- #1

Jules Winnfield

- 16

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Let's say I've got a very concentrated form of vacuum energy that creates an anti-gravity field (any repulsive field will do). The field generates an acceleration of ##a_{0}## directed radially outward that doesn't change with distance. If I place the source of the vacuum energy in the center of the Earth, having a mass of ##M##, what is the potential energy of an object with a mass of ##m## at a distance ##r## from Earth's center with the two fields?

My initial take on the formula is:

##PE = -m\left(\int_{\infty}^{r}-\frac {G M}{r^2}dr+\int_{r}^{0}a_0 dr\right)##

##PE = -m\left(\frac {G M}{r}-a_0 r\right)##

The rub (for me, anyway) is understanding how to integrate the second field. I'm guessing you'd integrate in the same direction as gravity (from infinity to 0) in order for the potential to work along the same vector.