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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?

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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.

On the other hand, the geodesic equation describes the evolution of a single point particle in a curved spacetime background. It is possible to extend this to a continuous fluid mechanical model that describes the evolution of an infinite continuum of fluid particles in a curved spacetime background with possibly various interactions included.

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  • $\begingroup$ I don't understand. In Newtonian Mechanics, space (and time) is continuous, but yet particles are discrete beings and we can formulate the theory as it applies to them. $\endgroup$
    – R S
    Oct 25, 2013 at 11:21
  • $\begingroup$ On one hand, EFE in an appropriate non-relativistic limit becomes Poisson's equation for the Newtonian gravitational potential (with continuous or discrete mass sources). On the other hand, the geodesic equation in an appropriate non-relativistic limit becomes Newton's second law for a point particle in a Newtonian gravitational potential background. $\endgroup$
    – Qmechanic ♦
    Oct 25, 2013 at 21:49
  • $\begingroup$ 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? $\endgroup$
    – R S
    Oct 26, 2013 at 8:03
  • $\begingroup$ That would be a stress-energy-tensor $T_{\mu\nu}$ build with the help of delta functions that model $N$ point particles. $\endgroup$
    – Qmechanic ♦
    Oct 26, 2013 at 9:27
  • $\begingroup$ @Qmechanic are you able to give any further reading for you claim that you can extend GR for a continuum? $\endgroup$
    – Ali Caglayan
    Apr 23, 2017 at 22:21

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