[ExI] Which direction does the arrow of time point in Conway's Game of Life?

[Stuart LaForge]

Quoting Dave Sill:

> On Sun, Jan 5, 2020 at 10:11 AM spike jones via extropy-chat <
> extropy-chat at lists.extropy.org> wrote:
>
>> Bill, it is still a big bang.  That didn?t actually change: everything
>> started from a point, the same point, which is why there is no center now,
>> ...
>>
>
> So if everything started from a point and expanded outward at the same
> rate, seems like that point would still be the center-of-mass of the
> universe, even if the universe is infinite.

Your "even if" clause complicates the answer to your question so let's  
deal with it first. Many people think of infinity as the largest  
possible number. Mathematically that is incorrect. The smallest  
possible infinity is defined as counting FOREVER meaning that it is a  
the literal non-existence of a largest number rather than any actual  
number. Such an infinity is called Aleph-0.

The continuum of real numbers on a mathematical line is an even bigger  
infinity because you can't even begin to count it. Pick a point and  
start counting the successive points. Between the point you call 1 and  
2 there are an uncountable number of other points that you could have  
easily chosen as number 2. Two mathematical points on a number line  
can be arbitrarily close to one another, therefore it is impossible to  
choose a 2 because any point you choose will have an infinite number  
of points between it and 1. That infinity is uncountable because you  
can't even choose a number 2 even if you had forever to choose it. The  
official name for this infinity Aleph-1.

So if the universe is infinite, then, regardless of whether it is flat  
or open, the notion of center of mass or any center really, is  
mathematically meaningless. Take an infinite geometric line. No matter  
which point you pick to be the center, the line goes on FOREVER to  
either side. An imaginary infinite beam that could support its own  
weight would be perfectly balanced no matter where you put the  
fulcrum. Therefore there is no center of mass to the universe.

Now let's address case of the very large but finite universe. For  
this, I can't rely on pure math and must invoke physics. If the  
universe is closed, then the overall shape of the universe is a  
hypersphere which can be visualized by analogy with the common sphere.  
Think of a sphere as a globe of the planet earth for example. The  
sphere consist of a 2-dimensional surface "wrapped" around a 3rd  
dimension. You can label points on the surface of the sphere by a pair  
of coordinates latitude and longitude. Every such point on the surface  
of a sphere is at an equal distance from the center of the sphere  
called the radius. The radius can be thought of the distance from the  
center of the sphere in the 3rd, inaccessible, dimension.

Take note that the coordinates 0 degrees latitude, 0 degrees longitude  
is completely arbitrary and in no way counts as the "center" of the  
sphere. It should be clear that the center of mass of  sphere does not  
lie anywhere on the 2-dimensional surface of the sphere but must  
instead lie in its 3rd dimension, completely hidden and unreachable by  
any flat-lander trapped on the surface of the sphere.

To apply this picture to the 4-dimensional universe, simply think of  
the "surface" of the hypersphere as 3-dimensional space and the radius  
as being the 13.8 billion years of elapsed time since the "center of  
the universe" otherwise known as the big bang. No matter where you go,  
embedded in the 3-dimensional "surface" of the hypersphere, you can  
assign yourself yourself the coordinates (0,0,0) but you are nowhere  
near the center because you would have move backwards in time to get  
there. And moving backwards in time is generally forbidden for normal  
matter.

So the location of the "big bang" and therefore, the center of mass  
lies in the past and not at any current location in space.

For a very large but finite flat or open universe, which would have an  
edge or boundary of some kind that lies beyond what we can observe  
through our best telescopes, the argument is a little more subtle.  
Center of mass is a gravitational concept and as such can only be  
meaningful for causally-connected space. The influence of gravity as  
demonstrated by LIGO, travels at the speed of light. Therefore the  
largest structure that can have a center of mass that is meaningful to  
us is our causal cell or Hubble sphere.

Anything outside of the Hubble radius is receding from us at faster  
than light speed due to the expansion of space. And because the Hubble  
radius is 13.7 biilion light-years away from us in every direction, we  
are, by definition, at the center of our causal cell even if we are  
not the center of the entire universe. Furthermore since the opposite  
edges of such a large but finite universe cannot causally affect one  
another, there can be no physically meaningful gravitational center or  
center of mass to such a universe.

Sorry for the long-winded answer, but your simple question was  
deceptively profound.

TLDR; The universe does not have center of mass whether the universe  
is infinite or not. But any point in the universe is an equally valid  
geometric origin for ones coordinate system provided one is not too  
close to the hypothetical boundary of a finite universe.

Stuart LaForge