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PROFESSOR: OK, in that
case, let's go on.

9
00:00:24,090 --> 00:00:26,030
At the end of last
class, we were

10
00:00:26,030 --> 00:00:28,490
beginning to talk about
the Doppler shift.

11
00:00:28,490 --> 00:00:30,985
And we defined our terms.

12
00:00:30,985 --> 00:00:33,360
And I guess I'll repeat the
definitions on the blackboard

13
00:00:33,360 --> 00:00:34,780
here.

14
00:00:34,780 --> 00:00:43,240
We are talking initially about
the case where the observer is

15
00:00:43,240 --> 00:00:56,950
stationary and the source is
moving with velocity v. We

16
00:00:56,950 --> 00:00:59,540
are talking
initially about sound

17
00:00:59,540 --> 00:01:04,910
waves, waves which have a fixed
speed relative to some medium.

18
00:01:04,910 --> 00:01:08,265
And the speed relative to
the medium will be called u.

19
00:01:22,310 --> 00:01:24,810
v will be the
velocity of recession,

20
00:01:24,810 --> 00:01:25,855
as shown on the diagram.

21
00:01:36,920 --> 00:01:41,890
Delta t sub s, s for source,
will be the time interval

22
00:01:41,890 --> 00:01:50,880
between wave crests as measured
at the source-- so period

23
00:01:50,880 --> 00:01:55,765
of wave at source.

24
00:01:59,290 --> 00:02:07,920
And delta t sub o
will be the period

25
00:02:07,920 --> 00:02:12,700
of the wave at the observer.

26
00:02:18,740 --> 00:02:20,550
And what we're
trying to calculate

27
00:02:20,550 --> 00:02:22,455
is the relationship
between delta t sub o

28
00:02:22,455 --> 00:02:23,320
and delta t sub s.

29
00:02:26,140 --> 00:02:33,480
OK, at this point, I'd
like to go to the screen

30
00:02:33,480 --> 00:02:37,960
and go through the
different stages of what

31
00:02:37,960 --> 00:02:42,680
happens as this
process takes place.

32
00:02:42,680 --> 00:02:48,130
We start in frame
one with an observer

33
00:02:48,130 --> 00:02:50,900
in some location, a source
at some different location.

34
00:02:50,900 --> 00:02:52,390
Source is moving to the right.

35
00:02:52,390 --> 00:02:54,700
Source is emitting
the first wave crest

36
00:02:54,700 --> 00:02:57,620
in this slide number one.

37
00:02:57,620 --> 00:03:00,780
Nothing too interesting so far.

38
00:03:00,780 --> 00:03:05,100
Next picture, the source
emits the second wave crest.

39
00:03:05,100 --> 00:03:07,340
But meanwhile, the
source has moved.

40
00:03:07,340 --> 00:03:11,560
The time between wave
crests as seen by the source

41
00:03:11,560 --> 00:03:13,510
is delta t sub s.

42
00:03:13,510 --> 00:03:17,100
So the distance that the source
will move during that time

43
00:03:17,100 --> 00:03:20,200
integral is v times
delta t sub s.

44
00:03:20,200 --> 00:03:22,220
And we'll call that delta l.

45
00:03:22,220 --> 00:03:26,620
And this really is
the important slide.

46
00:03:26,620 --> 00:03:28,990
I think I have it
highlighted here.

47
00:03:28,990 --> 00:03:31,880
That picture is what really
counts for the whole Doppler

48
00:03:31,880 --> 00:03:32,560
shift.

49
00:03:32,560 --> 00:03:35,250
It says that the
second wave crest

50
00:03:35,250 --> 00:03:38,080
has to travel a little further
than the first wave crest

51
00:03:38,080 --> 00:03:39,705
by this amount delta l.

52
00:03:39,705 --> 00:03:41,820
So delta l will be
the crucial quantity

53
00:03:41,820 --> 00:03:43,630
that will control the answer.

54
00:03:50,520 --> 00:03:54,470
Third slide, the
waves have traveled.

55
00:03:54,470 --> 00:03:58,130
Now the first crest
in this third frame

56
00:03:58,130 --> 00:03:59,545
has just hit the observer.

57
00:04:03,740 --> 00:04:07,290
Next frame, the last
crest, the second crest,

58
00:04:07,290 --> 00:04:09,840
has hit the observer.

59
00:04:09,840 --> 00:04:15,210
And to figure out
the Doppler shift

60
00:04:15,210 --> 00:04:18,829
from those images,
all we have to do

61
00:04:18,829 --> 00:04:24,180
is realize that if both
objects were stationary,

62
00:04:24,180 --> 00:04:30,079
there'd be no time difference
between observation and source.

63
00:04:30,079 --> 00:04:32,370
Each could occur later by
some fixed amount, the amount

64
00:04:32,370 --> 00:04:34,800
of time it would take
the sound wave to travel.

65
00:04:34,800 --> 00:04:36,870
But they will occur
later by the same amount

66
00:04:36,870 --> 00:04:38,370
if there was no motion.

67
00:04:38,370 --> 00:04:44,610
So if there's no
motion, delta t sub o

68
00:04:44,610 --> 00:04:50,270
would be equal to delta t sub
s, if there was no motion.

69
00:04:50,270 --> 00:04:52,780
But because there
is motion, we said

70
00:04:52,780 --> 00:04:54,670
that the wave, the
second crest, is

71
00:04:54,670 --> 00:04:58,840
going to have to travel
further by the amount delta l.

72
00:04:58,840 --> 00:05:00,680
So it'll be delayed
by the amount of time

73
00:05:00,680 --> 00:05:02,900
it takes for the wave
to travel delta l.

74
00:05:02,900 --> 00:05:13,490
And that's just delta
l divided by v--

75
00:05:13,490 --> 00:05:21,025
so plus delta l divided by v.
And we know what delta l is.

76
00:05:25,150 --> 00:05:28,740
Delta l is just v
times delta t sub s.

77
00:05:32,810 --> 00:05:34,225
I'm sorry, this is divided by u.

78
00:05:39,050 --> 00:05:40,360
u is the speed of the wave.

79
00:05:40,360 --> 00:05:43,130
v is the speed of the source.

80
00:05:43,130 --> 00:05:46,830
So here we have in the
numerator v times delta t sub s,

81
00:05:46,830 --> 00:05:49,950
and the denominator is u.

82
00:05:49,950 --> 00:05:52,610
This is just what
we said delta l is.

83
00:05:52,610 --> 00:05:54,260
So this really is our result.

84
00:05:54,260 --> 00:05:59,350
It tells us what delta o is
in terms of delta t sub s.

85
00:05:59,350 --> 00:06:03,460
And we can solve for the ratio.

86
00:06:03,460 --> 00:06:07,330
It tells us that delta
t sub o over delta t

87
00:06:07,330 --> 00:06:17,780
sub s is equal to
1 plus v over u.

88
00:06:21,080 --> 00:06:23,330
Solve that equation.

89
00:06:23,330 --> 00:06:24,760
Now there's a
standard definition

90
00:06:24,760 --> 00:06:28,250
that's used to describe
redshifts, which

91
00:06:28,250 --> 00:06:33,230
is that this ratio, which is
also the ratio of wavelengths,

92
00:06:33,230 --> 00:06:35,230
which is how I
usually think of it,

93
00:06:35,230 --> 00:06:37,440
the wavelength at the
observer divided by the wave

94
00:06:37,440 --> 00:06:39,610
length at the source--
the wave length

95
00:06:39,610 --> 00:06:43,030
being disproportional
to the delta t.

96
00:06:43,030 --> 00:06:47,040
The wave ones are just the
wave speed times delta t.

97
00:06:47,040 --> 00:06:53,550
This is defined to
be 1 plus z where

98
00:06:53,550 --> 00:06:55,625
z is what's called the redshift.

99
00:06:58,370 --> 00:07:00,030
And the astronomers
take out the one

100
00:07:00,030 --> 00:07:02,990
so that if both objects are
stationary, z is equal to 0.

101
00:07:02,990 --> 00:07:05,370
That corresponds
to know redshift.

102
00:07:05,370 --> 00:07:08,170
That means the wave length
is the same at the source

103
00:07:08,170 --> 00:07:09,830
and at the observer.

104
00:07:09,830 --> 00:07:12,430
So the ratio of the
wavelength as its observed

105
00:07:12,430 --> 00:07:16,240
to the wavelength of the source
is what's called 1 plus z.

106
00:07:16,240 --> 00:07:20,000
So in this case, it
follows immediately

107
00:07:20,000 --> 00:07:25,410
that the redshift for this
case is just v over u.

108
00:07:30,640 --> 00:07:34,360
So maybe I'll just write that
again in a box and label it.

109
00:07:34,360 --> 00:07:39,710
z is equal to v
over u corresponds

110
00:07:39,710 --> 00:07:47,580
to the nonrelativistic case,
or the sound wave case,

111
00:07:47,580 --> 00:07:49,150
with the source moving.

112
00:07:56,450 --> 00:07:57,020
OK?

113
00:07:57,020 --> 00:07:58,830
Everybody on board?

114
00:07:58,830 --> 00:08:00,530
Any questions?

115
00:08:00,530 --> 00:08:02,910
OK, straightforward
enough, I think.

116
00:08:02,910 --> 00:08:09,430
OK, now we will go on to do the
alternative simple case, where

117
00:08:09,430 --> 00:08:14,120
the observer is moving and
the source is stationary.

118
00:08:19,120 --> 00:08:23,520
So keep the source on the right
and the observer on the left.

119
00:08:23,520 --> 00:08:27,930
But this time it's the observer
that's moving again with speed

120
00:08:27,930 --> 00:08:31,700
v. So v is always for these
two cases in my notation

121
00:08:31,700 --> 00:08:34,995
the relative velocity between
the source and the observer.

122
00:08:38,041 --> 00:08:39,790
So now we have a new
sequence of pictures.

123
00:08:43,000 --> 00:08:45,700
So first picture again
is fairly trivial.

124
00:08:45,700 --> 00:08:49,185
The first wave crest is
being emitted by the source.

125
00:08:53,850 --> 00:08:56,460
Picture number two,
the second wave crest

126
00:08:56,460 --> 00:08:58,340
is emitted by the source.

127
00:09:02,740 --> 00:09:05,210
Picture number three,
the first wave crest

128
00:09:05,210 --> 00:09:06,430
arrives at the source.

129
00:09:09,290 --> 00:09:13,690
And picture number four
now, the second wave crest

130
00:09:13,690 --> 00:09:15,900
arrives at the source.

131
00:09:15,900 --> 00:09:20,300
And for this case,
it is this last frame

132
00:09:20,300 --> 00:09:22,470
where all the action is.

133
00:09:22,470 --> 00:09:27,150
The point is that between the
time when the first crest hits

134
00:09:27,150 --> 00:09:29,650
the source and the time
when the second crest hits

135
00:09:29,650 --> 00:09:33,490
the source, that is the time
between the third and fourth

136
00:09:33,490 --> 00:09:36,332
frame, the source has moved.

137
00:09:36,332 --> 00:09:37,790
And it's moved by
distance which is

138
00:09:37,790 --> 00:09:41,040
v times the time
between those images.

139
00:09:41,040 --> 00:09:42,630
And the time
between these images

140
00:09:42,630 --> 00:09:46,100
is just the time that
the source experiences

141
00:09:46,100 --> 00:09:48,531
between the receipt
of the two waves.

142
00:09:48,531 --> 00:09:50,780
And it's therefore what we've
been calling delta t sub

143
00:09:50,780 --> 00:09:54,130
o, the time interval as
measured by the observer.

144
00:09:54,130 --> 00:09:58,960
So the distance traveled
is just v times t sub o.

145
00:09:58,960 --> 00:10:02,760
And it's that last frame inside
that box where all the action

146
00:10:02,760 --> 00:10:05,580
happens for determining the
answer for this problem.

147
00:10:08,320 --> 00:10:13,210
So we can put that
into equations also.

148
00:10:13,210 --> 00:10:17,440
This time it's slightly
more complicated.

149
00:10:17,440 --> 00:10:19,510
It starts basically
with the same idea.

150
00:10:19,510 --> 00:10:23,560
Delta t sub o is
equal to delta t sub

151
00:10:23,560 --> 00:10:25,980
s, which is what it would
be if there was no motion.

152
00:10:25,980 --> 00:10:28,620
But it's a little bit longer
because of the extra distance

153
00:10:28,620 --> 00:10:31,450
the second pulse travels.

154
00:10:31,450 --> 00:10:34,710
And that extra distance
is again called delta l.

155
00:10:34,710 --> 00:10:40,370
So the time delay is again delta
l divided by u, the wave speed.

156
00:10:40,370 --> 00:10:43,030
But this time we have a
different formula for delta l.

157
00:10:43,030 --> 00:10:46,660
This time delta l is v times
delta t sub 0, instead of

158
00:10:46,660 --> 00:10:50,285
last time it was v
times delta t sub s.

159
00:11:06,260 --> 00:11:08,980
So this time the equation
is ever so slightly more

160
00:11:08,980 --> 00:11:10,950
complicated, because
delta t sub 0

161
00:11:10,950 --> 00:11:13,042
appears on both sides
of the equation.

162
00:11:13,042 --> 00:11:14,500
Last time we
immediately wrote down

163
00:11:14,500 --> 00:11:17,620
an equation for delta t sub o.

164
00:11:17,620 --> 00:11:20,710
But nonetheless, this is one
equation with one unknown,

165
00:11:20,710 --> 00:11:24,730
so of course it's trivial to
solve it for delta t sub o.

166
00:11:24,730 --> 00:11:29,820
And when we do that,
we get delta t sub o.

167
00:11:29,820 --> 00:11:33,704
I'll maybe divide
by delta t sub s.

168
00:11:33,704 --> 00:11:35,620
By just doing a little
bit of algebra on that,

169
00:11:35,620 --> 00:11:40,850
you discover it's 1
minus u over v inverse.

170
00:11:47,500 --> 00:11:57,170
And then z, the redshift, is
delta t sub 0 over delta t sub

171
00:11:57,170 --> 00:12:01,130
s minus 1 by definition.

172
00:12:01,130 --> 00:12:03,620
AUDIENCE: Is that
1 minus v over u?

173
00:12:03,620 --> 00:12:05,370
PROFESSOR: Oh, do I
have it wrong here?

174
00:12:05,370 --> 00:12:06,510
Yes, it's 1 minus v over u.

175
00:12:06,510 --> 00:12:07,420
Thank you very much.

176
00:12:16,450 --> 00:12:19,110
Right, coming from
the v over u here,

177
00:12:19,110 --> 00:12:20,790
obviously, as you noticed.

178
00:12:24,770 --> 00:12:29,220
OK, so to write down a
final equation for z,

179
00:12:29,220 --> 00:12:33,960
it's delta t0 over
delta t sub s minus 1.

180
00:12:33,960 --> 00:12:42,240
So it's 1 minus v over
u inverse minus 1.

181
00:12:42,240 --> 00:12:45,150
And then just putting things
over our common denominator,

182
00:12:45,150 --> 00:12:54,390
it ends up being v over u
divided by 1 minus v over u.

183
00:12:54,390 --> 00:12:58,480
And that then is
our final answer.

184
00:12:58,480 --> 00:13:02,680
So this now is the answer for
the nonrelativistic case--

185
00:13:02,680 --> 00:13:09,550
again, we haven't
done relativity yet--

186
00:13:09,550 --> 00:13:11,040
and the observer moving.

