Gravitational backreaction on a cosmic string: Formalism

We develop a method for computing the linearized gravitational backreaction for Nambu-Goto strings using a fully covariant formalism. We work with equations of motion expressed in terms of a higher dimensional analog of the geodesic equation subject to self-generated forcing terms. The approach allows arbitrary spacetime and world sheet gauge choices for the background and perturbation. The perturbed spacetime metric may be expressed as an integral over a distributional stress-energy tensor supported on the string world sheet. By formally integrating out the distribution, this quantity may be reexpressed in terms of an integral over the retarded image of the string. In doing so, one must pay particular attention to contributions that arise from the field point and from nonsmooth regions of the string. Then, the gradient of the perturbed metric decomposes into a sum of boundary and bulk terms. The decomposition depends upon the world sheet coordinates used to describe the string, but the total is independent of those considerations. We illustrate the method with numerical calculations of the self-force at every point on the world sheet for loops with kinks, cusps and self-intersections using a variety of different coordinate choices. For field points on smooth parts of the world sheet the self-force is finite. As the field point approaches a kink or cusp the self-force diverges, but is integrable in the sense that the displacement of the world sheet remains finite. As a consistency check, we verify that the period-averaged flux of energy-momentum at infinity matches the direct work the self-force performs on the string. The methodology can be applied to address many fundamental questions for string loop evolution.

Figure 1

Snapshot of the ACO string loop configuration in spacetime at time $\tau =0$. At later times the configuration can be obtained by a rigid rotation about the $z$-axis.

Figure 2

Tangent sphere representation of the ACO string loop configuration with ${\mathbf{a}}^{\prime }({\zeta }^{+})$ denoted by the two blue dots and ${\mathbf{b}}^{\prime }({\zeta }^{-})$ by the orange circle.

Figure 3

Corotating self-force for the ACO string.

Figure 4

Snapshots of the KT string loop ($\alpha =0$ and $\varphi =\pi /6$) configuration in spacetime, each labeled by time in units of $L$. All the boxes have the same size axes, $-1$ to 1 for $L=2\pi$, and fixed orientation.

Figure 5

Contributions to $F_1^{\mu }$ for the KT string ($\alpha =0$ and $\varphi =\pi /6$) when computed using the 1D integration method with integration with respect to $\zeta$. Each subfigure shows the relevant contribution to the force at all points on the string in the region $\tau \in (0,L/2)$, $\zeta \in (-L/2,L/2)$; all other points can be obtained from the standard periodic extension of the string. Each column corresponds to a different component of the force: $F_1^t$, $F_1^x$, $F_1^y$, and $F_1^z$. The rows correspond to the contributions from (i) the field point and (ii) the integral over $\zeta$ (ignoring distributional contributions at the field point). For the purposes of the plots, we have set the string tension, $\mu$, and Newton’s constant, $G$, equal to 1; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).

Figure 6

The two pieces of the self-force, $F_1^{\mu }$ (row 1) and $F_2^{\mu }$ (row 2), and the total self-force $F^{\mu }$ (row 3) for the KT string ($\alpha =0$ and $\varphi =\pi /6$) as a function of position on the string. The $F_1^{\mu }$ part can be obtained by summing the two rows in Fig. 5. For the purposes of the plots, we have set the string tension, $\mu$, and Newton’s constant, $G$, equal to 1; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).

Figure 7

Snapshots of the KT string loop ($\alpha =1/2$ and $\varphi =0$) configuration in spacetime, at equally spaced times during the fundamental period (as in Fig. 4). All the boxes have the same size axes, $-1$ to 1 for $L=2\pi$, and fixed orientation.

Figure 8

Contributions to $F_1^{\mu }$ for the KT string ($\alpha =1/2$ and $\varphi =0$) as otherwise described in Fig. 5.

Figure 9

The two pieces of the self-force, $F_1^{\mu }$ (row 1) and $F_2^{\mu }$ (row 2), and the total self-force $F^{\mu }$ (row 3) for the KT string ($\alpha =1/2$ and $\varphi =0$) as otherwise described in Fig. 6.

Figure 10

The four components ($t$, $x$, $y$ and $z$ [left to right, top to bottom]) of $\ln |{\mathcal{F}}_{\mathrm{conf}}^{\alpha }|$ on a small patch of the world sheet about the cusp. Four quadrants in $\{\log \tau ,\log \zeta \}$ are displayed with the smallest $|\tau |$ and $|\zeta |$ at the center of the picture, oriented in the same way as the usual linear system about $\{0,0\}$. Orange (blue) represent positive (negative) values.

