Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 2

Special Issue: In Honor of Robert Bartnik (Part 1 of 2)

Guest Editors: Piotr T. Chruściel, Greg Galloway, Jim Isenberg, Pengzi Miao, Mu-Tao Wang, and Shing-Tung Yau

Flatly foliated relativity

Pages: 707 – 747

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n2.a4


Hubert Bray (Duke University, Durham, North Carolina, U.S.A.)

Benjamin Hamm (Duke University, Durham, North Carolina, U.S.A.)

Sven Hirsch (Duke University, Durham, North Carolina, U.S.A.)

James Wheeler (Duke University, Durham, North Carolina, U.S.A.)

Yiyue Zhang (Duke University, Durham, North Carolina, U.S.A.)


Flatly Foliated Relativity (FFR) is a new theory which conceptually lies between Special Relativity (SR) and General Relativity (GR), in which spacetime is foliated by flat Euclidean spaces. While GR is based on the idea that “matter curves spacetime”, FFR is based on the idea that “matter curves spacetime, but not space”. This idea, inspired by the observed spatial flatness of our local universe, is realized by considering the same action as used in GR, but restricting it only to metrics which are foliated by flat spatial slices. FFR can be thought of as describing gravity without gravitational waves.

In FFR, a positive cosmological constant implies several interesting properties which do not follow in GR: the metric equations are elliptic on each euclidean slice, there exists a unique vacuum solution among those spherically symmetric at infinity, and there exists a geometric way to define the arrow of time. Furthermore, as gravitational waves do not exist in FFR, there are simple analogs to the positive mass theorem and Penrose-type inequalities.

Importantly, given that gravitational waves have a negligible effect on the curvature of spacetime, and that the universe appears to be locally flat, FFR may be a good approximation of GR. Moreover, FFR still admits many notable features of GR including the big bang, an accelerating expansion of the universe, and the Schwarzschild spacetime. Lastly, FFR is already known to have an existence theory for some simplified cases, which provokes an interesting discussion regarding the possibility of a more general existence theory, which may be relevant to understanding existence of solutions to GR.

H. Bray acknowledges the support of NSF Grant DMS-1406396.

Received 13 May 2019

Received revised 9 July 2019

Accepted 1 July 2019

Published 4 December 2019