187
00:13:37,434 --> 00:13:41,380
OK, now it's worth
pointing out that when

188
00:13:41,380 --> 00:13:44,182
velocity is small compared
to the wave speed,

189
00:13:44,182 --> 00:13:45,640
as is often the
case if we're going

190
00:13:45,640 --> 00:13:46,820
to be talking about
light which we'll

191
00:13:46,820 --> 00:13:48,500
be talking about in
a minute, but could

192
00:13:48,500 --> 00:13:50,880
be the case with respect
to sound as well.

193
00:13:50,880 --> 00:13:53,730
Then these two formulas
are almost the same.

194
00:13:53,730 --> 00:13:56,024
They're both
proportional to v over u,

195
00:13:56,024 --> 00:13:57,940
when I'm talking about
the case where v over u

196
00:13:57,940 --> 00:14:00,900
is a small quantity, where
the motion is slow compared

197
00:14:00,900 --> 00:14:03,880
to the sound speed
or wave speed.

198
00:14:03,880 --> 00:14:06,800
And the only difference
is that denominator.

199
00:14:06,800 --> 00:14:09,810
Here we have a denominator
of 1 minus v over u.

200
00:14:09,810 --> 00:14:12,690
And here the v over
u is the whole show.

201
00:14:12,690 --> 00:14:15,350
There's no denominator.

202
00:14:15,350 --> 00:14:19,500
And if v over u is small, the
denominator is close to 1.

203
00:14:19,500 --> 00:14:21,970
So the two answers are
going to be almost the same.

204
00:14:21,970 --> 00:14:28,390
And one can describe that a
little bit more succinctly

205
00:14:28,390 --> 00:14:36,800
perhaps, by saying that z with
the observer moving minus z

206
00:14:36,800 --> 00:14:52,680
with the source moving is equal
to v over u quantity squared

207
00:14:52,680 --> 00:14:57,360
times 1 minus v over
u in the denominator.

208
00:14:57,360 --> 00:14:59,695
Just a little bit of
algebra to get that result.

209
00:14:59,695 --> 00:15:01,070
And what that
shows explicitly is

210
00:15:01,070 --> 00:15:02,890
that second order in v over u.

211
00:15:02,890 --> 00:15:06,070
It's proportional to v over
u squared, not to v over u.

212
00:15:06,070 --> 00:15:09,420
So if v over u were
a part in 1,000,

213
00:15:09,420 --> 00:15:12,020
the difference would
be a part in a million.

214
00:15:12,020 --> 00:15:13,877
So for slow
velocities, it doesn't

215
00:15:13,877 --> 00:15:15,710
matter whether it's the
source that's moving

216
00:15:15,710 --> 00:15:17,156
or the observer that's moving.

217
00:15:17,156 --> 00:15:18,530
But these answers
can, of course,

218
00:15:18,530 --> 00:15:24,080
be very different if the
velocity v is comparable to u.

219
00:15:27,090 --> 00:15:30,130
OK, any questions about that?

220
00:15:30,130 --> 00:15:30,946
Yes?

221
00:15:30,946 --> 00:15:33,360
AUDIENCE: Does this violate
Galilean relativity?

222
00:15:33,360 --> 00:15:35,570
PROFESSOR: Does it violate
Galilean relativity?

223
00:15:35,570 --> 00:15:37,050
Not really.

224
00:15:37,050 --> 00:15:41,370
Although, one has to maybe be
careful about how one defines

225
00:15:41,370 --> 00:15:42,960
Galilean relativity.

226
00:15:42,960 --> 00:15:45,070
What makes it legitimate
as a classical mechanical

227
00:15:45,070 --> 00:15:48,900
calculation is that
unshown, but crucially

228
00:15:48,900 --> 00:15:51,350
important on the
blackboard, is the air

229
00:15:51,350 --> 00:15:54,120
that the sound wave
is moving through.

230
00:15:54,120 --> 00:15:56,310
So in one of these pictures,
the air was at rest.

231
00:15:56,310 --> 00:15:59,170
The other picture,
the air was moving.

232
00:15:59,170 --> 00:16:01,780
Or rather, well,
I said that wrong.

233
00:16:01,780 --> 00:16:03,680
The air was at rest
in both pictures.

234
00:16:03,680 --> 00:16:05,740
But if you want to make
a Galilean transformation

235
00:16:05,740 --> 00:16:07,645
to relate one
picture to the other,

236
00:16:07,645 --> 00:16:09,770
then after you've made the
Galilean transformation,

237
00:16:09,770 --> 00:16:11,728
the air would be moving
and it would not really

238
00:16:11,728 --> 00:16:13,220
be the same picture.

239
00:16:13,220 --> 00:16:16,056
So it is all consistent
with Galilean relativity.

240
00:16:16,056 --> 00:16:17,680
And one has to remember
that the air is

241
00:16:17,680 --> 00:16:20,050
playing a crucial role here.

242
00:16:20,050 --> 00:16:23,420
When we say that the observer
or the source is at rest,

243
00:16:23,420 --> 00:16:24,995
the full sentence
really is that it's

244
00:16:24,995 --> 00:16:28,330
at rest relative to the medium
that's transporting the wave.

245
00:16:32,690 --> 00:16:34,670
Any other questions?

246
00:16:34,670 --> 00:16:35,943
Yes?

247
00:16:35,943 --> 00:16:37,817
AUDIENCE: This is not
really a question, just

248
00:16:37,817 --> 00:16:38,775
kind of an observation.

249
00:16:38,775 --> 00:16:41,590
I just find it interesting
that in the case

250
00:16:41,590 --> 00:16:46,935
if v is greater than u that
this one is always positive,

251
00:16:46,935 --> 00:16:47,435
no problem.

252
00:16:47,435 --> 00:16:49,566
But if v is greater than
u in the other case,

253
00:16:49,566 --> 00:16:52,100
then you have a
negative number, which,

254
00:16:52,100 --> 00:16:57,900
I don't know if that--
I just find that weird.

255
00:16:57,900 --> 00:16:59,390
PROFESSOR: Right, let me think.

256
00:17:03,500 --> 00:17:13,079
If v is greater than u, then
the observer moving case

257
00:17:13,079 --> 00:17:14,150
becomes negative.

258
00:17:14,150 --> 00:17:18,360
And it presumably means
the wave never reaches it.

259
00:17:18,360 --> 00:17:21,180
If the observer is moving
faster than the wave speed,

260
00:17:21,180 --> 00:17:22,510
the wave never reaches them.

261
00:17:22,510 --> 00:17:25,079
And that's why the
answer is peculiar.

262
00:17:25,079 --> 00:17:27,230
If the source is moving
faster than the wave speed,

263
00:17:27,230 --> 00:17:29,570
the wave still
reaches the observer.

264
00:17:29,570 --> 00:17:32,970
So there is a nontrivial and
believable answer in that case.

265
00:17:38,380 --> 00:17:39,620
OK.

266
00:17:39,620 --> 00:17:46,320
What we're going to move on to
now is the relativistic case.

267
00:17:46,320 --> 00:17:52,510
And as I think I described in
the course handouts, relativity

268
00:17:52,510 --> 00:17:54,500
is has sort of an odd
role in this class.

269
00:17:54,500 --> 00:17:56,890
We need a little
bit of relativity.

270
00:17:56,890 --> 00:17:59,780
But at the same time, there
are the courses in relativity.

271
00:17:59,780 --> 00:18:02,570
So I don't want this to
be a course in relativity.

272
00:18:02,570 --> 00:18:04,200
In the old days,
when there weren't so

273
00:18:04,200 --> 00:18:05,900
many other courses
in relativity,

274
00:18:05,900 --> 00:18:08,110
we did in fact spend two
weeks of this course doing

275
00:18:08,110 --> 00:18:09,187
special relativity.

276
00:18:09,187 --> 00:18:11,020
But I don't think that's
worthwhile anymore.

277
00:18:11,020 --> 00:18:12,800
Well, I'll maybe ask
for a show of hands.

278
00:18:12,800 --> 00:18:14,258
How many of you
have had relativity

279
00:18:14,258 --> 00:18:15,400
in some other course?

280
00:18:15,400 --> 00:18:17,687
That's what I thought, most
of you, perhaps not all.

281
00:18:17,687 --> 00:18:19,520
Well, maybe I should
ask the other question.

282
00:18:19,520 --> 00:18:22,260
How many of you have
not had relativity?

283
00:18:22,260 --> 00:18:23,581
OK, a number.

284
00:18:23,581 --> 00:18:25,330
So I do want to make
the course completely

285
00:18:25,330 --> 00:18:28,380
intelligible to the people
in that second group.

286
00:18:28,380 --> 00:18:31,940
Special relativity is not a
prerequisite for this class.

287
00:18:31,940 --> 00:18:35,480
So what my goal will be
to tell you enough special

288
00:18:35,480 --> 00:18:39,270
relativity so that you'll be
able to follow what comes next.

289
00:18:39,270 --> 00:18:42,740
But I will not be
driving those results.

290
00:18:42,740 --> 00:18:44,765
I'll just leave
for other courses

291
00:18:44,765 --> 00:18:46,140
for people who
want to take them.

292
00:18:46,140 --> 00:18:48,306
And if you don't want to
take them, that's fine too.

293
00:18:48,306 --> 00:18:51,000
But I want to make
this course coherent.

294
00:18:51,000 --> 00:18:54,870
So what we're going to do
is discuss the consequences

295
00:18:54,870 --> 00:18:56,990
of special relativity
without trying

296
00:18:56,990 --> 00:18:58,950
to relate those
consequences directly

297
00:18:58,950 --> 00:19:02,860
to the underlying ideas
of special relativity.

298
00:19:02,860 --> 00:19:08,230
I will, however, mention where
special relativity comes from.

299
00:19:08,230 --> 00:19:12,580
It arose in the mind
of Albert Einstein,

300
00:19:12,580 --> 00:19:18,790
because he realized that the
physics that we knew well,

301
00:19:18,790 --> 00:19:21,560
basically Newtonian
physics at that time,

302
00:19:21,560 --> 00:19:25,080
possess this property, Galilean
relativity, which came up

303
00:19:25,080 --> 00:19:27,190
in a question just a minute ago.

304
00:19:27,190 --> 00:19:29,330
Galilean relativity
says that if you

305
00:19:29,330 --> 00:19:33,000
look at any given physical
process in a frame that's

306
00:19:33,000 --> 00:19:36,780
moving at a uniform velocity
relative to the first frame

307
00:19:36,780 --> 00:19:39,240
that you use to describe
it, it should also

308
00:19:39,240 --> 00:19:43,200
be consistent with the laws of
physics in the second frame.

309
00:19:43,200 --> 00:19:46,460
Incidentally, I-- maybe I'm
more ignorant about history

310
00:19:46,460 --> 00:19:48,820
than most-- only learned
a few years ago what

311
00:19:48,820 --> 00:19:51,441
Galileo had to do with this.

312
00:19:51,441 --> 00:19:53,190
It did actually play
a very important role

313
00:19:53,190 --> 00:19:54,900
in the history of
Galileo in the physics

314
00:19:54,900 --> 00:19:57,720
that he was debating about.

315
00:19:57,720 --> 00:19:59,782
A key issue in the
time of Galileo

316
00:19:59,782 --> 00:20:02,240
was whether the earth moved
around the sun or the sun moved

317
00:20:02,240 --> 00:20:04,750
around the earth.

318
00:20:04,750 --> 00:20:11,250
And that was something Galileo
was intensely involved in.

319
00:20:11,250 --> 00:20:13,060
And one of the
arguments that said

320
00:20:13,060 --> 00:20:14,710
that it must be the
sun that moves--

321
00:20:14,710 --> 00:20:16,277
it can't be the
earth that moves--

322
00:20:16,277 --> 00:20:17,860
was that if it's the
earth that moves,

323
00:20:17,860 --> 00:20:20,184
it means we move at
very high velocity.

324
00:20:20,184 --> 00:20:21,850
The velocity of the
earth around the sun

325
00:20:21,850 --> 00:20:24,870
is high by ordinary standards.

326
00:20:24,870 --> 00:20:28,680
And obviously, we'd feel
that motion, people thought.

327
00:20:28,680 --> 00:20:30,322
So it was a proof
that it must be

328
00:20:30,322 --> 00:20:32,280
that the earth is stationary
and the sun moves.

329
00:20:32,280 --> 00:20:35,250
Because otherwise, we'd feel
the effect of this high velocity

330
00:20:35,250 --> 00:20:36,610
motion.

331
00:20:36,610 --> 00:20:38,560
So it was crucial to
Galileo's point of view

332
00:20:38,560 --> 00:20:40,030
that it was really
the earth that

333
00:20:40,030 --> 00:20:42,980
was moving that you
don't detect motion.

334
00:20:42,980 --> 00:20:45,590
If you're in uniform
motion, the laws of physics

335
00:20:45,590 --> 00:20:49,030
are exactly the same as they
would be if you were at rest.

336
00:20:49,030 --> 00:20:51,680
And that's basically
what Galilean relativity

337
00:20:51,680 --> 00:20:52,490
is all about.

338
00:20:52,490 --> 00:20:55,890
And it was in fact very
clearly enunciated by Galileo

339
00:20:55,890 --> 00:20:58,080
in his writings.

340
00:20:58,080 --> 00:21:02,770
So that was the
case for mechanics.

341
00:21:02,770 --> 00:21:06,000
But at the same
time, in the 1860s,

342
00:21:06,000 --> 00:21:07,820
Maxwell invented
Maxwell's equations,

343
00:21:07,820 --> 00:21:10,860
or completed them is maybe
a more accurate description

344
00:21:10,860 --> 00:21:13,530
for what Maxwell really did.

345
00:21:13,530 --> 00:21:16,370
Most of those equations
already existed.

346
00:21:16,370 --> 00:21:19,230
And a prediction of
Maxwell's equations

347
00:21:19,230 --> 00:21:21,800
is that light travels at
some fixed speed, which

348
00:21:21,800 --> 00:21:24,630
could be calculated in terms
of epsilon naught and mu

349
00:21:24,630 --> 00:21:27,530
naught that appeared in
those equations, a speed

350
00:21:27,530 --> 00:21:29,790
that we call c.

351
00:21:29,790 --> 00:21:31,910
Now if light travels
at speed c, it

352
00:21:31,910 --> 00:21:34,380
would mean that if you
got into a spaceship

353
00:21:34,380 --> 00:21:37,610
and chased a light beam, it'd
say half the speed of light.

354
00:21:37,610 --> 00:21:40,537
The implication of the physics
that was known at the time

355
00:21:40,537 --> 00:21:42,370
would have been that,
from the point of view

356
00:21:42,370 --> 00:21:45,840
of that spaceship traveling
at half the speed of light,

357
00:21:45,840 --> 00:21:47,690
the light pulse would
only be receding

358
00:21:47,690 --> 00:21:49,040
at half the speed of light.

359
00:21:49,040 --> 00:21:52,060
You would have half
caught up with it.

360
00:21:52,060 --> 00:21:55,560
But that would mean that from
the frame of this rapidly

361
00:21:55,560 --> 00:21:57,680
moving spaceship,
the laws of physics

362
00:21:57,680 --> 00:21:58,860
must somehow be different.

363
00:21:58,860 --> 00:22:03,080
Maxwell's equations must not
hold in their standard form.

364
00:22:03,080 --> 00:22:08,070
So there was this tension
between Maxwell and Newton,

365
00:22:08,070 --> 00:22:10,180
if you like.

366
00:22:10,180 --> 00:22:12,150
The tension was not
a contradiction.

