Cosmic strings have been a source of fascination in cosmology since Tom Kibble first proposed their existence 40 years ago. Like an imperfection in a solidifying crystal, a cosmic string is a thread of energy that might have formed in the early universe during a symmetry breaking phase transition. Twenty years ago Ruth Gregory pointed out that a black hole could have a cosmic string as a single “hair”. Turning this idea around, in this article we have proposed that a cosmic string could thread one or more black holes, like beads on a necklace.
Specifically we considered two charged black holes placed in thermal equilibrium on a long cosmic string, where the tension of the string in between the black holes (called a strut) could be the same or different from the parts of the string that run to infinity. The black holes could be extremal or non-extremal. We built these solutions with the help of the C-metric and its extensions, which describes two black holes accelerating toward/away from each other. In contrast to previously existing solutions, the black holes will have at most one deficit angle on either the south or north poles (but not both), and the size of this angle sets the scale for the tension of the string. We investigated various configurations of this setup, with and without a cosmological constant and background magnetic field. Focusing on the configuration in which the tension of the strut and the portions of string running off to infinity are equal, we found that the only configuration in full equilibrium is that for which the background magnetic field is turned on.
We found two solutions for the background magnetic field strength that can sustain two black holes on the string: one proportional and the other inversely proportional to the acceleration parameter. Curiously, solar mass black holes on a cosmic string with cosmologically acceptable tension are admissible solutions. The equilibrium of this configuration is independent of the tension of the cosmic string and only relies on the black hole mass and charge parameters. A large massive black hole can be stabilized by a small tension cosmic string in the presence of an appropriate magnetic field. The fact that the cosmological constant cannot keep the system in equilibrium was another unexpected result.
When the strut and portions of string running off to infinity have different tensions, the configuration with unequal tensions can be kept in equilibrium with the proviso that the tensions of the outer strings are bigger than the tension of the strut in between the black holes. This result holds for zero and nonzero cosmological constant. Incorporating a background magnetic field, we find that the tension of the strut could be larger or smaller than the outer strings.
How could these beads form? One mechanism could be the trapping of a long cosmic string on binary black holes. Another could be the nucleation of two (or more) black holes on a long cosmic string, which for sub-Planckian holes could be quite substantial as Alex Vilenkin has noted. These objects can enhance the gravitational signatures expected from the cosmic string, such as gravitational wave signals or the lensing effects, and could in principle help us in finding these configurations in the universe.
So far we have assumed that the black holes are not rotating. It would be interesting to investigate how and if rotating black holes could be beads on a cosmic string.
Read the full article in Classical and Quantum Gravity:
Black holes as beads on cosmic strings
Amjad Ashoorioon and Robert B Mann
2014 Class. Quantum Grav. 31 225009
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