1
00:00:00,000 --> 00:00:02,465
[SQUEAKING]

2
00:00:02,465 --> 00:00:04,437
[RUSTLING]

3
00:00:04,437 --> 00:00:05,916
[CLICKING]

4
00:00:12,840 --> 00:00:15,167
MARKUS KLUTE: Welcome back
to 8.20 Special Relativity.

5
00:00:15,167 --> 00:00:16,500
So we're starting a new chapter.

6
00:00:16,500 --> 00:00:19,290
In this chapter, we
talk about some aspects

7
00:00:19,290 --> 00:00:22,650
of special relativity,
which are not crucially

8
00:00:22,650 --> 00:00:24,510
important to understand
the concepts,

9
00:00:24,510 --> 00:00:26,760
but they help you to
go a little bit deeper

10
00:00:26,760 --> 00:00:28,030
in your understanding.

11
00:00:28,030 --> 00:00:30,400
I hope this is
going to be useful.

12
00:00:30,400 --> 00:00:31,980
So we want to talk
about the algebra

13
00:00:31,980 --> 00:00:34,090
of known transformations.

14
00:00:34,090 --> 00:00:40,580
So we have seen that our gamma
factor is 1 over square root

15
00:00:40,580 --> 00:00:44,780
1 minus beta squared with beta
equal the relativistic velocity

16
00:00:44,780 --> 00:00:46,640
v/c.

17
00:00:46,640 --> 00:00:49,790
And thus, this we can rewrite
it as gamma squared minus beta

18
00:00:49,790 --> 00:00:53,090
squared gamma squared
is equal to 1, OK?

19
00:00:53,090 --> 00:00:56,960
So now, I would like you to
recall hyperbolic functions

20
00:00:56,960 --> 00:01:01,370
sinh and cosh and cosh
squared minus sinh

21
00:01:01,370 --> 00:01:02,630
squared is equal to 1.

22
00:01:02,630 --> 00:01:09,420
So the form here and here
are pretty much the same.

23
00:01:09,420 --> 00:01:11,520
And something squared minus
something else squared

24
00:01:11,520 --> 00:01:13,200
equal to 1, OK?

25
00:01:13,200 --> 00:01:14,040
Good.

26
00:01:14,040 --> 00:01:16,050
So let's see how
this looks like.

27
00:01:16,050 --> 00:01:19,140
As a reminder for us, the
hyperbolic functions as

28
00:01:19,140 --> 00:01:23,840
defined as 1/2 e to
the x power minus

29
00:01:23,840 --> 00:01:32,520
e to minus x and cosh equal to
1/2 e to x plus e to minus x.

30
00:01:32,520 --> 00:01:33,020
OK?

31
00:01:33,020 --> 00:01:36,205
The tangent is then
defined as a ratio.

32
00:01:36,205 --> 00:01:37,580
And you can plot
those functions,

33
00:01:37,580 --> 00:01:39,680
and you can see the
functional form as

34
00:01:39,680 --> 00:01:42,290
given in these two diagrams.

35
00:01:42,290 --> 00:01:45,200
Well, we want to come back to
those two equations looking

36
00:01:45,200 --> 00:01:46,170
very much the same.

37
00:01:46,170 --> 00:01:53,570
So we can define now
eta, the rapidity,

38
00:01:53,570 --> 00:01:58,280
as gamma equal to hyperbolic
function cosh eta and beta

39
00:01:58,280 --> 00:02:01,190
gamma equals sinh eta.

40
00:02:01,190 --> 00:02:03,170
So basically, we
have this rapidity,

41
00:02:03,170 --> 00:02:05,510
which is a measure on
how much the system is

42
00:02:05,510 --> 00:02:12,180
boosted as being equal to
this kind of hyperbolic angle,

43
00:02:12,180 --> 00:02:12,680
right?

44
00:02:12,680 --> 00:02:14,630
You can then
process again, where

45
00:02:14,630 --> 00:02:19,010
beta is equal to the tangents
of this hyperbolic angle.

46
00:02:19,010 --> 00:02:26,630
And just remember that the
slope in our space time diagram

47
00:02:26,630 --> 00:02:28,920
is 1 over the velocity.

48
00:02:28,920 --> 00:02:36,970
We find that angle again now
is being called rapidity, OK?

