Why is our best theory of gravity geometric?

06-21-2026

Context: I’m self-studying general relativity and was curious why of all the possible approaches to develop a relativistic theory of gravity, did Einstein choose to go with a geometric theory?

The equivalence of spacetime curvature and gravity isn't exactly obvious; why is it that we don't see the same effects with EM, and come to the conclusion that EM is spacetime curvature? Because there's no coordinate frame you can choose locally such that EM effects entirely disappear. EM acts on positive and negative charges differently, whereas gravity uniformly couples to all masses. As a result, there's no corresponding equivalence principle in EM as we have with gravity.

Combining the equivalence principle with special relativity leads to strange effects like clocks ticking faster or slower based on their position in a gravitational field. This makes the connection between gravity and spacetime somewhat unique, because it's the only force that appears to be directly related to spacetime.

If we were trying to derive general relativity from merely the special theory and the equivalence principle, we'd know that gravity is in some way related to spacetime. We'd notice that gravity can be locally transformed away but not globally transformed away (tidal effects), and remember that in curved spaces we can locally transform into flat space (a tangent space at a point), but not globally. So we have these two objects, the gravitational field and curved space, that share similar properties.

We might even go as far as to say they are equivalent, and do some fancy differential geometry to encode this fact. I’m going to skip over the derivation, but basically by taking certain physical conditions (like energy conservation and locality) into account, we are led to the Einstein Field Equations.