Spacetime's 5 Steps to Becoming Voluptuous

Do you ever wonder how spacetime became so seductively curvaceous? I've squashed the argument found on the Stanford Encyclopedia of Philosophy into 5 bullet points:

• By definition, inertial trajectories are the straight lines (geodesics) of spacetime
• By the Equivalence Principle of General Relativity, free-falling trajectories are indistinguishable from inertial trajectories
  
• By the General Covariance of General Relativity, free-falling trajectories ought to be identified as the inertial trajectories- and thus the geodesics- of spacetime
• Free-falling trajectories show relative accelerations- a defining feature of curved geometry.
• Thus, spacetime is curved.

QED

Now, that scant, skeletal version of the argument ought to be fleshed out if anyone really wants to understand it, but sometimes it's easier to memorize now, understand later. Also, I'm a huge fan of bullet-pointing when the logic behind the words is not so clear.

If I were to strip the argument down even further, it would look something like:

• geodesic
• inertial trajectory = geodesic
• freefalling trajectory = inertial trajectory = geodesic
• => freefalling trajectory (accelerates) = geodesic (curved)

Wow, that's pretty meaningless.

A third way of putting it would be:

Inertials used to be in, but now freefall is the new inertial, and freefall prefers curves. Thus, space is curved.