49
00:02:36,970 --> 00:02:41,050
And just as a reminder,
beta goes from minus 1 to 1,

50
00:02:41,050 --> 00:02:43,850
depending on the direction
and the speed of it

51
00:02:43,850 --> 00:02:45,820
is less than the speed of light.

52
00:02:45,820 --> 00:02:53,190
And then eta goes from
minus infinity to infinity.

53
00:02:53,190 --> 00:02:55,680
OK, so then we can rewrite
our Lorentz transformation.

54
00:02:55,680 --> 00:02:59,070
Instead of writing gamma and
beta gamma and minus beta

55
00:02:59,070 --> 00:03:02,220
gamma and so on,
we can write this

56
00:03:02,220 --> 00:03:06,470
through the hyperbolic angle.

57
00:03:06,470 --> 00:03:06,970
OK?

58
00:03:06,970 --> 00:03:11,360
So you should always
ask why is this useful.

59
00:03:11,360 --> 00:03:15,070
The first part is that
when we add velocities,

60
00:03:15,070 --> 00:03:18,580
we found this complicated
transformation

61
00:03:18,580 --> 00:03:22,570
where the new velocity is
equal to the first velocity

62
00:03:22,570 --> 00:03:25,990
times the second velocity over
1 plus the product of the two

63
00:03:25,990 --> 00:03:27,520
velocities.

64
00:03:27,520 --> 00:03:32,150
And this is much easier now as
we can just add the velocities.

65
00:03:32,150 --> 00:03:35,960
So the third velocity is equal
to the first plus the second.

66
00:03:35,960 --> 00:03:40,600
This is much, much easier
to actually calculate.

67
00:03:40,600 --> 00:03:43,370
And the proof of this
is coming directly

68
00:03:43,370 --> 00:03:47,190
from the proof of those
hyperbolic functions here.

69
00:03:47,190 --> 00:03:49,530
The second part where
this becomes useful

70
00:03:49,530 --> 00:03:53,040
is when you think about the
angle in your space time

71
00:03:53,040 --> 00:03:54,000
diagram.

72
00:03:54,000 --> 00:03:57,130
How does this now compare
to a normal rotation?

73
00:03:57,130 --> 00:03:58,780
So let's start here.

74
00:03:58,780 --> 00:04:00,210
So we have a normal rotation.

75
00:04:00,210 --> 00:04:02,820
We have a rotation at a angle,
and our coordinate system just

76
00:04:02,820 --> 00:04:05,040
rotates by specific angle.

77
00:04:05,040 --> 00:04:06,840
Let's call it phi here.

78
00:04:06,840 --> 00:04:08,940
And what we do now,
we have a similar

79
00:04:08,940 --> 00:04:11,610
but a hyperbolic
rotation in which

80
00:04:11,610 --> 00:04:16,370
the coordinate system in our
space time diagram rotates.

81
00:04:16,370 --> 00:04:17,329
All right?

82
00:04:17,329 --> 00:04:20,060
In the normal rotational
case, x squared

83
00:04:20,060 --> 00:04:22,820
plus y squared is invariant.

84
00:04:22,820 --> 00:04:24,860
And in our Lorentz
transformation,

85
00:04:24,860 --> 00:04:27,110
c squared t squared
minus x squared.

86
00:04:27,110 --> 00:04:28,850
All right?

87
00:04:28,850 --> 00:04:31,460
If you then have a more general
transformation, a rotation,

88
00:04:31,460 --> 00:04:32,960
and Lorentz
transformation, you find

89
00:04:32,960 --> 00:04:36,710
x squared plus y squared plus
z squared minus c squared

90
00:04:36,710 --> 00:04:39,460
t squared [INAUDIBLE].

91
00:04:39,460 --> 00:04:42,970
OK, so we have just
relabeled things,

92
00:04:42,970 --> 00:04:45,730
but now we can make
use of everything

93
00:04:45,730 --> 00:04:47,360
we know about
hyperbolic functions

94
00:04:47,360 --> 00:04:50,140
when we think about
adding velocities.

95
00:04:50,140 --> 00:04:54,340
Because the rapidity-- the
relative distance and speed

96
00:04:54,340 --> 00:04:57,160
between two reference frames
is basically the angle

97
00:04:57,160 --> 00:05:00,300
of the hyperbolic angle.