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Extensions of the Stability Theorem of the Minkowski Space in General Relativity (Hardcover, New ed.)
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Extensions of the Stability Theorem of the Minkowski Space in General Relativity (Hardcover, New ed.)
Series: AMS/IP Studies in Advanced Mathematics
Expected to ship within 12 - 19 working days
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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.
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