The relativistic Doppler effect

What we see is a yellow spaceship passing along a spacestation and its subsatellites. The top area is as seen from the spacestation and surroundings, which are standing still, while the spaceship moves to the right. The bottom area is the same happening, but now from the viewport of the captain. For him, his spacecraft is standing still, while the spacestations move to the left. The orange and lime dots are lightsignals, which we will discuss later.

(Drag the slider to the middle, timevalue= 0) When the captain, sitting on the bridge in the middle of his spaceship (yellow clock) passes along the traffic controller, sitting in the mainsatellite of her spacestation (yellow clock), both agree that is 12:00 exact and all the red armed clocks are supposed to have been adjusted to indicate that time too. The fact that this seems not to be the case for the moving clocks is a crucial point. There is no such thing as the "same time" in relativity for moving objects. Likewise the length of the spaceship and the distances between the subsatellites seems to be different in the two systems. There is no such thing as the "same length" in relativity for moving objects.

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Spacestation

Although the traffic controller sits in the main station, she has 6 subsatellites around her, all on regular distances of 30 lightminutes. The 2 farest (blue, magenta) are 90 lightminutes away (that is little bit more than from Earth to Saturn), the next 2 (cyan, red) are 1 lighthour away (from Earth to well beyond Jupiter), and the 2 closest (green, orange) are 30 lightminutes away (from Earth to halfway Mars and Jupiter). Remember that a lightminute is the distance covered by light in one minute. As the speed of light is almost 300 000 km/s and there are 60 seconds in an minute, that is 18 Gm (18 000 000 km).

How can the controller know that the clocks on all stations indicate the right time? Because she has a direct radio link with all of them, and continuously they send her their clock signals. The readings appear on her dashboard above her own clock as the blue armed clocks. All those clocks are lagging behind. Of course, radio waves go with the same speed as light, and we do not know of anything going faster than that. She knows the distance of the satellites farest out, that their light need 90 minutes to reach her, and as such the clocks are supposed to lag 90 minutes behind. The fact they indeed do, proves that the stations are on the right time. This trick is not working for the clocks in the spaceship. Although she can see any of its clocks there when they pass along, she cannot be sure about those elsewhere. Because their light needs time to reach her, and by that time the spaceship has moved, so what is then the distance? As it turns out: you can compare clocks which are on any distance away from you as long as they are standing still. Or you can compare clocks which pass along you with any speed. But you cannot directly compare clocks which are moving and at some distance away without additional information.

Spaceship

The captain of the spaceship also has a dashboard which shows him the clocks of the pilot in the forecastle (orange) and the engineer at the rear (green). Since both these clocks seem to him 30 minutes behind, while he has established that they are on time, he knows that his spaceship is one lighthour long. He can see the clocks of the satellite stations, but will only look at each of them when he passes along it, because he knows that the indication of the others makes no sense to him.

The speed of the ship is v = 0.6c

Both the controller and the captain can determine this by measuring the time needed for a particular distance. For example the traffic controller got the message from station cyan, 60 lightminutes away that the centre of the spaceship passed along at 10:20. Likewise station red, 1 lighthour in the other direction, will send her later a report that the ship's bridge came along at 13:40. Comparing that to her own 12:00 timing, the ship is covering 60 lightminutes in 100 minutes time, indeed a speed of 0.6 lightminute per minute, is 0.6c. The captain can ask his front pilot and rear engineer to report to him when they passed along the main station, which afterwards turns out to have been on 11:10 and 12:50. Again a 100 minutes timespan for his 60 lightminutes long vessel results in the same speed. Note that he cannot determine his speed by watching when he passes let say the -30 and +30 minutes subsatellites. Because their distances have been determined in a system in which they stand still, while from his viewpoint they move, and there is no such thing as same lengths in different systems. (However, he can do it by also using their time, seeing the clock of the +30 station on 12:50 when he passes along, even if his own watch tells it to be 12:40).

Length and time contraction

The relativistic correction to length and time is given by √(1-v²/c²), which gives 1.25 for v = 0.6 c. All moving clocks run 1.25x slower (or 80% as fast) as clocks standing still. All moving objects are 1.25x smaller (or 80% as big) in the movement's direction. This holds true for both the viewpoint of the captain and the controller. How can that be? If the controler sees the spaceship 80% compressed to be only 48 lightminutes long, and the captain sees her 80% compressed, should she then not be 64% totally compressed? Yet being 100% at the same time? If you think that, you make the assumption that the 2 compressions happen at the same time. But there is no such thing as simultaneity. Run the movie and check that although controller and captain may differ at any moment about lengths and times, in both systems they agree that, for example: the bridges passes substation 30 (yellow) at 12:50 substation time and 12:40 ship time; the bridge passes substation 60 (red) at 13:40 substation time and 13:20 ship time; and so forth. Likewise the controller sees the front of the spacecraft passing along at 11:20 her time or 11:10 ship time, and the tail at 12:40 her time, 12:50 his time. As these events, and many others, give the same reading in either system, there is no way to distinguish who is standing still and who moves. Indeed relativity is relativity because only a relative speed can be measured; it is no longer possible to define a source speed (u) and observer speed (v).

But be aware of a subtlety: when watching a moving clock, make sure you watch always the same clock, because we know that there is no simultaneity for moving clocks. And since that one clock is moving, you will have to ask your assistants, standing still along the line, to check that clock when it is coming along. Therefore when all the substation assistants report to the controller what was the reading of the ship's captain clock when it came along, she will see that the captain is loosing 10 minutes for every station passed. But if she on her own tries to see the different clocks of the craft when they pass by, they seem to run too fast. Because she is watching different moving clocks, they indicate different times and her measurements are worthless. But the captain can tell front and aft station to report to him when they pass along the controller's station. For him the clocks run the same, and he will find the the controller is loosing 10 minutes every time.

Doppler effect

If you insist on watching a moving clock which is not passing you, you have to consider that it takes time for the signal to reach. A wave travelling between moving observers, but that is nothing else than the Doppler effect!

The captain has ordered both the back and front officer to flash a light towards the bridge every hour; the orange and lime dots, which travel, of course, at speed c. Indeed it is easily checked that these messages reach the captain 30 minutes later, his distance from both officers being 30 lightminutes. Also, as they stand still compared to him, their successive messages reach him on every half hour, being on the same frequency of 1 wave per hour as they were emitted. This holds true in the coordinate system of the spaceship (of course) but also in that of the spacestation, or in that of any other constantly moving frame. There is another striking effect: in both coordinate systems, light moves forth with the same speed. Whatever the relative speed of any frame, they all measure the same value. In fact it is length and time which are changing in such a way that the constancy of the light speed is guaranteed.

Meanwhile, again in each, or any, coordinate system, the traffic controller sees the lightsignals of the approaching stern of the spacecraft arriving at 30 minutes intervals, twice the frequency of the orignal. Similarly the lightsignals of the receding bow come to her only once every 2 hours, half the frequency of the original. In complete accordance with the formula for the relativistic Doppler effect: √(1-v/c)/(1+v/c), which gives 2 or 0.5 for v = -0.6 c or +0.6 c respectively. This formula is essentially a multiplication of the relativity formula with the classical Doppler effect.