by wendy » Sat May 14, 2005 8:52 am
gravity works in any dimension.
It is best treated as a weak radiant-vector field, which means, that the size of the force is proportional to, and points to the centre, of masses. Because it is weak, it requires so much mass.
For euclidean geometries, it means roughly, that f = GMm/r^(n-1). For 1 and 3 dimensions, we can approximate a large object to its centre. For all others the approximation does not work. Where dr (changes in r) is relatively small to r, we can treat the measure GM/r^(n-1) as a constant g, and have f = gm (eg 1 lbf = g * 1 lb, or in metrics, 1 kg = 9.80665 N
For a given planet, it can be treated as a field radiating upwards at uniform density: that is g is 32.175 ft/s at the bottom and top of tall buildings. This allows us to implement gravity in the model without having to resort to doing calculations in integrals of gm1m2/r^3.
Gravity would no more crunch four-dimensions than it would crunch three or five. We evidnetly still are here, so gravity has not had its way with us.
As for ana and kata, the notions here is that because we have three naturally derivable directions in 3d, this is always the case. This is false.
One has only two dimensions that are set by nature: gravity (up/down) and motion (forward/backward). Plants, which don't move, have no obvious forward/backwards construction, while animals generally do.
Anything over that is purely in the domain of "equal valid directions of motion" Where we have left/right in a clear parity on a line [although people confuse these], in four dimensions, there is no intrinsic nature drawing the third and fourth directions out. All one has is a sense of "go clockwise 75 degrees from the church".
For those having problems following this, imagine that you lay a lot of clocks on the ground. In four dimensions, this is the across-space. While we can set face-of-clock = up or down, we have no reason to set the 12o'clock position to the north direction (or any other).
Correspoindingly, while up/down and back/forward intrinsically translate, the directions of left/right does not. ie ana and kata make no sense to a 4d being.
Wendy