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 dimensional problem. If the
exterior to the mass distribution is empty, we seek there a
solution to the vacuum Einstein equations
. But it is a theorem that in
dimensions any geometry which is Ricci flat must
also be Riemann flat:
! 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 -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,
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 [49, 48].
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 of two straight parallel strings
separated by a distance
. 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 [155] for the case of two
strings having constant mass-per-unit-lengths and
. The result obtained is (again
temporarily restoring the explicit powers of
and
)