I’m hoping not to confuse the matter any more than it already is, but make no promises. I’ll try to explain a couple concepts as best I understand
them, and believe they pretty much conform to current mainstream thinking.
1. The Nature of Space
Space, in and of itself, is only a geometric volume. It has no physical properties or energy to be warped, twisted, stretched, curved, etc. Statements
about “curved space”, etc are misleading in that it implies space has some set of physical properties of it’s own. Space is simply the geometric
volume which contains the existing energy/mass of the universe. To say that space expands only means that the volume has increased.
2. Spacetime and General Relativity
How particles and forces influence each other are expressed mathematically as geometric relationships, describing how the particles, etc being
measured occupy the volume of space. When GR refers to “spacetime curvature”, it’s describing how matter changes it’s geometric distribution
within space based on it’s energy and momentum as it moves through time. It doesn’t imply that space itself (sans time) has a curvature. Also, GR
is strictly a theory of geometry and does not state that space has a fabric or substance or any other physical property. In Einstein’s view, the
curvature of spacetime (not space) is a consequence of gravity’s influence on the way matter is distributed in it’s passage through time. In any
case, flat space and curved spacetime are not incompatible features of our universe, and work quite well together. But don’t confuse the 2 as being
the same thing, because they are very different concepts.
The difference between
space and
spacetime isn’t easy to visualize, which leads to a lot of confusion and misconceptions. I’m not
sure I have an easy answer, but I’ll give it a shot...
Space vs Spacetime
Spacetime is the arena in which all physical events take place - an event is represented as a point in spacetime and specified by its
time and
location. An event in classical relativistic physics is defined using coordinates (x,y,z,t), which is the location of an elementary (point-like)
particle at a particular time. A region of spacetime itself can be viewed as the union of all events taking place within it, much the same way that a
line is the union of all of its points. The ‘world line’ of a particle or light beam is the path that the particle or beam takes in spacetime and
represents the history of the particle or beam. The ‘world line’ of the orbit of the Earth in spacetime is usually depicted as two spatial
dimensions x and y (the plane of the Earth's orbit) and a time dimension (t) orthogonal to x and y, resulting in a helix. In space alone, however, the
time coordinate is dropped and the orbit of the Earth is represented as an ellipse (the most common representation).
Put another way, in a Euclidean space the seperation between 2 points is measured by the distance between two the 2 points . Simple enough. The
distance is a purely spacial measurement. In spacetime, however, the displacement (interval) between 2 events is a completely different calculation
and includes a temporal seperation factor of c^2dt^2 (the speed of light squared multiplied by the time difference squared). So, in the case of purely
time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas
in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points. The curvature of spacetime refers to the
non-Euclidean geometry used to describe it.
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold. The Lorentz metric
determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold,
coordinate charts are used to represent observers in reference frames, using Cartesian coordinates (x,y,z,t). The concept of geodesics is central in
general relativity, since geodesic motion is considered as pure inertial motion in spacetime, and is free from any external influences.
As far as space vs spacetime goes, I heard it (somewhere?) expressed as an old Chinese proverb:
“
An expanding universe demands spacetime curvature. However, it doesn’t demand space curvature.”
Now, to thoroughly confuse things, if you’re familiar with the Schwarzschild or Friedmann spatially flat solutions in GR, you’ll note that these
give motion to test bodies to account for ‘
curved time’. Chew on that one for awhile!
Finally, as successful and comprehensive as GR has been, the notion of curved spacetime as presented in GR isn’t the only rodeo in town. Since 3 of
the 4 ‘fundamental forces’ have been successfully expressed via quantum theory, many would like to complete the picture by deriving a workable
theory of quantum gravity. This would do away with curved spacetime and gravity would be treated as a force in 3D. So far it hasn’t panned out,
though. It seems GR can’t be renormalized and unacceptable infinities are encountered at high energy scales, and therefore a valid QFT for gravity
hasn’t yet been derived.
I didn’t intend to be so longwinded. On top of that, I probably just made matters worse. Carry on...
Peace