[peterclarke (OP)]

When the shape of space is discussed there are often three types of shape suggested - flat, positive curve (like the surface of a balloon), or a negative curve (like a Pringle!).
But I just can't get my head round what these metaphors imply. They look OK in a flat diagram on the printed page but when I look up and see the stars, milky way etc., it doesn't look like any of them. What do these concepts mean? What shape is space (or should it be space-time)?

Peter

[Solvay_1927]

Spacetime is (apparently) very very close to being flat overall. There may be local anomalies, but over large distances those in the know claim it to be either flat or only very slightly curved either positively or negatively. (A bit like saying the earth is positively curved, but locally - e.g. on the open sea - it looks flat, but can look negatively curved in some other localities - e.g. in a pringle-shaped valley.)

It's easy to visualise what a curved two-dimensional surface means, because we can see & think in 3 dimensions.  So when someone tells us the earth is round, we can visualise it (even though to us it locally looks flat).

The problem with visualising the curvature of 3-D space (or 4-D spacetime) is that you need to be able to visualise in 4-D (or 5-D) space in order to picture what it means - i.e. you need to be able to think in at least one higher number of dimensions than the surface/volume whose curvature you're trying to describe.

But it can be done mathematically.  In 2-D space, the angles of a triangle always add up to 180 degrees ... if the space is flat.  But if it's curved, and if your triangle is big enough, then the angles will add up to either more than (positive curvature) or less than (negative) 180 degrees.

And so the curvature of spacetime can be described mathematically by saying that the angles of a (very, very) large triangle plotted in spacetime would have angles either less than or greater than 180 is spacetime isn't flat.

Or I think that's the case, anyway. Maybe.

[DoctorBeaver]

That's as good a descrption of Riemannian space as you're gonna get []

[peterclarke (OP)]

Thanks for that response Solvay.
Maybe I am short of a mental dimension (or 5) - but what I am attempting to rationalise is the view I have of the night sky (which is very clear where I live, when it's not cloudy) and the theoretical situation that you describe. Am I looking at a flat(ish) bit of space-time? If so, why is it all around me, in all directions except down?

Peter

[DoctorBeaver]

It's all around you because you're in it. And you can't see it below you because the Earth is in the way.

[daveshorts]

I think you are short of a dimension Peter. If you imagine something that can only live in 2 dimensions like a 2 dimensional ant. It doesn't know anything about a third dimension apart from possibly stuff it has learnt in physics lessons. If it lives on a flat floor it's universe behaves as you would expect, but if it lives on a football, then the angles of a triangle don't add up to 180 and if it walks in one direction for far enough it will get back to where it started etc.

The hypothesis with the universe is that we are simple 3D (or 4D if you include time) beings living in a 4D (5D inc time) universe so the only way we can find out about this curvature is by measuring angles on triangles etc. If the universe was spacially closed if we went in a straight line for long enough in any direction we would end up where we started.

[peterclarke (OP)]

Well I had just about realised that the earth is in the way of the down view - but thanks anyway Doctor. Sorry if I asked a zero-dimesional question. Ill just start walking to see where I was when I started I suppose... (Walks off, confused, spaced out, lacking dimension, measuring triangles to see if they don't add up).
Thanks for your replies.


Peter