# Curvature in the Newtonian Gravity

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.

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

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 :)