Geometric Relativity

Geometric Relativity serves as a graduate-level textbook or a research-level reference book. As it states in the preface, this book looks at problems in general relativity that are geometric in nature, using theory and practices from Riemannian geometry and partial differential equations. In particular, the text focuses on concepts related to mass, scalar curvature, and minimal surfaces in smooth Riemannian manifolds.

The book assumes a relatively comprehensive background in Riemannian geometry, as well as a moderate background in PDEs (particularly elliptic linear PDEs and Sobolev spaces) and a light background in algebraic topology. The text is written with a mathematician in mind, so that only a shallow understanding of physics is necessary. Indeed, the manuscript is broken into two parts; the first part requires no physics knowledge at all (instead focusing on advanced Riemannian Geometry), and the second part requires very little working physics knowledge (offering a crash course in the most necessary elements of General Relativity). The text uses as focal points the Positive Mass Theorem and the Penrose Inequality, which are substantial results in this field. The first few chapters serve to offer sufficient background (in scalar curvature and minimal surface theory) to understand and appreciate these theorems, which are then introduced. The chapters that follow veer into results that are adjacent to, or spawn directly from, these theorems. The material discussed is all relevant to active research, abutting very recent results, and the author does a nice job of weaving a coherent narrative that ties together classic theorems, standard techniques, and recent results in this field.

Geometric Relatively is refreshing in its narrative approach to this topic. The author is open and honest about the material included and the material excluded in the text, explaining when certain material is omitted or glossed over. Indeed, oftentimes finer technical details will be omitted from a proof for the sake of narrative clarity. However, the author does not completely disregard these details – instead, he painstakingly refers to the papers in which these details can be found. This leads the text to be both readable and comprehensive in nature – it is easy to read, and there are direct references for the interested reader to explore all technical details.

Overall, this book is a nice textbook for a graduate student to study from or a great reference for a research mathematician. Anyone who is interested in exploring relativity from a geometry perspective, or simply interested purely in geometric analysis, can gain something from this text.

John Ross (rossjo@southwestern.edu) is an Assistant Professor of Mathematics at Southwestern University. His research interests lie in the field of geometric analysis.