General Relativistic Instability

Abstract

We can define the vertex function V v ia by = -/ / d 4 ud*u 'S a (x-u) giaT^ (u-z; z-u')S a (u'-y). (5,5) Taking the divergence and then the Fourier transform of this equation, we obtain the Ward-Takahashi identity 11 ' 12 for the field A i interacting with the current J a :5.-1 (#)-5.-»( ? )=(#g ),r,<«(M), (s.6) and thus, dS a~K p)/dp v =T v Hp,p)-(5-7)The renormalized functions T v ia , S a are given byZr\ a Y v^ {f,p) = Z2a-1 dS t r 1 /dp v , (5.10) but = (l+L ia )y v^( 2-Z lia )y P (5.11) andThen, from Eqs. (3.5) and (5.2), we deduce that gi«=#ngi«, a=l, •••,#.(5.14) This shows that the electric charges of all particles are changed by the same factor when renormalized, as we asserted. CONCLUSION We have shown that when mixing occurs, the renormalized fields must be taken as linear combinations of the bare fields. In order to calculate matrix elements, observed masses and coupling constants are used, but mixing parameters are not needed. We use a propagator whose pole terms are diagonal, and subtract self-energy parts at each relevant mass in order to calculate it. To lowest order this method gives the same results as the prescription of Feldman and Matthews, 1 and also calculates higher order corrections correctly.Finally we have shown that photon-vector-meson mixing still allows a zero bare mass for the photon, and that in such circumstances the electric charges of different particles are again renormalized by the same factor. ACKNOWLEDGMENTS

On the radial stability of ultra-compact Schwarzschild stars beyond the Buchdahl limit

“…Our results are shown in figure 3 (see also table A1 in the Appendix), where we plot γ c as a function of the ratio R/R S . We find that our results are in very good agreement with those reported by Chandrasekhar [2] and Wright [36], also presented in the figure for comparison. We note that γ c approaches the Newtonian value 4/3 when R/R S → ∞, as expected (in the Newtonian theory, the value of γ c is independent of the radius of the star).…”

“…In the limit when R/R S → ∞, γ c approaches the Newtonian value 4/3; the star becomes unstable as R/R S → 9/8. Our results are in good agreement with those found by Chandrasekhar [2] and Wright [36]. After having successfully tested our code with the results presented in the previous section, now we proceed to the analysis of the radial stability of an ultra compact Schwarzschild star in the negative pressure regime, 1 < R/R S < 9/8.…”

“…In the limit when R/R S → ∞, γ c approaches the Newtonian value 4/3; the star becomes unstable as R/R S → 9/8. Our results are in good agreement with those found by Chandrasekhar[2] and Wright[36].…”

Conformally coupled Abelian gauged Q balls

“…(2.33). The difference of the two would give the binding energy which would signal stability in a region where the binding energy increased with increases in R. The method is modeled after the original criteria for a Quasar stability as developed by WRIGHT (1964). The second method involves taking time dependent perturbations around the stationary solution we have found and looking for regions where they do not exhibit exponential growth which in turn would signal stability.…”