PHYSICAL REVIEW D VOLUME 39, NUMBER 4 15 FEBRUARY 1989 Gravitational fields of straight and circular cosmic strings: Relation between gravitational mass, angular deficit, and internal structure V. P. Frolov, * W. Israel, and W. G. Unruh Cosmology Program, Canadian Institute for Advanced Research, Department of Physics, University of British Columbia, Vancouver, British Columbia, Canada V6T 226 (Received 26 September 1988) The first part of the paper formulates general conditions (independent of a particular gauge- theoretic model) under which a cylindrical distribution of matter can be joined to a vacuum exterior with a conical geometry and exhibits the relation between angular deficit and internal structure. To bring out the relation to gravitational mass, the second part is devoted to a detailed study of solu- tions of the initial-value problem for circular loops of string at a moment of time symmetry. I. INTRODUCTION M«, =pH;, X(length), pH;, =b. tt /Ssr, (2) and the gravitational or Arnowitt-Deser-Misner (ADM) mass M determined by the asymptotic field at spatial infinity. Linearized theory, which one might reasonably hope to be adequate for grand-unified-theory (GUT) strings, which have p-(Higgs-boson mass) — 10 In general relativity, cosmic strings have gravitational fields with distinctly non-Newtonian characteristics. Their idiosyncrasies have been a snare for the unwary ever since strings were first introduced into cosmology by Kibble, Zel'dovich, and Vilenkin. This paper arose from an attempt to understand more clearly the general nature (irrespective of specific gauge-theoretic structural models) of the relation between the near and far fields of a string and its internal structure. From close up, the string may be considered straight and infinite. The exterior geometry is then conical, and linearized Einstein theory yields the relation b, P/gm = (inertial mass/unit length) for the angular deficit hP. The curvature and the force required to hold a test particle at rest both vanish. From the near-field point of view, the effective gravitational mass of the string is zero. The string is nevertheless a repository of positive energy, which must make its presence manifest in the total gravitational mass measured at spatial infinity. This suggests a schematic picture for the gravitational field of a loop of cosmic string. There is a near zone, whose size is small compared to the loop radius, in which the geometry is locally Aat and conical; a far zone, where the field is Schwarzschildean; and, sandwiched between these, a transition zone, of thickness comparable to the loop radius, in which the geometry evolves from one to the other configuration. The relations between the fields in these zones and the internal structure of the string can be neatly formulated for a newborn or momentarily static loop by defining three kinds of "mass": the inertial mass M;„„, = I( — To)( g)' d x the "Hiscock mass" (really a measure of angular deficit) predicts that the "masses" M;„„„MH;„and M are equal. Concerning the general relationship between them, how- ever, very little is rigorously known. To explore this question is one of the principal objectives of this paper. Since only local and near-field properties are involved in the definitions of M;„„, and MH;„ the idealization of an infinite straight string may be used in studying their in- terconnection. This is the subject of Sec. II, which is concerned with stationary cylindrical distributions and the conditions under which they can be joined to an exte- rior vacuum having a conical geometry. To deal with the gravitational mass M, the idealization of an infinitely ex- tended source is ineffective, and in Secs. III — VI we there- fore turn to a detailed study of the initial-value problem for circular loops of string at a moment of time symme- try. There is an infinite variety of solutions, depending on the amount of incoming gravitational radiation present on the initial time slice, a quantity that is difficult to bring under complete control without access to past lightlike infinity. However, it can be said in broad terms that for small angular deficits our results corroborate the equality M =MH;, =M, „„, (5/~0) . (4) But M dips below MH;, by a factor that grows to the or- der of 2 as b, P rises towards m. Beyond this, the loop is generally enclosed within a black hole. Specific properties and relations of this kind lead on to more general issues, of which the briefest mention may suffice here, since they have recently received attention elsewhere. Is the external geometry of a string a unique diagnostic of internal structure — i.e. , can a line distribu- tion of stress and energy be uniquely inferred from exter- nal properties such as angular deficit? ' Is a distribution- al description possible at all in general for line sources, stringlike or nonstringlike? In a preliminary reconnais- sance of these questions more than a decade ago, one of us concluded: "There exists no simple general prescrip- tion, analogous to [the well-known surface-layer formal- ism], for obtaining the physical characteristics of an arbi- trary line source. " A careful mathematical analysis by Geroch and Traschen reaffirms this pessimistic con- clusion. The difficulty is that Einstein s theory is non- linear, and a product is not definable for distributions more singular than step functions. It is nevertheless pos- 39 1084 1989 The American Physical Society