Does euclidean space actually exist then?

[questionposter]

Even though space time is naturally flat, there's a bunch of different curves in it and even enough in our local space to distort light in some way, so doesn't that mean euclidean space doesn't actually exist in the universe because there is no place where a completely straight line can be drawn?

[Sorcerer]

In regions of low gravity space won't be curved. I read somewhere recently astronomers discovered a great void http://en.wikipedia.org/wiki/Void_(astronomy), if this doesn't contain dark matter space there will be fairly flat. Also the shape of the universe is thought to be flat http://en.wikipedia.org/wiki/Shape_of_the_Universe.

 

Also it is worth noting that even if light/space curves around a massive object, the path it travels can still be interpreted as a straight line. Paradoxical I know, but for this reason - a straight line is the shortest distance between 2 points.

[DrRocket]

Even though space time is naturally flat, there's a bunch of different curves in it and even enough in our local space to distort light in some way, so doesn't that mean euclidean space doesn't actually exist in the universe because there is no place where a completely straight line can be drawn?

 

 

In regions of low gravity space won't be curved. I read somewhere recently astronomers discovered a great void http://en.wikipedia....Void_(astronomy), if this doesn't contain dark matter space there will be fairly flat. Also the shape of the universe is thought to be flat http://en.wikipedia....f_the_Universe.

 

Also it is worth noting that even if light/space curves around a massive object, the path it travels can still be interpreted as a straight line. Paradoxical I know, but for this reason - a straight line is the shortest distance between 2 points.

 

Spacetime is not naturally flat. In fact spacetime is flat only in the total absence of gravity, which is nowhere.

 

Euclidean space exists in the mathematical sense, and is necessary for the very definition of a manifold, and spacetime is a manifold. But it is very unlikely that there is any region of our spacetiime that is actually Euclidean.

 

Note that being flat and being Euclidean are not the same thing. There are in fact compact flat manifolds, and at least one is potential model for the topology of "space" (not spacetime) in cosmological models. That manifold is the flat 3-torus (called Pac Man space by Bryan Green in his latest book). IF this manifold represents space then spacetime is just the Cartesian product of a line segment with the 3-torus.

 

No one knows whether spacetime, on the very largest scales, is flat or not. So far there is no evidence of large scale curvature. But the topology of the universe is very sensitive to any curvature, no matter how small, even with the assumption of homogeneity and isotropy.

 

It is quite clear that when smaller scales are considered the spacetiime is most definitely not flat. If it were there would be no gravity whatever.

 

There is no such thing as a "straight line" in the general setting of a manifold. The closest thing to a straight line is a geodesic. But apart from a geodesic the notion of a "straight line" is meaningless in the general setting that one has in general relativity.

 

A geodesic, not a straight line, is the shortest local distance between two points on a Riemannian manifold. Straight lines are geodesics in Euclidean space.

 

However in spacetime light follows geodeesics, and geodesics in spacetime are the LONGEST distance between two points. This is because spacetime is a Lorentzian manifold, not a Riemannian manifold. Not only that but the distance along a timelike curve, as measured with the Lorentzian metric, is the time that is experienced by a body with that world line and that is how one explains the so-called "twin paradox" using general relativity (the stay-at-home twin in free fall follows a geodesic in spacetime and therefore experiences a greater proper time interval than does the accelerating, traveling twin who does not have a geodesic world line).