On a mass functional for initial data in 4+1 dimensional spacetime

TL;DR

This paper develops a mass functional for certain initial data in 4+1 dimensional spacetime, extending Dain's functional to five dimensions, and shows it is positive for specific symmetric solutions including Myers-Perry data.

Contribution

It introduces a new mass functional for 4+1D vacuum initial data with symmetries, extending previous work to higher dimensions and demonstrating its positivity for key solutions.

Findings

Mass functional evaluates to ADM mass for symmetric data.

Critical points of the functional correspond to vacuum Einstein solutions.

Positivity established for data including Myers-Perry initial conditions.

Abstract

We consider a broad class of asymptotically flat, maximal initial data sets satisfying the vacuum constraint equations, admitting two commuting rotational symmetries. We construct a mass functional for `$t-\phi^i$' symmetric data which evaluates to the ADM mass. We then show that $\mathbb{R} \times U(1)^2$-invariant solutions of the vacuum Einstein equations are critical points of this functional amongst this class of data. We demonstrate positivity of this functional for a class of rod structures which include the Myers-Perry initial data. The construction is a natural extension of Dain's mass functional to $D=5$, although several new features arise.