A new method to visualize the curvature of spacetime was recently proposed. This method finds the eigenvectors of the electric and magnetic components of the Weyl tensor and, in analogy to the field lines of electromagnetism, uses the eigenvectors’ integral curves to illustrate the spacetime curvature. Here we use this approach, along with well-known topological properties of fields on closed surfaces, to show that an arbitrary, radiating, asymptotically flat spacetime must have points near null infinity where the gravitational radiation vanishes. At the zeros of the gravitational radiation, the field of integral curves develops singular features analogous to the critical points of a vector field. We can, therefore, apply the topological classification of singular points of unoriented lines as a method to describe the radiation field. We provide examples of the structure of these points using linearized gravity and discuss an application to the extreme-kick black-hole-binary merger.

• Received 14 July 2011

DOI:https://doi.org/10.1103/PhysRevD.84.044037