Analytic Brill wave initial data with trapped surfaces

Abstract

Since Brill&#39;s work of 1959, the conformal factor of finite-amplitude Brill waves, the axisymmetric and time-symmetric vacuum initial data on $\mathbb{R}^3$ that describe pure gravitational radiation, has been obtained only numerically, up to the most recent studies. Here an explicit one-parameter family of such data is constructed: the first explicit real-analytic, non-trivial, asymptotically flat vacuum initial data with trivial topology and a single end. The free metric function is a closed-form combination of elementary functions and a dilogarithm. The data have the toroidal conformal symmetry of Beig and Husa, and a change of variable of Natanzon type reduces every mode equation to the Gauss hypergeometric equation, so that the conformal factor is a series with explicit Gamma-function coefficients. The quantities that have so far required numerical work are thereby obtained exactly. A ground-state identity gives a sharp existence criterion: complete, asymptotically flat data exist precisely for $k<k_c=0.8048016772\ldots$, and the explicit series is identified with the unique variational solution. The ADM mass is an absolutely convergent digamma series, with $m/a=πk^2/10+O(k^3)$ for small amplitudes and an explicit residue at its critical divergence. Sufficiently strong waves contain trapped surfaces: it is proved that for $k$ close to $k_c$ the rescaled conformal factor converges to an explicit function and explicit tori and spheres become outer trapped, so that the data contain an apparent horizon. A numerical search, restricted to equatorially symmetric star-shaped surfaces, finds marginally trapped (minimal) spheres, which appear in a fold near $k=0.7670$. No thin-torus or weak-field approximation is used.