Stationary rotating strings as relativistic particle mechanics

Stationary rotating strings can be viewed as geodesic motions in appropriate metrics in two-dimensional space. We obtain all solutions describing stationary rotating strings in flat spacetime as an application. These rotating strings have infinite length with various wiggly shapes. Averaged value of the string energy, the angular momentum, and the linear momentum along the string are discussed.

Figure 1

Parameter regions. The shaded areas are the parameter regions for the rigidly rotating strings, cases (i) and (ii). The dark shaded area is the parameter region for the stationary rotating strings, case (i).

Figure 2

Scalar curvature of the reduced two-dimensional space.

Figure 3

Helical strings: $\ell +q=1(q\ne 0)$. The three-dimensional snapshots are given in the left panel, and the projection of strings to the $x\mathrm{\text{−}}y$ plane, i.e., projected curves, are given in the middle, where $x$, $y$ are defined by (3.12). The dashed circle in the middle figure represents the light cylinder $\rho =1/\Omega$. In each figure, the strings with the parameters $(\ell ,q)=(0,1)$, $(9/10,1/10)$, $(1/2,1/2)$, $(1/10,9/10)$ are shown. The parameters on the $\ell \mathrm{\text{−}}q$ plane are plotted in the right panel.

Figure 4

Planar strings: $\ell =0$. The three-dimensional snapshots and projected curves are shown for $q=1/5$, $95/100$, and 1 as the same as Fig. 3.

Figure 5

Strings in the $x\mathrm{\text{−}}y$ plane: $q=0$. The dashed circle in the left figure is the light cylinder $\rho =1/\Omega$.

Figure 6

Strings in the cases $\ell =1/5$, $1/3$, and $1/2$.

Figure 7

Potentials for $q=1/5$, $2/5$, and $1/2$ in two-dimensional flat space.