Do Global String Loops Collapse to Form Black Holes?

Hawking has shown that the emission of gravitational radiation cannot prevent circular loops of gauged cosmic strings from collapsing into black holes. Here we consider the corresponding question for global strings: can Goldstone boson emission prevent circular loops of global cosmic strings from forming black holes? Our results show that for every value of the string tension there is a certain critical size below which the circular loop does not collapse to form a black hole. For GUT scale strings, this critical size is much larger than the current horizon. Some years ago, Hawking 1 proved that a circular loop of gauge cosmic string would eventually collapse to form a black hole. Furthermore, he showed that in the process of collapse, the loop would radiate at most 29% of its total energy in gravitational radiation before forming a black hole. The analysis Hawking gave used the singularity theorems of gravity and did not depend on the field theoretic details of the gauge string. In the present paper, we will consider a circular loop of global string. Here the loop primarily radiates Goldstone bosons during its collapse and the question arises if a black hole can eventually form. It is unfortunate that there are no corresponding singularity theorems that can give information about non-gravitational radiation and so we have to use a method which is much less elegant than that used by Hawking: we explicitly find the energy lost in Goldstone bosons as a function of time and check if the loop ever collapses to within its own Schwarzchild radius. If it does, then a black hole will form while if the loop never falls within its Schwarzchild radius then a black hole will not form. In order to find the energy lost by the loop into Goldstone boson emission, we have had to make a number of simplifying assumptions. For example, we have ignored the radiation back-reaction 2,3, the self-gravity of the loop and the energy lost to gravitational radiation. Because of these assumptions, our results cannot be considered rigorous. However, we feel that a substantially more complicated calculation without these simplifications would yield qualitatively similar results. The simplest field theoretic action that gives global strings is: