By using the flow of a vector field chosen to be complete, smooth, and timelike, it is elementary to prove that if a Cauchy surface S is Ck-smooth then the spacetime is Ck-diffeomorphic to the product S × R, and that any two such Cauchy surfaces are Ck-diffeomorphic.
[2] Various foundational textbooks, such as George Ellis and Stephen Hawking's The Large Scale Structure of Space-Time and Robert Wald's General Relativity,[3] asserted that smoothing techniques allow Geroch's result to be strengthened from a topological to a smooth context.
However, this was not satisfactorily proved until work of Antonio Bernal and Miguel Sánchez in 2003.
As a result of their work, it is known that every globally hyperbolic spacetime has a Cauchy surface which is smoothly embedded and spacelike.
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