Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 3

New structures in gravitational radiation

Pages: 531 – 594



Lydia Bieri (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


We investigate the Einstein vacuum equations as well as the Einstein-null fluid equations describing neutrino radiation. We find new structures in gravitational waves and memory for asymptotically-flat spacetimes of slow decay. These structures do not arise in spacetimes resulting from data that is stationary outside a compact set. Rather the more general situations exhibit richer geometric-analytic interactions displaying the physics of these more general systems. It has been known that for stronger decay of the data, including data being stationary outside a compact set, gravitational wave memory is finite and of electric parity only. In this article, we investigate general spacetimes that are asymptotically flat in a rough sense. That is, the decay of the data to Minkowski space towards infinity is very slow. As a main new feature, we prove that there exists diverging magnetic memory sourced by the magnetic part of the curvature tensor (a) in the Einstein vacuum and (b) in the Einstein-null-fluid equations. The magnetic memory occurs naturally in the Einstein vacuum setting (a) of pure gravity. In case (b), in the ultimate class of solutions, the magnetic memory contains also a curl term from the energy-momentum tensor for neutrinos also diverging at the aforementioned rate. The electric memory diverges too, it is generated by the electric part of the curvature tensor and in the Einstein-null-fluid situation also by the corresponding energy-momentum component. In addition, we find a panorama of finer structures in these manifolds. Some of these manifest themselves as additional contributions to both electric and magnetic memory. Our theorems hold for any type of matter or energy coupled to the Einstein equations as long as the data decays slowly towards infinity and other conditions are satisfied. The new results have a multitude of applications ranging from mathematical general relativity to gravitational wave astrophysics, detecting dark matter and other topics in physics.

The author thanks Demetrios Christodoulou for useful remarks on a draft of this article.

The author thanks the NSF and the Simons Foundation; the author was supported by the NSF Grant No. DMS-1811819 and the Simons Fellowship in Mathematics No. 555809.

Published 22 February 2023