Like Black holes, Buchdahl stars cannot be extremalized

TL;DR

This paper demonstrates that Buchdahl stars, like black holes, cannot be turned into extremal objects through test particle accretion, highlighting fundamental limits on their extremalization.

Contribution

It extends the extremalization impossibility result from black holes to Buchdahl stars, showing they cannot be extremalized by test particles.

Findings

Non-extremal Buchdahl stars cannot be extremalized by test particles.

Extremal limit for Buchdahl stars is greater than 1, allowing potential formation of extremal black holes.

The result parallels the known behavior of black holes regarding extremalization.

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.