A Mathematical Introduction to General Relativity

The book aims to give a mathematical presentation of the theory of general relativity (that is, spacetime-geometry-based gravitation theory) to advanced undergraduate mathematics students. Mathematicians will find spacetime physics presented in the definition-theorem-proof format familiar to them. The given precise mathematical definitions of physical notions help avoiding pitfalls, especially in the context of spacetime physics describing phenomena that are counter-intuitive to everyday experiences.In the first part, the differential geometry of smooth manifolds, which is needed to present the spacetime-based gravitation theory, is developed from scratch. Here, many of the illustrating examples are the Lorentzian manifolds which later serve as spacetime models. This has the twofold purpose of making the physics forthcoming in the second part relatable, and the mathematics learnt in the first part less dry. The book uses the modern coordinate-free language of semi-Riemannian geometry. Nevertheless, to familiarise the reader with the useful tool of coordinates for computations, and to bridge the gap with the physics literature, the link to coordinates is made through exercises, and via frequent remarks on how the two languages are related.In the second part, the focus is on physics, covering essential material of the 20th century spacetime-based view of gravity: energy-momentum tensor field of matter, field equation, spacetime examples, Newtonian approximation, geodesics, tests of the theory, black holes, and cosmological models of the universe.Prior knowledge of differential geometry or physics is not assumed. The book is intended for self-study, and the solutions to the (over 200) exercises are included.

這本書旨在向高年級本科數學學生提供對廣義相對論理論（即基於時空幾何的引力理論）的數學介紹。數學家們會發現這本書以定義-定理-證明的格式呈現時空物理學，這對他們來說是熟悉的。精確的數學定義有助於避免陷阱，特別是在描述對日常經驗具有反直覺性的時空物理現象時。在第一部分中，從頭開始發展了光滑流形的微分幾何，這是呈現基於時空的引力理論所需的。在這裡，許多示例都是後來用作時空模型的洛倫茲流形。這有兩個目的，一是使第二部分中即將出現的物理學更具關聯性，二是使第一部分中學到的數學不那麼枯燥。本書使用現代的無座標語言，即半黎曼幾何。然而，為了讓讀者熟悉用於計算的座標工具，並與物理文獻建立聯繫，通過練習和頻繁的備註，將座標與無座標語言相關聯。在第二部分中，重點是物理學，涵蓋了20世紀基於時空觀點的重力的基本材料：物質的能量-動量張量場、場方程、時空示例、牛頓近似、测地線、理論測試、黑洞和宇宙模型。不需要先備的微分幾何或物理知識。本書適合自學，並包含（超過200個）練習的解答。