I’d be curious where the factor of 2 comes from in the Newtonian approximation.
I can take a stab at explaining this. Both the Poisson equation and the Einstein equation have the general form
2nd order differential operator acting on some quantity F = Constant * Matter source
In the Newtonian case, F is the gravitational potential. In the Einstein case, it is the spacetime metric. This is a quantity with a simple, natural, purely “mathematical” definition that you cannot play with and change redifining constants; it measures the distance between events on a four-dimensional curved spacetime. “Matter source” in the Poisson equation stands for mass density, and in the Einstein equation it stands for a more complicated entitity that reduces to exactly mass density in the limit when Newtonian physics holds. So the ratio of the constants in each equation is determined by the ratio of how “spacetime metric” and “gravitational potential” are related in the Newtonian limit of GR.
In Newtonian physics, the gravitational potential is that whose first derivatives give the acceleration of a test particle:
gradient of potential = acceleration of particles
This is considered a phyiscs law combining both Newton’s law of gravity and Newton’s second law of motion. In GR, the spacetime metric also has the (purely mathematical) property that (in the limit where velocities are much smaller than the speed of light, and departures from flat space are small) its gradient is proportional, with a factor 2, to the acceleration of geodesic (minimum length) trajectories in spacetime:
gradient of metric = 2 acceleration of geodesics
So if we make the physical assumption that test particles in a gravitational field follow geodesics, then we can recover Newtonian gravity from GR. (The whole reason why this is possible is the equivalence principle, the observation that all forms of matter respond to gravity in the same way.) Since small perturbations to a flat metric have to be identified with twice the Newtonian potential, this is where the extra 2 in the Einstein equation comes from.
I can take a stab at explaining this. Both the Poisson equation and the Einstein equation have the general form
2nd order differential operator acting on some quantity F = Constant * Matter source
In the Newtonian case, F is the gravitational potential. In the Einstein case, it is the spacetime metric. This is a quantity with a simple, natural, purely “mathematical” definition that you cannot play with and change redifining constants; it measures the distance between events on a four-dimensional curved spacetime. “Matter source” in the Poisson equation stands for mass density, and in the Einstein equation it stands for a more complicated entitity that reduces to exactly mass density in the limit when Newtonian physics holds. So the ratio of the constants in each equation is determined by the ratio of how “spacetime metric” and “gravitational potential” are related in the Newtonian limit of GR.
In Newtonian physics, the gravitational potential is that whose first derivatives give the acceleration of a test particle:
gradient of potential = acceleration of particles
This is considered a phyiscs law combining both Newton’s law of gravity and Newton’s second law of motion. In GR, the spacetime metric also has the (purely mathematical) property that (in the limit where velocities are much smaller than the speed of light, and departures from flat space are small) its gradient is proportional, with a factor 2, to the acceleration of geodesic (minimum length) trajectories in spacetime:
gradient of metric = 2 acceleration of geodesics
So if we make the physical assumption that test particles in a gravitational field follow geodesics, then we can recover Newtonian gravity from GR. (The whole reason why this is possible is the equivalence principle, the observation that all forms of matter respond to gravity in the same way.) Since small perturbations to a flat metric have to be identified with twice the Newtonian potential, this is where the extra 2 in the Einstein equation comes from.