Figure 11

The components of the smooth integral contribution to ${\mathcal{F}}_{\mathrm{conf}}^{\mu }$ along two rays at angles $\theta =0$ and $\pi /2$ are shown. Red dots are numerical results and blue lines are fits of the form $\log |{\mathcal{F}}^{\mu }|=a\log |\log r|+b\log r+c$. For $\mu =t$, $(a,b,c)=(0.098,-0.99,2.45)$ for $\theta =0$ and $(0.046,-1,3.24)$ for $\theta =\pi /2$; likewise, for $\mu =x$, $(a,b,c)=(0.044,-1,2.52)$ and (0.018, $-1$, 3.28). These fits show that the dominant behavior in the direction of motion of the cusp as $r\to 0$ is $1/r$. In the other directions, the behavior is consistent with a logarithmic divergence at leading order: $\mu =y$, $(a,b,c)=(0.79,-0.011,3.35)$ and (1.23,0.013,2.39); $\mu =z$, $(a,b,c)=(0.43,-0.025,2.13)$ and (0.41, $-0.026$, 2.17).

Figure 12

${\mathcal{F}}_{\mathrm{conf}}^{\mu }$ for the nonintersecting KT string ($\alpha =1/2$ and $\varphi =0$) in the neighborhood of the cusp at $\tau =0=\zeta$. The $t$ and $x$ components have been scaled by the radial (Euclidean) distance from the cusp, and the $y$ and $z$ components have not been scaled at all. Each panel displays two separate contributions to the total force. For $t$ and $x$ components the upper (green solid and red dotted) lines give the field point contribution. It is antisymmetric in angle and integrates over the angle to give zero (in fact, when taken over the whole world sheet the integral also vanishes exactly). The lower (blue solid and orange dashed) lines give the smooth integral contribution. It is single signed and large where $\sqrt{-\gamma }$ is small. We plot scaled results for $r=0.02$ (solid lines) and $r=0.002$ (dashed/dotted lines) for each contribution to the total. These overlap and show in a qualitative fashion the dominant $1/r$ scaling near the cusp for both contributions to the $t$ and $x$ components. In the lower panels we display the $y$ and $z$ components. As before, the contribution from the field point is given by the green solid and red dotted curves. In this case it is independent of $r$ to lowest order (and integrates over the angle to give $4\pi$ at this order). The solid blue and dashed orange lines show that the integral contributions increase slowly as $r$ decreases, consistent with the $\log r$ type behavior.

Figure 13

The GV string loop configuration in spacetime at four equally spaced moments in the basic loop oscillation cycle $\tau =0$, $L/8$, $L/4$ and $3L/8$. Each box is the same size with fixed axes $-1$ to 1 and fixed orientation.

Figure 14

The arcs on the tangent sphere traced by ${\mathbf{b}}^{\prime }(x)$ (red line, a complete great circle passing through poles) and ${\mathbf{a}}^{\prime }(x)$ (two green segments symmetrically cut from a great circle) for the GV string. The line is made by a series of points, equally spaced in argument $x$ for the left- and right-moving modes. A blowup of the line in the figure would show equal intervals between the points. When the kink is “rounded off” (nonzero $\mathrm{\Delta }$ as described in the text) a few of these points will sit between the pictured green arcs and the green line is formally continuous. The ${\mathbf{a}}^{\prime }(x)$ tangent vector moves very rapidly from one side to the other.

Figure 15

Integrand used to compute the self-force for the Garfinkle-Vachaspati string. These correspond to the values in the $\zeta$ column of Table 5. Distributional contributions from the kinks and field point are denoted by dashed and solid arrows, respectively. The top panel shows all four components of the force vector, while the bottom panel only shows the $t$-component.

Figure 16

Contributions to $F_1^{\mu }$ for the Garfinkle and Vachaspati string when computed using the 1D integration method with integration with respect to $\zeta$. Each subfigure shows the relevant contribution to the force at all points on the string in the region $\tau \in (0,L/2)$, $\zeta \in (-L/2,L/2)$; all other points can be obtained from the standard periodic extension of the string. Each column corresponds to a different component of the force: $F_1^t$, $F_1^x$, $F_1^y$, and $F_1^z$. The rows correspond to the contributions from (i) the kink that passes through ($\tau =0$, $\zeta =0$); (ii) the kink that passes through ($\tau =0$, $\zeta =\pi$); (iii) the field point; and (iv) the integral over $\zeta$ (ignoring distributional contributions at the kinks and field point). The two kinks are denoted by diagonal black lines. For the purposes of the plots, we have set the string tension, $\mu$, and Newton’s constant, $G$, equal to 1; other values simply introduce an overall scaling.

