Various foundational textbooks, such as George Ellis and Stephen Hawking 's The Large Scale Structure of Space-Time and Robert Wald 's General Relativity, 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. As they proved in 2005, the diffeomorphism to S × R can be selected so that each surface of the form S × { a } is a spacelike smooth Cauchy surface and that each curve of the form { s } × R is a smooth timelike curve orthogonal to each surface S × { a }.