
36:59
then by matrix associativity, this matrix times this matrix is the matrix whose entries are SQ - RP, -RQ + SP, SP - QR, and -RP + QS or SQ - RP.
37:21
Now notice that they have the same kind of form as these ones. Here is one number, and here's the same number on the diagonal there, and here this number here is the same number as this number here. So this is the matrix that expresses the relationship between B's basis and C's basis. We're sort of given the relationship between A and B and between A and C, and now we're asking what's the corresponding relationship between B and C, and this is it.
37:54
This is an interesting matrix, and probably the most important thing to observe about it is that it's going to have a determinant of 1. It's going to be just like this matrix and this matrix, and that's just a consequence of the fact that the product of two matrices has a determinant, which is the product of the determinants of the two matrices. So the fact this has determinant 1 and this has determinant 1 means that their product will also have determinant 1.
38:28
So, in other words, the relationship between B's basis vectors and C's basis vectors is exactly the same kind as the relationship between A and B's or A and C's. There's something quite symmetrical about the relationship between these three bats and their relative bases and relative coordinate systems.
38:53
We might have thought that A plays some special role because it's sort of in the middle of our diagram, and it's corresponding to our frame of reference that we've used to set up the entire story. But now we see that there's a symmetry here, that the relationship between A and B and the relationship between B and C are essentially the same kind of relationships. The same rules are used to transfer from the coordinates of B to C as they are from A to B or from A to C. Okay, so
39:24
now we're going to turn our attention to velocity addition, which is the third sort of interesting and unintuitive consequence of Einstein's special theory of relativity that we're going to explore today. So let's talk a little bit about velocity. As we all know, velocity is defined in sort of a high school Euclidean setting as delta x over delta t. You measure a displacement, and you measure how much time that displacement took, and you divide the displacement
39:52
by the time. For bats, this quantity, this notion, is a relative quantity; it depends on your point of view, and that's not a surprise because we've seen that both
40:09
position and time are measured subtly differently between bats A and B, and so we would expect that they're not going to agree about what the velocity of some third object is.
40:24
Okay, so let's have a look first of all just at the velocity of B with respect to A. So we'll return just back to Bat B. Here she is with her world line, and here is her standard vector E_t', R, S representing the first tick of her clock.
40:45
And it's the vector R, S. Okay, in the diagram, what does that mean? Well, it means that in our coordinates or in A's coordinates, this is R, and this is S.
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Alright, this represents the displacement in position, and this S represents the displacement in time, and so Bat A would say that Bat B has velocity R/S,
41:16
and we'll write that this way: V_A(B) means the velocity of B with respect to A, as measured by A.
41:27
That's R/S, and let's call that V, and that's the same velocity that we, as the observers of the whole thing, would ascribe to B. We would say that B is moving along here at a speed of R/S.
41:45
So the relationship between R, S, and V can be squared: R^2/S^2 = V^2, and we can use the basic fact that S^2 - R^2 has been normalized to be one. I remind you that B normalized her clock so that
42:04
this first tick occurred on this hyperbola with the equation x^2 - t^2 = -1. So S^2 - R^2 = 1,
42:17
and we can replace the R^2 here with S^2 - 1. So we solve for R^2, you get S^2 - 1, and then we can rewrite this as 1 - 1/S^2. If we bring the -1/S^2 to the other side, we get 1/S^2 = 1 - V^2.
42:41
That then is the relationship between S and the velocity V.
42:49
So the same factor S that was involved in time dilation and in length contraction—I remind you that time was dilated by a factor of S and length was contracted by a factor of 1/S—that same factor S is pretty closely directly connected with the velocity of B as measured by us or by A: 1/S^2 = 1 - V^2. And if we flip that around, we can say that S^2 = 1/(1 - V^2).
43:24
Now, if you believe that you can take square roots—and I know many of you do, perhaps somewhat mistakenly—then you could say, "Well, why don't we take the square root of these expressions and write 1/S = √(1 - V^2) and S = 1/√(1 - V^2)," and then we can use these as the
43:49
factors of the time dilation and length contraction.
