# How does Equivalence Principle imply a Curved Space-Time?

First, consider spacetime without gravity. You notice that inertial objects have worldlines which are straight lines in spacetime and accelerometers measure how much a worldline bends in spacetime, with an accelerometer reading of zero corresponding to an inertial object’s straight worldline. Importantly, inertial objects at rest with respect to each other have parallel worldlines and they never intersect.

Now, an inertial frame consists of a coordinate grid of straight lines on the spacetime, and an accelerating frame consists of a coordinate grid of lines curving in the direction of acceleration. Describing the straight worldline of an inertial object in the accelerating coordinates produces a fictitious force (Christoffel symbols). By the equivalence principle that fictitious force is locally equivalent to gravity. Since fictitious forces do not change the fact that an inertial worldline (accelerometer reading 0) is straight, so also gravity cannot change a worldline from being straight. Otherwise, gravity would not be locally equivalent to a fictitious force.

Then, you extend those ideas to a global spacetime with gravity. Objects in free fall have accelerometers which read zero, so their worldlines are straight. But two objects in free fall initially at rest with respect to each other can eventually intersect. So we have straight lines which are initially parallel to each other but eventually intersect. This is impossible in a flat spacetime, but easily happens in a curved spacetime.

The equivalence principle only applies locally in small regions where gravity is approximately uniform and spacetime is approximately flat. Over large regions of spacetime gravity is non-uniform, and it is that non-uniform gravity (tidal gravity) that is spacetime curvature.

An observer in an uniform accelerated frame, that is the typical example for equivalence principle, has a flat spacetime.

But it is difficult to imagine a bunch of matter that can generate a field like that. Matter tends to concentrate in an approximately spherical form, generating a non uniform field, where the Riemann tensor is not zero.