Indices, Symmetrization, and Antisymmetrization

An index identifies a tensor slot or a component in a declared basis; its letter alone does not determine which. The site normally uses Greek letters for spacetime coordinate components and Latin letters for spatial components, while abstract, frame, and spinor alphabets are declared where they are used. This card fixes the summation and bracket rules needed to read those expressions. The full geometric treatment is Smooth Manifolds, Maps, and Tensor Fields. The metric-free construction is developed in Tensor Fields and Natural Operations, and specialized frame and spinor structures belong to Frames, Tetrads, Spinors, and Clifford Geometry.

Index families and tensor slots

An abstract index labels a tensor’s multilinear slots without choosing coordinates. A coordinate index labels components after a coordinate basis has been chosen. These are different readings even if an author happens to use the same letter for both. The distinction between invariant slots and basis components is developed systematically in Wendl 2021–2022, §§2–8, pp. 11–68.

Notation	Meaning on this site
$\mu,\nu,\rho,\ldots$	Spacetime coordinate components, normally ranging over $0,1,2,3$.
$i,j,k,\ldots$	Spatial components after a spatial subspace or foliation has been specified, normally ranging over $1,2,3$.
$a,b,\ldots$ as abstract indices	Tensor slots with no numerical range, only after the abstract-index use is stated.
$\hat a,\hat b,\ldots$, or locally declared $a,b,\ldots$	Components in an orthonormal, null, or other noncoordinate frame. A hat is preferred when it prevents ambiguity.
$A,B,\ldots$ and $A',B',\ldots$	Locally declared unprimed and primed spinor slots. The prime distinguishes the conjugate spin space in the declared two-spinor formalism; it is not a derivative.

This is a reading key, not a universal alphabet imposed on every specialist page. In particular, a bare Latin index may be spatial, abstract, or frame-valued; the surrounding declaration decides. Tong states the site’s default spacetime and spatial component ranges and develops component index operations in Tong 2019, Conventions, p. 2, and §2.3, pp. 67–75.

Free indices, dummy indices, and legal contractions

A free index labels the tensor type of the result. Every additive term and both sides of an equation must have the same free indices, with the same variance and index family; each free label occurs once in each monomial. Their order inside a tensor factor may change deliberately, as in $T_{\mu\nu}+T_{\nu\mu}$. Thus

$$A^\mu{}_{\nu}B^\nu=C^\mu$$

is well formed: $\mu$ is free and $\nu$ is summed. By contrast, $A_\mu+B_\nu$ does not define a covector component with a common free index.

A dummy index is summed and may be renamed without changing the expression. Under the site’s Einstein convention it appears exactly twice in a monomial, once upper and once lower, and both occurrences belong to the same vector space. The expressions $Q^\mu{}_{\mu\mu}$ and $A^\mu B^\mu$ are therefore not valid implicit contractions: the first repeats an index three times, while the second has no covector slot to pair with either vector slot. Dummy labels must also be distinct from every free label in the same term. These rules agree with the component calculus in Carroll 1997, §1, Eqs. (1.67)–(1.71).

Contraction itself needs no metric. It is the canonical pairing of one vector slot with one covector slot, as in $A^\mu{}_{\mu}$. Raising or lowering is different: it uses a specified nondegenerate bilinear form. For a covariant tensor,

$$S^\mu{}_{\mu}=g^{\mu\nu}S_{\nu\mu}$$

is a metric trace, whereas $S_{\mu\mu}$ is not an invariant trace. Coordinate spacetime indices are raised with $g_{\mu\nu}$; orthonormal-frame indices are raised with the declared frame metric, usually $\eta_{\hat a\hat b}$. Spinor indices require their locally declared antisymmetric spin metrics and sign conventions. Never infer any of these structures merely from the chosen alphabet.

