Spacetime

One of the most exciting aspects of general relativity is its prediction of black holes and the Big Bang, supported by Penrose's theorems on singularities. These theorems argue for the existence of black holes and suggest that the universe began with a Big Bang. However, a critical analysis reveals that the predictions of these theorems are often lacking. We typically provide a proof of a singularity theorem and use it to illustrate problems arising from violations of causality and the crossing of singularities. This adds weight to the view that singularity theorems alone are insufficient to predict the existence of physical singularities. The book emphasizes the importance of understanding mathematical theories through familiar non-mathematical concepts. Physical theories have significantly influenced the development of mathematics, while modern physical theories often require sophisticated mathematical formulations. Today, the complexity of both physics and mathematics makes it challenging to master theories in either discipline. However, the relationship between differential pseudo-Riemannian geometry and general relativity is particularly close, allowing for a fruitful interdisciplinary approach. This connection highlights the necessity of integrating mathematical and physical insights to deepen our understanding of these fundamental concepts.