367
00:22:12,150 --> 00:22:13,850
It would be perfectly
possible for there

368
00:22:13,850 --> 00:22:17,600
to be a fixed frame in which
Maxwell's equations had

369
00:22:17,600 --> 00:22:19,230
their simple form.

370
00:22:19,230 --> 00:22:20,910
But Newton's equations
could perhaps

371
00:22:20,910 --> 00:22:23,280
have the same form
in all frames.

372
00:22:23,280 --> 00:22:26,860
And that in fact was what
people thought at the time.

373
00:22:26,860 --> 00:22:31,570
To account for this situation,
physicists invented the idea

374
00:22:31,570 --> 00:22:35,210
of an ether which was a medium
through which light waves

375
00:22:35,210 --> 00:22:39,910
travelled, similar to the air
in which sound waves traveled.

376
00:22:39,910 --> 00:22:44,490
And the frame in which Maxwell's
equations had their simple form

377
00:22:44,490 --> 00:22:46,870
was the rest frame of the ether.

378
00:22:46,870 --> 00:22:49,790
And if you moved relative
ether, the equations

379
00:22:49,790 --> 00:22:50,860
would be different.

380
00:22:50,860 --> 00:22:54,704
And that's what people
thought in 1904.

381
00:22:54,704 --> 00:22:57,790
And it was a consistent
point of view,

382
00:22:57,790 --> 00:23:00,240
but it meant that there
was this dichotomy

383
00:23:00,240 --> 00:23:05,360
between electromagnetism
and mechanics.

384
00:23:05,360 --> 00:23:10,000
So Einstein thought maybe
physics is not so sloppy.

385
00:23:10,000 --> 00:23:14,520
Maybe there's a more elegant
way which all this plays out.

386
00:23:14,520 --> 00:23:19,290
And he realized that if you
modified the equations that

387
00:23:19,290 --> 00:23:23,310
are used to transform between
one frame and another,

388
00:23:23,310 --> 00:23:27,555
you could make Maxwell's
equations frame invariant.

389
00:23:27,555 --> 00:23:29,430
You could make it so
that Maxwell's equations

390
00:23:29,430 --> 00:23:31,390
are valid in all frames.

391
00:23:31,390 --> 00:23:34,240
And if we go back to our
example of the spaceship chasing

392
00:23:34,240 --> 00:23:38,100
the light beam, with these
new transformation equations

393
00:23:38,100 --> 00:23:41,780
that Einstein suggested,
it would turn out,

394
00:23:41,780 --> 00:23:44,660
even though it's very
contrary to intuition,

395
00:23:44,660 --> 00:23:47,752
that when the spaceship measures
the speed of the light pulse,

396
00:23:47,752 --> 00:23:50,210
it would still measure that
the light pulse was moving away

397
00:23:50,210 --> 00:23:54,360
at speed c, even though it
had moved at half c trying

398
00:23:54,360 --> 00:23:56,634
to catch up with
the light pulse.

399
00:23:56,634 --> 00:23:58,300
So it's not obvious
how that can happen.

400
00:23:58,300 --> 00:23:59,591
But it turns out it can happen.

401
00:23:59,591 --> 00:24:02,010
We'll be talking a little bit
more about how it happens.

402
00:24:02,010 --> 00:24:03,970
And that was basically
Einstein's proposal.

403
00:24:03,970 --> 00:24:06,580
It was a proposal that
there is no ether,

404
00:24:06,580 --> 00:24:10,180
that the laws of physics both
electromagnetism and mechanics

405
00:24:10,180 --> 00:24:12,080
are the same in all frames.

406
00:24:12,080 --> 00:24:14,890
And in order to do
that, he had to say

407
00:24:14,890 --> 00:24:17,150
that the equations
of transformation

408
00:24:17,150 --> 00:24:19,270
between one frame
and another are

409
00:24:19,270 --> 00:24:22,560
different from what
Galileo believed.

410
00:24:22,560 --> 00:24:25,640
So these are what we call
the Lorentz transformations.

411
00:24:25,640 --> 00:24:27,526
We might write them down
later in the course,

412
00:24:27,526 --> 00:24:29,400
but we're not going to
write them down today.

413
00:24:29,400 --> 00:24:32,800
But what goes into them
are three physical effects,

414
00:24:32,800 --> 00:24:34,840
which we will talk about.

415
00:24:34,840 --> 00:24:36,810
If we're talking about
the time dilation,

416
00:24:36,810 --> 00:24:39,430
we only need one of
those three effects.

417
00:24:39,430 --> 00:24:42,920
So I'm going to start by just
discussing that for a minute,

418
00:24:42,920 --> 00:24:45,940
and then we'll come back at the
end of the class, or perhaps

419
00:24:45,940 --> 00:24:48,600
next class, depending
on how timing works out,

420
00:24:48,600 --> 00:24:51,830
to discuss the other
two primary effects that

421
00:24:51,830 --> 00:24:54,500
are needed to make up the
theory of special relativity

422
00:24:54,500 --> 00:24:57,329
and explain how it could
be that the speed of light

423
00:24:57,329 --> 00:24:58,870
could look the same
to all observers,

424
00:24:58,870 --> 00:25:02,870
even if we're talking about
observers that might be moving.

425
00:25:02,870 --> 00:25:06,850
OK, so time dilation
is the simple statement

426
00:25:06,850 --> 00:25:11,830
that if I were to
watch a moving clock,

427
00:25:11,830 --> 00:25:17,490
the moving clock would appear
to me to be running slower.

428
00:25:17,490 --> 00:25:20,780
I'll just mention
for a moment now

429
00:25:20,780 --> 00:25:22,975
that I put the word
appear in quotation marks.

430
00:25:22,975 --> 00:25:25,350
That means we're going to come
back and discuss in detail

431
00:25:25,350 --> 00:25:27,920
what is meant by
the word appear.

432
00:25:27,920 --> 00:25:31,470
But to finish the
sentence first,

433
00:25:31,470 --> 00:25:36,450
the moving clock would appear
to me in my reference frame

434
00:25:36,450 --> 00:25:38,880
to always be running slower.

435
00:25:38,880 --> 00:25:42,930
And so by a very predictable
amount, a famous expression

436
00:25:42,930 --> 00:25:45,970
in special relativity,
gamma, where

437
00:25:45,970 --> 00:25:48,470
gamma is 1 over the
square root of 1

438
00:25:48,470 --> 00:25:52,715
minus beta squared, where beta
is just an abbreviation for v

439
00:25:52,715 --> 00:25:54,830
over c, the velocity
of the clock

440
00:25:54,830 --> 00:25:57,270
divided by the speed of light.

441
00:25:57,270 --> 00:26:00,550
So as long as v over c is
small, this is a small effect.

442
00:26:00,550 --> 00:26:01,850
Gamma is near 1.

443
00:26:01,850 --> 00:26:03,347
And running slowly
by a factor of 1

444
00:26:03,347 --> 00:26:04,760
means not running slowly at all.

445
00:26:04,760 --> 00:26:09,210
So the fact it was near 1
means it's a very small effect.

446
00:26:09,210 --> 00:26:12,830
But moving clocks will always
appear to be running slower.

447
00:26:12,830 --> 00:26:16,010
That's one of these three
effects of special relativity

448
00:26:16,010 --> 00:26:20,910
that we'll be discussing in the
course of lecture notes one.

449
00:26:20,910 --> 00:26:22,990
Now let me come back now
to talk about this word

450
00:26:22,990 --> 00:26:26,695
"appear," because that's
a little bit subtle.

451
00:26:30,680 --> 00:26:35,320
Might just add that
there was a series-- this

452
00:26:35,320 --> 00:26:38,290
is just an aside-- but I
guess broadcast last year

453
00:26:38,290 --> 00:26:41,080
there was a four-part
series of Brian Greene's

454
00:26:41,080 --> 00:26:44,820
Fabric of the Cosmos that
was broadcast on PBS.

455
00:26:44,820 --> 00:26:48,280
And the interesting thing about
that, which is relevant here,

456
00:26:48,280 --> 00:26:52,220
is that he tried to
illustrate time dilation.

457
00:26:52,220 --> 00:26:57,560
And he did it by having sort
of a parable of a person

458
00:26:57,560 --> 00:26:59,750
sitting in a chair
and somebody else

459
00:26:59,750 --> 00:27:01,960
carrying a clock over
his head, walking

460
00:27:01,960 --> 00:27:04,140
towards the person
sitting in the chair.

461
00:27:04,140 --> 00:27:07,130
And the camera showed what the
person sitting in the chair

462
00:27:07,130 --> 00:27:10,360
would see and showed the
clock running slowly.

463
00:27:10,360 --> 00:27:12,150
That's wrong.

464
00:27:12,150 --> 00:27:13,920
It's not what he
would actually see.

465
00:27:13,920 --> 00:27:17,490
And that's the crucial issue
of the word "appear" here.

466
00:27:17,490 --> 00:27:21,290
When we say that the clock
appears to run slowly,

467
00:27:21,290 --> 00:27:25,510
we're not talking about what
an observer would actually see.

468
00:27:25,510 --> 00:27:28,479
The complication
of literally seeing

469
00:27:28,479 --> 00:27:30,520
is that when you see
something, what you're doing

470
00:27:30,520 --> 00:27:33,990
is you're measuring the
light pulses as they arrive

471
00:27:33,990 --> 00:27:36,180
at your eyes at a given time.

472
00:27:36,180 --> 00:27:39,547
And since light has
a finite travel time,

473
00:27:39,547 --> 00:27:41,380
it means that you're
seeing different things

474
00:27:41,380 --> 00:27:43,480
at different times.

475
00:27:43,480 --> 00:27:45,920
In particular, if there's
a large object coming

476
00:27:45,920 --> 00:27:50,120
towards you, say this laser
pointer coming towards me,

477
00:27:50,120 --> 00:27:52,860
I would be seeing
the front of it

478
00:27:52,860 --> 00:27:56,840
at an earlier time than I
would be seeing the back of it.

479
00:27:56,840 --> 00:27:58,281
The other way around, actually.

480
00:27:58,281 --> 00:28:00,030
I'd be seeing the back
at an earlier time,

481
00:28:00,030 --> 00:28:02,238
because the light that leaves
here at an earlier time

482
00:28:02,238 --> 00:28:04,720
would take longer to
reach my eye, reach my eye

483
00:28:04,720 --> 00:28:09,230
the same time as light which
leaves the front of the object

484
00:28:09,230 --> 00:28:10,460
later.

485
00:28:10,460 --> 00:28:12,570
So the point is that as
this is coming toward me,

486
00:28:12,570 --> 00:28:14,630
I'm seeing different
pieces of it

487
00:28:14,630 --> 00:28:18,570
at different times in terms
of the actual existence

488
00:28:18,570 --> 00:28:21,730
of this laser pointer.

489
00:28:21,730 --> 00:28:24,260
And that makes
things complicated.

490
00:28:24,260 --> 00:28:26,940
So what you actually see
when you take into account

491
00:28:26,940 --> 00:28:29,226
special relativity is
fairly complicated.

492
00:28:29,226 --> 00:28:30,600
You can calculate
it, but there's

493
00:28:30,600 --> 00:28:31,935
no simple expression for it.

494
00:28:31,935 --> 00:28:33,560
You really just have
to calculate point

495
00:28:33,560 --> 00:28:37,090
by point what you'll be seeing
for every part of the object

496
00:28:37,090 --> 00:28:39,980
at a given time,
nothing very simple.

497
00:28:39,980 --> 00:28:41,730
So the simple expression,
which just says,

498
00:28:41,730 --> 00:28:43,950
clocks run slowly by
a factor of gamma,

499
00:28:43,950 --> 00:28:45,610
and we'll learn
later e expressions

500
00:28:45,610 --> 00:28:47,880
about how things
contract and how

501
00:28:47,880 --> 00:28:50,680
simultaneity changes,
those simple expressions

502
00:28:50,680 --> 00:28:54,630
are not based on what any
observer would actually see.

503
00:28:54,630 --> 00:28:57,180
But rather, they're
based on what

504
00:28:57,180 --> 00:29:00,310
ends up giving a simpler
picture, a picture in which you

505
00:29:00,310 --> 00:29:02,720
imagine that what
we're discussing

506
00:29:02,720 --> 00:29:05,030
is not what an individual
sees but rather what

507
00:29:05,030 --> 00:29:07,109
a frame of reference sees.

508
00:29:07,109 --> 00:29:09,150
So we're talking not about
what the observer sees

509
00:29:09,150 --> 00:29:13,170
but rather what is seen in the
observer's frame of reference.

510
00:29:13,170 --> 00:29:16,030
And a frame of
reference, I think

511
00:29:16,030 --> 00:29:18,110
you could think of it
pretty concretely, as kind

512
00:29:18,110 --> 00:29:22,170
of a structure of rulers
connected together

513
00:29:22,170 --> 00:29:24,690
to each other to
form a grid of rulers

514
00:29:24,690 --> 00:29:29,300
and with clocks located
everywhere along this grid.

515
00:29:29,300 --> 00:29:32,440
So all the observations
are made locally.

516
00:29:32,440 --> 00:29:35,440
That is, if you want to measure
a time in a given reference

517
00:29:35,440 --> 00:29:37,590
frame, you don't use a
central clock, waiting

518
00:29:37,590 --> 00:29:39,725
for the light pulse to
reach that central clock.

519
00:29:39,725 --> 00:29:41,350
Rather, you measure
the reference frame

520
00:29:41,350 --> 00:29:43,270
is filled with
clocks, all of which

521
00:29:43,270 --> 00:29:45,590
have been synchronized
to start with.

522
00:29:45,590 --> 00:29:47,970
And if you want to know
what time an event happens,

523
00:29:47,970 --> 00:29:50,480
you look at the clock
that's next to it.

524
00:29:50,480 --> 00:29:54,739
And that clock tells you what
time that event happened.

525
00:29:54,739 --> 00:29:56,280
So that's typically
what we draw when

526
00:29:56,280 --> 00:29:58,250
we draw a coordinate
systems and so on.

527
00:29:58,250 --> 00:30:01,802
It's really the way
we normally think.

528
00:30:01,802 --> 00:30:03,260
The point is,
though, if you really

529
00:30:03,260 --> 00:30:05,360
want to think what one
observer would see,

530
00:30:05,360 --> 00:30:06,550
it's more complicated.

531
00:30:06,550 --> 00:30:09,850
Then you have to take into
account the light travel time.

532
00:30:09,850 --> 00:30:12,190
So it's only after you take
out the way travel time

533
00:30:12,190 --> 00:30:15,550
and calculate how local
clocks would compare

534
00:30:15,550 --> 00:30:18,560
that you see this time
dilation in the simple form,

535
00:30:18,560 --> 00:30:20,959
that the clock
always runs slower.

536
00:30:20,959 --> 00:30:22,375
So in particular,
for this example

537
00:30:22,375 --> 00:30:24,090
of the person sitting
in a chair and the clock

538
00:30:24,090 --> 00:30:25,590
coming towards them,
that's exactly what

539
00:30:25,590 --> 00:30:26,340
we're going to be talking about.

540
00:30:26,340 --> 00:30:27,940
That is the Doppler shift.

541
00:30:27,940 --> 00:30:30,440
And what we'll find is when
the clock comes towards them,

542
00:30:30,440 --> 00:30:32,440
he will see the
blueshift not a redshift.

543
00:30:32,440 --> 00:30:34,930
He'll see the clock
running faster not slower,

544
00:30:34,930 --> 00:30:38,650
the opposite of what was
shown in the TV program.

