Saturday, July 1, 2017

FLATLAND AND CURVATURE OF UNIVERSE

Wikipedia says "Flatland: A Romance of Many Dimensions is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London."
I never actually read it but I know it really helps people to understand geometric dimensions.

I do not know if these ideas below ever occurred to anyone:

I think it does not matter if flatland (universe) had no curvature at all anywhere
or it had a (positive or negative) uniform global (universal) curvature of any (constant) value,
flatland would look flat to flatland people (any observer living in flatland who has the same dimensions as flatland).

Now imagine that flatland is the 2D surface of a 3D sphere which has a uniform positive curvature everywhere (which is 1/r^2).
The question is this:
Can Flatland people really measure the curvature of their universe or not?
I think most people maybe assuming that, since sum of internal angles of a triangle in Flatland would be greater than 180 degrees,
Flatland people could easily measure the (global and uniform) curvature of their universe.
But can they really do that, just like a 3D being would easily see that
sum of internal angles of a triangle in Flatland is really greater than 180 degrees obviously?
I think the answer is no.
Imagine a Flatland observer sends a laser beam straight ahead.
Imagine the view of the Flatland observer is like a camera moving along the photons of the laser beam, in front of the beginning (head) of the beam.
Imagine as the beam and camera is moving both would be following the curvature of their universe on the path of the beam.
If there are stars in Flatland universe and laser beam is moving towards stars, the view of the camera would be always a flat universe.
Realize that if the universal curvature of Flatland universe is uniform everywhere,
the Flatland observer would always think their universe is flat.
And I think this would be still the same no matter how many dimensions the Flatland universe really has.
But also think Flatland people can still measure non-uniform curvatures in their universe, like curvature created by the mass of a star.

So I think it is quite natural that global curvature of our universe looks very close to being flat.

If our universe started with a Big Bang from a point (singularity or a small spherical object?),
and uniformly expanding ever since, and if we combine Occam's Razor with observations of our universe,
I think the simplest global geometry for our universe would be a 3D spherical surface on a 4D sphere.
And just like a 2D spherical surface is curved in 3rd dimension of space,
our universe must be 3D spherical surface of 3 space dimensions curved in 4th dimension (time).

If so that implies we can calculate the global curvature of our universe at any time as 1/r^3.
(Where r is the radius of our universe at that time.)

Wikipedia says distance to Big Bang is "13.799±0.021 billion years" in time.
But I think if we take expansion of the universe since the Big Bang into consideration,
distance in space (radius of universe) is currently about 46 billion light-years.
This implies the current global curvature of our universe must be 1/(46 billion light-years)^3.

As for how to make sense of our visible universe around us we observe:
Imagine when we look at any direction in our universe, depending on how far we look,
for each point in the universe, we see light left from that point, that far in time.
(And current actual distance (in space) to that point can be calculated by applying what we know for expansion of the universe.)

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