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Mathematical Axioms of General Relativity

  1. Dec 15, 2011 #1
    What are the equations from which all of GR can be derived? Obviously one of the equations is Einstein's Field Equation: [itex]G^{\alpha\beta}=8\pi T^{\alpha\beta}[/itex]. I would also guess that you would need the Euler-Lagrange Equations: [itex]-\frac{d}{d\sigma}(\frac{\partial L}{\partial (dx^{\alpha}/d\sigma)}) + \frac{\partial L}{\partial x^{\alpha}} = 0[/itex]. 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)?
     
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  3. Dec 15, 2011 #2

    atyy

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    The Einstein-Hilbert action, the matter action, and the assumption that matter is minimally coupled to the metric.
     
    Last edited: Dec 15, 2011
  4. Dec 15, 2011 #3
    So the mathematical statement of the Einstein-Hilbert action is : [itex] I= \int_{V} dV (-g)^{1/2}R [/itex] (which as I understand is equivalent to the Einstein Field Equation), but what about the mathematical statements of the other two principles?
     
  5. Dec 16, 2011 #4

    atyy

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    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.
     
    Last edited: Dec 16, 2011
  6. Dec 16, 2011 #5
    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.
     
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