545
00:30:38,650 --> 00:30:41,220
But the difference
is that what causes

546
00:30:41,220 --> 00:30:43,820
it to look like
it's going faster

547
00:30:43,820 --> 00:30:46,320
is the fact that each pulse
travels a shorter distance

548
00:30:46,320 --> 00:30:48,920
if the clock is coming
towards the observer.

549
00:30:48,920 --> 00:30:50,960
And that becomes a bigger
effect than the fact

550
00:30:50,960 --> 00:30:54,360
the clock itself, if it were
measured relative to clocks

551
00:30:54,360 --> 00:30:56,900
that it passes, would
be running slowly.

552
00:30:56,900 --> 00:30:57,631
Yes?

553
00:30:57,631 --> 00:30:59,101
AUDIENCE: So in that case,
if the clock, let's say,

554
00:30:59,101 --> 00:31:00,517
was moving toward
you really fast,

555
00:31:00,517 --> 00:31:03,150
could you measure it when it was
directly perpendicular to you?

556
00:31:03,150 --> 00:31:05,397
And then that would be the
special relativity time

557
00:31:05,397 --> 00:31:06,791
dilation?

558
00:31:06,791 --> 00:31:08,790
PROFESSOR: Well, if it
was moving alongside you.

559
00:31:08,790 --> 00:31:11,070
If it was coming right at
you, it would just hit you.

560
00:31:11,070 --> 00:31:13,280
And that was the case,
Sonya, more or less.

561
00:31:13,280 --> 00:31:15,190
But yes, you're exactly right.

562
00:31:15,190 --> 00:31:18,190
If the clock were moving at
right angles to the observer

563
00:31:18,190 --> 00:31:23,000
so that the observer
saw it, what you really

564
00:31:23,000 --> 00:31:25,540
want for this to
be the pure effect

565
00:31:25,540 --> 00:31:29,380
is you want to have the
velocity of the clock

566
00:31:29,380 --> 00:31:33,450
to be perpendicular to
the velocity of the photon

567
00:31:33,450 --> 00:31:37,250
that the observer is seeing,
as measured in the observer's

568
00:31:37,250 --> 00:31:38,010
reference frame.

569
00:31:38,010 --> 00:31:39,900
Actually that matters.

570
00:31:39,900 --> 00:31:43,380
Then you see would see
pure time dilation effect.

571
00:31:43,380 --> 00:31:46,800
That's exactly right.

572
00:31:46,800 --> 00:31:55,000
OK, so might just add that Brian
Green and a bunch of people

573
00:31:55,000 --> 00:31:56,600
here at MIT, actually--
I was involved

574
00:31:56,600 --> 00:31:57,600
and so were some others.

575
00:31:57,600 --> 00:32:00,150
There's a number of MIT
people on this program.

576
00:32:00,150 --> 00:32:03,540
So we ended up having
a long conversation

577
00:32:03,540 --> 00:32:05,290
with Brian Green
about it by email.

578
00:32:05,290 --> 00:32:09,427
And everybody at MIT thought
it was just plain wrong.

579
00:32:09,427 --> 00:32:11,010
Brian Green actually
took the position

580
00:32:11,010 --> 00:32:12,730
that it was very
intentional on his part,

581
00:32:12,730 --> 00:32:15,550
and he was just trying to
illustrate the time dilation

582
00:32:15,550 --> 00:32:18,420
effect, and he didn't want to
talk about the Doppler shift.

583
00:32:18,420 --> 00:32:20,170
And since he didn't
want to talk about it,

584
00:32:20,170 --> 00:32:22,170
he could ignore the
fact that it was there.

585
00:32:22,170 --> 00:32:23,740
We all thought that
was bad pedagogy.

586
00:32:23,740 --> 00:32:25,950
But we never convinced
Brian, I should say.

587
00:32:30,460 --> 00:32:35,160
OK, now what we want to do is
go through these Doppler shift

588
00:32:35,160 --> 00:32:39,000
calculations again,
this time recognizing

589
00:32:39,000 --> 00:32:42,330
that moving clocks run
slowly by a factor of gamma.

590
00:32:42,330 --> 00:32:44,910
So we're now going to be doing
the relativistic case where

591
00:32:44,910 --> 00:32:46,970
the wave is sound
waves-- excuse me,

592
00:32:46,970 --> 00:32:48,820
where the wave is light waves.

593
00:32:48,820 --> 00:32:50,870
I'll get this straight.

594
00:32:50,870 --> 00:32:53,120
And the velocities might
be comparable to the speed

595
00:32:53,120 --> 00:32:53,660
of light.

596
00:32:53,660 --> 00:32:56,350
So this time dilation
effect is large

597
00:32:56,350 --> 00:32:59,410
enough so that we want
to take it into account.

598
00:32:59,410 --> 00:33:02,220
Now in this case,
what we are hoping,

599
00:33:02,220 --> 00:33:05,080
and everything would be wrong,
inconsistent, if we can find

600
00:33:05,080 --> 00:33:07,090
this, we're hoping
that in this case,

601
00:33:07,090 --> 00:33:09,020
the two answers
should be the same.

602
00:33:09,020 --> 00:33:11,830
It shouldn't matter whether
the source is moving

603
00:33:11,830 --> 00:33:13,960
or the observer is moving.

604
00:33:13,960 --> 00:33:17,590
Earlier it did matter, and
we said that was explicable,

605
00:33:17,590 --> 00:33:20,280
because we knew that
air was involved.

606
00:33:20,280 --> 00:33:24,680
And if we made a
velocity transformation

607
00:33:24,680 --> 00:33:27,874
to go from one
picture to the other,

608
00:33:27,874 --> 00:33:30,290
from the picture where the
source is moving to the picture

609
00:33:30,290 --> 00:33:32,800
where the observer
was moving, the air

610
00:33:32,800 --> 00:33:35,075
would have a different
velocity in the two pictures.

611
00:33:35,075 --> 00:33:37,730
It would be stationary in
one and moving in the other.

612
00:33:37,730 --> 00:33:40,230
So we would not expect
to get the same answer

613
00:33:40,230 --> 00:33:43,680
as we would have gotten
assuming the air was stationary.

614
00:33:43,680 --> 00:33:47,217
But in this case, if anything
has a different velocity when

615
00:33:47,217 --> 00:33:49,300
you go from the picture
where the source is moving

616
00:33:49,300 --> 00:33:51,460
to the picture where
the observer is moving,

617
00:33:51,460 --> 00:33:54,310
it would have to be the ether
that has a different velocity.

618
00:33:54,310 --> 00:33:56,580
But the basic axiom
of special relativity

619
00:33:56,580 --> 00:33:58,490
is that there is
no ether, at least

620
00:33:58,490 --> 00:34:00,620
there's no physical effects
coming from the ether.

621
00:34:00,620 --> 00:34:03,065
So you might as well
pretend it does not exist.

622
00:34:03,065 --> 00:34:04,690
You can't really
prove that it does not

623
00:34:04,690 --> 00:34:06,420
exist if it has no properties.

624
00:34:06,420 --> 00:34:07,990
It still exists.

625
00:34:07,990 --> 00:34:11,270
But the basic axiom
of special relativity

626
00:34:11,270 --> 00:34:12,870
is that there are
no physical effects

627
00:34:12,870 --> 00:34:14,196
coming from this either.

628
00:34:14,196 --> 00:34:16,570
So for special relativity, we
should get the same answer,

629
00:34:16,570 --> 00:34:18,111
whether it's the
source that's moving

630
00:34:18,111 --> 00:34:19,530
or the observer that's moving.

631
00:34:19,530 --> 00:34:21,699
It's really the same
situation, just viewed

632
00:34:21,699 --> 00:34:23,150
in different reference frames.

633
00:34:23,150 --> 00:34:26,139
And special relativity says it
cannot matter what reference

634
00:34:26,139 --> 00:34:29,590
frame we're doing
the calculations in.

635
00:34:29,590 --> 00:34:32,679
So it's the same
pictures, but this time we

636
00:34:32,679 --> 00:34:36,600
want to take into account the
fact that moving clocks run

637
00:34:36,600 --> 00:34:39,820
slowly by a factor of gamma.

638
00:34:39,820 --> 00:34:44,400
So looking at these
pictures, we can first

639
00:34:44,400 --> 00:34:47,060
start just glancing
at them and saying,

640
00:34:47,060 --> 00:34:48,560
where's there a
clock that's moving?

641
00:34:48,560 --> 00:34:53,719
That might be something
we need to think about.

642
00:34:53,719 --> 00:34:58,990
And maybe I'll ask you--
which of these four frames

643
00:34:58,990 --> 00:34:59,915
shows a moving clock?

644
00:35:06,134 --> 00:35:07,630
AUDIENCE: All of them?

645
00:35:07,630 --> 00:35:08,844
PROFESSOR: Sorry?

646
00:35:08,844 --> 00:35:09,760
AUDIENCE: All of them?

647
00:35:09,760 --> 00:35:11,280
PROFESSOR: All of
them, I guess so.

648
00:35:11,280 --> 00:35:13,440
But the time interval
measured on the clock

649
00:35:13,440 --> 00:35:16,612
is only actually
relevant to one of them.

650
00:35:16,612 --> 00:35:17,920
AUDIENCE: Two.

651
00:35:17,920 --> 00:35:21,070
PROFESSOR: Two, exactly, two.

652
00:35:21,070 --> 00:35:24,380
Here we said that
the source is moving,

653
00:35:24,380 --> 00:35:27,320
and we said the time measured
on the source's clock

654
00:35:27,320 --> 00:35:30,650
between the emission of
these two wave crests.

655
00:35:30,650 --> 00:35:32,720
Incidentally, we're
usually talking

656
00:35:32,720 --> 00:35:35,740
about continuous waves
like light waves.

657
00:35:35,740 --> 00:35:39,750
And then we're talking about
the time between successive wave

658
00:35:39,750 --> 00:35:40,720
crests.

659
00:35:40,720 --> 00:35:43,220
Well, if we just as well imagine
that the source is emitting

660
00:35:43,220 --> 00:35:45,590
a series of pulses where
each pulse represents a wave

661
00:35:45,590 --> 00:35:48,290
crest, and somehow to me
that sounds a little simpler

662
00:35:48,290 --> 00:35:49,840
to describe, because
you don't have

663
00:35:49,840 --> 00:35:53,330
to think about the sine wave
associated with the signal

664
00:35:53,330 --> 00:35:56,780
that the source is
actually creating.

665
00:35:56,780 --> 00:36:01,040
So in any case, the time
between these pulses,

666
00:36:01,040 --> 00:36:04,340
I'll call them, as measured
by the source clock,

667
00:36:04,340 --> 00:36:06,590
is what we call
delta t sub s, is

668
00:36:06,590 --> 00:36:09,650
the time that the source
would actually measure.

669
00:36:09,650 --> 00:36:12,630
And the source is
moving in this picture.

670
00:36:12,630 --> 00:36:15,570
So relative to
our frame, we want

671
00:36:15,570 --> 00:36:18,410
to think of this entire
series of pictures

672
00:36:18,410 --> 00:36:22,150
as all being consistent
in our frame.

673
00:36:22,150 --> 00:36:25,700
It's very important, since
transformations between frames

674
00:36:25,700 --> 00:36:27,750
are a little complicated
in special relativity.

675
00:36:27,750 --> 00:36:29,990
It's very important when
you're doing any problem

676
00:36:29,990 --> 00:36:33,246
to pick what frame you're going
to use for your description

677
00:36:33,246 --> 00:36:34,370
and be sure to stick to it.

678
00:36:34,370 --> 00:36:37,080
If anything is initially
described in another frame,

679
00:36:37,080 --> 00:36:39,890
you have to figure out what
it looks like in your frame

680
00:36:39,890 --> 00:36:42,100
in order to fit it together
with the other events

681
00:36:42,100 --> 00:36:45,690
that you're describing in your
own particular reference frame.

682
00:36:45,690 --> 00:36:47,160
So for this problem,
our frame will

683
00:36:47,160 --> 00:36:49,750
be the frame of the
slide, the frame, which

684
00:36:49,750 --> 00:36:51,910
is at rest relative
to the observer.

685
00:36:51,910 --> 00:36:54,770
So we could also call
it the observer's frame.

686
00:36:54,770 --> 00:36:57,510
And relative to that frame,
the source is moving.

687
00:36:57,510 --> 00:36:59,930
And therefore, the
clock-- a source

688
00:36:59,930 --> 00:37:01,390
is emitting a series of pulses.

689
00:37:01,390 --> 00:37:02,340
That's a clock.

690
00:37:02,340 --> 00:37:06,030
Anything that does anything at
regular intervals is a clock.

691
00:37:06,030 --> 00:37:08,320
So the source represents
a moving clock.

692
00:37:08,320 --> 00:37:11,810
And we need to take into account
the fact that the source will

693
00:37:11,810 --> 00:37:16,150
be running slowly by
a factor of gamma.

694
00:37:16,150 --> 00:37:17,920
And otherwise, nothing changes.

695
00:37:17,920 --> 00:37:20,350
The observer has a clock
also, which the observer

696
00:37:20,350 --> 00:37:23,169
is going to use to measure
the time between crests.

697
00:37:23,169 --> 00:37:25,210
But the observer's clock
is at rest in our frame.

698
00:37:25,210 --> 00:37:28,490
So there's no time dilation
associated with the observer's

699
00:37:28,490 --> 00:37:32,110
clock, only a time
dilation associated

700
00:37:32,110 --> 00:37:36,050
with the clock on the source.

701
00:37:36,050 --> 00:37:40,630
So again, the important issue
is all inside that yellow box.

702
00:37:40,630 --> 00:37:43,180
And what do I do now is
look at the equations

703
00:37:43,180 --> 00:37:45,305
and see how the
equations are modified.

704
00:37:58,120 --> 00:38:00,244
I guess I should start
back at the beginning

705
00:38:00,244 --> 00:38:01,035
of the blackboards.

706
00:38:24,462 --> 00:38:26,170
Maybe I should turn
the blackboard lights

707
00:38:26,170 --> 00:38:27,545
on when I work on
the blackboard.

708
00:38:36,680 --> 00:38:42,260
So the time integral
as seen by the observer

709
00:38:42,260 --> 00:38:48,510
will be equal to--
last time we just

710
00:38:48,510 --> 00:38:52,800
had delta t sub s as our first
term, which would be what it

711
00:38:52,800 --> 00:38:54,630
would be if there
was no velocity.

712
00:38:54,630 --> 00:38:58,300
And that's still true if
there was no velocity.

713
00:38:58,300 --> 00:39:01,710
But if the clock is going
to be running slowly

714
00:39:01,710 --> 00:39:06,130
by a factor of gamma, that would
mean that the time interval

715
00:39:06,130 --> 00:39:07,660
that we would
measure, even if there

716
00:39:07,660 --> 00:39:10,370
was no change in path length,
which will be the next term,

717
00:39:10,370 --> 00:39:12,310
if there was no
change in path length,

718
00:39:12,310 --> 00:39:14,870
the time that we would
measure as the observer would

719
00:39:14,870 --> 00:39:16,830
be different from
the time interval

720
00:39:16,830 --> 00:39:20,291
as measured by the source
by a factor of gamma.

721
00:39:20,291 --> 00:39:22,290
But one has to figure out
whether the gamma goes

722
00:39:22,290 --> 00:39:26,010
in the numerator
or the denominator.

