# Project Explicit minimizing sequences for Bartnik's mass functional

## Basic data

Title:

Explicit minimizing sequences for Bartnik's mass functional

Duration:

01/10/2017 to 30/09/2019

Abstract / short description:

This project deals with questions related to the denition and computation of the "mass" of a gravitating object such as a star, a galaxy, or a black hole. While defining

and computing the mass of an object is straightforward in Newton's classical theory of

gravity, it has proven elusive in Einstein's general theory of relativity. In 1968, Arnowitt--

Deser--Misner succeeded in defining a notion of mass for an "isolated" object (i.e. an

object not gravitationally interacting with any other objects), but we still do not know

how to properly capture the mass of an object that is not isolated.

Many suggestions have been made, e.g. by Hawking. They are called quasi-local (QL)

masses, in contrast to the total mass assigned to an isolated object. While certainly

mathematically very useful (e.g. for proving "Penrose-style" inequalities relating physical

and geometric properties of black holes), most QL masses are in conflict with physical

intuition: Hawking's QL mass, e.g., will be negative in almost all cases!

In 1989, Bartnik dened a QL mass that satises all physical criteria (as listed by

Christodoulou--Yau). However, it is not known whether Bartnik's mass is indeed "computable",

meaning that one can explicitly or approximately compute it for many examples

of non-isolated objects.

Associated with this are two famous conjectures by Bartnik that will be investigated in this project.

We will do so by explicitly constructing sequences of "asymptotically flat" Riemannian

3-manifolds with certain delicate features, generalizing work by Mantoulidis--Schoen and

Cabrera Pacheco--Miao. They will be used to test Bartnik's conjecture and to investigate

stability of the seminal (generalized) Penrose inequality in geometrically special situations.

and computing the mass of an object is straightforward in Newton's classical theory of

gravity, it has proven elusive in Einstein's general theory of relativity. In 1968, Arnowitt--

Deser--Misner succeeded in defining a notion of mass for an "isolated" object (i.e. an

object not gravitationally interacting with any other objects), but we still do not know

how to properly capture the mass of an object that is not isolated.

Many suggestions have been made, e.g. by Hawking. They are called quasi-local (QL)

masses, in contrast to the total mass assigned to an isolated object. While certainly

mathematically very useful (e.g. for proving "Penrose-style" inequalities relating physical

and geometric properties of black holes), most QL masses are in conflict with physical

intuition: Hawking's QL mass, e.g., will be negative in almost all cases!

In 1989, Bartnik dened a QL mass that satises all physical criteria (as listed by

Christodoulou--Yau). However, it is not known whether Bartnik's mass is indeed "computable",

meaning that one can explicitly or approximately compute it for many examples

of non-isolated objects.

Associated with this are two famous conjectures by Bartnik that will be investigated in this project.

We will do so by explicitly constructing sequences of "asymptotically flat" Riemannian

3-manifolds with certain delicate features, generalizing work by Mantoulidis--Schoen and

Cabrera Pacheco--Miao. They will be used to test Bartnik's conjecture and to investigate

stability of the seminal (generalized) Penrose inequality in geometrically special situations.

Keywords:

quasi-local mass

Bartnik capacity

minimizing sequences

explicit construction

## Involved staff

### Managers

Department of Mathematics

Faculty of Science

Faculty of Science

## Local organizational units

Department of Mathematics

Faculty of Science

University of Tübingen

University of Tübingen

## Funders

Stuttgart, Baden-Württemberg, Germany