SW VandeCarr said:
You're too fast Sylas. I was editing my post when you responded. Earlier, in a recent response to a post of mine on the topic "Flat universe" you stated that an infinite universe was always and will always be infinite. I don't see how this is consistent with a singularity at the beginning of time and a finite history.
Imagine a function of time. It should be smooth, monotonically increasing, and with f(0) approaching 0 as t reduces back towards 0. Here's the simplest example of such a function:
f(t) = t
For a more realistic function, I'll have to give the differential equation
\frac{df}{dt}= \sqrt{0.27 f^{-1} + 0.73 f^2}
The differential equation has a problem when t=0. In fact, it has a singularity. But that's okay. We can just let the differential equation define f for positive values of t, with boundary conditions so that f(t) approaches 0 in the limit as t approaches 0. When f is very small, df/dt is large, so it aligns very closely with
f(t) = (1.5 t \sqrt{0.27})^{2/3}
in the neighbourhood of 0.
We call this function "scale factor", and usually add a couple of constants to scale f and t as we choose, but the above functions will do.
Now. Imagine an infinite 3D cartesian space with specks laid out in an infinite 3D grid pattern, one at every point with integer co-ordinates. This represents a static infinite flat space, filled with evenly spaced galaxies.
To apply expansion, suppose that the specks are moving apart from each other, so that this is their position at a time t when f(t) = 1. Give each speck a name (x,y,z) corresponding to its co-ordinates at this time, and at all OTHER times, suppose that the speck is located at (x.f(t), y.f(t), x.f(t)).
Voila. This is expansion. The speck at location (0,0,0) never moves, and all the others move away from it, with a velocity that is the product of their distance when the scale factor was equal to 1, sqrt(x
2+y
2+z
2), and to the function f(t).
Of course, you can also convert the co-ordinates to see where all the specks are relative to any other speck; and curiously, no matter which speck you pick as your origin, the expansion looks exactly the same. Each speck would see itself as the center of the expansion.
Also, as you run time backwards towards 0, you always have a perfectly spaced rectangular grid, with distances of f(t) between the points.
At t = 0, of course, there's a singularity. Every point is at zero; which introduces a strange kind of discontinuity. At every other time, the grid is infinite and expanding.
That's the flat Big Bang model: flat meaning we can use nice simple cartesian co-ordinates like this.
Cheers -- sylas
PS. Sorry for being so fast. I've been monitoring the forum and happened to see your post. Yes, the Big Bang involves "superluminary" expansion, in the sense that the distance between widely separated specks (or galaxies) can be greater than the speed of light; in fact there's no limit on the speed of separation. Ignore this detail; it is not actually in conflict with relativity and why it isn't is a question for another time.
Finding the path of a photon moving between specks in this simple model can be fun; pick your velocity c and just make sure the photon is always moving at this speed
relative to any speck it is moving past. Don't worry about anything special in the way of relativistic conversions; they don't matter. The technique outlined in this postscript works.
