In this section we will provide just a quick description of the
remarkable cosmological features of cosmic strings.
Many of the proposed observational tests for the existence of cosmic
strings are based on their gravitational interactions. In fact, the
gravitational field around a straight static string is very unusual
[Vilenkin, 1981].
As is well known, the Newtonian limit of Einstein field equations with
source term given by *T*^{µ}_{} =
diag(,
-*p*_{1}, -*p*_{2}, -*p*_{3})
in terms of the Newtonian potential
is given by
^{2} =
4*G*
( +
*p*_{1} + *p*_{2} + *p*_{3}),
just a statement of
the well known fact that pressure terms also contribute to the
`gravitational mass'. For an infinite string in the *z*-direction one
has *p*_{3} =
-, *i.e.*,
strings possess a large relativistic
tension (negative pressure). Moreover, averaging on the string core
results in vanishing
pressures for the *x* and *y* directions yielding
^{2}
= 0
for the Poisson equation. This indicates that space is flat outside of
an infinite straight cosmic string and therefore test particles in its
vicinity should not feel any gravitational attraction.

In fact, a full general relativistic analysis confirms this and test particles in the space around the string feel no Newtonian attraction; however there exists something unusual, a sort of wedge missing from the space surrounding the string and called the `deficit angle', usually noted , that makes the topology of space around the string that of a cone. To see this, consider the metric of a source with energy-momentum tensor [Vilenkin 1981, Gott 1985]

(75) |

In the case with *T* = *µ* (a rather simple equation of
state) this is the effective energy-momentum tensor of an
unperturbed string with string tension *µ* as seen from distances
much larger than the thickness of the string (a Goto-Nambu string).
However, real strings develop small-scale structure and are therefore
not well described by the Goto-Nambu action. When perturbations are
taken into account *T* and *µ* are no longer equal and
can only be
interpreted as effective quantities for an observer who cannot resolve
the perturbations along its length. And in this case we are left without
an effective equation of state.
Carter [1990]
has proposed that these
`noisy' strings should be such that both its speeds of propagation of
perturbations coincide. Namely,
the transverse (wiggle) speed *c*_{T} = (*T* /
*µ*)^{1/2} for
extrinsic perturbations should be equal to
the longitudinal (woggle) speed *c*_{L} = (-*dT* /
*d**µ*)^{1/2} for
sound-type perturbations. This requirement yields the new equation of state

(76) |

and, when this is satisfied, it describes the energy-momentum tensor of a wiggly string as seen by an observer who cannot resolve the wiggles or other irregularities along the string [Carter 1990, Vilenkin 1990].

The gravitational field around the cosmic string [neglecting terms of
order (*G**µ*)^{2}] is found by solving the
linearized Einstein
equations with the above *T*_{µ}^{}. One gets

(77) | |

(78) |

where *h*_{µ}
= *g*_{µ}
- _{µ}
is the metric perturbation, the radial distance from the string is
*r* = (*x*^{2} + *y*^{2})^{1/2},
and *r*_{0} is a constant of integration.

For an ideal, straight, unperturbed string, the tension and mass per
unit length are *T* = *µ* = *µ*_{0} and
one gets

(79) |

By a coordinate transformation one can bring this metric to a locally flat form

(80) |

which describes a conical and flat (Euclidean) space with a wedge of
angular size =
8*G*
*µ*_{0} (the deficit angle) removed from
the plane and with the two faces of the wedge identified.

**Wakes and gravitational lensing**

We saw above that test particles
^{(15)} at rest in the
spacetime of the straight string experience no gravitational force,
but if the string moves the situation radically changes. Two particles
initially at rest while the string is far away, will suddenly begin
moving towards each other after the string has passed between
them. Their head-on velocities will be proportional to
or,
more precisely, the particles will get a boost *v* =
4
*G**µ*_{0} *v*_{s}
in the
direction of the surface swept out by the string.
Here, =
(1-*v*_{s}^{2})^{-1/2} is the
Lorentz factor and *v*_{s} the
velocity of the moving string. Hence, the moving string will built up
a *wake* of particles behind it that may eventually form the
`seed' for accreting more matter into sheet-like structures
[Silk & Vilenkin
1984].

Also, the peculiar topology around the string makes it act as a
cylindric gravitational lens that may produce double images of distant
light sources, *e.g.*, quasars. The angle between the two images
produced by a typical GUT string would be
*G**µ*
and of order
of a few seconds of arc, independent of the impact parameter and with
no relative magnification between the images [see
Cowie & Hu, 1987,
for a recent observational attempt].

The situation gets even more interesting when we allow the string to
have small-scale structure, which we called wiggles above, as in fact
simulations indicate. Wiggles not only modify the string's effective
mass per unit length, *µ*, but also built up a Newtonian
attractive term in the velocity boost inflicted on nearby test
particles. To see this, let us consider the formation of a wake behind
a moving wiggly string. Assuming the string moves along the *x*-axis,
we can describe the situation in the rest frame of the string. In this
frame, it is the particles that move, and these flow past the string
with a velocity *v*_{s} in the opposite direction. Using
conformally
Minkowskian coordinates we can express the relevant components of the
metric as

(81) |

where the missing wedge is reproduced by identifying the half-lines
*y* = ±4 *G*
*µ**x*, *x*
0.
The linearized geodesic equations in this metric can be written as

(82) | |

(83) |

where over-dots denote derivatives with respect to *t*.
Working to first order in *G**µ*, the second of these
equations can be
integrated over the unperturbed trajectory *x* =
*v*_{s} *t*, *y* = *y*_{0}.
Transforming back to the frame in which the string has a velocity
*v*_{s} yields the result for the velocity impulse in the
*y*-direction after the string has passed
[Vachaspati & Vilenkin,
1991;
Vollick, 1992]

(84) |

The second term is the velocity impulse due to the conical deficit
angle we saw above. This term will dominate for large string
velocities, case in which big planar wakes are predicted. In this
case, the string wiggles will produce inhomogeneities in the wake and
may easy the fragmentation of the structure. The `top-down' scenario
of structure formation thus follows naturally in a universe with
fast-moving strings. On the contrary, for small velocities, it is the
first term that dominates over the deflection of particles. The origin
of this term can be easily understood
[Vilenkin & Shellard,
2000].
From Eqn. (77), the gravitational force on a
non-relativistic particle of mass *m* is *F* ~ *m*
*G*(*µ*- *T*) /*r*. A
particle with an impact parameter *r* is exposed to this force for a
time *t*
~ *r* / *v*_{s} and the resulting velocity is *v*
~ (*F* / *m*)
*t*
~ *G*(*µ*- *T*) / *v*_{s}.

^{15} If one takes into account
the own gravitational field of the particle living in the spacetime
around a cosmic string, then the situation changes. In fact, the
presence of the conical `singularity' introduced by the string
distorts the particle's own gravitational field and results in the
existence of a weak attractive force proportional to
*G*^{2}*µ**m*^{2} /
*r*^{2}, where *m* is the particle's mass
[Linet, 1986].
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