Buchdahl compactness limit for a pure Lovelock static fluid star

We obtain the Buchdahl compactness limit for a pure Lovelock static fluid star and verify that the limit following from the uniform-density Schwarzschild's interior solution, which is universal irrespective of the gravitational theory (Einstein or Lovelock), is true in general. In terms of surface potential $\mathrm{\ensuremath{\Phi}}(r)$, it means at the surface of the star $r={r}_{0}$, $\mathrm{\ensuremath{\Phi}}({r}_{0})<2N(d\ensuremath{-}N\ensuremath{-}1)/(d\ensuremath{-}1{)}^{2}$, where $d$ and $N$ indicate spacetime dimensions and Lovelock order, respectively. For a given $N$, $\mathrm{\ensuremath{\Phi}}({r}_{0})$ is maximum for $d=2N+2$, while it is always $4/9$, Buchdahl's limit, for $d=3N+1$. It is also remarkable that for $N=1$ Einstein gravity, or for pure Lovelock in $d=3N+1$, Buchdahl's limit is equivalent to the criterion that gravitational field energy exterior to the star must be less than half its gravitational mass, having no reference to the interior at all.

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