Should have said last time that space-time "curvature" is relativity code-speak for "acceleration" or "gravitation." It's often something one can feel. Here's an imaginary example. Put a very precise atomic clock, call it A, on board a rocket. Put another, identical clock, call it B, in a lab on the ground. Start the clocks ticking. Now, let the rocket take off (ACCELERATION! Curvature difference! We can feel it!). Once the rocket stops accelerating, but while it's moving more-or-less in a straight line. let's do some experiments, say, measuring the decay rates of radioactive samples. A and B get the same answers, since at this time there is no curvature difference. Now, let A and B compare notes: let the ground clock A send some of its data to B. B sees A's clock running slow! Vice versa: if the rocket crew sends some data to A, A will reckon B's clock running slow. Can they both be right? YES. As long as the rocket keeps flying, they will both reckon each other's clocks running slow, but they're getting further and further apart and they can't compare notes forever unless the rocket turns around. ACCELERATION! Curvature difference! We can feel it! The rocket crew is actually, objectively, measurably different from the ground crew, and when they get back together, they will both agree that the clock on the rocket actually ran slower, over all, than then one on the ground.
Such experiments have been done many times with atomic clocks on planes, satellites, the space shuttle, and rockets. The details are complicated by the fact that clocks on the ground run slower because they're deeper in the Earth's gravitational field (CURVATURE! We can feel the difference!) and the fact that the projectiles travel non-straight-line paths (codespeak: "non-intertial" paths). In fact, the Earth-grav effect is easier to measure: put two clocks on different floors of a building and compare notes: they get different results because their curvature is different (grav field is stronger on lower floors, clocks actually run slower, things weigh more, we can feel the difference!). GPS satellites run at slightly different altitudes because of lumps (mountains, valleys) in the grav field, and it's necessary to correct for their mutual clock drifts to maintain accuracy.
This is really weird, but it's actually, objectively, measurably true. The theory, which is conceptually simple, though mathematically intricate, is very very accurate. No one has ever seen anything to refute the theory.
Weird as this is, it's NOTHING compared to the weirdness of quantum physics.