# Is gravity an attractive force

In general relativity, in general but we can consider the most simple case of a spherically symmetric gravitational field, gravity is always attractive as long as you are at rest with respect to the gravitational field. However, if you are moving with respect to the gravitational field, gravity can sometimes be considered to be "repulsive". If you are falling in radially, from initially being at rest at high altitude, you initially accelerate towards the central mass. However, you will reach a point, if the central mass is compact enough like a black hole, where you start to decelerate and when you are infinitely close to the "Schwarzschild radius" you will move infinitely slow. This is all if your movement is observed from a distant observer.

In the classical sense of the meaning, gravity will become "repulsive" as soon as you start to decelerate. The reason for this deceleration is that in contrast to "classical Newtonian mechanics", where the force only increases the momenta by increasing the velocity, and accelerating particles in an accelerator where the electromagnetic force increase the momenta by increasing the "lorentz factor" setting the speed of light to be the speed limit, in general relativity you have the third effect that the speed of light (again as measured by a distant observer) around a spherically symmetric mass distribution decreases with the radial distance.

The gravitational force will always be "attractive" if you are moving in towards a black hole in the sense that your velocity as a fraction of the local speed of light constantly increases but your velocity, as observed by a distant observer, will if you get close enought to a black hole inevitably start do decrease and it that sense gravitation will become "repulsive".

Also, in practice, Nasa/JPL is using the force on the right side of this expression to mimic relativistic effects in the weak fields of our solar system:

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

You can see that this becomes repulsive, even if you are not moving with respect to the central mass at a radial distance of $r=4GM/c^2$ which is twice the Schwarzschild radius in Schwarzschild coordinates.

If your moving radially inwards this becomes repulsive, if I am not doing any mistake, at: $$v=\frac{c}{\sqrt{3}}\sqrt{1-\frac{4GM}{rc^2}}$$

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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answered May 6 '19 at 12:58

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