Re: Creation (short - HA!)

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Tom Wayburn wrote:
> A comment on David's 3D Universe:  It seems that the Universe must be a
> 4D manifold as observers traveling at different velocities will observe
> S_3_i inside S_3_j and others will observe the reverse.  Thus, no
> particular S_3 is distinguished sufficiently to be called "The Universe".
> I realize this impacts upon free will, but I have a tentative
> explanation for that difficulty, but this is not METAPHYS-L.

I assume your S_3 notation means a 3-spherical surface?  If so, its not
my universe; it's Einstein's.  Once time is appended so the the radius
is time dependent then it is a particular finite Friedman-Robertson-Walker
universe.  Actually Einstein invented the notorious cosmological constant
just so such a universe could exist eternally/statically.  He called it his
greatest blunder once the Hubble expansion of the universe was confirmed.
 
Certainly the universe has a spacetime which *is* a 4-d manifold (barring
any extra compactified Kalusa-Klein-type extra dimensions for now).  The way
such a manifold can be imagined is that one takes a direct product of a line
segment (time axis) with the spatial S_3 sphere.  Then as one goes from the
middle of the line segment toward either end the radius of the sphere is
continually shrunk until the radius is zero at each end of the line segment.
One end of the segment is the Big Bang and the other end is the Big Crunch.
Once the 4-D manifold has been constructed in this way you can imagine
making various coordinate transformations to various moving frames which mix
up elements from different old S_3 surfaces at the same instant in the
transformed new time.

Another way to imagine this finite manifold of spacetime is to consider it a
(Lorentz signatured) version of S_4.  One picks one point on S_4 to be the
Big Bang and picks its antipodal point to be the Big Crunch.  As one travels
on S_4 radially outward in any direction from the BB point half way around
S_4 to the BC point one follows the increasing time parameter. At each fixed
radial value of this time parameter is an orthogonal S_3 manifold
representing space. The radius of this manifold increases from the BB point
until one has gone to the S_4's "equator" at which point further travel in
time shrinks the radius back to zero at the BC point.  An ordinary 2-d S_2
version of this is given as an illustrative example for a spacetime with 1
time and 1 space dimension.  The North pole (NP) of the S_2 is the BB and the
South pole (SP) is the BC.  Each circle of latitude represents a different
1-d space (which is compact, periodic, and simply connected).  The path of an
observer which is at rest (but moving through time into the future) follows a
world line which is a meridian of longitude from the NP to the SP.  The
equator represents the space of the world at its time of maximum expansion. 
Points in the Northern hemisphere are those of expansion phase of the
spacetime and the points in the Southern hemisphere are those of the
contraction phase.  Thus this space time consists of a space which is a
circle which radially expands with time from a BB-NP point until it has a
maximum radius (equator) and then this circle shrinks back to a point at a
BC-SP point.  This temporal sequence can be imagined the intersection of a
moving plane with the sphere where the plane moves along the polar axis and
is oriented parallel to the equator (perpendicular to its motion).

Any path with a fixed longitude is at rest in comoving coordinates and only
travels into the future.  All points with a fixed latitude have the same
value of the cosmic time parameter.  A material observer which is moving
through space as it/she/he moves from the BB-NP to the BC-SP will always be
moving in a mostly southerly direction but at an angle which is between 45
deg East of South and 45 deg West of South.  Any photon will travel either
due SE or due SW depending on which way around the circle it is traveling
as the spatial circle expands and later contracts.  Any change of local
Lorentz frame at some point on the surface is locally like making an ordinary
Lorentz transformation in the x-t plane for one dimensional motion in 
special relativity where we make the origin of both of the coordinate systems
at the local point where transformation is to be performed.  The original
t-axis goes from N to S and the x-axis goes from E to W.  After making the
transformation both the t'-axis and the x'-axis are tilted wrt the original
x and t axes.  If v is the boost velocity of the transformation then the
x'-axis runs from the NE to the SW at an angle of arctan(v/c) wrt the E-W
x-axis.  The t'-axis also runs from the NE to the SW but at an angle of
arctan(v/c) from the N-S t-axis.  As we follow these axes they follow paths
which keep the same local directions wrt N-E-S-W.  (A factor of c is included
in both the t-axis and the t'-axis so that intervals along such timelike
directions are c*[time interval in sec].)

In order to accomodate 3 mutually orthogonal directions in space rather than
just 1 spatial dimension we need to go from S_2 to S_4 to properly model
the finite expanding/contracting rubber balloon model of the universe.

David Bowman
dbowman@gtc.georgetown.ky.us