Like Black holes, Buchdahl stars cannot be extremalized

Abstract

It was shown long back in \cite{Dadhich97} that a non-extremal black hole cannot be converted into an extremal one by test particle adiabatic accretion. The Buchdahl star is the most compact object without horizon and is defined by $\Phi(R) = 4/9$, while black hole by $\Phi(R) = 1/2$. Here $\Phi(R)$ is the gravitational potential experienced by a particle, radially falling for static and axially for the rotating object. In this short note we examine the question of extremalization of the Buchdhal star and show that the same result holds good as for the black hole. That is, a non-extremal Buchdahl star cannot be extremalized by test particle accretion. Further since extremal limit for BS is $>1$, it could facilitate formation of extremal black holes by neutral and spinless accretion. That is perhaps the only way they could be formed.