I am looking to study physics as a mathematician. I am interested in books on special and/or general relativity, written in a style of rigor that would be acceptable to a mathematician. I would like a large number of examples of such books.
Here are some that might be of interest.
Sachs and Wu, General Relativity for Mathematicians.
( https://www.springer.com/gp/book/9781461299059 ) From their preface:
This is a book about physics, written for mathematicians. The readers we have in mind can be roughly described as those who:
1. are mathematics graduate students with some knowledge of global differential geometry
2. have had the equivalent of freshman physics, and find popular accounts of astrophysics and cosmology interesting
3. appreciate mathematical clarity, but are willing to accept physical motivations for the mathematics in place of mathematical ones
4. are willing to spend time and effort mastering certain technical details, such as those in Section 1.1.
Yvonne Choquet-Bruhat, General Relativity and Einstein's Equations.
From her preface:
The first five chapters are an exposition of the classical foundations of General Relativity and Einstein's equations. They can form the text of a major undergraduate or first-year graduate course. They are mathematically precise, but they require only calculus-level knowledge in mathematics. They include the links with physics and astronomy which make General Relativity such an interesting subject. The following chapters are more advanced.
Chapters VI-X prove global in space, local in time results for generic solutions of the Einstein equations and their classical sources.
Chapter XI treats progressive waves, also called high-frequency waves, a precise and rigorous approximation method which is applied to the study of relativistic fluids and electromagnetic and gravitational waves.
Chapter XII gives fundamental definitions and proves basic results in global Lorentzian geometry relevant for General Relativity,
Yvonne Choquet-Bruhat, Introduction to General Relativity, Black Holes, and Cosmology.
From her preface:
The aim of this book is to present with precision, but as simply as possible, the foundations and main consequences of General Relativity. It is written for an audience of mathematics students interested in physics and physics students interested in exact mathematical formulations— or indeed for anyone with a scientific mind and curious to know more of the world we live in. The mathematical level of the first seven chapters is that of undergraduates specializing in mathematics or physics; these chapters could be the basis for a course on General Relativity. The next three chapters are more advanced, though not requiring very sophisticated mathematics; they are aimed at graduate students, lecturers, and researchers. No a priori specialized physics knowledge is required. These chapters could serve as the text for a course for graduate students.
Demetrios Christodoulou, Mathematical Problems of General Relativity I (Zurich Lectures in Advanced Mathematics).
From the back cover,
... One of the mathematical methods analyzed and exploited in the present volume is an extension of Noether’s fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler–Lagrange system of partial differential equations. Another method is the study and systematic use of foliations by characteristic (null) hypersurfaces, and is in the spirit of the approach of Roger Penrose in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media.
The book is intended for advanced students and researchers seeking an introduction into the methods and applications of general relativity.