John Clark wrote:
On Sun, Oct 26, 2014 at 11:48 PM, Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
>>> it expands for ever even though closed). So you can
never see the back of your own head.
>> Obviously if it expands forever you could never see the back
of your head, and our universe is not only expanding its
accelerating; but how could you call such a universe closed?
> It has a finite extent, even if expanding exponentially.
I suppose a universe could be flat and finite at any given time and only
in the infinite future become infinite in extant do to the expansion; or
a flat universe could be infinite at any given time, but either way the
universe would be open not closed.
The terms, 'closed', 'flat', and 'open' for cosmologies in GR are
somewhat technical. They refer to the value of a particular parameter,
known as k, which can take on the values +1, 0, -1, for the three types
of cosmology respectively. So a closed universe has k = +1 and this is
the case whatever the extent of the universe, and whether it will expand
for ever or not. So an exponentially expanding universe might be closed,
and hence of finite extent at any particular value of coordinate time,
though that extent may increase with time without limit. Such a universe
is neither 'flat' nor 'open'.
> in the case under discussion, which is what we mean by a closed
universe.
A closed universe means that the observed expansion of the universe will
eventually reverse direction, so obviously if a universe expands forever
it will never reverse direction and thus is open.
As discussed above, a the expansion of a closed universe need not ever
reverse -- if there is a positive cosmological constant above some small
value (dark energy) then even the closed universe will expand without limit.
> The point is that the intrinsic curvature is positive
If the curvature is positive then the universe must be closed and
finite.
True, but that does not mean that it must re-contract at some point.
If the curvature is negative then the universe must be open and
infinite. If the curvature is exactly zero (flat) then the universe
could be either open and finite or open and infinite. My intuition says
its unlikely to be flat, open and finite but I could be wrong. The best
experimental evidence we have right now is that the universe is flat, or
if it is curved it's curved by less than one part in 100,000. So we do
know one thing, the universe is either infinitely larger than the
observable universe or just astronomically larger.
The current best cosmological model says that the universe is spatially
flat to a very good approximation, but expanding exponentially. This
means that the space-time is curved (in time), even though spatially
flat. Because of the presence of significant dark energy, it is not
completely determined whether k is zero or not, but taking k = 0 fits
the data adequately. Even so, some people claim that k = +1, ie., that
the universe is closed, because, theoretically, only k = 1 universes can
originate as bubbles formed from an existing vacuum.
Bruce
John K Clark
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