Curvy Spacetime
Gravity and acceleration aren't merely similar, according to the general theory of relativity. They are essentially the same thing. The astronaut in the accelerating space capsule can explain why things fall toward her feet by regarding herself as being at rest in a gravitational field, or she can explain the apparent downward force by regarding her space capsule as accelerating. Both explanations are correct, on Einstein's view. In other words, there's no fact of the matter whether the space capsule is really accelerating, or really in a gravitational field. The difference is just a matter of perspective.
Similarly, we can say that objects fall to the ground because Earth's gravitational field is pulling them downward, or we could instead explain why objects fall by regarding those objects as being at rest, while the ground accelerates upward toward them! From the latter point of view, Earth doesn't exert a gravitational force on anything. Rather, the surface of the earth is accelerating toward objects that are simply "floating" in spacetime.
Doesn't this mean opposite sides of the earth must be accelerating in opposite directions, and if so, why doesn't the earth fly apart? Well, perhaps it would, if spacetime were flat—that is, if the structure of spacetime obeyed the familiar Euclidean rules you learned in elementary school geometry: parallel lines never meet, the angles inside a triangle always add up to 180 degrees, and so on. But if spacetime is curved, then opposite sides of the Earth can accelerate in opposite directions without getting farther apart. General relativity implies that spacetime curves into the Earth from all directions, so that each part of Earth's surface is constantly accelerating (in a sense) with respect to spacetime itself.
In a way, Einstein's view of spacetime is similar to Newton's suggestion that space itself is a physical object. In Newton's view, however, space and time were supposed to be independent entities, and neither was affected by matter. According to Einstein, space and time are inseparable, and both are affected by matter.
In the general theory of relativity, spacetime itself is regarded as a real physical thing, a thing which affects and is affected by matter. The four-dimensional thing we call "spacetime" isn't just a mathematical model or conceptual abstraction. It's a real physical entity that interacts with matter. Specifically, the presence of mass (or energy) warps the geometric structure of spacetime, and this curvature of spacetime is what causes the phenomena we regard as gravitational forces.
To better understand how this works, let's reflect for a moment on Newton's first law of motion, which says that an object's velocity remains constant so long as the total force on the object is zero. This law implies that inertially-moving objects (i.e., objects that aren't being accelerated by any force) should have perfectly straight wordlines in a flat spacetime diagram, as we saw earlier in this chapter. But what if spacetime isn't flat? What if there's no such thing as a perfectly straight worldline through spacetime? In that case, all objects must have curved worldlines, and the worldlines of two inertially-moving objects might curve toward each other. In other words, the curvature of spacetime might make the objects accelerate toward each other, even though no force is pulling them together.
That's how gravity works, according to Einstein's general theory of relativity. Gravity isn't really a force, in the ordinary sense of a push or pull. Objects with mass accelerate toward each other not because of any force pulling them together, but rather because of the way in which mass affects the curvature of spacetime. Newton's law of universal gravitation is replaced by the Einstein field equations—a set of 10 equations that describe how mass (and energy) affect the curvature of spacetime. Although the math is complicated, the basic idea is simple: mass warps the geometry of spacetime in a way that brings the wordlines of inertially-moving objects together.
Curved Spacetime
To visualize how the curvature of spacetime brings wordlines together, imagine drawing a spacetime diagram on the surface of a ball. Two vertical worldlines will curve toward each other and meet at the top of the ball.
This means that Newton's first law of motion is incorrect, strictly speaking. Objects can and do accelerate when there is no force acting on them (except gravity, which isn't really a force at all). General relativity therefore replaces Newton's first law of motion with a slightly different law. Instead of following a perfectly straight wordline through spacetime, an inertially-moving object follows the closest thing to a straight line: a geodesic. In curved spacetime, geodesics are analogous to straight-line paths: they are the shortest paths between any two events in spacetime. (You can think of geodesics as the straightest lines that can be drawn, though they aren't perfectly straight because they're drawn on something curvy.) So, here is Einstein's replacement for Newton's first law:
The worldline of any inertially-moving object follows a geodesic through spacetime.
Free-falling objects, things in orbit, and things floating in space all follow geodesics through spacetime. Accelerating rockets do not, because there is an unbalanced force acting on them (the push of the rocket engine). You're not following a geodesic either, even while standing still. The ground is pushing upward on your feet, and this upward force isn't canceled out by any downward force (since gravity isn't really a force). So the total force on you isn't zero, and you aren't following a geodesic. In a sense, you are actually accelerating upward through spacetime.