Newton vs Einstein: Gravitational Self-Interaction

Newton's model of gravitation is Poisson's Equation: 

• $\Delta\phi (x,t)=\rho (x,t)$        (PE)

where $\rho$ is mass density, $\phi$ is gravitational potential depending on a 3d Euclidean space coordinate $x$ and a time coordinate $t$. It is a linear equation without self-interaction in the sense that there is no feed back from $\phi$ to itself, only input from $\rho$. 

Einstein's model is Einstein's equation:

• $G_{\mu\nu}=T_{\mu\nu}$               (EE)

stating that space-time curvature $G_{\mu\nu}$ equals stress energy $T_{\mu\nu}$, which is a coupled system of 10 non-linear partial differential equations for the components $g_{\mu\nu}$ of the metric tensor. This is a system with self-interaction. 

Formally (EE) reduces to (PE) in a limit of weak gravitation, low speed and slow variation in time, which covers all cases of practical importance. 

Physicists have agreed to view (EE) as the fundamental model and (PE) as a less fundamental reduction of (EE) covering all of practice.

But it is possible to shift perspective and view (PE) as fundamental covering all of practice and (EE) as a less fundamental extension covering certain extreme cases beyond practice, like collision of two black holes. 

So which is more fundamental (PE) or (EE)? Consider the following features of (PE) not shared by (EE)

• Simplicity of mathematical form including linearity without self-interaction.  
• Computable at low cost.
• Covers all of practice in low cost computation. 

We now ask if these features including in particular linearity without self-interaction, can be viewed to be fundamental? And lack thereof as non-fundamental?

Well, a system with self-interaction runs the risk of blow-up or extinction, which for the Universe would be catastrophic. 

Recall that there is no self-interaction in Schrödinger's equation of Quantum Mechanics, while there is in Quantum Field Theory which creates blow-up infinities as non-physics. 

Summary: PE appears to be more fundamental than EE. Thus classical physics appears as post-modern physics after a deviation into EE of modern physics