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5) (just ignoring the terms with Q21 ) we can infer that the time-like line r = r1 is complete toward future if r1 > 2M2 . 2. In the case of real scalar field φ = φ∗ we have Q = 0, and our R0 becomes the Schwartzschild radius independent of the form of nonlinearity in V (|φ|). Acknowledgments This research is supported partially by the grant no. 2000-2-10200-002-5 from the basic research program of the KOSEF and the SNU Research fund. Vol. 4, 2003 Solutions to the Coupled Einstein and Maxwell-Higgs System 61 References [1] L.
We consider the space and time oriented Lorentzian manifold diffeomorphic to R4 , on which the group SO(3) acts as an isometry, and the group orbits are the metric spacelike 2-spheres. The invariants of the group form a time-like curve in the space-time, which is the world line of the center of the spheres. In this spherically symmetric space-time it is convenient to introduce the function r, defined by r= A , 4π where A is the area of the 2-sphere. Thus the metric on the 2-sphere is given by ds2 = r2 dΣ2 = r2 (dθ2 + sin2 θdϕ2 ).
Vets never proved their assumption for any (non-product ) Gibbs state and until recently it has been an open question whether the above results really hold for any Gibbs states. In [12], we obtained a different proof of CLT of a single observable in quantum spin systems under a slightly different mixing condition. The advantages of our results in [12] are as follows. (i) We can prove our assumptions for several mixing states of one-dimensional systems. The assumption of [12] is valid for Gibbs states for finite range interactions, ground states of the (massive) XY model, quasifree states of CAR algebras.