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Re: [Phys-l] Big Bang density

Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu> writes:
As far as I can tell, the interesting part of this thread has
rather little to do with the big bang, and more to do with what
people mean when they talk about "infinity".



On 03/12/2009 10:00 AM, Jeffrey Schnick wrote:
How is it that infinite density implies zero volume?

It doesn't.

Obviously zero volume implies infinite density(*), but the
converse does not hold.

*) Assuming a nonzero amount of stuff......

In my original posting, I was assuming that the universe has a FINITE
amount of mass. Thus, the only way its density can go to infinity is if
its size goes to zero. Has it been "proven" that the universe is
infinitely large (and thus infinitely massive)? I doubt this can be done,
and we will never know. So I was thinking at the time of the Big Bang, the
universe was infinitesimally small and therefore infintiely dense. It is
still impossible to imagine what this was like.


Note that the wikipedia article
http://en.wikipedia.org/wiki/Big_bang
successfully talks about a universe with infinite density
without going anywhere near the notion of zero size.

I see the density
of numbers along the real number line as being infinite but that doesn't
keep the real number line from extending to infinity in both directions.

In accordance with that picture, there is a "doubly infinite"
amount of stuff: infinite density multiplied by infinite
spatial extent. Formally that's correct. It's a good answer
to those who have mathematical questions about the "big bang
density" at time t=0, i.e. at the moment of the big bang.

Meanwhile, there are physical, philosophical, and even mathematical
reasons for avoiding questions about what happens "at" a singularity.
It is usually best to avoid things that are "equal" to infinity and
to instead consider the limit as things approach infinity.

In accordance with the usual big bang model, start with the present
universe (which is expanding) and play the movie backwards. As a
function of reverse time,
-- the atoms do not get smaller
-- the stars do not get smaller
-- the galaxies do not get smaller
-- but the space they inhabit gets smaller. So the density goes up.

In particular, when the universe is smaller by a factor of z, the
average density is higher by a factor of z^D, where we live in D=3
spatial dimensions.

So we can make a little table

z density diameter
Now: 1 1 infinite
10 1000 infinite/10
100 1000000 infinite/100
1000 1000000000 infinite/1000

An this table can be continued very, very far. The consistence of
cosmology and elementary particle physics indicates that physics as
we know it continues to work back to z=10^20 or some such (I don't
know the best current value, but it's big).

As always when there is the potential for a double infinity, you need
to be careful about the order of limits. In this case, if you think
the universe is infinite in extent, then you can take that limit first,
as I have done in the table. Then you can run the movie back to some
early time where z is /almost/ infinite, and the density is truly
humongous, and the spatial extent is still infinite.

Although it may not be apparent at first glance, I am taking my own
advice about not worrying about what happens when something is "equal"
to infinity. In the table, when I say the extent is infinite, I mean
that it is very large, and _making it larger wouldn't make any difference_
to the observable physics.

To do cosmology, we do not need to consider what happens "at" the moment
of the big bang. We do not need to consider what happens when z is
"equal"
to infinity.

When people say the equation predicts infinite density at a finite time in
the past, that shouldn't be taken too literally. Things like that show up
in physics all the time. Consider for example the mechanical advantage of
a lever, as a function of where we attach the load. The equation predicts
that moving the load a finite amount will produce an infinite mechanical
advantage. That's what the equation says. Of course nobody takes that
seriously; we know that the equation doesn't model all the physics, and
at some point some physics that is not accounted for by the simple
equation
will come to dominate. Similarly nobody thinks the susceptibility of a
chunk
of iron really goes to infinity at the Curie point.

We are talking here about laws, typically scaling laws. Nobody expects
them
to be scalable infinitely far in either direction. But the interesting
thing,
the wonderful thing about physics, is that many of these laws can be
scaled
for many, many orders of magnitude before they break down. You can draw a
theoretical line that goes off to infinity, and the data sticks to the
line
over a huge range.


Also we should say something about the Markov property. In introductory
physics,
if we want to analyze what happens when a ball is dropped, we do not need
to
know much about what happened /before/ it was dropped. Knowing the
initial
position and initial velocity is sufficient. So it is with cosmology.
If we
know the properties of the universe at some early time, we do not
necessarily
need to know what happened before then in order to say interesting things
about
what happens after then. In particular, we definitely do not need to
know the
density (or anything else) "at" the moment of the big bang.

_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l