The angles in a triangle in traditional Euclidean geometry add up to 180. But that is true when space is flat. If space has other shapes, the angles can add up to something different. If space is spherical, then the angles will add up to more than 180. If space is saddle shaped, they will add up to less than 180.
Articles about our universe say our space is very flat. But articles about mass say that it curves the space around it. So even though our overall universe is flat, would the angles in triangle in the curved space around a massive object be different than 180? Are the angles the same in a triangle in the vacuum of space versus in a black hole?
Saying that space is curved is exactly saying that angles of a triangle won’t add up to 180º. For a triangle in the vicinity of the Sun or the Earth, the discrepancy will be very small and difficult to measure, but it’ll still be there. Near a black hole, it’ll be much easier to detect (though to correct a misconception there, so far as we can tell, an ordinary black hole consists entirely of “the vacuum of space”).
I think the root cause of your confusion is that, when we say that the Universe is flat, we mean on very large scales. But there’s still curvature on smaller scales, like within a solar system (or galaxy, or galactic supercluster). Think, for instance, of something like egg-crate foam: A big sheet of it is flat, but it still has a bunch of small-scale wiggles and wrinkles.
Think of the Earth. The prime meridian, the equator, and 90° longitude form a triangle (well, two, actually) with three 90° angles. The Earth’s surface is curved, so if space were curved in that way, it would happen there, too.
The curvature of a 2D surface is relatively simply described by associating each point on the surface with a single number. The sum of the angles of a triangle the is just 180 degrees plus the surface integral of this single-valued (scalar) filed in the area bounded by the triangle. However general relativity doesn’t deal with the curvature of surfaces, it deals with the curvature of 4D spacetime, which is described by associating each point in spacetime with 20 independent numbers.
You can slice cosmological spacetime nicely into neatly curved spaces of isotropic curvature, such that it becomes meaningful to talk about the deviation of the sum of the angles of a triangle from 180 degrees when talking about the curvature of space at a specific time. For a general description of curvature in general relativity though it becomes meaningless, as essentially your trying to describe something with 20 independent components with a description that only describes one component.
That doesn’t make sense, you buffoon! Surely you mean “The sum of the angles of a triangle is just 180 degrees plus the surface integral of this single-valued (scalar) field in the area bounded by the triangle.”
Hmm. I’m no mathematician, but I’m pretty sure that one of two things is true:
either curved space is a RELATIVE effect, and within it, all triangles still have angles that add up to 180; or, things that you draw on a curved space, cease to BE triangles to begin with, because no straight lines are possible.
Ideal triangles are conceptual. They don’t exist in the real universe and are therefore unaffected by things like gravity.
[QUOTigor frankensteen;20066376]Hmm. I’m no mathematician, but I’m pretty sure that one of two things is true:
either curved space is a RELATIVE effect, and within it, all triangles still have angles that add up to 180; or, things that you draw on a curved space, cease to BE triangles to begin with, because no straight lines are possible.
[/QUOTE]
Define straight line to mean shortest distance (aka geodesic). Light follows geodesics. Relativistically curved space will be hyperbolic, meaning the sum of the angles of a triangle is less than 180. But under ordinary circumstances you will not be able to measure the deviation; it will be too small. But when light was bent passing the sun during an eclipse, the deviation was certainly noticeable although we could not measure the three legs of a triangle.
The problem with interpreting spatial (as opposed to spacetime) curvature in relativity as being of physical consequence is that it can be achieved merely by transforming coordinates. For example flat Minkowski spacetime of special relativity which is usually parameterized into flat (i.e. zero curvature) spatial slices, can alternatively be parameterized as spatial slices with hyperbolic geometry as in the Milne model.