Tensor fields of type (r,s) on a Lorentzian four-manifold and their components

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $r,s\ge 0$ be integers.

(a) Tensors at a point. For $p\in M$, the tangent space $T_{p}M$ is the real vector space of derivations at $p$ (Derivations at a point and the tangent space) and the cotangent space is its algebraic dual, $T_p^{\ast }M=(T_{p}M{)}^{\ast }$ (Cotangent space and cotangent bundle as a disjoint union). A type $(r,s)$ tensor at $p$ is a multilinear map

$T_p:(T_p^{\ast }M{)}^r\times (T_{p}M{)}^s⟶\mathbb{R},$

in the sense of A type $(r,s)$ tensor on a finite-dimensional vector space: the first $r$ arguments are covectors and the last $s$ are vectors. In particular a type $(1,0)$ tensor is a vector, a type $(0,1)$ tensor is a covector and a type $(0,2)$ tensor is a bilinear form, so the covariant $2$-tensors of Smooth covariant 2-tensor fields on a smooth manifold are exactly the type $(0,2)$ tensors of this definition.

(b) The tensor bundle and smooth fields. The type $(r,s)$ tensor bundle $T_s^{r}M$ is the disjoint union of the fibres $T_s^{r}M{\mathrm{\mid }}_p=\mathrm{Mult}((T_p^{\ast }M{)}^r\times (T_{p}M{)}^s,\mathbb{R})$ over $p\in M$ (The type $(r,s)$ tensor bundle); it carries the smooth vector-bundle structure defined by the tensor-coordinate change rules (The tensor bundle on a smooth four-manifold from coordinate evaluations). A smooth tensor field of type $(r,s)$ on $M$ is a family $T=(T_p{)}_{p\in M}$ assigning to each $p$ a type $(r,s)$ tensor at $p$, such that for every smooth chart $(U,x)$ of the maximal smooth atlas the component functions

$T^{a_1\cdots a_r}{}_{b_1\cdots b_s}(q):=T_q(d{x}^{a_1}{\mathrm{\mid }}_q,\dots ,d{x}^{a_r}{\mathrm{\mid }}_q,{\mathrm{\partial }}_{b_1}{\mathrm{\mid }}_q,\dots ,{\mathrm{\partial }}_{b_s}{\mathrm{\mid }}_q),q\in U,$

are $C^{\mathrm{\infty }}$ on $U$; here the coordinate derivations ${\mathrm{\partial }}_b{\mathrm{\mid }}_q=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^b{\mathrm{\mid }}_q$ form a basis of $T_{q}M$ (Coordinate derivations form a basis of the tangent space) and the coordinate differentials $d{x}^a{\mathrm{\mid }}_q$ form the dual basis of $T_q^{\ast }M$ (Coordinate differentials form the dual cotangent basis). This is equivalent to $T$ being a smooth section of $T_s^{r}M$ in the sense of A smooth tensor field: the component functions are the frame coefficients of the section in the local frame induced by the chart. The smoothness condition is independent of the chosen smooth atlas and the components in two overlapping charts are related by the transformation law proved in the lemma on this page; the definition is therefore chart independent rather than merely chart-wise.

(c) Index conventions. Indices are written $a,b,c,\dots$ for components in a chart, with the Einstein summation convention: a repeated index, once upper and once lower, is summed over $0,1,2,3$. An upper index is a contravariant slot in which a covector is evaluated, a lower index a covariant slot in which a vector is evaluated. Slots are moved between these positions with the musical isomorphisms ${\mathrm{♭}}_p,{\mathrm{♯}}_p$ of The inverse metric, musical isomorphisms and index raising to change slot variance. For example, if $T$ is a type $(1,1)$ field, lowering its contravariant slot gives the type $(0,2)$ field with components $T_{ab}:=g_{ac}{T}^c{}_b$; if $S$ is a type $(0,2)$ field, raising its first covariant slot gives the type $(1,1)$ field with components $S^a{}_b:=g^{ac}{S}_{cb}$. Each operation changes the tensor type and is invertible because $g_p$ is nondegenerate.

(d) Scope. The definition is purely local and chart-based; no global triviality of $T_s^{r}M$, no global frame and no choice of metric-compatible fields is assumed or implied. All fields are smooth ($C^{\mathrm{\infty }}$); no lower-regularity or distributional tensor field is defined here. The smooth structure of the tangent bundle, and hence of $T_s^{r}M$ (The tensor bundle on a smooth four-manifold from coordinate evaluations), inherits the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) recorded with Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; the component formulation of smoothness itself makes no choice.