723
00:39:26,010 --> 00:39:27,470
And sometimes it's
a little tricky.

724
00:39:27,470 --> 00:39:30,880
It helps a lot, I think, to
just sort of imagine an example.

725
00:39:30,880 --> 00:39:32,750
Any example is
clear, it turns out.

726
00:39:32,750 --> 00:39:35,110
But when you try to write
down the answer in general,

727
00:39:35,110 --> 00:39:36,830
you sometimes get it wrong.

728
00:39:36,830 --> 00:39:40,020
That's my experience with
myself and with other people.

729
00:39:40,020 --> 00:39:42,410
So this clock is wanting slower.

730
00:39:42,410 --> 00:39:45,570
And say we're talking about
a second time interval.

731
00:39:45,570 --> 00:39:48,570
If this clock is running slower,
it means it takes longer for it

732
00:39:48,570 --> 00:39:49,870
to take off a second.

733
00:39:49,870 --> 00:39:52,660
We would see it as maybe running
slower by a factor of two.

734
00:39:52,660 --> 00:39:54,368
It would mean that it
would only take off

735
00:39:54,368 --> 00:39:56,530
a second every two seconds.

736
00:39:56,530 --> 00:39:59,010
So that means that the time
interval that we would see--

737
00:39:59,010 --> 00:40:00,760
and that's what we're
trying to calculate,

738
00:40:00,760 --> 00:40:03,060
the time interval
as the observer--

739
00:40:03,060 --> 00:40:08,310
is going to be longer than delta
t sub s by a factor of gamma.

740
00:40:08,310 --> 00:40:11,470
So the first term changes
from what it was before.

741
00:40:16,660 --> 00:40:18,960
And what we're doing is we're
rewriting this equation.

742
00:40:22,170 --> 00:40:24,510
So the first term
changes from what

743
00:40:24,510 --> 00:40:29,720
it was before by putting in a
factor of gamma in front of it.

744
00:40:29,720 --> 00:40:31,960
Because the source
clock is running slowly.

745
00:40:31,960 --> 00:40:39,610
And then the second term is
still delta l divided by u.

746
00:40:39,610 --> 00:40:41,670
But the equation for
delta l changes also,

747
00:40:41,670 --> 00:40:45,795
because delta l is
the time interval

748
00:40:45,795 --> 00:40:48,170
that it takes for the light
to travel the extra distance,

749
00:40:48,170 --> 00:40:50,770
where the extra
distance was because

750
00:40:50,770 --> 00:40:52,130
of the time between the clicks.

751
00:40:52,130 --> 00:40:54,470
And that is now changed
because of the time

752
00:40:54,470 --> 00:40:57,110
dilation of the source clock.

753
00:40:57,110 --> 00:41:01,100
So the second term also
changes by a factor of gamma.

754
00:41:01,100 --> 00:41:04,640
So it's gamma
times delta t sub s

755
00:41:04,640 --> 00:41:16,510
plus v times gamma times
delta t sub s divided by u.

756
00:41:21,690 --> 00:41:25,060
So the whole answer just
changes by a factor of gamma.

757
00:41:25,060 --> 00:41:32,120
So it's gamma times 1 plus v
over c times delta t sub s.

758
00:41:37,250 --> 00:41:39,550
And now if you do a
little bit of arithmetic

759
00:41:39,550 --> 00:41:42,160
here-- gamma, remember, is
1 over the square root of 1

760
00:41:42,160 --> 00:41:44,670
minus v squared over c squared.

761
00:41:44,670 --> 00:41:47,780
And a squared minus
b squared-- I'm now

762
00:41:47,780 --> 00:41:50,524
thinking of the denominator 1
minus v squared over c squared.

763
00:41:50,524 --> 00:41:51,940
Maybe I'll write
this on the side.

764
00:41:55,570 --> 00:41:58,060
1 minus v squared
over c squared--

765
00:41:58,060 --> 00:42:00,940
it's worth remembering here--
can be written as 1 plus v over

766
00:42:00,940 --> 00:42:03,340
c times 1 minus v over c.

767
00:42:08,120 --> 00:42:13,760
And we have the square
root of that appearing here

768
00:42:13,760 --> 00:42:15,490
in the denominator.

769
00:42:15,490 --> 00:42:18,000
Gamma is 1 over the
square root of this.

770
00:42:18,000 --> 00:42:22,190
So this becomes the square
root in the denominator,

771
00:42:22,190 --> 00:42:24,100
resulting the square
root of 1 plus v

772
00:42:24,100 --> 00:42:25,730
over c appearing in the answer.

773
00:42:25,730 --> 00:42:27,850
And this remains
in the denominator.

774
00:42:27,850 --> 00:42:35,720
And what we get simplifies
to simply 1 plus beta over 1

775
00:42:35,720 --> 00:42:40,470
minus beta inside the square
root times delta t sub s.

776
00:42:45,100 --> 00:43:03,214
So this is the special
relativity answer,

777
00:43:03,214 --> 00:43:06,830
the relativistic answer, and
this is for the source moving.

778
00:43:17,776 --> 00:43:19,650
Now we expect that the
answer won't depend on

779
00:43:19,650 --> 00:43:21,340
whether the source
is moving or not,

780
00:43:21,340 --> 00:43:23,180
but certainly the
calculations do.

781
00:43:23,180 --> 00:43:25,889
So this is what we got from the
calculation, which we assumed

782
00:43:25,889 --> 00:43:27,430
that it was a source
that was moving,

783
00:43:27,430 --> 00:43:29,717
and the observer was stationary.

784
00:43:29,717 --> 00:43:31,550
It's just corrected
from the previous answer

785
00:43:31,550 --> 00:43:33,820
by this factor of gamma.

786
00:43:33,820 --> 00:43:34,910
Any questions about that?

787
00:43:43,560 --> 00:43:47,550
OK, so to summarize on
the slide what's changed

788
00:43:47,550 --> 00:43:52,110
is that this time interval,
this extra distance delta l

789
00:43:52,110 --> 00:43:56,090
for the relativistic
problem, is not

790
00:43:56,090 --> 00:43:58,560
v times delta t sub s as it was.

791
00:43:58,560 --> 00:44:01,280
But rather what we said is
that it's gamma times v times

792
00:44:01,280 --> 00:44:03,800
delta s, because the
clock on the source

793
00:44:03,800 --> 00:44:06,040
is running slowly by
a factor of gamma.

794
00:44:06,040 --> 00:44:08,160
And it was this
difference that we just

795
00:44:08,160 --> 00:44:10,044
used on the calculation
on the blackboard

796
00:44:10,044 --> 00:44:10,960
to get the new answer.

797
00:44:15,150 --> 00:44:20,450
OK now the next
calculation, now we're

798
00:44:20,450 --> 00:44:22,690
going to do the same
calculation again,

799
00:44:22,690 --> 00:44:25,150
which we already did for
the nonrelativistic case.

800
00:44:25,150 --> 00:44:27,790
But this time it will
be the observer moving.

801
00:44:27,790 --> 00:44:31,600
And we're going to try to
do the relativistic case.

802
00:44:31,600 --> 00:44:34,950
So this time it's
the clock carried

803
00:44:34,950 --> 00:44:37,350
by the observer
that's running slowly.

804
00:44:37,350 --> 00:44:38,900
And remember, this
is running slowly

805
00:44:38,900 --> 00:44:42,080
relative to us, relative to
our frame of reference, where

806
00:44:42,080 --> 00:44:44,510
our frame of reference is
by definition the frame

807
00:44:44,510 --> 00:44:46,540
of reference of the slide
that we're looking at.

808
00:44:49,250 --> 00:44:51,540
Source is stationary,
so delta t sub

809
00:44:51,540 --> 00:44:55,330
s is just an honest time
as we would measure it.

810
00:44:55,330 --> 00:44:59,000
But times as measured by
the observer, delta t sub o,

811
00:44:59,000 --> 00:45:00,970
are going to be different.

812
00:45:00,970 --> 00:45:04,690
So to correct the
calculation for relativity,

813
00:45:04,690 --> 00:45:06,950
that's the crucial
box as it was even

814
00:45:06,950 --> 00:45:09,076
in the nonrelativistic case.

815
00:45:09,076 --> 00:45:10,450
And we're going
to have to change

816
00:45:10,450 --> 00:45:14,920
that equation by
replacing delta l.

817
00:45:14,920 --> 00:45:21,210
Instead of v times delta t sub
0 is v times delta t prime.

818
00:45:21,210 --> 00:45:26,630
Now delta t prime is not
exactly delta t sub 0

819
00:45:26,630 --> 00:45:28,680
or gamma times delta t sub 0.

820
00:45:28,680 --> 00:45:30,981
It's a little trickier.

821
00:45:30,981 --> 00:45:31,480
Let's see.

822
00:45:31,480 --> 00:45:32,660
Hold on.

823
00:45:32,660 --> 00:45:35,400
The definition I want
to use for delta t prime

824
00:45:35,400 --> 00:45:37,090
is that it's the
time as measured

825
00:45:37,090 --> 00:45:39,310
between the third frame
and the fourth frame.

826
00:45:39,310 --> 00:45:41,152
But it's the time
as we measure it.

827
00:45:41,152 --> 00:45:42,610
I still want to
describe everything

828
00:45:42,610 --> 00:45:45,120
in terms of the way
we would measure it.

829
00:45:45,120 --> 00:45:47,120
So delta t sub prime
is not measured either

830
00:45:47,120 --> 00:45:48,390
by the source or the observer.

831
00:45:48,390 --> 00:45:49,320
It's measured by us.

832
00:45:57,540 --> 00:46:01,060
And it's related to the time as
the observer would measure it

833
00:46:01,060 --> 00:46:04,540
by a factor of gamma,
because relative to us,

834
00:46:04,540 --> 00:46:08,252
the observer's clock is running
slowly by a factor of gamma.

835
00:46:08,252 --> 00:46:10,325
AUDIENCE: So we're the source?

836
00:46:10,325 --> 00:46:11,700
PROFESSOR: Well,
we're stationary

837
00:46:11,700 --> 00:46:14,210
relative to the source, because
the source is stationary.

838
00:46:14,210 --> 00:46:15,580
But we're the frame of
reference of the slide,

839
00:46:15,580 --> 00:46:17,165
is the way I like
to think about it.

840
00:46:17,165 --> 00:46:18,540
But it is the
frame of reference,

841
00:46:18,540 --> 00:46:20,590
same as the frame of
reference as the source.

842
00:46:20,590 --> 00:46:23,500
That's right.

843
00:46:23,500 --> 00:46:26,655
So delta t sub 0 is
related to delta t

844
00:46:26,655 --> 00:46:28,334
sub prime by a factor of gamma.

845
00:46:28,334 --> 00:46:30,000
And again, one has
to think a little bit

846
00:46:30,000 --> 00:46:32,470
to make sure one gets the
gamma in the right place,

847
00:46:32,470 --> 00:46:34,760
the numerator or
the denominator.

848
00:46:34,760 --> 00:46:37,796
We're saying that the
observer clock appears

849
00:46:37,796 --> 00:46:40,690
to be running slowly
relative to us.

850
00:46:40,690 --> 00:46:45,800
And that means that
during the amount of time

851
00:46:45,800 --> 00:46:50,620
it would take for it to
say take off one second,

852
00:46:50,620 --> 00:46:55,190
since it's running slowly,
should take more than a

853
00:46:55,190 --> 00:46:57,184
second relative to us.

854
00:46:57,184 --> 00:46:58,850
And that's what this
formula would give,

855
00:46:58,850 --> 00:47:00,010
if we multiply by gamma.

856
00:47:00,010 --> 00:47:02,400
It would say that delta t
prime is equal to gamma times

857
00:47:02,400 --> 00:47:03,870
delta t sub 0.

858
00:47:03,870 --> 00:47:07,390
So we would say that the time
that the observer clock ticks

859
00:47:07,390 --> 00:47:10,650
off one second we might
measure two seconds.

860
00:47:10,650 --> 00:47:13,680
That's the direction
of the time shift,

861
00:47:13,680 --> 00:47:15,990
all given by that formula.

862
00:47:15,990 --> 00:47:18,080
And those are all the pictures.

863
00:47:18,080 --> 00:47:20,810
Now we just have to write
down the equations that

864
00:47:20,810 --> 00:47:21,944
go with those pictures.

865
00:47:21,944 --> 00:47:23,860
And the hope is that
we'll get the same answer

866
00:47:23,860 --> 00:47:26,176
as we got last time.

867
00:47:26,176 --> 00:47:28,410
I think I'll leave
that on the board.

868
00:47:28,410 --> 00:47:28,910
Let's see.

869
00:47:28,910 --> 00:47:31,774
What do I do?

870
00:47:31,774 --> 00:47:33,190
No, I won't leave
it on the board.

871
00:47:48,996 --> 00:47:50,870
OK, this time the
calculation we're mimicking

872
00:47:50,870 --> 00:47:52,640
is the calculation up here.

873
00:47:52,640 --> 00:47:55,540
This was the calculation we had
for the nonrelativistic case

874
00:47:55,540 --> 00:47:57,900
where the observer was moving.

875
00:47:57,900 --> 00:48:00,470
Now we want to put in
the time dilation that

876
00:48:00,470 --> 00:48:04,090
would correct that calculation.

877
00:48:04,090 --> 00:48:11,920
And the key equation is
that the time interval

878
00:48:11,920 --> 00:48:15,710
between the receipt
of the two wave crests

879
00:48:15,710 --> 00:48:17,690
as we would measure
it, which is why

880
00:48:17,690 --> 00:48:20,244
I call it delta t
sub prime, so prime

881
00:48:20,244 --> 00:48:21,660
is not the source
or the observer.

882
00:48:21,660 --> 00:48:22,360
It's us.

883
00:48:22,360 --> 00:48:24,430
It's the same frame of
reference as the source,

884
00:48:24,430 --> 00:48:26,510
but the source
isn't located there.

885
00:48:26,510 --> 00:48:28,160
So we think of it
as the time interval

886
00:48:28,160 --> 00:48:31,950
as measured in the
frame of reference.

887
00:48:31,950 --> 00:48:35,200
That's the frame of the
slide, which is our frame.

888
00:48:35,200 --> 00:48:43,460
And it is equal to
gamma times delta t sub

889
00:48:43,460 --> 00:48:46,110
0, where delta t sub 0 is
the time as it would actually

890
00:48:46,110 --> 00:48:49,006
be measured on the
observer's clock.

891
00:48:49,006 --> 00:48:52,070
So, we're keeping
their labels s and 0

892
00:48:52,070 --> 00:48:53,780
to mean what would
actually be measured

893
00:48:53,780 --> 00:48:57,180
on the clocks at the
source and at the observer.

894
00:48:57,180 --> 00:48:58,800
That's what we're
trying to establish

895
00:48:58,800 --> 00:48:59,758
a relationship between.

896
00:49:02,190 --> 00:49:04,630
So the basic formula
that we have up

897
00:49:04,630 --> 00:49:15,670
top there would first
become delta t sub 0

898
00:49:15,670 --> 00:49:21,370
is equal to-- hold on.

899
00:49:25,991 --> 00:49:27,990
Now actually, we would
not start by writing down

900
00:49:27,990 --> 00:49:29,964
an equation for delta t sub 0.

901
00:49:29,964 --> 00:49:31,380
Rather what we
want to do is start

902
00:49:31,380 --> 00:49:33,504
by writing down the
relationship as we would see it

903
00:49:33,504 --> 00:49:36,100
in our frame, which is going
involve delta t sub prime.

