Why hyperbolic geometry in spacetime if it is flat?

[closet mathemetician]

This is driving me crazy. Consider a two-dimensional spacetime, with coordinates (t,x). If this is a flat spacetime, we can just imagine a regular-old two-dimensional plane. On that plane I could just as easily map a Cartesian/Euclidean coordinate system as a hyperbolic system of coordinates. The coordinate system is a choice.

Now, suppose I have a flat rubber sheet. I map a Euclidean coordinate chart to the sheet. Now I bend the sheet so that it has constant negative curvature. Now, I'm "forced" into a hyperbolic coordinate system because of the geometric shape of the space. The Euclidean coordinates I drew onto the sheet have become warped. I now don't have a choice of coordinate systems.

If spacetime is really flat, why are we forced into a hyperbolic geometry?

 

[JesseM]

What do you mean by "hyperbolic coordinate system"? The curvature is determined by the metric, the coordinate system itself doesn't tell you anything about the curvature. It is true that in a flat spacetime, if you pick different coordinate systems and then use the metric to figure out the http://www.bun.kyoto-u.ac.jp/~suchii/embed.diag.html for some info.

 

[closet mathemetician]

What do you mean by "hyperbolic coordinate system"? The curvature is determined by the metric, the coordinate system itself doesn't tell you anything about the curvature.

That makes sense, ok a metric that gives a hyperbolic motion?

 

[JesseM]

That makes sense, ok a metric that gives a hyperbolic motion?

What do you mean by hyperbolic motion? And do you not like my idea of rephrasing the question in terms of the geometry of space (not spacetime) at a particular instant in time according to the time coordinate?

 

[Dale]

If spacetime is really flat, why are we forced into a hyperbolic geometry?

I think you are thinking that the Minkowski metric has the same form as a hyperbola:
ds²=-c²dt²+dx²+dy²+dz²

That does not imply a curved hyperbolic geometry any more than the Euclidean metric (ds²=dx²+dy²+dz²) implies curved spherical geometry.

 

[closet mathemetician]

What do you mean by hyperbolic motion? And do you not like my idea of rephrasing the question in terms of the geometry of space (not spacetime) at a particular instant in time according to the time coordinate?

I mean motion that would follow the geodesic hyperplanes of a hyperbola. And no, its not that I didn't like your rephrasing, I was just trying to express the idea of a hyperbolic curvature associated with a geometry other than just by coordinates, i.e., the metric, in agreement with what you said. I apologize if my use of terms was not rigorous or mathematically precise. I'm just a noob who's interested in physics.

 

[closet mathemetician]

I think you are thinking that the Minkowski metric has the same form as a hyperbola:
ds²=-c²dt²+dx²+dy²+dz²

That does not imply a curved hyperbolic geometry any more than the Euclidean metric (ds²=dx²+dy²+dz²) implies curved spherical geometry.

That's a good point DaleSpam. I'll ponder that some more.

 

[JesseM]

I mean motion that would follow the geodesic hyperplanes of a hyperbola.

But isn't that a coordinate-dependent notion? A given geodesic may have the equation of a hyperbola in one coordinate system and the equation of a straight line in a different coordinate system.

And no, its not that I didn't like your rephrasing, I was just trying to express the idea of a hyperbolic curvature associated with a geometry other than just by coordinates, i.e., the metric, in agreement with what you said.

Well, the spatial curvature of a given 3D spacelike slice of 4D spacetime is determined by the metric (which determines the length of any spacelike curve that lies within that slice), it's just that the choice of how to slice 4D spacetime into a stack of 3D slices depends on how simultaneity is defined in your coordinate system.

I apologize if my use of terms was not rigorous or mathematically precise. I'm just a noob who's interested in physics.

No need to apologize! I'm just trying to pin down what your question means a little better...can you explain what you read or were thinking about that made you link hyperbolic geometry with flat spacetime? Was it about cosmology and how the universe is supposed to be hyperbolic if the mass density is below a certain critical value (which implies space is hyperbolic if the mass density is zero, which should just be flat SR spacetime), or about the thing DaleSpam pointed out with the metric resembling the equation for a hyperbola, or something else?