**AMS/IP Studies in Advanced Mathematics**

Volume: 45;
2009;
491 pp;
Hardcover

MSC: Primary 83;
Secondary 58; 53

**Print ISBN: 978-0-8218-4823-4
Product Code: AMSIP/45**

List Price: $125.00

Individual Member Price: $100.00

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#### Supplemental Materials

# Extensions of the Stability Theorem of the Minkowski Space in General Relativity

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*Lydia Bieri; Nina Zipser*

A co-publication of the AMS and International Press of Boston, Inc.

This book consists of two independent works: Part I is "Solutions of the
Einstein Vacuum Equations", by Lydia Bieri. Part II is "Solutions of the
Einstein-Maxwell Equations", by Nina Zipser.

A famous result of Christodoulou and Klainerman is the global
nonlinear stability of Minkowski spacetime. In this book, Bieri and
Zipser provide two extensions to this result. In the first part,
Bieri solves the Cauchy problem for the Einstein vacuum equations with
more general, asymptotically flat initial data, and describes
precisely the asymptotic behavior. In particular, she assumes less
decay in the power of \(r\) and one less derivative than in the
Christodoulou–Klainerman result. She proves that in this case, too,
the initial data, being globally close to the trivial data, yields a
solution which is a complete spacetime, tending to the Minkowski
spacetime at infinity along any geodesic. In contrast to the original
situation, certain estimates in this proof are borderline in view of
decay, indicating that the conditions in the main theorem on the decay
at infinity on the initial data are sharp.

In the second part, Zipser proves the existence of smooth, global
solutions to the Einstein–Maxwell equations. A nontrivial solution of
these equations is a curved spacetime with an electromagnetic field.
To prove the existence of solutions to the Einstein–Maxwell equations,
Zipser follows the argument and methodology introduced by
Christodoulou and Klainerman. To generalize the original results, she
needs to contend with the additional curvature terms that arise due to
the presence of the electromagnetic field \(F\); in her case the Ricci
curvature of the spacetime is not identically zero but rather
represented by a quadratic in the components of \(F\). In particular the
Ricci curvature is a constant multiple of the stress-energy tensor for
\(F\). Furthermore, the traceless part of the Riemann curvature tensor
no longer satisfies the homogeneous Bianchi equations but rather
inhomogeneous equations including components of the spacetime Ricci
curvature. Therefore, the second part of this book focuses primarily
on the derivation of estimates for the new terms that arise due to the
presence of the electromagnetic field.

Titles in this series are co-published with International Press of Boston, Inc., Cambridge, MA.

#### Table of Contents

# Table of Contents

## Extensions of the Stability Theorem of the Minkowski Space in General Relativity

#### Readership

Graduate students and research mathematicians interested in general relativity.

#### Reviews

Both parts are well written. ...the book should be of interest to anyone who is doing research in mathematical relativity.

-- Mathematical Reviews