904
00:49:36,100 --> 00:49:39,000
We'll worry about
delta t sub 0 later.

905
00:49:39,000 --> 00:49:41,350
So our equation
is going to become

906
00:49:41,350 --> 00:49:50,710
delta t prime is equal to
delta t sub s plus v times

907
00:49:50,710 --> 00:49:54,522
delta t sub prime divided
by the wave speed,

908
00:49:54,522 --> 00:49:55,980
which now we'll
call c, since we're

909
00:49:55,980 --> 00:49:57,188
doing a relativistic problem.

910
00:50:00,141 --> 00:50:02,140
So this is the basic
equation for the time delay

911
00:50:02,140 --> 00:50:05,240
as we would see it in our frame,
where delta t sub prime is

912
00:50:05,240 --> 00:50:08,525
the time between the receipt
of the first and second crests,

913
00:50:08,525 --> 00:50:11,880
the time between
frames three and four

914
00:50:11,880 --> 00:50:13,198
as we would measure it.

915
00:50:15,950 --> 00:50:20,070
And now we can do the same
thing that we did over there,

916
00:50:20,070 --> 00:50:22,050
first for delta t sub prime.

917
00:50:22,050 --> 00:50:26,180
And we discover that
delta t sub prime

918
00:50:26,180 --> 00:50:30,920
can be written as 1
minus v over c inverse,

919
00:50:30,920 --> 00:50:36,140
just doing algebra on this
equation, times delta t sub s.

920
00:50:36,140 --> 00:50:39,570
So that equation relates delta
t prime to delta t sub s.

921
00:50:39,570 --> 00:50:42,860
And now we use the
equation up top here

922
00:50:42,860 --> 00:50:47,510
to see what the observer
himself would actually measure.

923
00:50:47,510 --> 00:50:51,970
And that becomes just 1 over
gamma times delta t sub prime.

924
00:50:51,970 --> 00:50:59,720
So delta t sub observed is
equal to 1 over gamma times 1

925
00:50:59,720 --> 00:51:07,650
minus v over c inverse
times delta t sub source.

926
00:51:11,514 --> 00:51:13,680
And now since it's important
we get the same answer,

927
00:51:13,680 --> 00:51:15,820
I'm going to write in
some intermediate algebra

928
00:51:15,820 --> 00:51:19,730
here just so we can all
really see that it works out.

929
00:51:19,730 --> 00:51:26,390
The 1 over gamma, I am going
to write as-- remember gamma is

930
00:51:26,390 --> 00:51:29,270
1 over the square root
of 1 minus beta squared--

931
00:51:29,270 --> 00:51:32,770
so 1 over gamma is the square
root of 1 minus beta squared.

932
00:51:32,770 --> 00:51:36,640
And I'm going to write that
as the square root of 1

933
00:51:36,640 --> 00:51:41,240
plus beta times 1 minus beta.

934
00:51:41,240 --> 00:51:44,060
So this is the
factor 1 over gamma.

935
00:51:44,060 --> 00:51:50,600
Now we have explicitly here a
factor of 1 over 1 minus beta.

936
00:51:50,600 --> 00:51:52,952
The inverse makes it 1 over.

937
00:51:52,952 --> 00:51:55,825
Did I write something wrong?

938
00:51:55,825 --> 00:51:57,783
AUDIENCE: From the second
until the third time?

939
00:52:00,472 --> 00:52:02,180
PROFESSOR: Second to
third, you're there?

940
00:52:02,180 --> 00:52:03,090
AUDIENCE: I thought it wasn't--

941
00:52:03,090 --> 00:52:04,965
PROFESSOR: Yeah, this
goes to the other side.

942
00:52:04,965 --> 00:52:05,740
It is minus.

943
00:52:09,380 --> 00:52:12,480
OK, finally, the 1
minus beta, you see

944
00:52:12,480 --> 00:52:16,020
occurs to the first
power in the denominator.

945
00:52:16,020 --> 00:52:19,250
And the 1/2 power
to the numerator.

946
00:52:19,250 --> 00:52:21,980
So we do indeed get
exactly what we wanted.

947
00:52:32,200 --> 00:52:36,920
Delta t sub 0 is equal
to the square root of 1

948
00:52:36,920 --> 00:52:42,900
plus beta divided by 1 minus
beta times delta t sub s.

949
00:52:48,390 --> 00:52:54,450
And now we know that this
is the relativistic answer

950
00:52:54,450 --> 00:52:56,525
for either source
or observer moving.

951
00:53:13,210 --> 00:53:18,320
And then if we want to
write down what z is,

952
00:53:18,320 --> 00:53:25,670
this is delta t observer divided
by delta t source minus 1.

953
00:53:25,670 --> 00:53:29,630
So that becomes just
the square root of 1

954
00:53:29,630 --> 00:53:36,840
plus beta over 1
minus beta minus 1.

955
00:54:08,140 --> 00:54:10,820
So we found what
we expected, what

956
00:54:10,820 --> 00:54:13,670
we wanted to find to be
consistent with the basic ideas

957
00:54:13,670 --> 00:54:14,380
of relativity.

958
00:54:14,380 --> 00:54:15,880
It's the same answer
no matter which

959
00:54:15,880 --> 00:54:17,720
one is moving, because
it doesn't matter

960
00:54:17,720 --> 00:54:21,380
what frame of reference
we do the calculation in.

961
00:54:21,380 --> 00:54:24,270
OK, questions about either
the calculations or the ideas

962
00:54:24,270 --> 00:54:24,780
behind them?

963
00:54:36,220 --> 00:54:38,260
OK in that case, let's move on.

964
00:54:38,260 --> 00:54:42,470
I do want to come back and
talk about the other two

965
00:54:42,470 --> 00:54:46,100
kinematic effects of
special relativity, which

966
00:54:46,100 --> 00:54:49,290
are Lorentz contraction
and a change

967
00:54:49,290 --> 00:54:51,205
in the notion of simultaneity.

968
00:54:51,205 --> 00:54:52,730
But before I get
to that, there's

969
00:54:52,730 --> 00:54:55,120
one other issue I
want to discuss first,

970
00:54:55,120 --> 00:54:57,600
mainly because it's something
you need to understand

971
00:54:57,600 --> 00:55:00,900
to do the problem set
that's due tomorrow.

972
00:55:00,900 --> 00:55:04,730
And that is the
situation that's needed

973
00:55:04,730 --> 00:55:08,570
to describe clocks which
might be accelerating.

974
00:55:08,570 --> 00:55:11,810
Special relativity really only
describes inertial reference

975
00:55:11,810 --> 00:55:14,510
frames and how things change
from one inertial frame

976
00:55:14,510 --> 00:55:16,460
to another.

977
00:55:16,460 --> 00:55:19,270
So if you know how
a clock behaves

978
00:55:19,270 --> 00:55:21,560
if it's at rest in
one reference frame,

979
00:55:21,560 --> 00:55:25,706
special relativity completely
dictates without any ambiguity

980
00:55:25,706 --> 00:55:27,330
what it would look
like at a frame that

981
00:55:27,330 --> 00:55:30,310
was moving at a
uniform velocity,

982
00:55:30,310 --> 00:55:33,490
relative to the original frame,
which means it completely

983
00:55:33,490 --> 00:55:37,860
dictates how that clock is
going to behave if it is moving

984
00:55:37,860 --> 00:55:41,270
at a constant uniform velocity.

985
00:55:41,270 --> 00:55:43,430
But nonetheless,
in the real world,

986
00:55:43,430 --> 00:55:48,650
we have very few clocks around
that are completely inertial.

987
00:55:48,650 --> 00:55:50,075
Every clock that
we see around us

988
00:55:50,075 --> 00:55:51,450
from the clock on
the wall, which

989
00:55:51,450 --> 00:55:54,940
is moving with the earth or my
wrist watch which moves more,

990
00:55:54,940 --> 00:55:58,040
is constantly undergoing
accelerations.

991
00:55:58,040 --> 00:56:00,400
So we want to be able
to talk about clocks

992
00:56:00,400 --> 00:56:02,770
which are accelerating.

993
00:56:02,770 --> 00:56:05,480
So we need to say a little
bit about how we would do that

994
00:56:05,480 --> 00:56:08,510
and how we do it if the clocks
were moving at relativistic

995
00:56:08,510 --> 00:56:12,700
speeds , which also happens
with satellites, for example.

996
00:56:12,700 --> 00:56:15,690
The GPS system as you've
probably been told

997
00:56:15,690 --> 00:56:18,470
wouldn't actually work
unless the calculations were

998
00:56:18,470 --> 00:56:21,010
done in such a sophisticated
way that they even

999
00:56:21,010 --> 00:56:23,740
take into account the
effects of general relativity

1000
00:56:23,740 --> 00:56:25,500
as well special relativity.

1001
00:56:25,500 --> 00:56:27,610
So moving clocks
and how they behave

1002
00:56:27,610 --> 00:56:31,480
is a crucially important
topic technologically.

1003
00:56:31,480 --> 00:56:34,910
So what do we say that a
clock that's accelerating?

1004
00:56:34,910 --> 00:56:38,620
There's I think a common myth
that to describe acceleration

1005
00:56:38,620 --> 00:56:43,259
you need general
relativity, and therefore we

1006
00:56:43,259 --> 00:56:45,300
have to put off talking
about accelerating clocks

1007
00:56:45,300 --> 00:56:47,640
until we take a course
in general relativity.

1008
00:56:47,640 --> 00:56:51,350
That's actually totally false.

1009
00:56:51,350 --> 00:56:54,640
What general relativity does
is provide a theory of gravity

1010
00:56:54,640 --> 00:56:58,170
which basically says the gravity
and acceleration are intimately

1011
00:56:58,170 --> 00:56:59,730
linked.

1012
00:56:59,730 --> 00:57:03,000
And that's where
acceleration gets

1013
00:57:03,000 --> 00:57:05,970
pushed into general relativity.

1014
00:57:05,970 --> 00:57:08,340
But special relativity
alone is enough for us

1015
00:57:08,340 --> 00:57:11,320
to describe any system that
could be described by equations

1016
00:57:11,320 --> 00:57:13,950
that are consistent
with special relativity.

1017
00:57:13,950 --> 00:57:16,980
Special relativity does
not describe gravity.

1018
00:57:16,980 --> 00:57:19,560
So in any situation where
gravity is important,

1019
00:57:19,560 --> 00:57:22,800
special relativity is at a
loss to make real predictions

1020
00:57:22,800 --> 00:57:24,840
about what should happen.

1021
00:57:24,840 --> 00:57:27,200
But as long as gravity is
absent, as long as we're only

1022
00:57:27,200 --> 00:57:30,310
dealing with
electromagnetic forces

1023
00:57:30,310 --> 00:57:33,440
that we think we
understand, there's

1024
00:57:33,440 --> 00:57:36,060
nothing that prevents
us from using

1025
00:57:36,060 --> 00:57:37,800
the equations of
special relativity

1026
00:57:37,800 --> 00:57:39,170
to describe what happens.

1027
00:57:39,170 --> 00:57:41,840
We have to use the dynamical
equations of special relativity

1028
00:57:41,840 --> 00:57:44,490
that talk about how
things respond to forces.

1029
00:57:44,490 --> 00:57:46,740
And whenever there's a force,
there's an acceleration.

1030
00:57:46,740 --> 00:57:48,640
But there really
are such equations.

1031
00:57:48,640 --> 00:57:50,800
We can combine, for
example, electromagnetism

1032
00:57:50,800 --> 00:57:56,920
with relativistic mechanics to
describe a system of particles

1033
00:57:56,920 --> 00:57:59,480
that are interacting
electromagnetically, completely

1034
00:57:59,480 --> 00:58:01,530
consistent with
special relativity.

1035
00:58:01,530 --> 00:58:03,280
And even those particles
are accelerating,

1036
00:58:03,280 --> 00:58:05,930
we could say everything
we want to say about them.

1037
00:58:05,930 --> 00:58:08,572
So in particular, if
there's a physical clock,

1038
00:58:08,572 --> 00:58:11,030
to the extent that we could
describe that clock as made out

1039
00:58:11,030 --> 00:58:14,290
of particles whose
physics we understand,

1040
00:58:14,290 --> 00:58:15,860
special relatively
will still tell us

1041
00:58:15,860 --> 00:58:19,970
what that clock will do even
when that clock accelerates.

1042
00:58:19,970 --> 00:58:23,820
The answer, however, from that
calculation, you might imagine,

1043
00:58:23,820 --> 00:58:24,970
is very, very complicated.

1044
00:58:24,970 --> 00:58:28,210
Because the physics
of any actual clock,

1045
00:58:28,210 --> 00:58:31,484
my wrist watch as an example,
is pretty damn complicated.

1046
00:58:31,484 --> 00:58:33,150
And we're not really
going to write down

1047
00:58:33,150 --> 00:58:36,510
the equations that describe
my wrist watch to figure out

1048
00:58:36,510 --> 00:58:39,630
how it's going to behave
when it accelerates.

1049
00:58:39,630 --> 00:58:41,890
So what are we going to do?

1050
00:58:41,890 --> 00:58:43,770
Let me point out
that you already

1051
00:58:43,770 --> 00:58:46,354
have pretty much experience
with accelerating clocks,

1052
00:58:46,354 --> 00:58:48,020
because all of you--
well, many of you--

1053
00:58:48,020 --> 00:58:50,500
wear wrist watches like I
do that are accelerating

1054
00:58:50,500 --> 00:58:51,890
all the time.

1055
00:58:51,890 --> 00:58:52,870
And they tend to work.

1056
00:58:52,870 --> 00:58:55,530
You basically assume that even
though they're accelerating

1057
00:58:55,530 --> 00:58:58,250
they've been designed well
enough so they can withstand

1058
00:58:58,250 --> 00:59:01,510
the accelerations that
your wrist gives them

1059
00:59:01,510 --> 00:59:04,060
and still read the right time.

1060
00:59:04,060 --> 00:59:07,050
On the other hand, one could
imagine contrary situations.

1061
00:59:10,909 --> 00:59:12,700
Probably my wrist watch
would survive this,

1062
00:59:12,700 --> 00:59:16,130
but if you take a mechanical
clock, a windup clock,

1063
00:59:16,130 --> 00:59:19,060
and heave it against
the wall and let

1064
00:59:19,060 --> 00:59:21,320
it smash against the
wall and come to a stop,

1065
00:59:21,320 --> 00:59:22,810
as it smashes
against the wall, it

1066
00:59:22,810 --> 00:59:25,304
wanted to go a very
large acceleration.

1067
00:59:25,304 --> 00:59:26,970
And if the acceleration
is large enough,

1068
00:59:26,970 --> 00:59:30,090
we can predict the effect
it will have the clock, even

1069
00:59:30,090 --> 00:59:32,391
though it's a
complicated interaction.

1070
00:59:32,391 --> 00:59:33,890
If the acceleration
is large enough,

1071
00:59:33,890 --> 00:59:36,740
it'll simply break the
clock and it will stop.

1072
00:59:36,740 --> 00:59:39,040
And that's one possible
effect that acceleration

1073
00:59:39,040 --> 00:59:41,920
can have a clock.

1074
00:59:41,920 --> 00:59:44,530
And other effects are
similar in nature.

1075
00:59:44,530 --> 00:59:47,240
If there's any effect
that the motion of my hand

1076
00:59:47,240 --> 00:59:49,532
has on the wrist watch, it
would be a mechanical effect

1077
00:59:49,532 --> 00:59:51,073
that you'd calculate
by understanding

1078
00:59:51,073 --> 00:59:52,680
the mechanics of
how the watch works,

1079
00:59:52,680 --> 00:59:54,300
not by understanding
any principles

1080
00:59:54,300 --> 00:59:56,630
of general relativity.

