George F. W. Ellis (PhD, 1964) and Stephen Hawking (PhD, 1966) coauthored the 1973 book, The Large Scale Structure of Space-Time (404 pages). Though their intent was to cite the axiomatic foundations for the geometry of four-dimensional spacetime, the book became a classic and set standards:

“The subject of this book is the structure of space–time on length-scales from 10^–13 cm, the radius of an elementary particle, up to 10²⁸ cm, the radius of the universe. For reasons explained in chapters 1 and 3, we base our treatment on Einstein’s General Theory of Relativity. This theory leads to two remarkable predictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a ‘black hole’ which will contain a singularity; and secondly, that there is a singularity in our past which constitutes, in some sense, a beginning to the universe. “

Planck units were not part of those formulations. Irrational numbers were never mentioned. The physical and mathematical constants are also not of their formulations.

George F. W. Ellis remembers those postdoc days at Cambridge University.  He tells us (private correspondence) that there was no consideration of Planck Temperature in their earliest discussions about the structure of space-time, “The Planck base units were not a concern.”

They started their work, as proclaimed in the opening chapter, “The subject of this book is the structure of space-time on length scales from 10^-13 cm, the radius of an elementary particle, up to 10²⁸ cm, the radius of the universe.”

The range from the Planck base units  (l_P ≈ 1.616×10^-35m, t_P ≈ 5.391×10^-44s) up to the wave-particle duality is not considered.

Also, there is within this book, The Large Scale Structure of Space-Time, Chapters 8 &10, consideration of space-time singularities, particularly what they call “the initial singularity of the universe.” It is a most speculative concept.

Of course, within our model with its emphasis on an open universe that is constantly exponentially growing, the issues around a singularity are placed on hold. So, there will, of course, be much more to come!

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