# Can we describe all forces as a curvature in space-time?

There are some reasons why this will not work.

The obvious point is that we can write gravity as curvature because gravity is universal, i.e. gravitationaly interactions act in the same way on everything (roughly speaking). In the end, that's a reason why it makes sense to consider gravity as a property of spacetime itself.

For example, all free particles, that is, particles that only feel gravity, move on geodesics determined by the metric, so all particles "pervceive" the curvature. Furthermore, the energy-momentum tensor on the right-hand side of your first equation contains contributions from all fields and/or particles in your theory, charged an uncharged.

On the other hand, electromagnetism is not universal: The path of a particle in an electromagnetic field depends on the charge. Hence, you shouldn't have an energy-momentum tensor of uncharged matter influnce the electromagnetic force on a chraged particle.

Second, gravity is always attractive -- for example,there are no "repulsive" geodesics in Schwarzschild spacetime.

Third, upon quantisation, the electormagnetic (also weak, strong) force become spin-1 fields, while the metric is (presumably) described by a spin-2 field (graviton). Also, you would expect that there can be only one graviton field in a consistent theory.

Finally, note that there is a formulation of electordynamics in a manner very similar to gravity in a certain way via bundles -- here, the correspondence is between, e.g., the electromagnetic field strength and the Riemann tensor, not between photon and graviton field directly.