**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

**Electronic ISBN: 978-1-4704-1747-5
Product Code: AMSIP/45.E**

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.

#### Readership

Graduate students and research mathematicians interested in general relativity.

#### Reviews & Endorsements

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

-- Mathematical Reviews

# Table of Contents

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

- Cover Cover11
- Title page ii3
- General table of contents iv5
- General introduction vi7
- Solutions of the Einstein vacuum equations, by Lydia Bieri 126
- Dedication 227
- Abstract 328
- Acknowledgments 530
- Contents 732
- Introduction 1136
- Preliminary tools 4368
- Main theorem 5176
- Comparison 6994
- Error estimates 105130
- Second fundamental form ๐: estimates for the components of ๐ 167192
- Second fundamental form ๐: estimating ๐ and ๐ 189214
- Uniformization theorem 221246
- ๐ on the surfaces ๐-changes in ๐ and ๐ 241266
- The last slice 247272
- Curvature tensor-components 283308
- Uniformation theorem: standard situation, cases 1 and 2 285310
- Bibliography 291316
- Index 293318
- Solutions of the Einstein-Maxwell equations, by Nina Zipser 297322
- Abstract 299324
- Acknowledgments 301326
- Contents 303328
- Introduction 307332
- Norms and notation 321346
- Existence theorem 337362
- The electromagnetic field 343368
- Error estimates for ๐น 363388
- Interior estimates for ๐น 413438
- Comparison theorem for the Weyl tensor 425450
- Error estimates for ๐ 439464
- Second fundamental form 453478
- The lapse function 465490
- Optical function 471496
- Conclusion 485510
- Bibliography 491516
- Index 493518
- Back Cover Back Cover1522