# Connecting spacetime curvatures

#### Loren

Can the connected curvatures of the universe be more than one of flat, positive or negative?

#### Country Boy

Math Team
I really have no idea what you are thinking here. "Curvature" is, by definition, a real number. All real numbers are one and only one of "zero", "positive", or "negative". There is no other possibilities!

#### romsek

Math Team
I'm thinking

this

might be a good place to start.

Perhaps you can refine your question.

If you're asking can spacetime locally take on any value of curvature the answer is probably yes though it might take theoretical negative mass to achieve it.

• 1 person

#### Loren

"The curvature of space is a geometric description of length relationships in spatial coordinates. In mathematics, any geometry has three possible curvatures, so the geometry of the universe has the same three possible curvatures.

Flat (A drawn triangle's angles add up to 180Â° and the Pythagorean theorem holds)

Positively curved (A drawn triangle's angles add up to more than 180Â°)

Negatively curved (A drawn triangle's angles add up to less than 180Â°)"

__________

I am asking whether two or all three of such spaces can exist as one. I.e., can different curvatures connect continuously? (E.g., saddles to spheres to planes or combinations thereof?)

#### Maschke

In mathematics, any geometry has three possible curvatures, so the geometry of the universe has the same three possible curvatures.
That statement jumped out at me. It doesn't seem valid. Just because math says something doesn't mean nature couldn't be different. Maybe the universe has some other geometry entirely that we just haven't discovered yet. Maybe the concept of geometry needs to be absorbed into some higher concept we don't yet have a glimmer of.

What if mathematics itself tells us more about us than it does the universe? The ant crawling on a leaf on a tree in a forest has a theory too, at its level of being able to form theories. Arguably same as us.

#### Loren

Maschke,

The statement was from the wiki article that romsek referred to, and which I glanced at.

Can one construct microscopically any of the three curvatures from the others? Could a fractal universe arise with two or three curvatures?

How about if spatial curvature was relative to the observer's curvature? Maybe then a saddle universe would seem flat to a saddle observer.

Four-dimensionality (space-time) has been explored extensively with regard to curvature. Do higher dimensions like Calabi-Yau involve more concepts of curvature?

A peculiar universe (like that of polyhedra) with singularities might require a discontinuous treatment.