The spatial dimensions are related to each other via the Euclidean metric (AKA the Pythagorean formula): d2 = x2+y2+z2.
The time dimension is related to the spatial dimensions via the Minkowski metric, which differs only by one sign: d2 = x2+y2+z2-t2. So it’s kind of the same thing as a spatial dimension, but with a difference.
Among other things, the change in sign means that, where spatial rotations result in circular transformations, spacetime rotations with a time component (AKA acceleration) result in hyperbolic transformations. And the asymptotes of the underlying hyperbola are the light cone of the center of the transformation—which is why no amount of acceleration can cause an object’s future path to leave its current light cone, and why faster-than-light travel is impossible.
The spatial dimensions are related to each other via the Euclidean metric (AKA the Pythagorean formula): d2 = x2+y2+z2.
The time dimension is related to the spatial dimensions via the Minkowski metric, which differs only by one sign: d2 = x2+y2+z2-t2. So it’s kind of the same thing as a spatial dimension, but with a difference.
Among other things, the change in sign means that, where spatial rotations result in circular transformations, spacetime rotations with a time component (AKA acceleration) result in hyperbolic transformations. And the asymptotes of the underlying hyperbola are the light cone of the center of the transformation—which is why no amount of acceleration can cause an object’s future path to leave its current light cone, and why faster-than-light travel is impossible.
That…actually made sense