At 04:26 PM 5/15/2017, Keith F. Lynch wrote:
But there's no way to confuse 1 with -1 or with any of the imaginary units. Exactly like our t axis. There's a unique "arrow of time." Relativity can mix space and time to an extent, but every interval between two events is unabiguously either spacelike or timelike, depending on whether the invariant, x^2 + y^2 + z^2 - ct^2, is positive or negative. And if it's timelike, it's unambiguous which event is earlier and which is later. There's no frame of reference in which Macron's inauguration came before Trump's.
Not 100% true. We can't tell "from inside" whether we're in a left-handed universe or a right-handed universe. Thus, there's no way "from inside" to distinguish +t from -t. Yes, there's entropy, which is the usual way to tell a normal movie from a time-reversed movie, but all the usual mechanical laws are time-symmetric.
(Disclaimer: That's special relativity. In general relativity, reverse causality appears to be possible under extreme conditions.)
With GR, gravitational waves from coalescing black holes seem to distinguish +t from -t; it might be possible to focus gravitational waves onto a black hole to get it to split in two, but that might be asking a little too much from our current technology.
By contrast, in Egan's universe the four coordinates are completely interchangable. Once their spaceship reaches infinite speed, one of their planet's space axes is their spaceships's time axis, and one of their spaceship's space axes is their planet's time axis. There's no unique arrow of time. Each object carries its own arrow of time along with it. Hence it's possible to land on a time-reversed planet. But you had better bring a flashlight, since you won't be able to see by the light of its time-reversed sun. Also, watch out for the time-reversed dust that was in your spaceship before you landed, and you weren't able to remove it then. The good news is that it was all gone by the time you took off, as you had, without trying, tracked it all out. You also removed (by walking in them) all the footprints that were around your landing site when you landed.
Perhaps I should read these books; I'm having a hard time (!) understanding how one could consider one of the space coordinates as a time coordinate. As you point out, how do you set up a differential equation in "quaternion space" such that a wave can propagate in the other space coordinates?