### Mathematical Prerequisites

There are three paths through this book, with different levels of mathematical sophistication.

#### Basic

The geometry of the spacetimes discussed in this book can be understood as geometric models without knowing anything about Einstein's field equation. This path requires only elementary manipulations starting from the line element, together with a single symmetry principle, but does not require any further knowledge of differential forms.

With these basic tools, a detailed study of the Schwarzschild geometry is possible, including its black hole properties, as is the study of simple cosmological models. However, the fact that these solutions solve Einstein's equation must be taken on faith, and the relationship between curvature, gravity, and tidal forces omitted.

**Omit**:

- Sections 1.5–1.6 (except for the definition of geodesic);
- Sections 2.4–2.5 (and only skim Sections 2.1–2.3);
- Chapters 6 and 7;
- The Appendix.

#### Complete

This path represents the primary route through the book, covering all of the content, but leaving out some of the details. Familiarity with differential forms is assumed, up to the level of being able to compute connection and curvature forms. However, familiarity with (other) tensors is not necessary, provided the reader is willing to treat the metric and Killing's equation informally, as simple products of infinitesimals.

Some further advanced mathematical topics can be safely skipped on this path, such as the discussion of the divergence of the metric and Einstein tensors in the Appendix. The reader who chooses this path may also choose to omit some computational details, such as the calculations of curvature given in the Appendix; such computations can also easily be done using computer algebra systems.

**Omit**:

- Sections 2.4–2.5;
- The Appendix.

#### Expert

This path includes everything.