# Why is general relativity only formulated in continuum terms?

So, when we are discussing Newtonian mechanics, we treat particles as point particles. In continuum mechanics, which I understand to be a version in which mass is continuously distributed, we have equivalent formulations.

In Special Relativity, we again formulate everything discretely (a particle has its worldline, its four-velocity, etc.). But, we also have the Stress-Energy tensor, and we can use it to formulate some conservation rules (e.g. ${T^{\mu\nu}}_{,\nu}=0$).

However, in General Relativity, the Einstein equation is formulated in terms of a continuum, while the geodesic equation deals with a point particle. Is there a point-particle version of the Einstein equation? Is there a continuum version of the geodesic equation? If so, why not?

On one hand, Einstein Field equations (EFE) describe the evolution of the metric tensor $g_{\mu\nu}$. If we assume that space-time is continuous (i.e. excluding discretizations and Regge-type formulations), then EFE is necessarily a (continuous) field theory (FT). As long as we consider classical gravity (i.e. GR as opposed to quantum gravity with gravitons), it will stay this way.
• Poisson's equation related to continuous mass distribution. The version that related to discrete mass distribution is simply one that defines the potential field as the sum of potential fields $-mG/r$ generated by each discrete point particle. What is the GR equivalent? – R S Oct 26 '13 at 8:03
• That would be a stress-energy-tensor $T_{\mu\nu}$ build with the help of delta functions that model $N$ point particles. – Qmechanic Oct 26 '13 at 9:27