Gravitational instability and curvature effect at finite temperature

We investigate the gravitational instability which is indicated by the appearance of an imaginary mass proportional to the squared temperature. We show in the perturbative calculations that the time component of the graviton's self-energy ${\Pi }_{00,00}$, which was calculated by Gross, Perry, and Yaffe, depends on the definitions of the graviton field, while ${\Pi }_{00,00}-{{\Pi }^{\mu }}_{\mu ,00\mathrm{}}$ does not depend on them. However, the mass derived from it is found to differ from the one in the classical argument by a factor 3. The effect of space-time curvature on this instability is qualitatively studied.