Figure: Spacetime curvature explicated

Caption: Einstein's general relativity shows that mass-energy causes spacetime to curve---in way that is NOT easy to perceive for relatively small mass objects like the Earth---but that can be measured.

Further explication:

1. The image illustrates the curvature vastly exaggerated with a 2-dimensional surface and the Earth as the mass-energy clump causing curvature.

   It's easy to understand curved 2-dimensional spaces like the surface of a sphere. The curvature is really in 3-dimensions which very hard for humans to visualize though mathematically it can be treated.

2. Now general relativity is essentially a theory of gravity AND motion under gravity.

   For the observable universe a whole gravity is so much the dominant force that NO other forces matter significantly. In fact, the expansion of the universe can only be understood fundamentally using general relativity. Newtonian gravity is inadequate.

   For the large-scale structure of the universe gravity and pressure force are the dominant forces. However, for the most part Newtonian gravity is adequate for understanding the large-scale structure.

3. Gravity in general relativity is, in fact, NOT a force like other forces. It is a curvature of spacetime which is relativity speak for the 3 spatial dimensions plus the 1 time dimension of reality.

4. The shorthand for describing how general relativity works is that mass-energy tells spacetime how to curve and the curvature tells mass-energy how to move under insofar as gravity is concerned.

   Note, mass-energy is the jargon term in relativity speak for mass and energy which in relativistic physics are the same thing, just with different emphasis as we know from E=mc**2 (see Relativity file: e_mc2.html).

5. It is very difficult to solve problems with general relativity because, among other things, of the interlocking of curvature and motion.

   The interlocking means one has to determine self-consistent general relativity solutions for physical systems dominated by gravity. This is NOT easy and, except for a few simple cases, requires masssive numerical solutions on the computer.

6. There are, in fact, only a small set of exact analytic solutions in general relativity: only 6 high level ones by one expert's count (see Relativity file: general_relativity_exact_solutions.html; Malcolm A. H. McCallum, 2013, Exact solutions of Einstein's equations; Wikipedia: Exact solutions in general relativity). Exact analytic solution are solutions specified by exact formulae.