Analytic Brill wave initial data with trapped surfaces
Abstract
Since Brill's work of 1959, the conformal factor of finite-amplitude Brill waves, the axisymmetric and time-symmetric vacuum initial data on ℝ3 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<kc=0.8048016772…, and the explicit series is identified with the unique variational solution. The ADM mass is an absolutely convergent digamma series, with m/a=πk2/10+O(k3) 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 kc 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.
Related
- The Segre invariant and Brill-Noether theory on Hirzebruch surfaces
- Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant
- On the Brill--Noether Theory of Sheaves on Regular Surfaces
- Canonical parameters on marginally trapped surfaces in the Minkowski 4-space
- Formation of Trapped Surfaces for the Einstein-Euler System
BibTeX
@misc{koc2026analytic,
title = {{Analytic Brill wave initial data with trapped surfaces}},
author = {Piotr Koc},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16773},
eprint = {2610.16773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16773},
}
RIS
TY - UNPB
TI - Analytic Brill wave initial data with trapped surfaces
AU - Piotr Koc
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16773
UR - https://arxiv.org/abs/2610.16773
AB - Since Brill's work of 1959, the conformal factor of finite-amplitude Brill waves, the axisymmetric and time-symmetric vacuum initial data on 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…, 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.
ER -