So the issue here is that we might have three points x, y and z, where x and y have a space-like separation, y and z have a time-like or null separation (which is future pointing from y to z) and x and z have a space-like separation. Further, d(x,y) < 6 (measured in light hours), but d(x,z) > 6.
If so, then the principle I described would prevent information passing from x to z. So it either prevents information transmission from x to y; or if a Time-Turner has already been used to get info from x to y, prevents the further transmission from y to z. The last would be a very interesting effect, because either there is no communication attempt from y to z at all, or the usual communication methods—like light-beams—are attempted, but fail for some reason.
Alternatively, suppose the principle I described is wrong, and info can move from x back in time to x’, then forward to y, then to z by the usual physical means. Then I believe we can make d(x,z) arbitrarily large and this opens up some even more exciting possibilities.
Consider a two-dimensional example, with one space and one time co-ordinate—space first then time. Fix a particular inertial reference frame. Point x is (0,0), point x’ is (0, -delta) i.e. a tiny bit in the past of x. Point y is (delta, 0) i.e. a tiny bit spatially separated from x, and then point z is (delta+t, t) i.e. t hours in the future from y along a light beam through x’ and y.
Then d(x,z) = Sqrt( (delta+t)^2 - t^2 ) = Sqrt(2 x delta x t + delta^2) which of course grows without limit as t gets arbitrarily large. And there is some reference frame in which x and z are that distance apart spatially. So now here’s a neat trick. Imagine that Harry wants to send a message to the Andromeda galaxy, but without it taking two millions years to arrive. Let’s say that x is the Earth now, and z is somewhere in Andromeda which is approximately “now” in the Earth’s reference frame, and in cosmological terms (e.g. at z, measurements of the temperature of the background radiation and the Hubble constant are the same as at x). Then Harry finds a suitable alternative reference frame in which to pick the points x’ and y, uses a Time Turner to send a message from x to x’, then uses ordinary light to send it through y to z. Instant magical faster-than-light signalling out to any distance! And indeed faster-than-light space travel as well, provided he can transfigure rockets that travel at near-light speed...
So the issue here is that we might have three points x, y and z, where x and y have a space-like separation, y and z have a time-like or null separation (which is future pointing from y to z) and x and z have a space-like separation. Further, d(x,y) < 6 (measured in light hours), but d(x,z) > 6.
If so, then the principle I described would prevent information passing from x to z. So it either prevents information transmission from x to y; or if a Time-Turner has already been used to get info from x to y, prevents the further transmission from y to z. The last would be a very interesting effect, because either there is no communication attempt from y to z at all, or the usual communication methods—like light-beams—are attempted, but fail for some reason.
Alternatively, suppose the principle I described is wrong, and info can move from x back in time to x’, then forward to y, then to z by the usual physical means. Then I believe we can make d(x,z) arbitrarily large and this opens up some even more exciting possibilities.
Consider a two-dimensional example, with one space and one time co-ordinate—space first then time. Fix a particular inertial reference frame. Point x is (0,0), point x’ is (0, -delta) i.e. a tiny bit in the past of x. Point y is (delta, 0) i.e. a tiny bit spatially separated from x, and then point z is (delta+t, t) i.e. t hours in the future from y along a light beam through x’ and y.
Then d(x,z) = Sqrt( (delta+t)^2 - t^2 ) = Sqrt(2 x delta x t + delta^2) which of course grows without limit as t gets arbitrarily large. And there is some reference frame in which x and z are that distance apart spatially. So now here’s a neat trick. Imagine that Harry wants to send a message to the Andromeda galaxy, but without it taking two millions years to arrive. Let’s say that x is the Earth now, and z is somewhere in Andromeda which is approximately “now” in the Earth’s reference frame, and in cosmological terms (e.g. at z, measurements of the temperature of the background radiation and the Hubble constant are the same as at x). Then Harry finds a suitable alternative reference frame in which to pick the points x’ and y, uses a Time Turner to send a message from x to x’, then uses ordinary light to send it through y to z. Instant magical faster-than-light signalling out to any distance! And indeed faster-than-light space travel as well, provided he can transfigure rockets that travel at near-light speed...