Figure 17

The two pieces of the self-force, $F_1^{\mu }$ (row 1) and $F_2^{\mu }$ (row 2), and the total force $F^{\mu }=F_1^{\mu }+F_2^{\mu }$ (row 3), for the Garfinkle and Vachaspati string as a function of position on the string. The $F_1^{\mu }$ part can be obtained by summing the four rows in Fig. 16. For the purposes of the plots, we have set the string tension, $\mu$, and Newton’s constant, $G$, equal to 1; other values simply introduce an overall scaling.

Figure 18

The Kibble string loop configuration for $p=1/2$ in spacetime at six equally spaced moments $\tau =j\pi /6$ for $j=0$ to 5 (invariant length $2\pi$) in the basic loop oscillation cycle. The blue dashed and dotted loops self-intersect at the central red dot. The solid blue lines are nonintersecting configurations.

Figure 19

The arcs on the tangent sphere traced by ${\mathbf{a}}^{\prime }(x)$ and $-{\mathbf{b}}^{\prime }(x)$ for the Kibble string resemble the seams on a baseball. The green and red lines are smooth and continuous and do not intersect each other. They satisfy an integral condition such that the loop has zero total momentum.

Figure 20

Contributions to $F_1^{\mu }$ for the Kibble string when computed using the [1D] integration method with integration with respect to $\zeta$. Each subfigure shows the relevant contribution to the force at all points on the string in the region $\tau \in (0,L/2)$, $\zeta \in (-L/2,L/2)$; all other points can be obtained from the standard periodic extension of the string. Each column corresponds to a different component of the force: $F_1^t$, $F_1^x$, $F_1^y$, and $F_1^z$. The rows correspond to the contributions from (i) the field point and (ii) the integral over $\zeta$ (ignoring distributional contributions at the field point). For the purposes of the plots, we have set the string tension, $\mu$, and Newton’s constant, $G$, equal to 1; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).

Figure 21

The two pieces of the self-force, $F_1^{\mu }$ (row 1) and $F_2^{\mu }$ (row 2), for the Kibble string as a function of position on the string. The $F_1^{\mu }$ part can be obtained by summing the two rows in Fig. 20. For the purposes of the plots, we have set the string tension, $\mu$, and Newton’s constant, $G$, equal to 1; other values simply introduce an overall scaling. Note that we have used a logarithmic scale and denoted positive (negative) values by coloring the plot orange (blue).

Figure 22

The four components ($t$, $x$, $y$, and $z$ [left to right, top to bottom]) of $\ln |{\mathcal{F}}_{\mathrm{conf}}^t|$ for a small patch of the world sheet about the crossing point. Four quadrants in $\{\delta \tau ,\delta \zeta \}$ are displayed in log absolute value coordinates (oriented to match the usual linear system about $\{\tau ,\zeta \}=\{0,\pi /2\}$). Blue (orange) represents negative (positive) values. The small dots have been added for $\delta \tau =0$ and $\delta \zeta =0$.

Figure 23

The red dot shows the field point for case I for the KT string loop. The boxes enclose the regions with rms integrands $>{10}^{-8}$ for $m=0$ and $n=5$.

Figure 24

The difference between smoothed Green function based 2D grid evaluation of $F_1^{\rho }$ and the exact delta function [1D] evaluation as a function of $n$ for loop I. The [2D] results should approach the [1D] results as $n\to \infty$ when the Gaussian tends to the delta function. These results have no over-retardation ($m=0$) and use the exact $\mathrm{\Theta }$ function for causality. The colors label results for the four spacetime components of $F_1$; red, yellow, green and blue correspond to components 0, 1, 2 and 3, respectively. The ordinate, ${\log }_{10}|F_{1,2\text{D}}^{\rho }-F_{1,1\text{D}}^{\rho }|$, quantifies the difference, component by component, as a function of $n$. The delta function contribution at the field point is of order unity and far larger than the 1D-2D differences. Table 2 gives detailed information on individual contributions.

Figure 25

Same as Fig. 24 except for $F_2^{\rho }$.

Figure 26

Same as Fig. 24 except for loop II.

Figure 27

Same as Fig. 24 except for $F_2^{\rho }$ and loop II.

Figure 28

Same as Fig. 24 except for loop III.

Figure 29

Same as Fig. 24 except for $F_2^{\rho }$ and loop III.