# The Cauchy Problem in General Relativity

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*Hans Ringström*

A publication of the European Mathematical Society

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 without prior background in 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in mathematical physics.