Chess may not really be mathematics, but it is in some way mathematical, so I think I can get away with it here.
Nine years ago Gary K. Gifford published a chess variant called Time Travel Chess, in which pieces and pawns can travel forward in time, and the King (and only the King) can travel backward in time. Backward time travel creates a new timeline.
Gifford was not the first to think about time travel variants of chess; in the comments Larry Smith described another. Smith’s version avoids new timelines, but thereby leaves itself open to paradox; a player loses if a piece that is due to travel backward in time gets captured. (Gifford’s version has quasi-paradoxes to deal with too; a piece that cannot be legally placed after a trip into the future is regarded as “lost in time”.)
The rules for these chess variants have been tuned to make them feasible and enjoyable to play as games. But what if we want to create, not a good game, but a full time travel model? Then a different approach is called for.
First, let’s think a little more about what it means to travel in time.
There’s an underlying assumption in your standard Hollywood time machine type time travel, which is that when you initiate time travel, you travel to a specific endpoint — a specific time (and place). You vanish from time t0 and reappear at time t1. Now, of course, we all are time travellers in reality: we travel into the future at one second per second. And we don’t vanish and reappear… at least not in classical physics. Instead, we occupy all time points in the interval [t0..t1].
We don’t have control over this “time travel”; but we do have control over “space travel”. We can go from x0 to x1, or not, as we choose; but to do so we occupy all points in the interval [x0..x1]. If instead of occupying all points in the interval [x0..x1] we just vanished from x0 and reappeared at x1 we would not be travelling, we’d be teleporting! Likewise, vanishing from time t0 and reappearing at t1 isn’t time travel, it’s time teleporting. In the real world, if time travel were possible, I would not expect time teleporting to happen any more than space teleporting happens, at least macroscopically.
But teleportation is the rule in chess! That is, the simplest way to think about space in chess is to regard it as quantized; only 64 points exist, and by definition space travel is discontinuous, teleporting from one point to the next. Knights are famous for teleporting from e.g. b1 to c3 without going through the pawns in between, but in fact everything moves by hops. Kings and Pawns move directly from the center of one square to the center of an adjacent square. As for Rooks, Bishops, and Queens (and Kings when castling, and sometimes Pawns on their first move), you can regard them as undergoing a series of teleports — e.g. from a1 to a2 to a3 to a4 to a5 in sequence.
Likewise time is quantized, in units of half moves. All time travel must in fact be time teleportation. However, if time teleports are limited to one time point away — one half move at a time — then forward time travel is pretty boring, because teleporting forward one half move at a time is what all pieces normally do! So for time travel to be interesting, there must be backward teleporting or there must be forward teleporting by more than a half move; preferably both.
Paradoxes have to be avoided. Multiple timelines are needed for this, as in Gifford’s rules. But in Gifford’s rules, when a King goes backward in time, not only does a new timeline start, but the old one stops! There is no further play from that position. Of course there isn’t a King to checkmate, so there isn’t much point from the standpoint of game play… unless there’s a way for the time travelling King to return to his original timeline, perhaps at a later time. But for a model of time travel, we need the original timeline to continue, King or no King.
Getting “lost in time” isn’t really a paradox, but it is untidy. One way to deal with it would simply be to eliminate Kings from the game entirely — or at least the concepts of check and checkmate. Then there’d be nothing to prevent a move into the future from taking place; no getting “lost in time”. (Though you’d have to allow capture of a friendly piece if the rules were such that the destination square was pre-specified, about which see below.) There’d be no game, either, but those are the breaks.
On the other hand, looking more closely at the “lost in time” problem might give us some better insight into how time teleportation would work. Supposing the laws of physics allowed time travel but only in ways that obeyed conservation laws, e.g. energy conservation, baryon number conservation, etc. How would that work? Time teleportation appears to violate baryon number conservation: if it disappears from one time and appears at another time, don’t baryons get destroyed at the first time and created at the second? Leaving those questions aside for the moment, let’s just consider that time teleportation from time t0 to a later time t1 might or might not violate some conservation law, i.e. it might or might not be impossible, and it might not be evident at time t0 whether conditions at t1 will allow the trip. So what happens if you try? Do you find you can’t, in which case you’ve received information from the future? Do you get “lost in time”, whatever that means? Do you teleport to a different time?
Well, what does “lost in time” mean? As for teleporting to a different time, what if there are no times to which you can teleport without violating conservation laws?
