Radiation from a $D$-Dimensional Collision of Shock Waves: A Remarkably Simple Fit Formula

Recently we estimated the energy radiated in the head-on collision of two equal $D$-dimensional Aichelburg-Sexl shock waves, for even $D$, by solving perturbatively, to first order, the Einstein equations in the future of the collision. Here, we report on the solution for the odd $D$ case. After finding the wave forms, we extract the estimated radiated energy for $D=5$, 7, 9, and 11 and unveil a remarkably simple pattern, given the complexity of the framework: (for all $D$) the estimated fraction of radiated energy matches the analytic expression $1/2-1/D$, within the numerical error (less than 0.1%). Both this fit and the apparent horizon bound converge to $1/2$ as $D\to \infty$.

Figure 1

Wave Forms for $D$ odd.—The panels contain wave form curves for the radiation signal seen by an observer close to the axis for various (large) $r$ as a function of time. The horizontal axis coordinate has been rescaled and shifted so that the times for the first and second optical rays coincide for the different curves.

Figure 2

Limiting fractions—The panels contain, for each $D$, curves of $ϵ(r)$ for some values of $\theta$ that can be used to extract the limiting ${ϵ}_{1\mathrm{st}\mathrm{order}}$. Some asymptotic curves that fit the numerical data to a high accuracy are indicated in each panel. The best estimates (relative error less than ${10}^{-3}$) are indicated by the constant terms in the asymptotic fits of the red solid curves.

Figure 3

Apparent horizon bound (blue dashed line and points) compared with the first order result (red points) and our fit—Eq. (1)—matching the result perfectly (red solid curve).