1
00:00:00,000 --> 00:00:02,468
- We've spent several
videos now getting familiar

2
00:00:02,468 --> 00:00:04,488
with the Lorentz transformations.

3
00:00:04,488 --> 00:00:06,828
What I want to do now, instead
of thinking of what X prime

4
00:00:06,828 --> 00:00:09,108
and C T prime is in terms of X and C T,

5
00:00:09,108 --> 00:00:11,748
I wanna think about, what
is the change in X prime

6
00:00:11,748 --> 00:00:14,228
and the change in C T prime
going to be in terms of

7
00:00:14,228 --> 00:00:16,187
change in X and change in C T.

8
00:00:16,187 --> 00:00:18,188
And we'll see it's just
going to involve some

9
00:00:18,188 --> 00:00:21,448
fairly straight-forward
algebraic manipulation.

10
00:00:21,448 --> 00:00:22,988
So, let's think about it.

11
00:00:22,988 --> 00:00:25,968
Change in X prime is going to be

12
00:00:25,968 --> 00:00:29,118
X prime final minus

13
00:00:29,118 --> 00:00:33,100
X prime initial.

14
00:00:33,100 --> 00:00:35,110
Well, X prime final,

15
00:00:35,110 --> 00:00:36,810
let me just pick a
suitable color for that,

16
00:00:36,810 --> 00:00:39,170
X prime final is going to be

17
00:00:39,170 --> 00:00:44,170
gamma times X final

18
00:00:44,307 --> 00:00:49,293
minus beta times C T final.

19
00:00:49,293 --> 00:00:51,623
All I did is use this formula up here.

20
00:00:51,623 --> 00:00:53,743
If I want to figure out my final X prime,

21
00:00:53,743 --> 00:00:57,243
I'm just going to think of
my final X and my final C T.

22
00:00:57,243 --> 00:00:59,483
So, that's that.

23
00:00:59,483 --> 00:01:04,163
And from that, I'm going to
subtract the initial X prime.

24
00:01:04,163 --> 00:01:07,943
Well, X prime initial is just going to be,

25
00:01:07,943 --> 00:01:09,601
let me get another color here,

26
00:01:09,601 --> 00:01:14,301
Lorentz factor gamma times X initial

27
00:01:14,301 --> 00:01:18,941
minus beta times C T initial.

28
00:01:18,941 --> 00:01:22,901
So, now, let's see, we
can factor out the gamma,

29
00:01:22,901 --> 00:01:24,861
so this is going to be equal to,

30
00:01:24,861 --> 00:01:26,561
and I'll do it in my color for gamma.

31
00:01:26,561 --> 00:01:28,401
If we factor out the
gamma, we're gonna get

32
00:01:28,401 --> 00:01:32,919
gamma times, we're gonna have X final,

33
00:01:32,919 --> 00:01:35,059
let me do this in a,

34
00:01:35,059 --> 00:01:38,299
so we have, do that in white actually.

35
00:01:38,299 --> 00:01:43,259
We're gonna have X final,
and then if we distribute

36
00:01:43,259 --> 00:01:47,567
this negative sign, minus X initial,

37
00:01:47,567 --> 00:01:49,647
and then, let's see,

38
00:01:49,647 --> 00:01:53,507
if we distribute this negative
sign, well, I don't want

39
00:01:53,507 --> 00:01:56,087
to skip too many steps
here, so that's that.

40
00:01:56,087 --> 00:01:58,586
And then we're going to have

41
00:01:58,586 --> 00:02:03,076
negative beta C T final,

42
00:02:03,076 --> 00:02:06,941
negative beta C T final,

43
00:02:06,941 --> 00:02:10,621
and then we have plus, we
distributed this negative,

44
00:02:10,621 --> 00:02:12,874
plus beta C T initial

45
00:02:12,874 --> 00:02:17,193
plus beta C T initial

46
00:02:18,185 --> 00:02:19,580
and so what can we do here,

47
00:02:19,580 --> 00:02:24,310
well this, that's just
going to be change in X.

