Einstein field equation

In classical mechanics, gravitational potential is

And it is sourced by the mass density $\rho_m$ following

We want to find a relativistic equivalent. There are several non-relativistic issues with the equation that we must address.

1. First, $\nabla^2$ is a spatial derivative in a particular reference

frame. We want a covariant representation. Perhaps we can use the relativistic wave operator $\square$?

1. Second, we have shown that gravity acts on energy, not rest mass, so

we need some covariant representation of energy. The stress-energy tensor $T^{\mu\nu}$ comes to mind.

The left hand side of the classical equation $\nabla^2 \Phi$ can be written as $-\nabla \cdot (-\nabla \Phi)$, where $-\nabla \Phi$ is the gravitational field. This makes it clear that the left side represents the divergence of the gravitational field, i.e. gravitational tides.

Doing some derivations I don’t know how to do yet, we reach the Einstein field equation

Where the Einstein curvature tensor $G^{\mu\nu}$ represents the curvature of spacetime.

In Cartesian coordinates, the curvature tensor has units of $\text{length}^{-2}$. The numerical factor relating energy and curvature is

It takes an enormous amount of energy to create even a small curvature in spacetime.