Geometry of Our Universe

The following are my comments recently published at:
http://scienceblogs.com/startswithabang/2017/10/14/ask-ethan-is-the-universe-finite-or-infinite-synopsis/

@Ethan:
“If space were positively curved, like we lived on the surface of a 4D sphere, distant light rays would converge.”
Think of surface of a 3d sphere first:
It is a 2d surface curved in the 3rd dimension.
Now think of surface of a 4d sphere:
It is a 3d surface curved in the 4th dimension.
What if Universe is surface of a 4d sphere where 3d surface (space) curved in the 4th dimension (time)?
So is it really not possible, 3d space we see using our telescopes, could be flat in those 3 dimensions of space, but curved in time dimension?

First let me try to better explain what I mean exactly:
Let’s first simplify the problem:
Assume our universe was 2d, as the surface of a 3d sphere. Now latitude and longitude are our 2 space dimensions. Our distance from the center of the sphere is our time dimension.

Since our universe is the surface of a 3d sphere, it has a general uniform positive curvature, depending on our time coordinate, anytime.

Now the big question is this:
As beings of 2 dimensions now, can we directly measure the global uniform curvature of our universe in any possible way? Or asking the same question in another way would be this: Our universe would look curved or flat to us?

If speed of light was high enough, and if we had an astronomically powerful laser, we could send a beam in any direction, and later see it came back from exact opposite direction, sometime later.
Then we would know for certain our universe if finite.
But I claim, we still would not know what is the general curvature of our universe.

Could we really find/measure it by observing the stars or galaxies around, in our 2d universe?

For answer, first realize we don’t know any poles for our universe. We can use any point in our 2d universe as our North Pole, would it make any difference for coordinates/measurements/observations?
Then why not take our location in our 2d universe as the north pole of our universe.

Now try to imagine all longitude lines coming into our location (the north pole our coordinate system) as the star/galaxy lights.
Can we really see/measure the general curvature of our universe from those light beams coming to us from every direction we can see?
I claim the answer is no.

Why? I claim, as long as we are making all observations and experiments, to calculate the general curvature, using only our space dimensions (latitude and longitude),
we would always find it to be perfectly flat in those 2 dimensions. I also claim, we could calculate the general curvature of our 2d universe (latitude and longitude), only if we include the precise time coordinates in the measurements/experiments, as well as precise latitude and longitude coordinates.

So I really claim, our universe looks flat to us, because we are making all observations/measurements in 3 space dimensions. But if we also include time coordinates, then we can calculate true general curvature of our universe.

And I further claim:

Curvature of circle (1d curved line on 2d space):
1/r

Curvature of sphere (2d curved plane on 3d space):
1/r^2

Curvature of sphere (3d curved space on a 4d space):
1/r^3

So if our universe was 2d space and 1 time (2d curved plane on 3d space):
Its general curvature at any time would be:
1/r^2=1/(c*t)^2 (where c is the speed of light and t time passed since The Big Bang in seconds)

And so if our universe is 3d space and 1 time (3d curved space on 4d space):
Then its general curvature at any time is:
1/r^3=1/(c*t)^3 (where c is the speed of light and t time passed since The Big Bang in seconds)

And I further claim:

If astrophysicists recalculated general curvature of our universe, by including all space and time coordinate information correctly, then they should be able to verify, the calculation results always match to the theoretical value which is 1/(c*t)^3 .

The raw data to use for those calculations would be the pictures of universe, for the same direction, looking at views there from different times.

I realized this value for the current general curvature of our universe (1/(c*t)^3) would be correct only if we ignore the expansion of the universe. To get correct values for any time, we need to use current radius of the universe for that time, including effect of the expansion until that time.

Wikipedia says:
“it is currently unknown whether the observable universe is identical to the global universe”

From what I claimed above, I claim they are identical.

(So if the current radius of observational universe is 46 Bly, then I claim it means current global curvature of our universe is 1/(46 Bly in meters)^3.)