Is There a Centre? The Geometry of the Universe

Can We Be in a Finite Universe that has No Center or Edges? Is this possible? It sounds bizarre… Yes; but you have to wrap your mind once again around the concept of the curvature of space. The universe may be ‘folded on itself’ with no boundaries and no ‘outside’. A two-dimensional analogy may help!

A Starting Point Consider the everyday experience of living on the curved surface of the Earth. Locally, on small scales, it is flat. On large scales, however, the curvature matters.

Is Greenland Really That Big? It looks nearly as large as the whole continent of Africa!

An ‘Equal-Areas’ Projection

Havana to Moscow https://www.flightradar24.com

The Shortest Path! (a Geodesic)

Two Kinds of Curved Surfaces 1. Positive Curvature Imagine standing on the top of a sphere. The surface ‘falls away’ in front and behind, and also to the left and right.

Two Kinds of Curved Surfaces 2. Negative Curvature Imagine standing ‘in the saddle.’ The surface rises in front and behind, but falls to the left and right.

Measuring the Curvature Can we measure the curvature of the Earth with local measurements? (say, as opposed to walking all the way around it)

Curved Surfaces Quantified

To Quadruple the Crop, Double the Radius of the Garden

It’s Different on a Positively-Curved Surface!

Curved Surfaces – Your Experience

Hence The ‘Torn’ Maps

…and the Imperfect Fit to a Pie Pan

Consider Only the Surface Forget the ‘third dimension’ - the local ‘up and down’ A person on the surface could walk an infinite distance in any direction, never encountering an edge. It is unbounded. There is no edge. Yet the total area to be explored is finite.

Moreover… …no particular point on the surface is any more important than any other. The surface has no ‘special point’ -- no ‘center.’ [On Earth, we think of the poles as special. But if the Earth simply a ball of rock in empty space, they would have no particular significance.]

The Reason The positively-curved surface closes back on itself.

A Digression: Fun With Curved/Folded Surfaces A Moebius strip.

The Moebius Strip It has only one surface! http://1.bp.blogspot.com/-D6mjxLue61M/Uh9epCYX9FI/AAAAAAAACrQ/tJgN0wljLig/s200/moebius_escher_anim.gif

What If Light Itself Follows the Surface? We would have no notion of ‘up and down,’ a direction perpendicular to the surface! We would live in a curved ‘flatland’

You Might See The Back of Your Own Head!

But Does Light ‘Follow the Surface?’ Clearly not along the surface of the Earth! The curvature (‘puckering’) of space is too small. But light does follows the curvature of space! Given the totality of matter (including dark matter) in the universe, it is possible that space ‘folds back’ on itself. If so, light will eventually return to its place of origin (unless it runs into something en route) like a nomad wandering over the surface of the Earth.

In Short As we have seen, light can be deflected slightly by a star like the Sun, and significantly ‘lensed’ by the existence of large concentrations of matter (such as a cluster of galaxies) -- but also governed overall by the total distribution of matter in the entire [finite] universe itself. If the density of matter is high enough, the universe can be ‘closed’ – that is, finite, and folded back on itself – with no ‘outside’, no centre, and no edges.

Now Add Expansion (first cover the balloon uniformly with dots)

The Dots (Galaxies!) Do Not Expand - they are merely carried along by the expanding space

Result: A Finite Universe… Observers located on any individual dot see all other dots recede from them as the ‘universe’ expands. No dot is the center of the expansion There are a finite, countable number of dots The entire observable universe is the surface of the balloon (and the ‘dots’ on it).

Now the Real Challenge! Convert this understanding into one higher dimension! That is, imagine a fully 3-dimensional space that curves back on itself so there is no outside, no edge, no boundary, no center. The finite universe is not ‘inside’ some greater volume, surrounded by some vast void. It is all there is. And we can’t travel to ‘the edge’ and look beyond.