Excising das All: Evolving Maxwell waves beyond scri

We study the numerical propagation of waves through future null infinity in a conformally compactified spacetime. We introduce an artificial cosmological constant, which allows us some control over the causal structure near null infinity. We exploit this freedom to ensure that all light cones are tilted outward in a region near null infinity, which allows us to impose excision-style boundary conditions in our finite-difference code. In this preliminary study we consider electromagnetic waves propagating in a static, conformally compactified spacetime.

Figure 1

Coordinate speeds for the de Sitter light cone: inward ($v_{\mathrm{in}}$, blue, lowest curve), normal to the spacelike time slices ($-\beta$, green, middle curve), and outward ($v_{\mathrm{out}}$, red, highest curve). For the Minkowski metric the graph looks very similar, except that $v_{\mathrm{in}}$ is never positive there.

Figure 2

Coordinate speeds for the modified de Sitter [Eq. (10)] light cone: inward light rays ($v_{\mathrm{in}}$, blue, lowest curve), motion normal to the spacelike time slices ($-\beta$, green, middle curve), and outward light rays ($v_{\mathrm{out}}$, red, highest curve).

Figure 3

Coordinate speeds inward ($v_{\mathrm{in}}$) for light rays in the modified [Eq. (10)] de Sitter metric.

Figure 4

World line of a light ray on the inward side of the light cone in the modified de Sitter metric.

Figure 5

The L2 norms (over the region $r\le 2$) of the constraints plotted vs time. The higher resolution data is multiplied by the factor appropriate to demonstrate second-order convergence. The initial values arise from $\mathfrak{B}$ and $\mathfrak{D}$ fields which are exact analytic solutions of the constraints, evaluated at the grid points and then differenced to form the constraints.

Figure 6

A spacetime graph of $r^2{\mathfrak{B}}^z$. Notice that the wave passes through ${\mathcal{}}^{+}$, located at $r=2$ in our coordinate system.

Figure 7

The time evolution of ${\mathfrak{B}}^z$ at a point on ${\mathcal{}}^{+}$. We show both a numerical solution generated by our code using the modified de Sitter (engineered light cones beyond ${\mathcal{}}^{+}$) and the analytic solution with a zero cosmological constant. The close agreement of the lines suggests that we are able to accurately capture the behavior of the fields as they pass through ${\mathcal{}}^{+}$. Other numerical evolutions based on the original (compactified, analytically continued) Minkowski metric ($s/L=0.0$) and the similar de Sitter metric ($s/L=0.1$), for which the grid boundaries were not spacelike, fail after $t\equiv u=4$.

Figure 8

Samples of the time evolution of the constraint violation $|\mathrm{div}\mathfrak{D}|$ at $z=0$ within our computational cube $|x^i|<2.1$ whose edges are at $r=2.97$.

Figure 9

The time evolution of an L2 norm of the constraint violations over the unphysical region $r>2$ beyond ${\mathcal{}}^{+}$ within our computational cube $|x^i|<2.1$ whose corners are at $r=3.64$.

Figure 10

A continuation of the test wave pulse to later times at ${\mathcal{}}^{+}$.

Figure 11

The damping factor for constraints in the de Sitter metric.

Figure 12

The damping factor for constraints in the modified de Sitter metric. Where $\mathrm{div}\beta$ is positive, the constraints should grow exponentially at the plotted $e$-folding rate.

Figure 13

For the analytic solution in Minkowski spacetime, the magnetic field at $z=0$, after the pulse from Fig. 7 has exited the computational grid, is plotted against the radius $r$. The (red) curve, which reaches large values $\sim 1$ beyond ${\mathcal{}}^{+}$ (which is $r=2$), is the solution in which both $U$ and $V$ terms are retained in Eqs. (15, 16, 17). The (green) curve, which remains below ${10}^{-3}$, is a solution in which the $V$ terms have been deleted outside $r=2$.

Figure 14

A field component plotted along a line $z=0,x=1$ parallel to the $y$ axis. At the grid limit $y=2.1$ one has $r=2.33$, while ${\mathcal{}}^{+}$ is met at $y=1.73$. Inside ${\mathcal{}}^{+}$ the field values appear to be convergent to zero after $u=4$, while in the region beyond $y=1.85$ or $r=2.1$ a slowly moving pulse appears to remain independent of the selected resolution.