43:58
Well, my point of view is that taking a square root is, in fact, a highly problematic operation if you want to do something more than approximate. It's no problem taking an approximate square root, but it's very difficult, in general, to take an exact square root, and so in my whole setup of things, the way I want to look at mathematics, I want to avoid, if I can, taking square roots. Or if I am going to take square roots, I want to make sure that I have a prior algebraic system
44:27
that has been carefully constructed to incorporate the particular square roots that I might be using. It means that, for me, these two formulas are much preferable to these ones. Okay, and so I'm going to encourage you to think about these as the fundamental formulas, okay, relating our factors of time dilation and length contraction and relating them to the velocity, at least the perceived velocity of B with respect to A. Also, we get this sort of sense here that perhaps this
45:09
V, this velocity, yes, it's important in our everyday descriptions of mechanics and so on, of course, velocity is very important. But here in this context, maybe the V is not quite as
45:26
important or natural as perhaps some other related quantities because it's a relative quantity; it depends on the point of view. Okay, but in any case, these are very important formulas. These are also used if you want to convert what I'm doing to the standard formulas that you find in the textbooks. Then you will want to use these to convert back and forth. But I urge physicists to try to let go of the square roots. Try to reformulate
45:56
your thinking and your ideas so that they are square root free. I claim this will orient you to
46:04
a mathematics that is ultimately more pleasant and natural and more effective, ultimately. Alright, so we're interested now in our three bats, and they each have their own point of view. They each have their own frame of reference, their way of measuring x and t coordinates. We're interested in the relationship between the various velocities that they ascribe to each other, and so to get at that, let me just remind you of the relationship
46:32
between the three sets of basis vectors. So here's the relationship between A's basis and B's basis with this S, R, R, S matrix. Here's the relationship between A's basis and C's basis with the Q, P, P, Q matrix, and then here's this third relation that we derived relating B's basis and C's basis involving this new matrix N, M, N, where you may recall from an equation I called star a few slides back that N was SQ - RP and M was -RQ + SP.
47:18
Okay, so what are the various velocities
47:21
that they're going to measure? Okay, so here are some velocities or relative velocities that
47:28
are involved. So A is going to measure B's velocity. How? Well, remember we take this matrix, and we take R/S—that was the velocity of B with respect to A—R/S, let's call that V.
47:46
Similarly, the velocity of C with respect to A is given by this relation here. We do correspondingly P/Q, the same as R/S there, and let's call that U. Right, so that's the velocity of C with respect to A. So these are both velocities with respect to A, in other words, the same as our
48:09
observations of things. And then finally, here is the velocity of C with respect to B obtained from this third relation. There's B's vectors, and here's C's vectors, and so we take M/N, exactly parallel to what we did before, and we might call that W. So we might be interested in what is W in terms of V and U. Well, W is M/N, and here is M: -RQ + SP, and here is N: SQ - RP.
48:45
Now, what I'm going to do to simplify this is I'm going to divide by S times Q. Okay, if I divide this by S times Q, I'm going to get -R/S. If I divide this by S times Q, I'm going to get P/Q.
49:03
Here, I'm going to get 1, and here I'm going to get R/S times P/Q.
49:12
And that's useful because R/S is V, so this thing here is V, and that thing there is V, and P/Q is U, so that's U, and that's U. And so we get U - V, there's U - V on top, and 1 - the product UV
49:31
on the bottom. And this is Einstein's formula for the addition of relativistic velocities in this setting, but without his assumptions. So again, just at the level of bats in a Newtonian world where only sound is being used to make measurements.
49:52
Okay, and without the assumption that sound is the same in all reference frames, without the assumption that the laws of physics are the same with regard to all these observers, we're simply making a mathematical derivation of the relationship between the three velocities that they each reckon the other one has. Okay, and so we're finding that the velocity of C with respect to B is obtained by taking U - V over 1 - UV.
50:25
So, for example, a stationary—imagine that C is going relatively fast, let's say at 5
50:35
miles an hour, and B is going slower, say at 2 miles an hour. Okay, so
50:42
C is going 5 miles an hour in that direction, B is going 2 miles an hour in that direction.
50:47
Alright, if B measures C, what is she going to measure? What's the velocity of C with respect to B? Well, one's going 5, B is going 2, so the difference would be 5 - 2, that would be 3. We would ordinarily suspect that the answer should be 3; it should be just U - V.
51:07
It's not U - V in this case; it's U - V modified by a factor. The factor is 1 - UV.
51:16
Okay, now, in fact, these velocities that we're considering here are not with respect to miles an hour, they're with respect to the various units of length and time that each of the
51:28
individual bats has set up for their own coordinates.
51:33
But nevertheless, this is some kind of modification. If U and V are very, very small velocities, then we expect UV is very negligible compared to U and V, and this will then reduce, sort of in a limiting case, to the formula that would be used, U - V.
51:53
So people used to say, "Oh, it reduces to the Newtonian
51:57
setup." But from my point of view, that's not really accurate because this is a Newtonian formula. It's a Newtonian formula, okay. What is the case is this is
52:09
a formula where we have Newtonian physics, but again, the observers are all point observers. Okay, and the U - V formula is really more a formula where you allow yourself the assumption that you have whole grids of observers with calibrated rulers and clocks.
52:31
Again, that's a much more complicated setup to define physically, and in fact, in practice, the setup is essentially a Newtonian formula right off the bat and could have been discovered anytime after the year 1700 if people had thought about bats.
