1 00:00:06,810 --> 00:00:09,210 MARKUS KLUTE: Welcome back to 8.20, Special Relativity. 2 00:00:09,210 --> 00:00:10,710 In this section, we're going to talk 3 00:00:10,710 --> 00:00:13,080 about the relativistic Doppler effect. 4 00:00:13,080 --> 00:00:15,990 And we make good use of our space-time diagrams, 5 00:00:15,990 --> 00:00:18,190 which we discussed earlier. 6 00:00:18,190 --> 00:00:20,130 So the situation is as follows-- 7 00:00:20,130 --> 00:00:22,920 to simplify this, we have a source 8 00:00:22,920 --> 00:00:24,960 which is emitting pulses. 9 00:00:24,960 --> 00:00:26,550 So the waves are pulses. 10 00:00:26,550 --> 00:00:29,820 Every now and then there is a beep, and another beep, 11 00:00:29,820 --> 00:00:31,380 and another beep. 12 00:00:31,380 --> 00:00:34,230 And those pulses travel with their velocity-- 13 00:00:34,230 --> 00:00:37,650 with their wave velocity. 14 00:00:37,650 --> 00:00:39,570 And they have a world line represented here 15 00:00:39,570 --> 00:00:40,800 in the space-time diagram. 16 00:00:40,800 --> 00:00:44,640 This is pulse number one, and this is pulse number two. 17 00:00:44,640 --> 00:00:46,620 The distance between those two pulses 18 00:00:46,620 --> 00:00:51,070 is our period, the period of our wave, which we call tau. 19 00:00:51,070 --> 00:00:53,500 The question now is, how is this being observed 20 00:00:53,500 --> 00:00:56,530 by an observer which is moving with a relative velocity 21 00:00:56,530 --> 00:00:59,350 v with respect to the source? 22 00:00:59,350 --> 00:01:01,820 So let's analyze this. 23 00:01:01,820 --> 00:01:06,130 So if we want to characterize or find our position x1 and x2, 24 00:01:06,130 --> 00:01:11,140 we can do this by saying x1 is equal to ct1 25 00:01:11,140 --> 00:01:15,590 or equal to x0, which is the distance of the observer 26 00:01:15,590 --> 00:01:18,470 to the source plus c times t1. 27 00:01:18,470 --> 00:01:22,430 v is the velocity in which the source is moving. 28 00:01:22,430 --> 00:01:27,990 And similarly for t x2, we find c times t2 minus tau. 29 00:01:27,990 --> 00:01:33,930 And that's also equal to x0 plus v times t2. 30 00:01:33,930 --> 00:01:36,720 So the distance in time-- 31 00:01:36,720 --> 00:01:39,300 we're still in the reference frame as of the source-- 32 00:01:39,300 --> 00:01:43,800 is given by c times tau over c minus v. 33 00:01:43,800 --> 00:01:46,410 And the distance in space is given by v times 34 00:01:46,410 --> 00:01:49,990 c times tau over c minus v. 35 00:01:49,990 --> 00:01:52,040 So the question is not how this observed-- 36 00:01:52,040 --> 00:01:55,330 how this is seen by the source but how this 37 00:01:55,330 --> 00:01:57,040 is being seen by the observer. 38 00:01:57,040 --> 00:02:00,470 So we have to apply Lorentz transformation. 39 00:02:00,470 --> 00:02:04,210 So in the s prime frame, which is the observer frame, 40 00:02:04,210 --> 00:02:08,650 we find delta t prime is equal to gamma delta t minus v 41 00:02:08,650 --> 00:02:11,020 over c squared delta x. 42 00:02:11,020 --> 00:02:12,850 And then we just fill in the information 43 00:02:12,850 --> 00:02:14,065 as we discussed before. 44 00:02:19,400 --> 00:02:25,610 Tau prime is then gamma times c tau over c minus v times 45 00:02:25,610 --> 00:02:28,580 1 minus v square over c square. 46 00:02:28,580 --> 00:02:32,180 And then you make use of delta equal v of over c. 47 00:02:32,180 --> 00:02:36,450 And we make use of gamma equals 1 48 00:02:36,450 --> 00:02:39,270 over square root of 1 minus beta square. 49 00:02:39,270 --> 00:02:41,020 And we find then-- 50 00:02:41,020 --> 00:02:43,770 this is a little bit of an algebra exercise here-- 51 00:02:43,770 --> 00:02:48,090 that the period now is given by 1 plus 52 00:02:48,090 --> 00:02:53,100 beta over 1 minus beta square root of that times tau. 53 00:02:53,100 --> 00:02:55,530 And the frequency is the inverse. 54 00:02:55,530 --> 00:02:58,770 We'll have 1 minus beta over 1 plus beta square root 55 00:02:58,770 --> 00:03:01,990 of that [? times ?] the frequency. 56 00:03:01,990 --> 00:03:04,560 So we just calculated relativistically 57 00:03:04,560 --> 00:03:08,940 how the period and the frequency of a wave 58 00:03:08,940 --> 00:03:12,200 is Lorentz transformed.