I exchanged a bunch of emails a week or two ago with a journalist who was working on a story involving the possibility of faster-than-light travel. He wanted me to check some statements about the relationship between FTL and causality. FTL creates problems for causality, because if you have an object moving faster than light, there will be pairs of observers who see events involving the FTL object happening in different orders, which means somebody will see an effect happen before its cause.
I talk about this is How to Teach Relativity to Your Dog using the example of a stationary dog, a moving cat, and an alien zipping by at four times the speed of light. Here’s a figure showing how this appears to the dog:
In this “spacetime diagram,” the left-right axis indicates the position along the direction of the cat’s motion, while time marches upward into the future. Vertical lines are equally spaced position markers according to the dog, while horizontal lines are equally space instants of time according to the dog. The dashed red lines are rays of light sent out at the instant the cat and dog had the same position, and set the scale for everything.
The dog sees the cat moving left to right at half the speed of light. The alien comes in from the left, passes the dog first (event 1), and then the cat (event 2). Perfectly sensible, and the dog could, for example, hand the passing alien a water balloon which the alien could then use to soak that pesky cat.
This looks very different if you replace the dog’s grid of position and time markers with the cat’s, though. In that case, you get something that looks like this:
Both space and time look different to the cat, due to her high speed. Equally spaced position markers according to the cat have to tip to the right, parallel to the cat’s trajectory, while the cat’s time instants tip up. The regular squares of the dog’s grid become rhombuses in this representation.
According to the cat, then, the alien passes the cat, and only later passes the dog. Which makes the whole water balloon thing kind of problematic– the alien would appear to the cat to come in from the right empty-handed, soak the cat with a water balloon and then carry that balloon on to the left, and later on hand it to the dog.
This reversal of ordering obviously screws up causality, and is one of the best reasons why FTL travel is impossible. It’s even possible to create paradoxes by using FTL communications, so sending information faster than the speed of light is ruled out.
My journalist friend had this basically correct, but we went back and forth a bit about a subtle issue of perception. This is always a tricky business in relativity, as it’s very tempting to attribute weird effects to the finite travel time for light to get from one place to another, and say things like “The events according to the cat appear to be in the opposite order.” That’s not what’s really going on, though– relativity isn’t about optical illusions. There is no measurement the cat can do that will tell her anything other than that the alien passed her first, and only later encountered the dog.
I think we got this cleared up– I don’t think the story has appeared yet, which is also why I’m being coy about the identity of the journalist, who is free to identify himself in comments. But it did raise the question in my mind of what this would actually look like from the point of view of the dog. That is, what would the dog see?
You see, the diagrams above are a sort of “God’s eye” view of the scenario, not the kind of thing any of the participants could directly record. The dog could draw this diagram of events, but only after the fact, after either compiling the records of lots of individual observers at the different position markers, or by looking at the light she sees, and working backwards to correct for the travel time of the light emitted by different objects.
So, what would the “dog’s eye” view of the scenario really look like? How would the dog perceive her interaction with the alien? Well, we can understand this by adding some extra events to the diagram:
Here, I’ve added events “a,” “b,” and “c,” so there are two events on either side– while the alien is approaching, and while the alien is heading away. To work out what the dog sees, we need to add lines corresponding to the light emitted by the alien as it passes each of these. Looking at the approaching side first, we see that things are a little weird:
The alien, moving at four times the speed of light, arrives well ahead of the light from earlier in its trajectory. Thus, the dog would have absolutely no warning of the alien’s approach– it would just suddenly be there. Then the light from nearby would arrive (event b), and then the light from farther away (event a). If we add in the receding side, we have:
Again, the alien outraces its own light, so it’s gone just as suddenly as it appears. The light from its departure lags well behind the actual events, with nearby events appearing only after some delay (event c) and more distant events much later (event 2, the soaking of the cat).
So, the answer to the question “What does the dog see?” is “Some weird stuff.” Adding markers for the arrival of the light from each of the events gives you the idea:
From the dog’s point of view, the alien appears without warning (event 1), then seems to move away in both directions simultaneously– like two identical aliens headed in opposite directions. Light from a given distance on the approaching side will arrive a bit ahead of light from the receding side (event b is seen before event c, though they’re the same distance away), so it will look sort of like the alien zipping off to the left is heading away a bit faster than the one heading off to the right. The “dog’s eye” sequence of events is not the “a-b-1-c-2” sequence of the “God’s eye” view, but “1-b-c-a-2.” It’s only after the fact, when she’s had time to say “What the hell was that?” and do a bit of math that the dog can construct the global picture shown in the diagrams.
So, there’s the answer. You could extend this to the cat’s scenario by a similar process of event-adding and line-drawing, but I’m not going to. We’ll call that homework– draw and label your diagrams neatly, and send them to Rhett for grading.
And don’t even think about moving faster than the speed of light. Seriously, it’ll mess with your head.