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Suppose we do not have yet General Relativity conclusions (like, Schwarzschild Gemetry and Weak Field Approximation) , but rather, just Minkowski space-time, newtonian gravity, principle of equivalence and special relativity on accelerated frames (i.e. special relativity on non-inertial frames).

First, we have then the Minkowski spacetime without any gravitational influence:

$$ds^{2} = -c^{2}dt^{2} + dx^{2}+dy^{2} + dz^{2} \equiv \eta_{\mu\nu}^{(Far-from-Gravitational-field)}dx^{\mu}dx^{\nu} \tag{1}$$

Secondly we then have a spacetime, which descrives the effects of Newtonian Gravity:

$$ ds^{2} = -\Big(1+\frac{2\Phi(x',y',z')}{c^{2}}\Big)c^{2}dt^{2}+\Big(1-\frac{2\Phi(x',y',z')}{c^{2}}\Big)(dx'^{2}+dy'^{2} + dz'^{2})\equiv g_{\mu\nu}^{(Under-the-Gravitational-Field-near-Earth's- Surface)}dx'^{\mu}dx'^{\nu} \tag{2}$$

Now, is it possible to say that the spacetime which describes Newtoninan Gravity is obtained by just a coordinate transformation between an inertial frame to an non-inertial frame (Much like from Minkowski spacetime to Rindler Spacetime)? I.e. is the Newtonian Gravity just another effect of a "accelerated reference frame" (then here we see the principle of equivalence)? :

$$ g_{\mu\nu}^{(Under-the-Gravitational-Field-near-Earth's- Surface)} = \frac{\partial x^{\alpha}}{\partial x'^{\mu}}\frac{\partial x^{\beta}}{\partial x'^{\nu}}\eta_{\alpha\beta}^{(Far-from-Gravitational-field)} $$

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Concepts like global frames of reference exist in special relativity and newtonian gravity, but not in GR. If $\Phi/c^2$ is small, then you're not describing the limit of newtonian mechanics, you're describing the limit of special relativity (small curvature). The limit of newtonian mechanics can't be obtained in this way, because there is no spacetime metric in newtonian spacetime. Instead you have a spatial metric and a time metric.

I know you're trying to outline a minimal possible theory that isn't necessarily full GR, but what you've presented is already GR. There is a nice discussion of this sort of thing in ch. 17 of Misner, Thorne, and Wheeler. You have a (1) metric theory and (2) the equivalence principle. Because you want to incorporate newtonian gravity, you have sources of the field, and therefore the field is not fixed, i.e., you have (3) no prior geometry. Once you have all those things, I'm pretty sure it logically implies all of general relativity, and therefore all you can talk about is the weak-field limit of GR.

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  • $\begingroup$ I opened here Gravitation and, yes (as we expected), the line element $(2)$ you can obtain neatly by just applying a metric just like: $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$. But, in fact, my question is just: Is it possible to construct the metric $(2)$ just with special relativity and equivalence principle? $\endgroup$ – M.N.Raia Oct 24 '19 at 10:50
  • $\begingroup$ Because what I want is a logical path like: "Oh hey, here is special relativity and here is equivalence principle. If we merge the two we obtain a aproximate theory of gravity (aka Newtonian Gravity). Now, if we want to generalize this we need....GR" $\endgroup$ – M.N.Raia Oct 24 '19 at 10:52

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