by wendy » Thu Feb 04, 2010 7:16 am
Hyperspace, is simply "space over solid-space". The idea of relating 4d to 3d as 3d to 2d is that we can see 2d's hyperspace as real space, and that by analogy, we might reason 4d space as 3d space.
You should be wary of space-time diagrams, and other "4d" stuff from physics.
Space-time, either newtonian or einstein, is merely a matter of overlaying pictures, rather like the flip-cartoons kids might do in books.
The diagrams where you see curvature, like deep holes around stars, is not space-time. It is intended to be a billiard ball table, where the ball is pulled into deep holes by (real) gravity. In practice, all space is curved, but it's not curved in anything. Instead, the length of arc varies with angle at every point, and that in the presence of large masses, the lengths of arc is longer. Since space is in tension, the "straight path" across any point divides the length equally, which means there could be fewer degrees on one side than the other. Straight lines, then bend in towards masses, because the arc-length is greater there.
In four dimension, there is indeed time and gravity. Where we use 4d space-time or 4d space-curvature, the four-dimensionals use 5-d space-time or 5d space-curvature diagrams. The pictures we draw for books show space-time or space-curvature for two-dimensional worlds.