1081
00:59:56,630 --> 00:59:58,250
What's at stake
underlying this--

1082
00:59:58,250 --> 01:00:00,730
you might wonder what
the real difference is--

1083
01:00:00,730 --> 01:00:04,610
special relativity can make
precise predictions about how

1084
01:00:04,610 --> 01:00:08,150
a clock will behave if it
moves at a uniform velocity,

1085
01:00:08,150 --> 01:00:10,210
even without knowing
anything about the details

1086
01:00:10,210 --> 01:00:12,145
of that clock.

1087
01:00:12,145 --> 01:00:14,020
Special relativity could
make that prediction

1088
01:00:14,020 --> 01:00:17,750
because there's a symmetry,
Lorentz symmetry, which

1089
01:00:17,750 --> 01:00:19,290
relates those two situations.

1090
01:00:19,290 --> 01:00:21,200
And that's an exact
symmetry of nature.

1091
01:00:21,200 --> 01:00:23,580
So no matter what the
clock is made out of,

1092
01:00:23,580 --> 01:00:25,820
if it's moving at
a uniform velocity,

1093
01:00:25,820 --> 01:00:27,320
special relativity
tells you there's

1094
01:00:27,320 --> 01:00:32,490
no doubt it would run
slowly by a factor of gamma.

1095
01:00:32,490 --> 01:00:37,360
On the other hand,
there's no such principle

1096
01:00:37,360 --> 01:00:39,080
of any kind, either
in special relativity

1097
01:00:39,080 --> 01:00:42,660
or general relativity,
about acceleration.

1098
01:00:42,660 --> 01:00:44,740
So if you want to know
the effect of acceleration

1099
01:00:44,740 --> 01:00:48,000
on a clock, it really depends
in detail on how large

1100
01:00:48,000 --> 01:00:50,280
the acceleration is of
course and the detailed

1101
01:00:50,280 --> 01:00:52,710
physics of the clock and how
much acceleration it takes

1102
01:00:52,710 --> 01:00:57,080
to effect it and in what
way it affects, precisely.

1103
01:00:57,080 --> 01:00:58,820
So what's the bottom line?

1104
01:00:58,820 --> 01:01:00,620
The bottom line is
that when we want

1105
01:01:00,620 --> 01:01:03,030
to talk about an accelerated
clock, which we really

1106
01:01:03,030 --> 01:01:06,290
do all the time, what
we always do is simply

1107
01:01:06,290 --> 01:01:08,920
assume that the clock
is built well enough

1108
01:01:08,920 --> 01:01:11,680
so that the acceleration
does not affect its speed.

1109
01:01:11,680 --> 01:01:15,520
And that really can be
said very precisely.

1110
01:01:15,520 --> 01:01:17,790
The assumption that
we're going to be making

1111
01:01:17,790 --> 01:01:20,450
is that these are
ideal clocks, meaning

1112
01:01:20,450 --> 01:01:22,812
that they're built well.

1113
01:01:22,812 --> 01:01:24,520
And when we say the
acceleration does not

1114
01:01:24,520 --> 01:01:28,140
affect the speed of the
clock, what we're saying

1115
01:01:28,140 --> 01:01:31,260
is that the clock will run
at precisely the same speed

1116
01:01:31,260 --> 01:01:34,650
as another clock that's,
say, moving instantaneously

1117
01:01:34,650 --> 01:01:36,820
alongside it with
the same velocity

1118
01:01:36,820 --> 01:01:39,640
but with no acceleration.

1119
01:01:39,640 --> 01:01:43,260
So at any point in the
motion of my arm here,

1120
01:01:43,260 --> 01:01:45,810
my wrist watch will have
some specific velocity.

1121
01:01:45,810 --> 01:01:47,960
The velocity will
affect in some tiny way

1122
01:01:47,960 --> 01:01:50,750
the speed of the clock by
this factor gamma, which

1123
01:01:50,750 --> 01:01:53,390
will be very close
to 1 for that case.

1124
01:01:53,390 --> 01:01:59,380
But we're going to assume, if
we call my watch an ideal clock,

1125
01:01:59,380 --> 01:02:01,820
that in the even time,
it will be running

1126
01:02:01,820 --> 01:02:04,290
at exactly the same speed
as a clock which is not

1127
01:02:04,290 --> 01:02:05,940
accelerating, but
which is moving

1128
01:02:05,940 --> 01:02:08,170
with the same velocity
as the wrist watch.

1129
01:02:08,170 --> 01:02:10,410
Therefore, the factor
of gamma will be there,

1130
01:02:10,410 --> 01:02:13,024
but there'll be no
effect of acceleration.

1131
01:02:13,024 --> 01:02:14,940
The speed of the clock
will be determined only

1132
01:02:14,940 --> 01:02:18,990
by its velocity relative
to our reference frame.

1133
01:02:18,990 --> 01:02:20,530
OK, is that clear enough?

1134
01:02:23,684 --> 01:02:26,100
And that's what you need to
assume about some accelerating

1135
01:02:26,100 --> 01:02:28,430
clocks that show up
on the problem set.

1136
01:02:28,430 --> 01:02:30,690
That's why I wanted
to get it in today.

1137
01:02:30,690 --> 01:02:33,591
OK, if there are no other
questions about that,

1138
01:02:33,591 --> 01:02:35,590
I'd like to come back and
talk a little bit more

1139
01:02:35,590 --> 01:02:40,164
about special relativity
and its consequences.

1140
01:02:40,164 --> 01:02:42,580
Sometime later in the course,
we'll talk a little bit more

1141
01:02:42,580 --> 01:02:45,240
about what I would call
the dynamical consequences

1142
01:02:45,240 --> 01:02:47,590
of special relativity,
which include

1143
01:02:47,590 --> 01:02:51,970
well known equations like e
equals mc squared, for example.

1144
01:02:51,970 --> 01:02:55,090
But before one talks
about energy and momentum,

1145
01:02:55,090 --> 01:02:59,000
which are quantities which
I will dub dynamical,

1146
01:02:59,000 --> 01:03:02,470
there kinematic effects
of special relativity

1147
01:03:02,470 --> 01:03:04,460
of which this time
dilation is one.

1148
01:03:04,460 --> 01:03:06,250
And by kinematic
I really just mean

1149
01:03:06,250 --> 01:03:07,950
the consequences of
special relativity

1150
01:03:07,950 --> 01:03:11,680
for the measurements
of times and distances.

1151
01:03:11,680 --> 01:03:16,150
And if one limits oneself
to discussing consequences

1152
01:03:16,150 --> 01:03:19,610
for times and distances,
kinematic consequences,

1153
01:03:19,610 --> 01:03:22,750
there really are precisely
three and no more consequences

1154
01:03:22,750 --> 01:03:24,040
of special relativity.

1155
01:03:24,040 --> 01:03:26,520
And really all special
relativity in some sense

1156
01:03:26,520 --> 01:03:28,537
is embodied by these
three statements

1157
01:03:28,537 --> 01:03:30,870
that we're going to be talking
about, the first of which

1158
01:03:30,870 --> 01:03:33,050
was time dilation, which
we've already seen.

1159
01:03:33,050 --> 01:03:34,356
I'll just remind you.

1160
01:03:34,356 --> 01:03:35,980
Time dilation says
that any clock which

1161
01:03:35,980 --> 01:03:39,330
is moving at speed d relative
to a given reference frame

1162
01:03:39,330 --> 01:03:43,100
will appear in quotation
marks to an observer using

1163
01:03:43,100 --> 01:03:46,619
that reference frame to run
slower than normal by a factor

1164
01:03:46,619 --> 01:03:48,035
denoted by the
Greek letter gamma.

1165
01:03:51,680 --> 01:03:53,930
Turn out the board lights
in case they're distracting.

1166
01:03:58,770 --> 01:04:02,990
And again, "appear" refers
not to how it would actually

1167
01:04:02,990 --> 01:04:07,096
look to any particular observer,
because any particular observer

1168
01:04:07,096 --> 01:04:08,470
in a particular
location is going

1169
01:04:08,470 --> 01:04:11,201
to be waiting for light
rays to reach that observer.

1170
01:04:11,201 --> 01:04:12,950
And they'll take
different amounts of time

1171
01:04:12,950 --> 01:04:15,170
depending on where they start.

1172
01:04:15,170 --> 01:04:17,630
"Appear" refers to measurements
made in the reference

1173
01:04:17,630 --> 01:04:19,770
frame of the observer,
where we assume

1174
01:04:19,770 --> 01:04:23,690
that all the actual measurements
are made by on the spot clocks

1175
01:04:23,690 --> 01:04:27,090
and rulers who measure
where the object actually

1176
01:04:27,090 --> 01:04:29,670
were at the time these
events happened and not

1177
01:04:29,670 --> 01:04:31,760
what it looks like at
sometime later when

1178
01:04:31,760 --> 01:04:33,340
light rays reach some observer.

1179
01:04:36,230 --> 01:04:38,370
OK, the second consequence,
and again all these

1180
01:04:38,370 --> 01:04:39,661
will involve the word "appear."

1181
01:04:39,661 --> 01:04:41,500
And I'll always write
it in quotation marks

1182
01:04:41,500 --> 01:04:44,420
to remind you that it's not
exactly what a person would

1183
01:04:44,420 --> 01:04:45,300
see.

1184
01:04:45,300 --> 01:04:48,450
The second one is another famous
effect of special relativity,

1185
01:04:48,450 --> 01:04:50,440
Lorentz contraction,
or sometimes called

1186
01:04:50,440 --> 01:04:53,420
Lorentz-Fitzgerald contraction.

1187
01:04:53,420 --> 01:04:57,220
Any rod which is moving at
a speed v along its length

1188
01:04:57,220 --> 01:04:58,840
relative to a given
reference frame

1189
01:04:58,840 --> 01:05:01,290
will appear-- and again
appear to an observer

1190
01:05:01,290 --> 01:05:04,130
using that reference
frame-- to be shorter

1191
01:05:04,130 --> 01:05:07,570
than its normal length by
the same factor, gamma.

1192
01:05:07,570 --> 01:05:10,060
A rod which is moving
perpendicular to its length

1193
01:05:10,060 --> 01:05:12,470
does not undo that change
in apparent length.

1194
01:05:12,470 --> 01:05:14,900
So these pictures
kind of show it all.

1195
01:05:14,900 --> 01:05:18,180
A rod, that bar is
the rod, a rod which

1196
01:05:18,180 --> 01:05:21,984
is moving at speed v
will look contracted,

1197
01:05:21,984 --> 01:05:26,560
will appear to be contracted,
by a factor of gamma.

1198
01:05:26,560 --> 01:05:28,590
And a rod which is
moving perpendicular

1199
01:05:28,590 --> 01:05:32,130
to its length, a rod which
is like this moving this way,

1200
01:05:32,130 --> 01:05:37,000
has no such effect, we'll appear
to have its natural length.

1201
01:05:37,000 --> 01:05:38,670
And this is a very
famous consequence

1202
01:05:38,670 --> 01:05:39,586
of special relativity.

1203
01:05:39,586 --> 01:05:41,630
It means that moving
rocket ships shorter

1204
01:05:41,630 --> 01:05:44,311
and shorter and faster
they go, and so on.

1205
01:05:44,311 --> 01:05:45,810
And again, you
should remember, it's

1206
01:05:45,810 --> 01:05:46,976
not what you'd actually see.

1207
01:05:46,976 --> 01:05:50,490
It's what you'd measure if you
had on the spot local observers

1208
01:05:50,490 --> 01:05:54,840
making these measurements,
which you then later compile.

1209
01:05:54,840 --> 01:05:59,680
OK, that's actually all I want
to say about the contraction.

1210
01:05:59,680 --> 01:06:00,410
Any questions?

1211
01:06:04,590 --> 01:06:09,040
OK, next and last is an effect
we just talked about less

1212
01:06:09,040 --> 01:06:11,640
because it's a little bit
more intricate to describe.

1213
01:06:11,640 --> 01:06:14,700
But the other crucially
important effect-- these

1214
01:06:14,700 --> 01:06:17,560
would not be consistent if
you didn't have all three.

1215
01:06:17,560 --> 01:06:19,370
The other crucially
important effect

1216
01:06:19,370 --> 01:06:24,800
is the changing simultaneity,
the relativity of simultaneity.

1217
01:06:24,800 --> 01:06:26,730
And it takes more
words describe it,

1218
01:06:26,730 --> 01:06:29,429
so there's more words on the
slide than on the other slides.

1219
01:06:29,429 --> 01:06:31,720
And the pictures are a little
bit more complicated too.

1220
01:06:31,720 --> 01:06:34,380
But the pictures do
say it all, really.

1221
01:06:34,380 --> 01:06:40,290
The point is that if you have a
system consisting of two clocks

1222
01:06:40,290 --> 01:06:44,182
which have been synchronized
in their own rest frame--

1223
01:06:44,182 --> 01:06:46,140
they're both at rest with
respect to each other

1224
01:06:46,140 --> 01:06:49,140
and they've been synchronized
in their own rest frame--

1225
01:06:49,140 --> 01:06:50,990
and they're connected
by say a rod,

1226
01:06:50,990 --> 01:06:55,020
which has some length which
we'll call l sub 0 in the rest

1227
01:06:55,020 --> 01:06:57,600
frame of the two clocks .

1228
01:06:57,600 --> 01:07:00,980
If that whole system
moves relative to us

1229
01:07:00,980 --> 01:07:04,260
by speed v along
the length, those

1230
01:07:04,260 --> 01:07:07,090
clocks, even though they were
synchronized in the rest frame,

1231
01:07:07,090 --> 01:07:09,040
would not look
synchronized to us

1232
01:07:09,040 --> 01:07:12,070
but rather would look like
they're out of synchronization.

1233
01:07:12,070 --> 01:07:15,350
And in particular, it will
look like the trailing clock,

1234
01:07:15,350 --> 01:07:18,860
the one on the back of
this combination of clocks,

1235
01:07:18,860 --> 01:07:21,040
will look a little bit
like it reads a little bit

1236
01:07:21,040 --> 01:07:24,250
later in the day
by a factor which

1237
01:07:24,250 --> 01:07:27,490
is beta times l sub 0 times c.

1238
01:07:27,490 --> 01:07:30,380
Beta remember is
v over c. l sub 0

1239
01:07:30,380 --> 01:07:32,110
is the distance
between the clocks

1240
01:07:32,110 --> 01:07:35,230
as measured in the rest
frame of the clocks.

1241
01:07:35,230 --> 01:07:37,670
And c is of course
the speed of light.

1242
01:07:37,670 --> 01:07:40,012
On the other hand,
if these two clocks

1243
01:07:40,012 --> 01:07:41,470
were moving in a
direction which is

1244
01:07:41,470 --> 01:07:43,860
perpendicular to the
line that joins them,

1245
01:07:43,860 --> 01:07:45,776
then there's no change
in the synchronization.

1246
01:07:51,850 --> 01:07:54,250
Now I should mention that
this really is crucially

1247
01:07:54,250 --> 01:07:58,220
important to the consistency
of the whole picture.

1248
01:07:58,220 --> 01:08:00,830
And actually showing that
the picture is consistent

1249
01:08:00,830 --> 01:08:05,120
is more than we're going to do.

