You got tourettes or somthing? spouting meaningless defensive rhetoric in sheer desperation that nothing exists but your own microcosm. PATHETIC
http://en.wikipedia.org/wiki/Theodor_Kaluza
http://en.wikipedia.org/wiki/Oskar_Klein
http://en.wikipedia.org/wiki/Why_10_dimensions%3F
http://en.wikipedia.org/wiki/Dimension
cylindrical space represented by the perimeter of a circle.
X has the natural embedded in two-dimensional space
(x1,x2)
x1/R = cos(X/R)x2/R = sin(X/R)
The length of this cylindrical space is 2pR.
From dx1/dX = -sin(X/R) and dx2/dX = cos(X/R) the Pythagorean thing, (dx1)2 + (dx2)2 = (dX)2.
Cartesian product of n
radii R1, R2, ..,Rn respectively, to give an n-dimensional space that is cylindrical in all directions, with a total "volume" of
V = (2p)n R1 R2 ... Rn
For example, a three-dimensional space that is everywhere locally Euclidean and yet cylindrical in all directions can be constructed by embedding the three spatial dimensions in a six-dimensional space according to the parameterization
x1 = R1 cos(X/R1)x2 = R1 sin(X/R1)
x3 = R2 cos(Y/R2)x4 = R2 sin(Y/R2)
x5 = R3 cos(Z/R3)x6 = R3 sin(Z/R3)
so the spatial Euclidean line element is
dx12 + dx22 + dx32 + dx42 + dx52 + dx62 = dX2 + dY2 + dZ2
giving a Euclidean spatial metric in a closed three-space with total volume (2p )3R1R2R3. Subtracting this from an ordinary temporal component gives an everywhere-locally-Lorentzian spacetime that is cylindrical in the three spatial directions, i.e.,
ds2 = c2 dT2 - (dX2 + dY2 + dZ2) However, this last step seems somewhat half-hearted. We can imagine a universe cylindrical in all directions, temporal as well as spatial, by embedding the entire 4D spacetime in a space of eight dimensions, two of which are purely imaginary, as follows:
x1 = R1 cos(X/R1)x2 = R1 sin(X/R1)
x3 = R2 cos(Y/R2)x4 = R2 sin(Y/R2)
x5 = R3 cos(Z/R3)x6 = R3 sin(Z/R3)
x7 = i R4 cos(T/R4)x8 = i R4 sin(T/R4)
This leads again to a locally Lorentzian 4D metric
(ds)2 = (dX)2 + (dY)2 + (dZ)2 - (dT)2
but now all four of the dimensions X,Y,Z,T are periodic. So here we have an everywhere-locally-Lorentzian manifold that is closed and unbounded in every spatial and temporal direction. Obviously this manifold contains closed time-like worldlines, although they circumnavigate the entire universe. Whether such a universe would appear (locally) to possess a directional causal structure is unclear. We might imagine that a flat, closed, unbounded universe of this type would tend to collapse if it contained any matter, unless a non-zero cosmological constant is assumed. However, it's not clear what "collapse" would mean in this context. For example, it might mean that the Rn parameters would shrink, but they are not strictly dynamical parameters of the model. The 4-dimensional field equations of general relativity operate only on X,Y,Z,T, so we have no context within which the Rn parameters could "evolve". Any "change" in Rn would imply some meta-time parameter t, so that all the Rn coefficients in the embedding formulas would actually be functions Rn(t).Interestingly, the local flatness of the cylindrical 4-dimensional spacetime is independent of the value of R(t), so if our "internal" field equations are satisfied for one set of Rn values they would be satisfied for any other values. The meta-time t and associated meta-dynamics would be independent of the internal time T for a given observer unless we imagine some "meta field equations" relating t to the internal parameters X,Y,Z,T. We might even speculate that these meta-equations would allow (require?) the values of Rn to be "increasing" versus t, and therefore indirectly versus our internal time T = f(t), in order to ensure stability. (One interesting question raised by these considerations locally flat n-dimensional spaces embedded in flat 2n-dimensional spaces is whether every orthogonal basis in the n-space maps to an orthogonal basis in the 2n-space according to a set of formulas formally the same as those shown above, and, if not, whether there is a more general mapping that applies to all bases.)The above totally-cylindrical spacetime has a natural expression in terms of "octonion space", i.e., the Cayley algebra whose elements are two ordered quaternions
x1 = i R1 cos(X/R1)x2 = i R1 sin(X/R1)
x3 = j R2 cos(Y/R2)x4 = j R2 sin(Y/R2)
x5 = k R3 cos(Z/R3)x6 = k R3 sin(Z/R3)
x7 = R4 cos(T/R4)x8 = R4 sin(T/R4)
Thus each point (X,Y,Z,T) in 4D spacetime represents two quaternions
q1 = x1 + x3 + x5 + x7 q2 = x2 + x4 + x6 + x8
To determine the absolute distances in this 8D space we again consider the eight coordinate differentials, exemplified by
d x1 = i R1 (-sin(X/R1)) (1/R1) (dX)
(using the rule for total differentials) so the squared differentials are exemplified by
(d x1)2 = - sin2(X/R1) (dX)2
Adding up the eight squared differentials to give the square of the absolute differential interval leads again to the locally Lorentzian 4D metric
(ds)2 = (dT)2 - (dX)2 - (dY)2 - (dZ)2
Edited, Mon Jul 18 23:56:50 2005 by Kelvyquayo