Dynamics of local cosmic strings
77
• c.o
Figure 3.6. Temperature discontinuity due to a cosmic string. 0 is the axis of the string,
w is the observer and PI and P2 are two images of the same point.
't
Figure 3.7. World sheet for a cosmic string.
neglecting the radius of the string core, it is much simpler to use the equations of
motion for a relativistic string of zero radius which derive from the Nambu-Goto
action and are identical to the equations of motion for the fundamental bosonic
string.
We denote by XII- the position in spacetime of points on the axis of the
cosmic string (where the Higgs field cP is zero.) Whereas a point particle may be
described by degrees of freedom XII-(T) depending only on a timelike coordinate
r, to describe a string we need, in addition, a spacelike coordinate a which we
may take, for convenience, to be in the range 0 ~ a ~ 7r. Then, the string degrees
offreedom XII-(T, a) trace out a curve (see figure 3.7) as a varies at fixed T. The
action S for a relativistic string propagating in Minkowski spacetime is of the
form
S = -2" T 1
rl dT la
1f
da (-deth)I/2hafll1l1-vaaXII-aflxv
(3.55)
r/
0
where T is the string tension,
1111-v = diag(l, -I, ... , -I)
(3.56)