If (in real life) we attempt to travel (continuously) from x0 to x1 and can’t get there because it violates some conservation law, then we don’t get there, but it doesn’t necessarily mean we can’t start the trip. We might get halfway there, to the point x0.5, and stop, in the process going through all points in the interval [x0..x0.5]. But that’s travel, not teleportation.
In chess, a piece cannot space teleport (move) to a square occupied by a friendly piece. It doesn’t get “lost in space”, and it doesn’t end up on a square other than its intended destination; it simply can’t attempt the move at all. Information passes from the destination to the source to let the knight know it can’t move there, if you like. Likewise it seems to me time teleporting must be impossible if the result is an illegal position; in that case information passes from the future to the past, telling the past that the game situation in the future doesn’t allow the teleport.
In terms of game mechanics, how does that work? A forward teleport attempt requires knowledge of the future. One way to handle this is to say that since we necessarily don’t have knowledge of the future of the timeline in which the teleport originates, it follows that (unlike in Gifford’s rules) time travel into the future cannot be into the same timeline! An exception might be made if the moves up to the end of the teleport are forced, or at least are constrained by the rules sufficiently to show the teleport cannot result in an illegal position. Better yet, maybe, we could allow the teleport into an as-yet-unspecified future of the same (or another) timeline if and only if it can be shown that there exist one or more sequences of moves resulting in the teleport’s producing a legal position. Then the players would be constrained to play such a move sequence in that timeline. In other words, a move (for either player) resulting in a piece becoming “lost in time” would be an illegal move.
Or how about this? In a forward teleport, the destination timestream is unknown until the destination time is reached! For example: let’s say you teleport your King forward to move 7. Now, the next time move 7 is reached in any timestream, if the King can legally appear, it does and the time teleport is completed. If not, then check again the next time move 7 is reached in another timestream. Now a piece can be “lost in time” if no legal destination timestream ever appears — but you won’t know if that’s happened until the game is fully played out. (Hmm, it’d be amusing if it reappeared in some other game.) I think I like this best.
Some other considerations. In Gifford’s rules, a time teleporting piece can land on any square it can legally occupy. This again may be a good game design decision, but is questionable time travel theory. It’s as if time teleporting automatically bestows the power to do a space teleport not normally available to that piece. But recall that in normal chess, pieces time teleport all the time — a half move into the future — but their space teleportation is tightly constrained. It seems to me more reasonable to do one of two things. First possibility: Keep the same space teleportation restrictions for all time teleports. In other words, when a piece teleports, it also may move according to the usual rules: a time teleporting King may reappear on a square adjacent to its origin square, and so forth. But what about, say, a Rook? Does it hop through multiple space points and then time teleport, or vice versa? In other words, must the path be clear in the origin position or the destination position? Or both? Or both plus all intervening positions?
I’m not sure. The last of these kind of makes logical sense to me, for teleports within the same timeline, but what if teleportation is to a different timeline? Messy. Let’s just go with the second possibility: for teleports other than the default (one half move into the future in the same timeline), no simultaneous space teleport is possible. The destination square must be the origin square.
Summing up, the extensions or modifications to Gifford’s rules would include:
- Forward time travel may or may not be into the same timeline. It is into the timeline where the destination move is next reached and the resulting teleport is legal.
- Backward time travel is always into a new timeline, as in Gifford’s rules. When a new timeline is created, both it and the old timeline continue to be played until the game ends.
- Each player may choose any incomplete timeline in which it’s their turn to move to make their move in. So for example if the White King teleports back in time from the original timeline, creating a new timeline, then Black can play a move in either timeline. White would have to respond in the same timeline since it’s Black’s turn to move in the other. Then Black can play another move in either timeline, and so on until the next backward time teleport occurs.
- A piece that time teleports (other than by the default of a half move into the future of the same timeline) must reappear on the same square it disappeared from.
Is that everything? I’ve probably overlooked some complications. As far as e.g. restrictions on which pieces can teleport, by how many moves, and so on, that’s a matter of taste.
Now, does this have anything to do with (hypothetical!) real world time travel? Probably not much. Real world time travel probably has to do with general relativity, with travellers taking different paths and finding on coming back together again their proper time interval is different — teleporting from one moment to another without passing through intervening ones not only is unlikely but I think doesn’t even make any sense in our laws of physics. Still, it’s kind of fun to think about how time teleportation might work, if it could work, and it’s easier to do so in this little 64-points-of-space microcosm.