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Hans Ringström (KTH Mathematics, Stockholm, Sweden)
The Cauchy Problem in General Relativity
ISBN print 978-3-03719-053-1, ISBN online 978-3-03719-553-6DOI 10.4171/053
June 2009, 307 pages, softcover, 17 x 24 cm.
42.00 Euro
The general theory of relativity is a theory of manifolds equipped with
Lorentz metrics and fields which describe the matter content. Einstein’s
equations equate the Einstein tensor (a curvature quantity associated
with the Lorentz metric) with the stress energy tensor (an object constructed
using the matter fields). In addition, there are equations
describing the evolution of the matter. Using symmetry as a guiding
principle, one is naturally led to the Schwarzschild and
Friedmann–Lemaître–Robertson–Walker solutions,
modelling an isolated system and
the entire universe respectively. In a different approach, formulating
Einstein’s equations as an initial value problem allows a closer study of
their solutions. This book first provides a definition of the concept of
initial data and a proof of the correspondence between initial data and
development. It turns out that some initial data allow non-isometric
maximal developments, complicating the uniqueness issue. The second
half of the book is concerned with this and related problems, such as
strong cosmic censorship. The book presents complete proofs of several classical results that play a
central role in mathematical relativity but are not easily accessible to
those wishing to enter the subject. Prerequisites are a good knowledge
of basic measure and integration theory as well as the fundamentals of
Lorentz geometry. The necessary background from the theory of partial
differential equations and Lorentz geometry is included.
Keywords: General relativity, Cauchy problem, strong cosmic censorship, non-linear wave equations
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