Mathematical Axioms of General Relativity

[learypost]

What are the equations from which all of GR can be derived? Obviously one of the equations is Einstein's Field Equation: $G^{\alpha\beta}=8\pi T^{\alpha\beta}$. I would also guess that you would need the Euler-Lagrange Equations: $-\frac{d}{d\sigma}(\frac{\partial L}{\partial (dx^{\alpha}/d\sigma)}) + \frac{\partial L}{\partial x^{\alpha}} = 0$. Are those all the necessary equations, ie, if given a set initial conditions could you correctly calculate the entire history of the system using only these two equations and a lot of math (assuming of course that gravity is the only force)?

 

[atyy]

The Einstein-Hilbert action, the matter action, and the assumption that matter is minimally coupled to the metric.

 

[learypost]

The Einstein-Hilbert action, the matter action, and the assumption that matter is minimally coupled to the metric.

So the mathematical statement of the Einstein-Hilbert action is : $I= \int_{V} dV (-g)^{1/2}R$ (which as I understand is equivalent to the Einstein Field Equation), but what about the mathematical statements of the other two principles?

 

[atyy]

The matter action is the action of matter in special relativity, but with the Minkowski metric replaced by the metric in the Einstein-Hilbert action. Minimal coupling means that the matter action does not contain derivatives of the metric.

Take a look at Eq 2.33 in http://www.cpt.univ-mrs.fr/~rovelli/book.pdf.

 

[dcpo]

A group of Hungarian mathematicians has done quite a lot of work recently on axiomatizing relativity (special and general) in first order logic (see e.g. http://www.renyi.hu/~turms/phd.pdf). I'm not overly familiar with their work but it may be of interest, if you like things formal.