Conventions and Notation

This page fixes the conventions that every other note in the sequence silently assumes. The aim is a modern Weinberg: Weinberg’s physical narrative — equivalence principle first, gravity as a field, relentless contact with experiment — expressed in the geometric language and global methods of Wald. Where the two texts disagree on signs, we follow Wald, and the disagreements are flagged below so that formulas lifted from Weinberg can be translated on sight.

1. Metric Signature

The spacetime metric is Lorentzian with the mostly-plus signature

sign (g_{ab}) = (-, +, +, +) .

In a local inertial frame the metric reduces to $η_{ab} = diag (- 1, + 1, + 1, + 1)$ . Consequences used throughout:

a vector $v^{a}$ is timelike, null, or spacelike according as $g_{ab} v^{a} v^{b}$ is negative, zero, or positive;
for a timelike worldline the proper time satisfies $d τ^{2} = - g_{ab} d x^{a} d x^{b} = - d s^{2}$ ;
the four-velocity is normalized to $g_{ab} u^{a} u^{b} = - 1$ .

This is the convention of Wald and MTW, and — perhaps surprisingly for a particle physicist — of Weinberg as well. It is opposite to the mostly-minus $(+, -, -, -)$ convention common in QFT coursework, so four-vector dot products carry the reverse overall sign relative to that habit.

2. Units

We work in geometrized units with

G = c = 1.

Mass, length, and time then share a single dimension, and the Einstein equation carries no dimensional prefactor beyond $8 π$ . The quantum and thermal constants are kept explicit:

$ℏ$ is retained as the bookkeeping parameter for the semiclassical expansion, so that “order $ℏ$ ” cleanly labels quantum effects on the classical background;
$k_{B}$ is retained so that temperatures (e.g. the Hawking temperature) display with their proper dimension.

To restore SI factors in any geometrized formula, reinsert powers of $c$ and $G$ by dimensional analysis; the standard replacements are $M \to GM / c^{2}$ (mass as length), $t \to c t$ , and surface gravity $κ \to κ c^{?}$ restored by matching $T_{H} = ℏ κ /2 π k_{B} c$ .

3. Indices

We use abstract-index notation (Wald) as the primary language:

Latin indices $a, b, c, \dots$ are abstract — they label the tensor type of a geometric object, not its components in any chart. An equation written in Latin indices is a basis-independent statement.
Greek indices $μ, ν, ρ, \dots = 0, 1, 2, 3$ denote components in a coordinate (or other named) basis. They appear only when we deliberately pass to a chart.
Latin mid-alphabet $i, j, k, \dots = 1, 2, 3$ denote spatial components, used in the $3 + 1$ split and in Newtonian limits.

Round brackets symmetrize and square brackets antisymmetrize with weight $1/ n!$ :

T_{(ab)} = \frac{1}{2} (T_{ab} + T_{ba}), T_{[ab]} = \frac{1}{2} (T_{ab} - T_{ba}) .

Indices are raised and lowered with $g_{ab}$ and its inverse $g^{ab}$ , with $g^{ab} g_{b c} = δ^{a}_{c}$ . The Einstein summation convention is in force.

4. Derivative Operators

$\nabla_{a}$ denotes the unique torsion-free, metric-compatible derivative operator (the Levi-Civita connection),

\nabla_{a} g_{b c} = 0, \nabla_{[a} \nabla_{b]} f = 0 for scalars f .

In a coordinate basis its action is encoded by the Christoffel symbols

Γ^{ρ}_{μν} = \frac{1}{2} g^{ρ σ} (\partial_{μ} g_{ν σ} + \partial_{ν} g_{μ σ} - \partial_{σ} g_{μν}),

symmetric in $μν$ because the connection is torsion-free.

5. Curvature (Wald conventions)

The Riemann tensor is defined by its action as the commutator of derivatives on a dual vector:

(\nabla_{a} \nabla_{b} - \nabla_{b} \nabla_{a}) ω_{c} = R_{ab c}^{d} ω_{d} .

The Ricci tensor and scalar are the contractions

R_{a c} = R_{ab c}^{b}, R = g^{a c} R_{a c} .

With these signs a round 2-sphere has positive scalar curvature. The Einstein tensor is

G_{ab} = R_{ab} - \frac{1}{2} R g_{ab},

and the field equation, including the cosmological constant, reads

G_{ab} + Λ g_{ab} = 8 π T_{ab} .

Translation from Weinberg. Weinberg defines the Ricci tensor with the opposite sign, so his field equation appears as $R_{μν} - \frac{1}{2} g_{μν} R = - 8 π G T_{μν}$ . When importing a Weinberg formula, flip the sign of every standalone $R_{μν}$ (the Riemann and Einstein tensors then match ours). The metric signature itself needs no translation — it already agrees.

6. Differential Forms and Orientation

For the frame-based and integral arguments we use the exterior calculus with the conventions:

the wedge product is antisymmetrized without a combinatorial prefactor on basis forms, so $d x^{a} \land d x^{b} = - d x^{b} \land d x^{a}$ ;
the exterior derivative satisfies $d^{2} = 0$ and the graded Leibniz rule;
the spacetime is oriented, and the volume form / Levi-Civita tensor is

ϵ_{ab c d} = - g [ab c d], ϵ_{0123} = + - g

in a right-handed coordinate chart, where $[ab c d]$ is the totally antisymmetric symbol with $[0123] = + 1$ .

These coexist with the index notation: forms are the natural language for the connection and curvature 2-forms, for Stokes-type arguments, and for asymptotic charges, while abstract indices remain primary for the bulk field equations.

7. Fourier and Quantum Conventions

In the field-theory and semiclassical notes, a mode of positive frequency $ω > 0$ with respect to a given time coordinate $t$ carries the dependence

e^{- iω t},

and annihilation operators are associated with positive-frequency modes via the Klein–Gordon inner product. This is the convention already used in the Hawking-radiation note and should be held fixed across the QFT-in-curved-spacetime material.

8. Summary Table

A compact reference for the choices above.

Choice	Convention	Notable contrast
Signature	$(-, +, +, +)$	opposite to $(+, -, -, -)$ QFT habit
Units	$G = c = 1$ ; $ℏ, k_{B}$ explicit	Weinberg keeps all constants
Tensor indices	abstract Latin; coordinate Greek	Weinberg is component-only
Connection	torsion-free, metric-compatible $\nabla_{a}$	—
Riemann	$(\nabla_{a} \nabla_{b} - \nabla_{b} \nabla_{a}) ω_{c} = R_{ab c}^{d} ω_{d}$	Wald sign
Ricci	$R_{a c} = R_{ab c}^{b}$	opposite sign to Weinberg
Field equation	$G_{ab} + Λ g_{ab} = 8 π T_{ab}$	Weinberg carries $- 8 π G$

9. Suggested Continuations

These conventions are first exercised in the geometric-foundations notes of I. Preliminaries, where manifolds, the derivative operator, and curvature are built intrinsically.
The curvature sign and field-equation normalization are put to work in Einstein’s Field Equations.
The geometrized-unit and Fourier conventions are assumed throughout the Hawking-radiation note in the Black Holes branch.

Hypomnemata

Notebook

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