1/r potential in higher dimensions

In Einstein gravity, gravitational potential goes as $$1/r^{d-3}$$ in d non-compactified spacetime dimensions, which assumes the familiar 1 / r form in four dimensions. On the other hand, it goes as $$1/r^{\alpha }$$ , with $$\alpha =(d-2m-1)/m$$ , in pure Lovelock gravity involving only one mth order term of the Lovelock polynomial in the gravitational action. The latter offers a novel possibility of having 1 / r potential for the non-compactified dimension spectrum given by $$d=3m+1$$ . Thus it turns out that in the two prototype gravitational settings of isolated objects, like black holes and the universe as a whole – cosmological models, the Einstein gravity in four and mth order pure Lovelock gravity in $$3m+1$$ dimensions behave in a similar fashion as far as gravitational interactions are considered. However propagation of gravitational waves (or the number of degrees of freedom) does indeed serve as a discriminator because it has two polarizations only in four dimensions.

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