52:54
Speaking of bats, it's interesting to speculate to what extent the
53:02
consequences that we've extracted here from this linear algebra might actually be reflected in the mental makeup or the brain structure of a bat. Bats are reasonably complicated; they're mammals,
53:18
so we might not be surprised to find that their brains have somehow some kind of built-in factor of this kind of relativistic aspect because they're dealing with sound signals, right? And they're making very precise and remarkably accurate measurements of where things are, things are moving around fast, they're moving fast, and they're making these calculations, of course, mostly subconsciously. But still, their brains are perhaps oriented somehow to incorporate
53:56
these kinds of relativistic effects because somehow they're pretty close to the actual world the bats live in. I'm not really making any claim here that they actually are, but I wouldn't be surprised if you could somehow magically look inside a bat's brain and decode the software there that you would find some capacity to make some kind of relativistic calculations, perhaps along the lines that these equations suggest. But let me iterate: this is a purely mathematical formula.
54:30
I'm not really insisting upon the real world being any one which way. This is a very Newtonian setup; whether the real world is Newtonian or not, we're not in a position to say. But assuming that we have a very Newtonian setup with bats with sound, then Einstein's relativistic addition of velocities is going to be a necessary part of the story. Alright, so a few exercises for you to think about. So the first one is just to rewrite that addition formula
54:57
in what's sometimes a more standard way, which is to express U in terms of V and W rather than W in terms of U and V. So V is the velocity of B with respect to A, and W is the velocity of C with respect to B, then what is the velocity of C with respect to A? Well, so it's U = (V + W)/(1 + VW).
55:26
And if we want to make things a little bit more symmetrical, you can show that you can rewrite either of these formulas in the following way: that the velocity of B with respect to A plus the velocity of C with respect to B plus the velocity of A with respect to C is - the product of the same three velocities. And you might like to compare this with the triple-turn formula, which is Theorem 61 of my book "Divine Proportions: Rational Trigonometry to Universal Geometry,"
55:57
which has a lot of connections with the geometry that's sort of unfolding here, even though it's mostly oriented to the Euclidean setup. In fact, we're going to see that rational trigonometry is really crucial to understanding the further geometry of this relativistic world in which we actually live. Okay, so very important for physicists to start to reorient themselves so that rational trigonometry is part of their toolbox.
56:29
And the last exercise: show that if V does not equal one and W does not equal one, kind of going back up to this equation here, then U does not equal one.
56:43
So if you have one velocity, which is pretty big, approaching the speed of sound—remember, the speed of sound is one in all observers' measurement frames because they have all set up their rulers and clocks so that the speed of sound is one—so if you have something that's going, say, a reasonable fraction of the speed of sound, say 4/5 of the speed of sound, and then from that bat, another bat is going fast, say another 4/5 of the speed of sound,
57:15
you might ask, "Well, what about the cumulative velocity of the first one to the third one?" That's exactly this formula here, and the statement is that if the two initial velocities are not equal to one, then there's no way that this final velocity, U, can ever be equal to one.
57:30
So you cannot compose finite non-sound velocities to get a velocity of sound. It's not saying that sound is some kind of limit, really; it's just the way the mathematics works. It's quite interesting. You might like to ponder this. And as such, we have one more exercise to show that if V = 1, so if one of these velocities is one, then U = 1. That's a very easy exercise but still entertaining, perhaps, to think about these things. Okay, we have actually now the tools
58:10
for you to explore many of the standard consequences of special relativity that you find in textbooks. In particular, those of you who are physically oriented might have a lot of fun now having a look at, say, standard examples where you have a pole supposedly going through a barn, and
58:30
somebody tries to close the two doors of the barn and try to capture the pole. Does the pole fit inside the barn? Yes or no, etc. So all those examples and sort of exercises or dilemmas or puzzles can be explored in the context of our bats here. We have most of the requisite formulas for those, including the twin paradox. So you can think about the twin paradox.
58:54
What we don't have yet is a theory of dynamics, so we have not yet talked about momentum and energy, and that's also a very interesting additional story, but we're not going to go there yet. In fact, maybe that's more appropriate in some other context. What I want to do is address the question of, you know, we have all these bats moving around uniformly, and they seem to disagree about a lot of things. Is there anything that they do agree on? You know, is there some aspect
59:23
of their world or some aspects of their world that they can come together on and say, "Yes, that's happening"? Well, there is, and of course, that's the relativistic dot product that's really sort of floating around behind the scenes here, and it's time that we brought that to the fore and explored the consequences of that. And we're going to do a little bit more than that in our next video. We're going to look at the relativistic dot product, but also a very
59:54
interesting algebraic structure associated with it. We're going to look at the usual complex numbers, and we're going to look at sort of a relativistic analog of the complex numbers. And there will be, I think, novel aspects in this sort of comparison between Euclidean and relativistic algebraic structures that are closely connected to the geometry of Euclidean and relativistic space-time. So I hope you'll join me for that. I'm Norman Wildberger, thanks for listening.
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