Effect of a low density dust shell on the propagation of gravitational waves

Abstract

Using the Bondi-Sachs formalism, the problem of a gravitational wave source surrounded by a spherical dust shell is considered. Using linearized perturbation theory, the geometry is found in the regions: in the shell, exterior to the shell, and interior to the shell. It is found that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust.

A Description of computer algebra scripts

The computer algebra scripts used in this paper are written in Maple in plain text format, and are available as Supplementary Material. Note that the output files may be viewed using a plain text editor provided line-wrapping is switched off.

The scripts gamma.out, initialize.map, lin.map, ProcsRules.map are not used directly, but are called by the other scripts described below. gamma.out contains formulas for the Bondi-Sachs metric, its inverse and the metric connection coefficients. lin.map constructs the Einstein equations linearized about a given background. ProcsRules.map contains various procedures and rules that are used by other scripts.

The script initialize.map initializes various arrays etc., and sets the density profile of the matter as given in Eq. (4); other density profiles tested were

$$\begin{aligned} \rho&=\rho _c\left( \frac{1}{r^3}-\frac{r_0}{r^4}\right) \left( \frac{(r_0+\varDelta )^2}{r^5}-\frac{1}{r^3}\right) ,\qquad \rho =\rho _c\left( \frac{1}{r^4}-\frac{r_0}{r^5}\right) \left( \frac{r_0+\varDelta }{r^4}-\frac{1}{r^3}\right) ,\nonumber \\ \rho&=\rho _c\left( \frac{1}{r^2}-\frac{r_0}{r^3}\right) \left( \frac{(r_0+\varDelta )}{r^5}-\frac{1}{r^4}\right) ,\qquad \rho =\rho _c\left( \frac{1}{r^2}-\frac{r_0}{r^3}\right) \left( \frac{r_0+\varDelta }{r^6}-\frac{1}{r^5}\right) . \end{aligned}$$

(28)

The script backgroundShell.map constructs the background (spherically symmetric) solution for the given density profile; it also checks that the metric functions are sufficiently smooth at the interface $r=r_0$, and that the solution satisfies all 10 Einstein equations. The output is in backgroundShell.out

The script paperEqs.map, with output in paperEqs.out, generates the formulas given in Eqs. (12), (14) and (16); note that some manual simplifications have been applied to the formulas generated by the computer algebra.

The script shell.map uses the divergence-free condition on the energy-momentum tensor, $\nabla _a T_{bc}g^{ac}=0$, to determine the fluid properties, i.e. density and velocity perturbations. It then constructs the metric in $r<r_0,r_0<r<r_0+\varDelta$ and $r>r_0$, and checks that the solutions obtained satisfy all 10 Einstein equations. Finally, it constructs and solves the continuity conditions at $r=r_0,r=r_0+\varDelta$, and then evaluates the gravitational wave strain. The output is in shell.out.

The script shellCinI_0.map is the same as shell.map except that: on line 399 CinI is hard-coded to be 0, and some output has been suppressed. The output is in shellCinI_0.out.

The script regular_0_IncomingGW.map is an adaptation of the script regular_0.map used in [30] to calculate the GWs emitted by an equal mass binary. The adaptations are: (a) an incoming wave, as a free parameter, is included; and (b) the coefficient names have been changed to be consistent with those used in shell.map. The output is in regular_0_IncomingGW.out.

The file formulas.pdf is in pdf format and contains the Maple output for

$$\begin{aligned}&\beta ^{[B]},W^{[B]}, \beta ^{[2,2,I]},J^{[2,2,I]},U^{[2,2,I]},W^{[2,2,I]},C_{1I},C_{5I}, \rho ^{[2,2]},V^{[2,2]}_0,V^{[2,2]}_1,V^{[2,2]}_{ang} \\&\beta ^{[2,2,S]},J^{[2,2,S]},U^{[2,2,S]},W^{[2,2,S]},C_{1S},C_{5S}, \beta ^{[2,2,E]},J^{[2,2,E]},U^{[2,2,E]},\\&\quad W^{[2,2,E]},C_{1E},C_{5E}, \\&{{\mathcal {N}}}^{[2,2]},b_{0E},C_{3E},C_{4E},C_{inS}, b_{0S},C_{3S},C_{4S},C_{inI}. \end{aligned}$$

The formulas for the above are generated during the execution of shell.map and written to the file formulas.out.

The file Eqs.pdf is in pdf format and includes the content of paperEqs.out with annotations, together with the formulas for $\beta ^{[B]},W^{[B]}$ within the matter shell extracted from backgroundShell.out.