What is curvature?
Naty1 said:
DrGreg said:
Inertial observers (a.k.a. free-falling observers) in the presence of gravitation do not see "all spacetime" as straight -- it only looks straight locally. The curvature or flatness of spacetime is an intrinsic property of spacetime itself and does not depend on the motion of observers. An inertial observer can tell if spacetime is flat or curved by looking at other, non-local inertial observers and seeing if they all move at constant velocity or if some don't.
I have been puzzled by the boldface part for sometime...and I think it does depend on frame...Let me disagree for discussion: A free falling inertial observer in the presence of gravitation sees local spacetime as flat; all other frames will observe the same local space as curved.
In fact, when light is observed from a distance approaching a black hole and slowing, blue shifting, as viewed say from earth, and never reaching the event horizon, (curved space) while a local free falling observer sees that same light proceed to the event horizon and disappear (flat space)...isn't that an example of different observers seeing different shape spacetime??
What we call the "curvature of spacetime" has a technical meaning; the equations that describe it are very similar to the equations that describe, say, the curvature of the Earth's surface in terms of latitude and longitude coordinates, or any other pair of coordinates you might choose. This "curvature" need not manifest itself as a physical curve "in space".
For the rest of this post let's restrict our attention to 2D spacetime, i.e. 1 space dimension and 1 time dimension, i.e. motion along a straight line. This is adequate for many scenarios, even for black holes provided we are interested only in particles moving radially.
In the absence of gravitation, an inertial frame corresponds to a flat sheet of graph paper with a square grid. If we switch to a different inertial frame we "rotate" to a different square grid, but it is the same flat sheet of paper.
(The words "rotation" and "square" here are relative to the Minkowski geometry of spacetime, which doesn't look quite like rotation to our Euclidean eyes, but nevertheless it preserves the Minkowski equivalents of "length" (spacetime interval) and "angle" (rapidity).)
If we switch to a non-inertial frame (but still in the absence of gravitation), we are now drawing a curved grid, but still on the same flat sheet of paper. Thus, relative to a non-inertial observer, an inertial object seems to follow a curved trajectory through spacetime, but this is due to the curvature of the grid lines, not the curvature of the paper which is still flat.
When we introduce gravitation, the paper itself becomes curved. (I am talking now of the sort of curvature that cannot be "flattened" without distortion. The curvature of a cylinder or cone doesn't count as "curvature" in this sense.) Now we find that it is impossible to draw a square grid to cover the whole of the curved surface. The best we can do is draw a grid that is approximately square over a small region, but which is forced to either curve or stretch or squash at larger distances. This grid defines a
local inertial frame, where it is square, but that same frame cannot be inertial across the whole of spacetime.
So, to summarise, "spacetime curvature" refers to the curvature of the graph paper, regardless of observer, whereas visible curvature in space is related to the distorted, non-square grid lines drawn on the graph paper, and depends on the choice of observer.