1250
01:08:05,120 --> 01:08:07,500
It's not impossible to do
at this level by any means.

1251
01:08:07,500 --> 01:08:09,624
But it's more than we're
going to do in this class,

1252
01:08:09,624 --> 01:08:12,780
since we're not focusing
on special relativity.

1253
01:08:12,780 --> 01:08:15,460
But when you hear about
special relativity

1254
01:08:15,460 --> 01:08:17,240
and look at these
postulates, you

1255
01:08:17,240 --> 01:08:20,120
might realize that there seems
to be a pretty obvious tension

1256
01:08:20,120 --> 01:08:24,569
between the idea that
moving clocks run slowly

1257
01:08:24,569 --> 01:08:28,500
and that all
observers experience

1258
01:08:28,500 --> 01:08:30,284
the same laws of physics.

1259
01:08:30,284 --> 01:08:32,700
Because it means that if you
and I are moving with respect

1260
01:08:32,700 --> 01:08:35,649
to each other, I would
claim that your clock

1261
01:08:35,649 --> 01:08:37,398
was running slowly.

1262
01:08:37,398 --> 01:08:38,939
But at the same
time, you would claim

1263
01:08:38,939 --> 01:08:40,314
that my clock was
running slowly.

1264
01:08:40,314 --> 01:08:42,260
Because from your
point of view, you're

1265
01:08:42,260 --> 01:08:44,880
at some fixed velocity and
therefore an inertial frame.

1266
01:08:44,880 --> 01:08:47,270
And I'm moving relative to you.

1267
01:08:47,270 --> 01:08:50,200
So as you would describe it,
I would be a moving clock.

1268
01:08:50,200 --> 01:08:52,714
And my clock would
be running slowly.

1269
01:08:52,714 --> 01:08:54,380
So I think your clock
is running slowly.

1270
01:08:54,380 --> 01:08:57,229
You think my clock
is running slowly.

1271
01:08:57,229 --> 01:08:58,270
That seems to contradict.

1272
01:08:58,270 --> 01:08:59,870
What happens if we hold the
clocks next to each other

1273
01:08:59,870 --> 01:09:01,495
and really just watch
how they compare?

1274
01:09:01,495 --> 01:09:02,670
Which one gets ahead?

1275
01:09:02,670 --> 01:09:05,054
How can we disagree on that?

1276
01:09:05,054 --> 01:09:07,470
Well of course, we can't hold
the clock next to each other

1277
01:09:07,470 --> 01:09:09,511
and also have them moving
relative to each other.

1278
01:09:09,511 --> 01:09:12,950
That's part of how you
get out of this conflict.

1279
01:09:12,950 --> 01:09:16,170
But let's think in a
little bit more detail

1280
01:09:16,170 --> 01:09:17,710
about what we're
really saying when

1281
01:09:17,710 --> 01:09:20,850
I say that your clock
is running slowly.

1282
01:09:20,850 --> 01:09:23,040
Remember, I want to
make all my observations

1283
01:09:23,040 --> 01:09:25,760
not by watching you, because
then there's this time delay

1284
01:09:25,760 --> 01:09:27,979
effect which complicates things.

1285
01:09:27,979 --> 01:09:29,770
I make all of my
observations by having

1286
01:09:29,770 --> 01:09:33,000
a family of local observers
surrounding me all at rest

1287
01:09:33,000 --> 01:09:34,609
relative to me.

1288
01:09:34,609 --> 01:09:36,302
And they report back to me.

1289
01:09:36,302 --> 01:09:38,260
And only when I receive
those reports and piece

1290
01:09:38,260 --> 01:09:40,210
them together do I
get the simple picture

1291
01:09:40,210 --> 01:09:42,540
of what happened
where when, which

1292
01:09:42,540 --> 01:09:44,859
is the simple picture
that is described

1293
01:09:44,859 --> 01:09:49,470
by these "appear" relationships.

1294
01:09:49,470 --> 01:09:53,120
So when I say your clock is
running slowly, what I mean

1295
01:09:53,120 --> 01:09:55,590
is that if I have a network
of clocks all at rest relative

1296
01:09:55,590 --> 01:09:58,060
to me, and your
clock comes shooting

1297
01:09:58,060 --> 01:10:02,700
by, I would measure what time
it reads as it goes past all

1298
01:10:02,700 --> 01:10:04,780
my local clocks.

1299
01:10:04,780 --> 01:10:08,170
Rather, they would measure
what time your clock reads

1300
01:10:08,170 --> 01:10:09,920
as it goes past
each local clock.

1301
01:10:09,920 --> 01:10:12,840
And then they would
all report back to me.

1302
01:10:12,840 --> 01:10:15,230
So if your clock is running
slowly, for example,

1303
01:10:15,230 --> 01:10:17,450
let's say by a factor
of two, it would

1304
01:10:17,450 --> 01:10:20,420
mean that when you're
clock passes my clock

1305
01:10:20,420 --> 01:10:23,690
and my clock reads one
second, your clock would only

1306
01:10:23,690 --> 01:10:25,794
read half a second, because
it's running slowly.

1307
01:10:25,794 --> 01:10:27,210
It hasn't ticked
off as much time.

1308
01:10:27,210 --> 01:10:31,770
When it passes some later clock
of my sequence of clocks, where

1309
01:10:31,770 --> 01:10:33,410
my clock reads two
seconds, your clock

1310
01:10:33,410 --> 01:10:36,330
will read one second, and so on.

1311
01:10:36,330 --> 01:10:39,542
So in that sense, I would say
your clock was running slowly.

1312
01:10:39,542 --> 01:10:41,500
Now that has to be
consistent with you thinking

1313
01:10:41,500 --> 01:10:45,400
that my clocks are all
running slowly as well.

1314
01:10:45,400 --> 01:10:49,250
So if you agree that my
clocks were all synchronized,

1315
01:10:49,250 --> 01:10:52,150
then you would conclude that
my clocks must be running fast.

1316
01:10:52,150 --> 01:10:53,990
Because when your clock
reads a half second,

1317
01:10:53,990 --> 01:10:55,050
my clock reads one second.

1318
01:10:55,050 --> 01:10:56,424
When your clock
reads one second,

1319
01:10:56,424 --> 01:10:58,114
my clock reads two seconds.

1320
01:10:58,114 --> 01:11:00,030
You would say that my
clocks were reading fast

1321
01:11:00,030 --> 01:11:02,444
if we just made that
direct comparison.

1322
01:11:02,444 --> 01:11:04,860
But at the same time, we know
that's not the right answer.

1323
01:11:04,860 --> 01:11:07,210
You should see the
same physics and I see.

1324
01:11:07,210 --> 01:11:09,450
If you and I are moving
with respect to each other,

1325
01:11:09,450 --> 01:11:12,680
you should see my
clocks running slowly.

1326
01:11:12,680 --> 01:11:17,350
So the way out of that is
this question of simultaneity.

1327
01:11:17,350 --> 01:11:19,380
From the point of
view of your clocks

1328
01:11:19,380 --> 01:11:22,410
going past all my
clocks, if you just

1329
01:11:22,410 --> 01:11:24,694
looked at the time on my
clocks as you passed them,

1330
01:11:24,694 --> 01:11:26,110
you would actually
think that they

1331
01:11:26,110 --> 01:11:28,640
were running fast
relative to your clock.

1332
01:11:28,640 --> 01:11:30,237
But you would also,
however, not think

1333
01:11:30,237 --> 01:11:32,320
of those clocks were
synchronized with each other.

1334
01:11:32,320 --> 01:11:34,300
So you don't
determine what speed

1335
01:11:34,300 --> 01:11:37,309
they're running by looking
at two different clocks.

1336
01:11:37,309 --> 01:11:39,100
If you want to figure
out whether my clocks

1337
01:11:39,100 --> 01:11:42,590
running fast or slow, you want
to look at one of my clocks

1338
01:11:42,590 --> 01:11:44,590
and see how it
changes with time,

1339
01:11:44,590 --> 01:11:45,882
not comparing different clocks.

1340
01:11:45,882 --> 01:11:47,506
Because the different
clocks would just

1341
01:11:47,506 --> 01:11:50,020
be out of synchronism with
respect to each other as you

1342
01:11:50,020 --> 01:11:52,300
would see it.

1343
01:11:52,300 --> 01:11:54,680
And we're not going to
go through the details.

1344
01:11:54,680 --> 01:11:58,070
But if you do look at
my clocks consistently

1345
01:11:58,070 --> 01:12:01,990
using a family of
your clocks that

1346
01:12:01,990 --> 01:12:04,440
are stationary
relative to you, just

1347
01:12:04,440 --> 01:12:06,482
as I thought about
a network of clocks

1348
01:12:06,482 --> 01:12:08,690
when I was trying to measure
the speed of your clock,

1349
01:12:08,690 --> 01:12:09,898
then everything's consistent.

1350
01:12:09,898 --> 01:12:12,490
You would see all my
clocks running slowly.

1351
01:12:12,490 --> 01:12:14,700
I would see all of your
clocks running slowly.

1352
01:12:14,700 --> 01:12:17,330
And because we disagree
on what's simultaneous,

1353
01:12:17,330 --> 01:12:20,250
there are no contradictions.

1354
01:12:20,250 --> 01:12:21,834
So simultaneity is
absolutely crucial

1355
01:12:21,834 --> 01:12:23,250
to get out of what
would otherwise

1356
01:12:23,250 --> 01:12:25,670
be a glaring contradiction
in the whole system.

1357
01:12:31,720 --> 01:12:34,400
OK, that's about all I
really wanted to say today.

1358
01:12:34,400 --> 01:12:36,280
But let me just give
a preview of things

1359
01:12:36,280 --> 01:12:38,446
that we'll talk about later
in the course concerning

1360
01:12:38,446 --> 01:12:40,617
relativity.

1361
01:12:40,617 --> 01:12:42,700
So far I think we've said
all we are going to say,

1362
01:12:42,700 --> 01:12:45,324
unless the questions.

1363
01:12:45,324 --> 01:12:46,740
I think we've said
all we're going

1364
01:12:46,740 --> 01:12:48,860
to say about the
kinematic consequences

1365
01:12:48,860 --> 01:12:51,260
of special relativity.

1366
01:12:51,260 --> 01:12:54,070
And we're not going to be trying
to derive them, as I said.

1367
01:12:54,070 --> 01:12:56,730
If you're interested,
by the way,

1368
01:12:56,730 --> 01:12:58,530
the notes recommend
several references,

1369
01:12:58,530 --> 01:13:00,860
including the lecture
notes from eight to 86

1370
01:13:00,860 --> 01:13:03,430
from earlier years when
special relativity was

1371
01:13:03,430 --> 01:13:05,916
included as a real topic.

1372
01:13:05,916 --> 01:13:07,790
So certainly if you're
interested in learning

1373
01:13:07,790 --> 01:13:09,600
about this and you
haven't already seen it,

1374
01:13:09,600 --> 01:13:11,430
I'm happy to help you.

1375
01:13:11,430 --> 01:13:14,180
But otherwise, it will
not be part of this course

1376
01:13:14,180 --> 01:13:17,460
to discuss how these
three consequences

1377
01:13:17,460 --> 01:13:20,740
of special relativity arise
from the basic postulates

1378
01:13:20,740 --> 01:13:22,530
of special relativity.

1379
01:13:22,530 --> 01:13:25,740
But later we will be saying
things that follow further

1380
01:13:25,740 --> 01:13:29,720
along the line by
pursuing the consequences

1381
01:13:29,720 --> 01:13:33,060
of special relativity
for momentum and energy,

1382
01:13:33,060 --> 01:13:35,420
which will be important to us.

1383
01:13:35,420 --> 01:13:38,530
The connection is the
important connection

1384
01:13:38,530 --> 01:13:42,330
and simple connection
that energy and momentum

1385
01:13:42,330 --> 01:13:44,110
are only interesting
to us if they're

1386
01:13:44,110 --> 01:13:47,380
defined in a way which makes
them conserved quantities.

1387
01:13:47,380 --> 01:13:49,960
That's why energy and momentum
are important in physics.

1388
01:13:49,960 --> 01:13:52,250
Because for a closed
system, the total energy

1389
01:13:52,250 --> 01:13:54,730
and the total momentum
do not change.

1390
01:13:54,730 --> 01:13:56,740
Energy and momentum
can be transferred

1391
01:13:56,740 --> 01:13:59,130
from one part of a
system to another.

1392
01:13:59,130 --> 01:14:03,830
But energy and momentum cannot
be either made nor destroyed.

1393
01:14:03,830 --> 01:14:07,760
Now if we took Newton's
definitions of energy

1394
01:14:07,760 --> 01:14:14,700
and momentum, and used
relativistic kinematics, what

1395
01:14:14,700 --> 01:14:17,070
we would find is
that if we looked

1396
01:14:17,070 --> 01:14:19,750
at, say, a collision
of two particles,

1397
01:14:19,750 --> 01:14:22,910
the Newtonian definitions
of energy and momentum,

1398
01:14:22,910 --> 01:14:25,720
if we took those
seriously, would tell us

1399
01:14:25,720 --> 01:14:29,115
what might happen
in a collision.

1400
01:14:29,115 --> 01:14:30,990
Usually there's an angle
that's undetermined.

1401
01:14:30,990 --> 01:14:33,315
But given an angle, it
determines everything else.

1402
01:14:36,140 --> 01:14:39,490
If we used special
relativity to then describe

1403
01:14:39,490 --> 01:14:43,147
what that same collision would
look like in a different frame,

1404
01:14:43,147 --> 01:14:45,480
we would find that these
Newtonian definitions of energy

1405
01:14:45,480 --> 01:14:48,240
and momentum would not be
conserved in the other frame,

1406
01:14:48,240 --> 01:14:50,510
if they were conserved
in the first frame.

1407
01:14:50,510 --> 01:14:53,440
The conservation laws
are dependent on what

1408
01:14:53,440 --> 01:14:55,860
reference frame you're using.

1409
01:14:55,860 --> 01:15:00,500
So what Einstein
amended was slightly

1410
01:15:00,500 --> 01:15:04,950
modified definitions of
energy and momentum, which

1411
01:15:04,950 --> 01:15:10,250
are determined by the criterion
that these slightly modified

1412
01:15:10,250 --> 01:15:13,590
definitions of energy
and momentum should,

1413
01:15:13,590 --> 01:15:15,950
if they are conserved
in one frame

1414
01:15:15,950 --> 01:15:18,140
be also concerned in
any other frame, which

1415
01:15:18,140 --> 01:15:21,710
are related by the relationships
to special relativity.

1416
01:15:21,710 --> 01:15:24,040
So that's why it was
essential once one changed

1417
01:15:24,040 --> 01:15:26,180
the kinematics of going
from one frame to another

1418
01:15:26,180 --> 01:15:30,440
to also change the definitions
of energy and momentum

1419
01:15:30,440 --> 01:15:34,540
so that the conservation laws
would hold in all frames using

1420
01:15:34,540 --> 01:15:38,530
the new transformation equations
to get from frame to frame.

1421
01:15:38,530 --> 01:15:40,120
And that's why
later in the course

1422
01:15:40,120 --> 01:15:43,930
we will be introducing a
slightly modified, slightly

1423
01:15:43,930 --> 01:15:47,100
non-Newtonian, definitions
of energy and momentum

1424
01:15:47,100 --> 01:15:49,009
of moving particles.

1425
01:15:49,009 --> 01:15:50,050
OK, that's all for today.

1426
01:15:50,050 --> 01:15:53,900
I will see folks next Tuesday.