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Let me give a little bit of insight. I was trying to calculate the geodesic of different curves when I realised some relation (if I can call it like that), between General Relativity and Newton's Law of gravitation in the equation: $$-\frac{k}{\sqrt{x^2 + y^2}}$$

$k$, representing an arbitrary number to be able to change a little bit the shape of the curve.

If $k = 0$, then we have an empty space, similar to the Minkowski Space-Time.

Flat Space

If $k > 0$, then we have something similar to a Schwarzschild Space-Time or a Flamm's Paraboloid.

Curved Space

Then I noticed, that if a some "imaginary" force exists that pulls, for example, a rocket inside the hole, would be proportional, or related to the gradient of the function. So that was exaclty what I did. I found the gradient; and suprisingly, it matches exactly with something that I have previously seen.

$$f(x, y) = -\frac{k}{\sqrt{x^2 + y^2}}$$ $$ \boldsymbol \nabla f(x, y) = \frac{k}{\sqrt{\left (x^2 + y^2 \right)^3}} (x, y) = \frac{k}{|x_i|^3} \vec{x_i} = \frac{k}{|x_i|^2} \hat{x_i}$$ So I realised, that $k$ is nothing but $\mu = G M$, $G$ being Newton's gravitational constant and $M$, are the mass of a body.

More relations to take in account: $f(x, y)$ is equal to the potential $\phi(\vec{r})$, and that's why $-\boldsymbol \nabla \phi(\vec{r}) = -\frac{G M}{r^2} \hat{r}$. Also $\sqrt{x^2 + y^2}$, is equal to the distance, to the centre of a body, $r$.

My questions are:

  1. This is somehow a relationship between Newton's Law of Gravitation and GR (relationship between in quotes)? Is this a way of representing Newton's Laws as a distorsion of space-time caused by mass (in a similar way in which the energy-momentum tensor is related to the curvature of the space-time itself). Is this a valid interpretation?
  2. Secondly, if I get the geodesic that a body around this well would make, I could get the trajectory equation of an orbit?
  3. Finally, this equation is pretty similar to the Flamm's Paraboloid, except in one thing, this one isn't "cut" at the Schwarzschild radius. My question is, I could determine the Schwarzschild radius from this equation or get the corrected equation for the Schwarzschild radius?

I hope this are not silly questions and make sense :)

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  1. No, not really. This is not a distortion of spacetime, only of space, and the time part is essential in GR. It is a way to represent two-dimensional motion in Newton's theory as space curvature, but the laws are those of Newtonian gravity, not general relativity. GR is not only a conceptual change - it has different equations of motion, and the resulting orbits are different from those in the Newtonian theory.

  2. Yes, but like I said in the previous point, they will be the orbits we know from Newton's theory, not the relativistic versions.

  3. No, because there's no actual GR in your model, it's just a different representation of Newtonian gravity, which doesn't (and can't) include black holes. And the Flamm paraboloid is only roughly similar to your sheet - it doesn't have a horizontal asymptote at large distances like your model, and the slope becomes infinite at the event horizon. Plus, it's not even meant to give the geodesics - it just has the same spatial curvature as the Schwarzschild metric, no more.

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  • $\begingroup$ And what if I aproximate, at infinitely small sizes, to a Minkowski space-time similar to the pseudo-Riemannian manifold in GR? It will give me an aproximation of the curvature of space-time (not only space, as before)? $\endgroup$ Commented Sep 14, 2021 at 8:59
  • $\begingroup$ What do you mean to approximate at infinitely small sizes to a Minkowski spacetime? Where does size enter the picture? $\endgroup$
    – Javier
    Commented Sep 14, 2021 at 16:27
  • $\begingroup$ I mean, transform that space, into an space that follows the laws of Special Relativity. $\endgroup$ Commented Sep 14, 2021 at 16:47
  • $\begingroup$ Well, I can't really give an answer unless you give a more explicit proposal, with math and everything. $\endgroup$
    – Javier
    Commented Sep 14, 2021 at 18:04

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