# Which specific smooth structure are we using in general relativity?

In this lecture by Fredric Schuller it is said that in the case of a non compact four dimensional manifold there is a non countable infinity of differentiable or smooth manifolds that are NOT diffeomorphic.

Differentiable structures definition and classification - Lec 07 - Frederic Schuller

My question is that how this fact from math can be related to or affect the study of black holes, say finding the Schwarzschild solution or the study of cosmology, say solving for the FLRW metric.

I mean in which part of the calculations we specify which specific smooth structure, i.e. $C^{\infty}$-compatible maximal atlas are we using to take the chart from it and put a coordinate system?

• It's almost always the "standard" smooth structure on manifolds that are used (ie the trivial smooth structure for $\mathbb R^4$ for instance). For more details on the specific choice of smooth structures in GR try this book : maths.ed.ac.uk/~v1ranick/papers/exoticsmooth.pdf – Slereah Jun 8 '18 at 12:21
• Could you tell please on which page of the book you suggest the answer to the question can be found? – user56963 Jun 8 '18 at 13:04
• The whole book is on the topic, but the section specific to GR is chapter 10. There's also a much simpler toy model using electromagnetism starting on p. 10. – Slereah Jun 8 '18 at 13:27
• Great source. Thanks a lot. It seems that on page 270 my question is answered: there is no explicit coordinate system of an exotic four manifold available! – user56963 Jun 8 '18 at 13:58
• Well there is one for $S^7$, but it's not a great example of a spacetime. – Slereah Jun 8 '18 at 13:59