Weighted symmetrization and antisymmetrization

Parentheses and square brackets carry unit weight: symmetrization or antisymmetrization over $k$ indicated slots includes $1/k!$,

$$\begin{aligned} T_{(\mu_1\ldots\mu_k)} &=\frac{1}{k!}\sum_{\pi\in S_k} T_{\mu_{\pi(1)}\ldots\mu_{\pi(k)}},\\ T_{[\mu_1\ldots\mu_k]} &=\frac{1}{k!}\sum_{\pi\in S_k}\operatorname{sgn}(\pi) T_{\mu_{\pi(1)}\ldots\mu_{\pi(k)}}. \end{aligned}$$

For two indices this gives

$$T_{(\mu\nu)}=\frac12\left(T_{\mu\nu}+T_{\nu\mu}\right), \qquad T_{[\mu\nu]}=\frac12\left(T_{\mu\nu}-T_{\nu\mu}\right).$$

For three indices the full expansions are

$$\begin{aligned} T_{(\mu\nu\rho)}=\frac16\big(&T_{\mu\nu\rho}+T_{\mu\rho\nu} +T_{\nu\mu\rho}+T_{\nu\rho\mu}\\ &+T_{\rho\mu\nu}+T_{\rho\nu\mu}\big), \end{aligned}$$

and

$$\begin{aligned} T_{[\mu\nu\rho]}=\frac16\big(&T_{\mu\nu\rho}+T_{\nu\rho\mu} +T_{\rho\mu\nu}-T_{\mu\rho\nu}\\ &-T_{\nu\mu\rho}-T_{\rho\nu\mu}\big). \end{aligned}$$

Vertical bars exclude an index from the nearest bracket operation. For example,

$$T_{[\mu|\alpha|\nu]} =\frac12\left(T_{\mu\alpha\nu}-T_{\nu\alpha\mu}\right),$$

so $\alpha$ stays in the middle slot while $\mu$ and $\nu$ are exchanged. These weighted conventions, including excluded-index notation, are given explicitly in Tong 2019, §2.3, printed p. 74. A source using unweighted brackets differs by $k!$ and must be translated before its formulas are combined with the site convention.

Coordinate, frame, and spinor seams

Coordinate and frame components can occur in one expression only when a frame or coframe supplies the bridge. For a vector and a covariant rank-two tensor,

$$V^{\hat a}=\theta^{\hat a}{}_{\mu}V^\mu, \qquad T_{\hat a\hat b} =e_{\hat a}{}^\mu e_{\hat b}{}^\nu T_{\mu\nu}.$$

On the right-hand sides, the Greek indices are coordinate dummy indices and the hatted indices remain free frame labels. Writing $V^{\hat a}=V^\mu$ without the coframe is not a change-of-basis formula. Carroll gives the tetrad conversion rules in Carroll 1997, §3, pp. 88–92, Eqs. (3.114)–(3.129); their geometric interpretation is worked out in Orthonormal Tetrads and Measured Components.

Likewise, a spinor index cannot be contracted with a tensor or frame index merely because both use Latin letters. A specialist page must declare its spin representation, priming convention, spin metrics, and raising/lowering order before the notation is usable. The connection between frame and spinor indices is developed in Spin Connections and Spinor Curvature.

Common failures

Alphabet mistaken for type. A letter does not prove that an index is spatial, abstract, frame-valued, or spinorial. Find the local declaration and the object that converts between index families.

Illegal visual trace. Repeating two lower indices does not create an invariant contraction. Supply the metric explicitly, or retain the two covariant slots.

Lost bracket weight. Expanding site brackets without $1/k!$ changes every antisymmetric projection and differential-form normalization. Check the two-index case before importing a longer identity.

Mixed bases without a bridge. Greek and hatted components are not numerically identical in general. Insert the tetrad or coframe and verify that every dummy index is paired within one family.

This lookup card summarizes the linked geometry treatments under the site’s global conventions.

References

• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019v1, 1997. Open PDF.
• Tong, David. General Relativity. University of Cambridge Part III lecture notes, 2019. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022. Open PDF.