Re: Creation (long)

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

Tom Wayburn complained:

>   I can't parse the locution "the big bang occurred at a certain point
> in time".   Presumably, time did not exist when the BB occurred; so it
> couldn't have done anything at a point in time.  It was over at every
> point in time save on the boundary if the Universe be closed and bounded
> (compact).  (If the Universe were an open set, I think we would be in
> trouble.   Physically *and* metaphysically.)   I choose "finite without
> boundary" and I think I'll play these cards, dealer.  (Everyone's saying,
> "He's got a straight - not a flush.")

A problem with using mathematical continuum concepts to model physical
situations is that we can never actually fully 'take the limit'.  We are not
really sure of the resolution of Zeno's paradox.  We do know that the
continuum concept of classical dynamical phase space breaks down at the
quantum level and the state of a real dynamical system cannot be thought of
as a point in classical phase space because the state is really better
described as a projection operator onto a one dimensional subspace of an
infinite dimensional complex Hilbert space (effectively equivalent to to the
idea of a wave function).  Similarly we have no reason to trust our classical
continuum manifold model for spacetime on a scale finer than about the
resolution of the Planck length (~10^-35 m) and the Planck time (~10^-43 s).
If GR is ever to be properly quantized and made fully compatible with quantum
mechanical principles then such a continuum concept *must* break down at such
a fine level.  Thus the question is really moot speculation as to whether we
define the limiting time of the BB singularity as a point *in time* or as an
external limiting boundary *of time*.

In doing the continuum mathematics of differential geometry singularities are
carefully excised from the manifolds and it is important for the proof of
theorems as to whether a manifold includes any boundary that it may have or
not.  Such fine distinctions tend to be lost in the world of physics.  In
physics we not only consider all distinct sets and functions that differ on a
set of Lesbegue measure zero as equivalent, we even consider all sets and
functions that differ on a set which is too small to measure the difference
to be equivalent.  Physics is more sloppy than mathematics since we have no
way to verify or falsify such very fine distinctions in our mathematical
models.  For us two things are essentially the same, not if they necessarily
match exactly point-by-point, but if a weak measure of their difference is
too small to care about.  It is this sloppiness of physics that gives those
in pure mathematics fits.  This is why physicists tend to assume that
whatever function or set they are interested in is sufficiently smooth and
differentiable so that any theorem they wish to use is valid.  There is no
way to tell if the real set or function is not really that smooth on a fine
enough length scale where our continuum ideas for our model may break down
anyway.

>  Also, I am confused by all these distances that light travels.  From
> the viewpoint of the corpuscle,  it hasn't gone anywhere as all of its
> traveling is where it is.   Are not two events zero interval apart the
> same event?   Notice, that question is *not* rhetorical.  In space-time
> do we not have events rather than points?   Is a point defined even?

Its true that we call the points of spacetime events.  If two events have
a null interval then they are *not* the same event unless they have the
same coordinates (and the coordinate system is not singular at these events).
(The indefinite metric of relativity is not technically a 'metric' as is
used in the sense of 'metric spaces' where the definition of a metric
requires that two elements to be identical if their metric 'distance' is
zero.)  If the interval between the events is null then the proper spatial
separation between the events is always equal to the proper temporal
separation between them in all frames, but this does not make those
separations go to zero in any allowed frame.  We are not allowed to transform
to a frame in which a massless corpuscle (i.e. photon) is at rest since such a
transformation is intrinsically singular, because it is impossible to have the
photon at rest if the photon travels at speed c in *all* allowed frames.  It
can't both be travelling at speed c and be at rest in the same frame.  If we
tried to make a Lorentz boost up to a speed of c then the transformation,
being singular, collapses our whole 4-d spacetime manifold into a 2-d spatial
surface since in this case (from the point of view of the photon) all of space
along the direction of motion of the photon is length-contracted to a flat
plane and all of time is dilated so that the photon experiences just one
eternal now point of time.  For the photon the whole universe exists for one
moment and only extends in the 2-d transverse spatial directions to its
motion.

David Bowman
dbowman@gtc.georgetown.ky.us