### 13.4 The relativistic problem

The theoretical framework for studying stellar stability in general relativity was mainly
developed during the 1970s, with key contributions from Chandrasekhar and Friedman [31, 32]
and Schutz [102, 103]. Their work extends the Newtonian analysis discussed above. There
are basically two reasons why a relativistic analysis is more complicated than the Newtonian
one. Firstly, the problem is algebraically more complex because one must solve the Einstein
field equations in addition to the fluid equations of motion. Secondly, one must account for the
fact that a general perturbation will generate gravitational waves. The work culminated in a
series of papers [43, 44, 45, 42] in which the role that gravitational radiation plays in these
problems was explained, and a foundation for subsequent research in this area was established. The
main result was that gravitational radiation acts in the same way in the full theory as in a
post-Newtonian analysis of the problem. If we consider a sequence of equilibrium models, then a
mode becomes secularly unstable at the point where its frequency vanishes (in the inertial
frame). Most importantly, the proof does not require the completeness of the modes of the
system.