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 $m$th 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
$m$th 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.
