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I was wondering if it would be consistent with empirical data if gravity became extremely repulsive at short distances.

For example (please bear with my use of classical formulas, I know little of relativity), we could define the force of gravity as

$$F_g = \dfrac{GMm}{r^2}\left(1-\varepsilon\left(\dfrac{r_0}{r}\right)^n\right)\hat r$$

where $n$ is very large and $\varepsilon$ is very small.

I was thinking that if these values were made extreme, the impact on anything measurable would be basically zero. However, I was thinking that this might have some impact on black holes, etc. that would rule this theory out (apart from Occam's razor, of course).

Note: I have read some of the many questions asking about repulsive gravity; this question is specifically discussing repulsive gravity at very short distances.

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Note that the value of Newton's constant, $G$, is measured in the lab at distances on order of 1 meter.

This puts strict limits on your $\epsilon$ and $n$, and immediately rules out significant effects on astrophysical black holes ( whose Schwartzchild radii are much larger than that).

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As a matter of fact, NASA/JPL are using an expression looking a little bit like the one you are suggesting to handle relativistic effects on orbits in the solar system. In the case of one light object moving in a spherically symmetric gravitational field the expression looks like:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\left(1-\frac{4GM}{rc^2}+\frac{v^2}{c^2}\right)\hat{r} +\frac{4GM}{r^2}\left(\hat{r}\cdot \hat{v}\right)\frac{v^2}{c^2}\hat{v}$$

If you ignore the velocity-dependent terms you see that this is the same as your expression with $n=1$ and $\epsilon r_0 = 4GM/c^2$. It is easy to see that this expression becomes repulsive (again ignoring the velocity-dependent terms) for $r=4GM/c^2$ which is twice the Schwarzschild radius in Schwarzschild coordinates.

Note that this expression is based on a first order expansion of the Schwarzschild solution expressed in isotropic coordinates. Taken to the third post-Newtonian order the expression become a little bit more complicated as seen in this post.

Notice that JPL is only using the expression above in the weak fields of our solar system, it gives the right value of the so called "anomalous precession of perihelion" but it is not supposed to work in the strong field limit. The expression is provided as number 4-61 on page 4-42 in the official documentation, Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation.

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