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Äthertheorie Up: Kurs
Previous: Kausalität
I have told this story once before, but without supporting it with a space-time diagram. This is done with the image below. The dark blue coordinate system is the reference system of the sun and also of an asteroid (the black chunk), which is at rest with respect to the sun. Let our asteroid be 10000 light years away. The sinusoidal line is the projection of the world line of the earth on the xct plane. It is not drawn to scale, because otherwise it would only appear as a straight line on the scale of 10000 light-years, which corresponds to the distance to the asteroid, and that line would coincide with the ct-axis. However, I have taken care that the topology of the drawing remains correct despite the scale violations and also the metric, as far as possible.
The green rocket represents Perry Rhodan's space ship, which departs from the earth at a time when it moves towards the asteroid (this event is at the origin of the coordinates). The frame of reference of the earth is then the x'ct'-system. Its speed relative to the inertial frame of the sun is the speed of the earth around the sun, i.e. 30 km/s. If Rhodan's spacecraft travels to the asteroid with respect to this frame of reference in zero time (or a very short time, say, 1 day), it moves (practically) along the x'-axis. The slope of the x'-axis in the xct-system is v/c, i.e. (30km/s)/(300000km/s) = 1/10000. Since the asteroid is 10000 light years away, the x'-axis intersects its "world line" (the gray corridor with a dashed border) at ct = 1 light-year. The earth has now orbited the sun once (one oscillation of the sinus). In the system of the sun (and of the asteroid), Perry Rhodan's flight took a year.
Now Rhodan is attacked and killed on the asteroid. He succeeds, however, to make a hyperspace emergency call. That signal moves, in the system xct (which is also that of the asteroid) in (almost) zero time to the earth, hence arrives there a year after Rhodan's departure. I have drawn the signal as a jagged line only for it to look more interesting. Its world-line should simply be a parallel to the x-axis.
Atlan is now equipping a rescue team. However, he has time, because he knows that the earth's state of motion is unfavorable for an immediate start. If he were to fly immediately, he would need a year relative to the frame of reference of the sun as Rhodan did, because the earth is just moving back towards the asteroid (the x'ct'-system is the relevant one). So he rather waits for six months. Then the earth moves away from the asteroid and its reference system is the x''ct'' system. In it Atlan flies at full whack, his world line is practically the x''-axis, which has a slope of -1/10000. Therefore, as measured in the reference frame of the sun, he will arrive a year *before* his departure. Because this is also the system of the asteroid, he will reach the latter half a year before Rhodan's arrival, enough time to save him.
We have not said anything about *how* the superluminal flight is going on. Whether it happens via hyperspace, linear space, sextadim-half-track or whatever kind of superluminal drive, it does not matter. The crucial point is that the initial and end points of the trip are such that the average speed is significantly greater than the speed of light (e.g., 10000 light years/day). Hence, one can deduce from the validity of special relativity that if superluminal trips are possible, causal loops are possible, too, the principle of causality is thus violated. Therefore, one rather concludes that superluminal journeys are impossible.
At the time, I did emphasize in the discussion that this result already follows from the validity of special relativity at subluminal speed. This, of course, is a negligent formulation. One can not separate between the validity in different speed ranges. Relativity is a theory of the structure of space-time. If this structure is correct, then the conclusions mentioned can be drawn. The correctness of the structure of space-time, however, has nothing to do with the velocities used by a traveler to connect his sojourns in different places, as long as he is within it (*between* his sojourns he may stay wherever he likes to, even outside space-time). The only speed that explicitly enters into the calculations of the Perry Rhodan story is the speed of the earth on its orbit around the sun.
Rene Haustein has objecteed that the violation of causality in my example is done by the message that Rhodan sends. This message moves to the past already. Well, you see in the picture that it is not so in the frame of reference of the sun and the asteroid. If you forget about the zigzag representation of the drawing, it is clear that this message can go ''a little bit'' to the future (up to half a year) without changing the consequences (we only have assumed zero time transport for the sake of simplicity). That Atlan moves into the past relative to the sun is *independent* of the message. In fact, you could omit the message from the story and let Atlan fly only because of the idea that he needs to prepare a few things on the asteroid before Rhodan's arrival there. The result would be the same.
This argument is directed only against considering the message as the true non-causal element in the chain. One may also identify Atlan's flight as this element. To create a *causal loop*, *four* world lines are needed: that of the asteroid between Atlan's and Rhodan's arrival, that of the message, that of the earth between the arrival of the message and Atlan's departure and that of Atlan. And one can not say that Atlan's or the message's world line alone is responsible for the loop, because the omission *of each* of these pieces destroys the loop.
If you do *not* have a *causal loop* without message, then it may be assumed that in this case there must be a "harmless" interpretation of the chain of events. Indeed, this is the case. For the observer in the green coordinate system x'''ct''', who moves slightly faster to the left than the earth at Atlan's takeoff, the sequence of events reads simply as follows: Rhodan flies towards the asteroid, but quite slowly, Atlan (unsuspecting) starts one-and-a-half years later, and flies much faster than Rhodan, which is why he arrives much earlier than the latter. For this green observer, of course, the existence of Rhodan's message would be a great annoyance because it breaks down all his interpretation. Insofar, the *possibility* of sending a message is important for the construction of a time paradox, but not for the question of whether Atlan flies into the past. He does so without a time paradox in the frame of reference of the asteroid.
The Perry Rhodan cosmos does not work that way, of course. There, implicitly a universal time for all civilizations of the universe is assumed. (For example, in ''When worlds go silent'', no. 1941, catastrophic consequences are predicted, if Jii'Nevever could jump back and forth between two galaxies, because a *simultaneous* attack on both her base planets would become practically impossible. From the viewpoint of special relativity, this is absurd, since simultaneity is observer-dependent and, given a distance of millions of light-years between galaxies, can differ by millions of years for different observers.) Obviously, the authors can not afford to let their cosmos work like this, because it would impose an enormous complexity on the story lines.
In the last section of the course, I shall discuss Peter Holzer's objection, which actually shows a possibility of imagining such a cosmos without the predictions of special relativity being touched upon by experiments. However, this possibility would mean that the relativity principle, and thus the statements of special relativity about the structure of space-time, are wrong (and, strangely enough, the physics is still described correctly for subluminal phenomena).
Next: Lorentzsche Äthertheorie Up: Kurs Previous: Kausalität