Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Extensions of the Stability Theorem of the Minkowski Space in General Relativity
 
Lydia Bieri Harvard University, Cambridge, MA
Nina Zipser Harvard University, Cambridge, MA
A co-publication of the AMS and International Press of Boston
Extensions of the Stability Theorem of the Minkowski Space in General Relativity
Hardcover ISBN:  978-0-8218-4823-4
Product Code:  AMSIP/45
List Price: $140.00
MAA Member Price: $126.00
AMS Member Price: $112.00
eBook ISBN:  978-1-4704-1747-5
Product Code:  AMSIP/45.E
List Price: $132.00
MAA Member Price: $118.80
AMS Member Price: $105.60
Hardcover ISBN:  978-0-8218-4823-4
eBook: ISBN:  978-1-4704-1747-5
Product Code:  AMSIP/45.B
List Price: $272.00 $206.00
MAA Member Price: $244.80 $185.40
AMS Member Price: $217.60 $164.80
Extensions of the Stability Theorem of the Minkowski Space in General Relativity
Click above image for expanded view
Extensions of the Stability Theorem of the Minkowski Space in General Relativity
Lydia Bieri Harvard University, Cambridge, MA
Nina Zipser Harvard University, Cambridge, MA
A co-publication of the AMS and International Press of Boston
Hardcover ISBN:  978-0-8218-4823-4
Product Code:  AMSIP/45
List Price: $140.00
MAA Member Price: $126.00
AMS Member Price: $112.00
eBook ISBN:  978-1-4704-1747-5
Product Code:  AMSIP/45.E
List Price: $132.00
MAA Member Price: $118.80
AMS Member Price: $105.60
Hardcover ISBN:  978-0-8218-4823-4
eBook ISBN:  978-1-4704-1747-5
Product Code:  AMSIP/45.B
List Price: $272.00 $206.00
MAA Member Price: $244.80 $185.40
AMS Member Price: $217.60 $164.80
  • Book Details
     
     
    AMS/IP Studies in Advanced Mathematics
    Volume: 452009; 491 pp
    MSC: Primary 83; Secondary 58; 53;

    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.

  • Table of Contents
     
     
    • Solutions of the Einstein vacuum equations, by Lydia Bieri
    • Introduction
    • Preliminary tools
    • Main theorem
    • Comparison
    • Error estimates
    • Second fundamental form $k$: estimates for the components of $k$
    • Second fundamental form ${\mathbf {\chi }}$: estimating ${\mathbf {\chi }}$ and ${\mathbf {\zeta }}$
    • Uniformization theorem
    • $\boldsymbol {\chi }$ on the surfaces $\boldsymbol {S}$-changes in $\boldsymbol {r}$ and $\boldsymbol {s}$
    • The last slice
    • Curvature tensor-components
    • Uniformation theorem: standard situation, cases 1 and 2
    • Solutions of the Einstein-Maxwell equations, by Nina Zipser
    • Introduction
    • Norms and notation
    • Existence theorem
    • The electromagnetic field
    • Error estimates for $F$
    • Interior estimates for $F$
    • Comparison theorem for the Weyl tensor
    • Error estimates for $W$
    • Second fundamental form
    • The lapse function
    • Optical function
    • Conclusion
  • Reviews
     
     
    • Both parts are well written. ...the book should be of interest to anyone who is doing research in mathematical relativity.

      Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 452009; 491 pp
MSC: Primary 83; Secondary 58; 53;

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.

  • Solutions of the Einstein vacuum equations, by Lydia Bieri
  • Introduction
  • Preliminary tools
  • Main theorem
  • Comparison
  • Error estimates
  • Second fundamental form $k$: estimates for the components of $k$
  • Second fundamental form ${\mathbf {\chi }}$: estimating ${\mathbf {\chi }}$ and ${\mathbf {\zeta }}$
  • Uniformization theorem
  • $\boldsymbol {\chi }$ on the surfaces $\boldsymbol {S}$-changes in $\boldsymbol {r}$ and $\boldsymbol {s}$
  • The last slice
  • Curvature tensor-components
  • Uniformation theorem: standard situation, cases 1 and 2
  • Solutions of the Einstein-Maxwell equations, by Nina Zipser
  • Introduction
  • Norms and notation
  • Existence theorem
  • The electromagnetic field
  • Error estimates for $F$
  • Interior estimates for $F$
  • Comparison theorem for the Weyl tensor
  • Error estimates for $W$
  • Second fundamental form
  • The lapse function
  • Optical function
  • Conclusion
  • Both parts are well written. ...the book should be of interest to anyone who is doing research in mathematical relativity.

    Mathematical Reviews
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.