The inverse metric, musical isomorphisms and index raising

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and let $p\in M$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). The cotangent space $T_p^{\ast }M$ is the algebraic dual of $T_{p}M$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$).

Musical isomorphisms. Define the flat map

${\mathrm{♭}}_p:T_{p}M⟶T_p^{\ast }M,{\mathrm{♭}}_p(v):=g_p(v,\cdot ),$

so that ${\mathrm{♭}}_p(v)(w)=g_p(v,w)$; it is $\mathbb{R}$-linear by bilinearity of $g_p$. Its kernel is $\{v:g_p(v,w)=0\text{ for all }w\}=\{0\}$ because $g_p$ is nondegenerate, and ${\dim }_{\mathbb{R}}{T}_{p}M={\dim }_{\mathbb{R}}{T}_p^{\ast }M=4$ since the coordinate derivations of a chart form a basis of $T_{p}M$ (Coordinate derivations form a basis of the tangent space); hence ${\mathrm{♭}}_p$ is a linear isomorphism and has an inverse ${\mathrm{♯}}_p:T_p^{\ast }M\to T_{p}M$.

Inverse metric. The inverse metric at $p$ is the symmetric bilinear form

$g_p^{-1}:T_p^{\ast }M\times T_p^{\ast }M⟶\mathbb{R},g_p^{-1}(\alpha ,\beta ):=g_p({\mathrm{♯}}_p\alpha ,{\mathrm{♯}}_p\beta )\mathrm{.}$

It is symmetric because $g_p$ is, and it is nondegenerate: if $g_p^{-1}(\alpha ,\beta )=0$ for all $\beta$, then $g_p({\mathrm{♯}}_p\alpha ,\cdot )=0$, so ${\mathrm{♯}}_p\alpha =0$ and hence $\alpha =0$.

Components in a chart. Let $(U,x)$ be a smooth chart and $g_{ab}(q)=g_q({\mathrm{\partial }}_a{\mathrm{\mid }}_q,{\mathrm{\partial }}_b{\mathrm{\mid }}_q)$ the components of $g$; let $d{x}^a{\mathrm{\mid }}_q$ be the differential of the coordinate function $x^a$ at $q$ (The differential of a smooth map), so that $d{x}^a{\mathrm{\mid }}_q({\mathrm{\partial }}_b{\mathrm{\mid }}_q)={\mathrm{\partial }}_b{\mathrm{\mid }}_q(x^a)={\delta }^a{}_b$. Define the components of the inverse metric in this chart by

$g^{ab}(q):=g_q^{-1}(d{x}^a{\mathrm{\mid }}_q,d{x}^b{\mathrm{\mid }}_q),q\in U,$

so that ${\mathrm{♯}}_q(d{x}^a{\mathrm{\mid }}_q)={\sum }_b{g}^{ab}(q){\mathrm{\partial }}_b{\mathrm{\mid }}_q$; equivalently $(g^{ab}(q))$ is the inverse matrix of $(g_{ab}(q))$ at every $q$, that is

$g^{ac}(q)g_{cb}(q)={\delta }^a{}_b,g_{ac}(q)g^{cb}(q)={\delta }_a{}^b,$

because $g_q({\mathrm{♯}}_{q}d{x}^a{\mathrm{\mid }}_q,{\mathrm{\partial }}_c{\mathrm{\mid }}_q)=d{x}^a{\mathrm{\mid }}_q({\mathrm{\partial }}_c{\mathrm{\mid }}_q)={\delta }^a{}_c$ and also $g_q({\mathrm{♯}}_{q}d{x}^a{\mathrm{\mid }}_q,{\mathrm{\partial }}_c{\mathrm{\mid }}_q)={\sum }_b{g}^{ab}(q)g_{bc}(q)$. In particular $g^{ab}=g^{ba}$. The components $g^{ab}$ depend smoothly on $q$; this is proved in Smoothness and transformation of the inverse-metric components ↗.

Raising, lowering and metric trace. Let $A$ be a smooth symmetric covariant $2$-tensor field on an open set $V\subseteq M$ (Smooth covariant 2-tensor fields on a smooth manifold). At each $p\in V$ define the endomorphism

$A_p^{\mathrm{♯}}:T_{p}M\to T_{p}M,A_p^{\mathrm{♯}}(v):={\mathrm{♯}}_p(A_p(v,\cdot )),$

characterized by $g_p(A_p^{\mathrm{♯}}(v),w)=A_p(v,w)$ for all $w\in T_{p}M$; it is self-adjoint for $g_p$ because $A_p$ is symmetric. Its metric trace is

${\mathrm{tr}}_g(A)(p):=\mathrm{tr}\left(A_p^{\mathrm{♯}}\right)$

with the trace of an endomorphism as in The basis-independent trace of an endomorphism of a finite-dimensional vector space. In a chart the raised, mixed and doubly raised components are

$A^{ab}:=g^{ac}{g}^{bd}{A}_{cd},A^a{}_b:=g^{ac}{A}_{cb},\text{and}{\mathrm{tr}}_g(A)=g^{ab}{A}_{ab},$

the last identity obtained by evaluating in a $g_p$-orthonormal basis $(e_0,e_1,e_2,e_3)$, where $g_p(e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ by The timelike set has exactly two convex components, described in orthonormal coordinates so that $g^{ab}{A}_{ab}={\sum }_a{g}^{aa}{A}_{aa}=-A(e_0,e_0)+A(e_1,e_1)+A(e_2,e_2)+A(e_3,e_3)$, which is the trace of the endomorphism $v\mapsto {\mathrm{♯}}_p(A_p(v,\cdot ))$ in that basis. All these operations are pointwise and depend only on $g$, $A$ and the duality pairing, not on a chart.

Conventions. Raising and lowering apply to one index at a time and preserve index order; $A_{ab}$, $A^a{}_b$ and $A^{ab}$ denote the same tensor $A$ with indices positioned by $g$. The musical isomorphisms and the inverse metric are defined for every smooth Lorentzian four-manifold of this library; no choice principle is used in this definition, every ingredient being pointwise linear algebra over $\mathbb{R}$ together with the pointwise nondegeneracy of $g_p$.