48
00:02:27,130 --> 00:02:29,159
So I can rewrite this as being equal to

49
00:02:29,159 --> 00:02:34,049
gamma times change in X...

50
00:02:40,709 --> 00:02:43,860
Let me factor out a negative beta.

51
00:02:43,860 --> 00:02:47,958
So I'l say, minus beta times,

52
00:02:50,698 --> 00:02:55,447
well then you're going to have C T final

53
00:02:57,957 --> 00:03:01,186
minus C T initial.

54
00:03:01,572 --> 00:03:04,092
And what's C T final minus C T initial?

55
00:03:04,092 --> 00:03:06,952
I think I'm skipping
too many steps already.

56
00:03:06,952 --> 00:03:09,532
Well that's just going
to be change in C T.

57
00:03:09,532 --> 00:03:13,053
So we get, this is all
going to be equal to

58
00:03:13,053 --> 00:03:16,432
gamma, our Lorentz
factor, times change in X

59
00:03:20,232 --> 00:03:24,680
minus beta times change in C T.

60
00:03:28,590 --> 00:03:29,990
And since C isn't changing,
it could also be viewed

61
00:03:29,990 --> 00:03:32,510
as C times change in T, either way.

62
00:03:32,510 --> 00:03:34,610
So, there you have it.

63
00:03:34,610 --> 00:03:36,750
Notice, it takes almost
the exact same form.

64
00:03:36,750 --> 00:03:40,590
X prime is equal to gamma
times X minus beta C T

65
00:03:40,590 --> 00:03:43,910
and change in X prime is going to be gamma

66
00:03:43,910 --> 00:03:48,890
times change in X minus
beta time change in C T.

67
00:03:48,890 --> 00:03:50,831
And, I'm not going to do it in this video,

68
00:03:50,831 --> 00:03:52,558
but you can make the exact same algebraic

69
00:03:52,558 --> 00:03:55,239
argument for your change in C T prime,

70
00:03:55,239 --> 00:03:57,519
as you'll see and I encourage
you to do this on your own,

71
00:03:57,519 --> 00:04:02,498
change in C T prime, which
you could also view as,

72
00:04:02,498 --> 00:04:06,999
since C isn't changing,
C times delta T prime,

73
00:04:06,999 --> 00:04:09,698
these are equivalent,
is going to be equal to

74
00:04:13,898 --> 00:04:18,898
gamma times change in C T

75
00:04:20,447 --> 00:04:25,004
minus beta times change in X,

76
00:04:25,440 --> 00:04:26,280
and I encourage you,

77
00:04:26,280 --> 00:04:28,100
right after this video,
actually do this one too.

78
00:04:28,100 --> 00:04:31,940
Delta X prime is going to be X prime final

79
00:04:31,940 --> 00:04:34,000
minus X prime initial and
then do what I just did here,

80
00:04:34,000 --> 00:04:35,300
just a little algebraic manipulation.

81
00:04:35,300 --> 00:04:37,960
You can make the same
exact argument over here

82
00:04:37,960 --> 00:04:41,220
to get to the result
that I just wrote down.

83
00:04:41,220 --> 00:04:43,960
You see, well our change
in C T prime is going to be

84
00:04:43,960 --> 00:04:47,500
C T prime final minus C T
prime initial and then you can

85
00:04:47,500 --> 00:04:49,200
substitute with this, do
a little bit of algebraic

86
00:04:49,200 --> 00:04:51,940
manipulation and you'll
get that right over there.

87
00:04:51,940 --> 00:04:53,360
And the whole reason I'm doing this is,

88
00:04:53,360 --> 00:04:54,760
well now, we can think in terms of change

89
00:04:54,760 --> 00:04:56,360
in the coordinates, which will allow us to

90
00:04:56,360 --> 00:04:57,920
think about what velocities would be

91
00:04:57,920 --> 00:05:00,500
in the different frames of reference,

92
00:05:00,500 --> 00:00:00,000
which is going to be pretty neat.