Reading guide & conventions · General Relativity

Begin with measurements#

For a reader with basic calculus and linear algebra. Chapter 0 builds the needed mechanics, partial derivatives, differential equations, and flux accounting. You do not need a prior course in physics or relativity. The geometric language is developed as it becomes useful. Later chapters extend the same measurement questions into graduate material; each new mathematical tool needs its own explanation.

Begin with a ruler, a clock, and a moving object. By the end, we want to calculate what clocks record near stars, how falling bodies move relative to one another, and what light can tell us about the universe. Einstein’s equation will become useful after we have built the ideas it connects.

Start with measurements and motion in Chapter 0, or use the route guide below to find a refresher.

How to travel through this book#

Read with a pencil and occasionally pause to predict the next step. What quantity should the answer measure? Which direction should an object move? As new notation is introduced, use it to check your reasoning.

Begin with Chapter 0 if mechanics or multivariable calculus is unfamiliar. Use its five checks to decide which refreshers you need. The recommended course runs through Chapters 0–19 and finishes with Chapter 24. Chapters 20–23 are optional deeper trails; their introductions explain the physical questions and identify additional mathematical or quantum input. Chapter 24 is the main course’s synthesis, not an optional prerequisite for those trails.

Route	Read	What it builds
The main course	Chapters 0–19, then 24; add 20–23 when their questions interest you	A connected foundation, applications, and a final metric-to-measurement calculation
The equation route	Chapter 0’s checks, then Chapters 1–15 and 24	The meaning of every term and the action derivation; Chapter 1 supplies the motivation
The applications route	Chapters 0–12, then 15–19 and 24; return to 13–14 for the full action derivation	Clocks, tides, matter, black holes, waves, and cosmology with their tensor and curvature prerequisites intact

For the applications route, the opening action argument in §15.2 may be read as a preview; §15.4’s symmetry and conserved-current calculation is the immediate tool for later applications. A shorter tour through Chapters 1, 3–5, and 16–19 can give you the physical questions, but it skips mathematical dependencies. Treat that as a preview, not as a promise that every displayed calculation will already be within reach.

Do some exercises while the corresponding ideas are fresh rather than saving all of Appendix A for the end. After Chapter 2, try A.1; after Chapter 4, A.2; after Chapter 7, A.4 and A.6; after Chapter 10, A.9. Each problem asks you to use an operation, which is a stronger check than recognizing its finished formula.

Analogies are scaffolding. A good analogy reveals a relationship. It does not provide a license to import every feature of the familiar object. A map can help distinguish a coordinate choice from the place being described; a rotating compass can help explain changing components. Each comparison has limits that the calculation must make clear.

“Derive” has several meanings. We can derive a consequence from assumptions, derive an equation from a chosen action, or motivate why that action is a good low-energy model. These are different accomplishments. General relativity is not forced on us by pure logic or by the equivalence principle alone. Its assumptions must meet experiment.

About the sources. The explanations, analogies, and worked calculations are written as an independent tutorial. Links identify historical evidence, research results, and places to pursue particular ideas; the book is not a paraphrase of a single textbook. The final reading guide distinguishes foundational notes from original research. Exact contemporary parameter estimates and speculative claims are deliberately unnecessary to the main argument.

Notation reference for returning readers

Conventions used in later chapters#

Books differ in their sign and unit conventions. Compare those choices before comparing component formulas. This book uses the following conventions.

Symbol or convention	Meaning
Metric signature	$(-,+,+,+)$
Coordinates	$x^\mu=(ct,x,y,z)$ when using standard SI component formulas; special charts explicitly say when they use $t$
Indices	Greek $\mu,\nu,\ldots=0,1,2,3$ ; spatial Latin $i,j,\ldots=1,2,3$
Summation	A repeated upper–lower pair is summed unless stated otherwise
$g_{\mu\nu}$ and $g^{\mu\nu}$	Metric matrix and its inverse, not componentwise reciprocals
$g$	$\det(g_{\mu\nu})$ , negative for a nondegenerate Lorentzian metric in four dimensions
$G_N$	Newton’s gravitational constant
$G_{\mu\nu}$	Einstein tensor, $R_{\mu\nu}-\tfrac12Rg_{\mu\nu}$
$\epsilon$	Rest-frame energy density, in joules per cubic metre
$\rho$	Mass-equivalent density $\epsilon/c^2$ , when that notation is used
$p$	Pressure; it has the same dimensions as energy density
$u^\mu$	Four-velocity $dx^\mu/d\tau$ , with $u^\mu u_\mu=-c^2$
$\Lambda$	Cosmological constant, with dimensions inverse length squared
Natural units	A section using $c=1$ says so; powers of $c$ return for physical numerical results

Our curvature convention is

R^\rho{}_{\sigma\mu\nu} =\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} +\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma},

with

[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma, \qquad R_{\mu\nu}=R^\rho{}_{\mu\rho\nu}.

This reference collects notation taught in Chapters 2–14. With these conventions, a round two-sphere has positive scalar curvature.

The physical point-particle action is $S=-mc^2\int d\tau$ . In coordinates with $x^0=ct$ , our gravitational action is

S_{\mathrm{EH}}= \frac{c^3}{16\pi G_N} \int d^4x\,\sqrt{-g}\,(R-2\Lambda),

and the stress tensor is normalized by

\delta S_m=-\frac{1}{2c}\int d^4x\,\sqrt{-g}\, T_{\mu\nu}\,\delta g^{\mu\nu}.

The action is called Einstein–Hilbert, after Einstein and David Hilbert. The determinant, inverse-metric variation, boundary terms, and factors of $c$ will all get their own explanations.

Coordinate units: angular coordinates are dimensionless. In $ds^2=dr^2+r^2d\theta^2$ , $g_{\theta\theta}=r^2$ has dimensions of length squared. It is the whole line element that must have the correct units, not every coordinate component separately.

Begin with measurements#Begin with measurements

How to travel through this book#How to travel through this book

Conventions used in later chapters#Conventions used in later chapters

Begin with measurements#

How to travel through this book#

Conventions used in later chapters#