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4.2 Co-dimension two and cosmic strings

A second explicit calculation of quantum corrections within general relativity has been done for the gravitational field of a cosmic string, which for our purposes is a line distribution of mass characterized by a mass-per-unit-length ρ. This system naturally suggests itself as a theoretical laboratory for computing quantum effects because its classical gravitational field is extremely simple.

The classical field due to a line distribution of mass is simple for the following reason. Because of the symmetry of the mass distribution, the calculation of the gravitational field it produces is effectively a 2 + 1 dimensional problem. If the exterior to the mass distribution is empty, we seek there a solution to the vacuum Einstein equations R μν = 0. But it is a theorem that in 2 + 1 dimensions any geometry which is Ricci flat must also be Riemann flat: R μνλρ = 0! Superficially this appears to lead to the paradoxical conclusion that long, straight cosmic strings should not gravitate.

This conclusion is not quite correct, however. Although it is true that the vanishing of the Riemann tensor implies no tidal forces for test particles which pass by on the same side of the string, test particles are influenced to approach one another if they pass by on opposite sides of the string. The reason for this may be seen by more closely examining the spacetime’s geometry near the position of the cosmic string. The boundary conditions at this point require that spacetime there to resemble the tip of a cone, inasmuch as an infinitely thin cosmic string introduces a δ-function singularity into the curvature of spacetime. This implies that the flat geometry outside of the string behaves globally like a cone, corresponding to the removal of a defect angle, Δ θ = 8πG ρ radians, from the external geometry. This conical geometry for the external spacetime is what causes the focussing of trajectories of pairs of particles which pass by on either side of the string [4948].

The above considerations show that the gravitational interaction of two cosmic strings furnishes an ideal theoretical laboratory for studying quantum gravity effects near flat space. Since the classical gravitational force of one string on the other vanishes classically, its leading contribution arises at the quantum level. Consider, for instance, the interaction energy per-unit-length uint of two straight parallel strings separated by a distance a. This receives no contribution from the Einstein–Hilbert term of the effective action, for the reasons just described. Furthermore, just as for point gravitational sources, higher-curvature interactions only generate contact interactions, and so are also irrelevant for computing the strings’ interactions at long range. The leading contribution therefore arises at the quantum level, and must be ultraviolet finite.

These expectations are borne out by explicit one-loop calculations, which have been computed [154] for the case of two strings having constant mass-per-unit-lengths ρ1 and ρ2. The result obtained is (again temporarily restoring the explicit powers of ℏ and c)

( 24ℏ ) G2 ρ1ρ2 uint(a) = ----3 ----2--, (61 ) 5πc a
whose sign corresponds to a repulsive interaction.

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