I see Eliezer has responded in a way which kills my theory stone cold. (Though it was dead anyway if the 6 light-hour separation by Pioneer wasn’t reached in 1992.)
But basically what I was thinking was this. Consider any two space-time points x and y. Either they have a time-like separation, or a space-like separation or a null separation. If they have a space-like separation then there is a particular inertial reference-frame in which they are only separated in space, not in time. If the spatial separation in that frame is > 6 light-hours, then information cannot travel from x to y. (Or, if you want to think of it in terms of a causal graph, and Pearl’s intervention calculus, then every intervention to the graph at x will leave events at y unaltered.)
Incidentally, this formulation implies the rule that “information can’t go back in time more than 6 hours” and implies it in any inertial reference frame. For if information could travel from x to a point z, more than 6 hours in the past of x (but at the same place) in some reference frame, then it could be sent further along a future-pointing null vector from z to y (by an ordinary light-beam), where y is > 6 light-hours from x in the same reference frame. So the restriction of “no spatial jumps > 6 light-hours” neatly implies “no temporal jumps back > 6 hours”.
Basically, this looks something like the Minkowski interval formulation: there is no privileged reference frame, just a new constant of nature (i.e. whatever 6 light-hours translates to in Planck lengths).
But basically what I was thinking was this. Consider any two space-time points x and y. Either they have a time-like separation, or a space-like separation or a null separation. If they have a space-like separation then there is a particular inertial reference-frame in which they are only separated in space, not in time. If the spatial separation in that frame is > 6 light-hours, then information cannot travel from x to y. (Or, if you want to think of it in terms of a causal graph, and Pearl’s intervention calculus, then every intervention to the graph at x will leave events at y unaltered.)
The problem is that points that 6 light-hours away from X have points more than 6 light-hours away from X in their future light cone.
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...
Returning to this thread after a few months...
I see Eliezer has responded in a way which kills my theory stone cold. (Though it was dead anyway if the 6 light-hour separation by Pioneer wasn’t reached in 1992.)
But basically what I was thinking was this. Consider any two space-time points x and y. Either they have a time-like separation, or a space-like separation or a null separation. If they have a space-like separation then there is a particular inertial reference-frame in which they are only separated in space, not in time. If the spatial separation in that frame is > 6 light-hours, then information cannot travel from x to y. (Or, if you want to think of it in terms of a causal graph, and Pearl’s intervention calculus, then every intervention to the graph at x will leave events at y unaltered.)
Incidentally, this formulation implies the rule that “information can’t go back in time more than 6 hours” and implies it in any inertial reference frame. For if information could travel from x to a point z, more than 6 hours in the past of x (but at the same place) in some reference frame, then it could be sent further along a future-pointing null vector from z to y (by an ordinary light-beam), where y is > 6 light-hours from x in the same reference frame. So the restriction of “no spatial jumps > 6 light-hours” neatly implies “no temporal jumps back > 6 hours”.
Basically, this looks something like the Minkowski interval formulation: there is no privileged reference frame, just a new constant of nature (i.e. whatever 6 light-hours translates to in Planck lengths).
The problem is that points that 6 light-hours away from X have points more than 6 light-hours away from X in their future light cone.
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...