Metric compatibility, torsion freedom and the metric-selected connection class

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 $\mathrm{\nabla }$ be an affine connection on $M$ (Affine connection on a smooth manifold). Smooth vector fields are smooth sections of the tangent bundle (A smooth vector field is a smooth section of the tangent bundle); as there, their use rests on the canonical smooth structure of $TM$, which assumes the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). This is the only use of that axiom in this definition.

(a) Metric compatibility. The connection $\mathrm{\nabla }$ is $g$-compatible, or compatible with the metric $g$, when for all smooth vector fields $X,Y,Z$ on $M$ the identity of smooth functions

$X(g(Y,Z))=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z)$

holds on $M$. Here $g(Y,Z)$ is the smooth function $p\mapsto g_p(Y_p,Z_p)$ and $X$ acts on functions as a derivation; the identity uses only the action of $\mathrm{\nabla }$ on vector fields and requires no extension of $\mathrm{\nabla }$ to tensor fields. When such an extension is available, this identity is the statement ${\mathrm{\nabla }}_{X}g=0$, and the compatibility of $\mathrm{\nabla }$ with the inverse metric $g^{-1}$ follows; neither statement is part of this definition.

(b) Torsion freedom. The torsion of $\mathrm{\nabla }$ is the tangent-valued $2$-tensor $T(X,Y)={\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y]$ of Torsion tensor of an affine connection, with bracket as in The Lie bracket of smooth vector fields. The connection is torsion-free when $T(X,Y)=0$ for all smooth $X,Y$, that is, when ${\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]$ identically.

(c) Metric-selected connections. A metric-selected connection for $(M,g)$ is an affine connection on $M$ that is both torsion-free and $g$-compatible. Equivalently, it is an element of the class

$\mathrm{LC}(M,g):=\{\mathrm{\nabla }:\mathrm{\nabla }\text{ is an affine connection on }M,\mathrm{\nabla }\text{ is torsion-free and }g\text{-compatible}\}\mathrm{.}$

The name records that the pair of conditions selects the connection out of all affine connections; it is the pair of conditions imposed on the connection that conventional general relativity uses, and the connection so selected is called the Levi–Civita connection of $g$.

(d) Intrinsic construction before physical use. For smooth local vector fields define the Koszul expression

$K(X,Y,Z):=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)+g([X,Y],Z)+g([Z,X],Y)-g([Y,Z],X)\mathrm{.}$

Define ${\mathrm{\nabla }}_X^{g}Y$ by $2g({\mathrm{\nabla }}_X^{g}Y,Z)=K(X,Y,Z)$ for every smooth local $Z$. The proof below establishes existence, smoothness, the affine-connection rules, metric compatibility, torsion freedom and uniqueness; it uses symmetry and nondegeneracy, not positive definiteness. Thus $\mathrm{LC}(M,g)$ has this unique element, called the Levi–Civita connection of $g$. The later nondegenerate-metric theorem retains its fuller tensor-derivative statement. Its Christoffel symbols in any chart are defined by ${\mathrm{\nabla }}_{{\mathrm{\partial }}_b}^g{\mathrm{\partial }}_c={\mathrm{\Gamma }}^a{}_{bc}{\mathrm{\partial }}_a$ (Christoffel symbols of an affine connection) and obey

${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{bd}-{\mathrm{\partial }}_d{g}_{bc}),a,b,c,d=0,1,2,3.$

(e) Basic curvature and contractions. The curvature of this connection is the operator of Curvature of an affine connection, with the fixed sign convention

$R^g(X,Y)Z:={\mathrm{\nabla }}_X^g{\mathrm{\nabla }}_Y^{g}Z-{\mathrm{\nabla }}_Y^g{\mathrm{\nabla }}_X^{g}Z-{\mathrm{\nabla }}_{[X,Y]}^{g}Z\mathrm{.}$

The general affine-connection theorem Curvature is a type (1,3) tensor proves that it depends pointwise only on $X,Y,Z$ and is a smooth tensor. Define its chart components by $R^g({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$. Define the Ricci tensor and scalar curvature pointwise by

${\mathrm{Ric}}_p(v,w):=\mathrm{tr}(z\mapsto R_p^g(z,w)v),S(p):=\mathrm{tr}({\mathrm{♯}}_p\circ {\mathrm{Ric}}_p^{\mathrm{♭}}),{\mathrm{Ric}}_p^{\mathrm{♭}}(v):={\mathrm{Ric}}_p(v,\cdot )\mathrm{.}$

Here $\mathrm{♯}$ is the inverse metric isomorphism and traces are the basis-independent trace of The basis-independent trace of an endomorphism of a finite-dimensional vector space. Equivalently, ${\mathrm{Ric}}_{bd}={\sum }_a{R}^a{}_{bad}$ and $S={\sum }_{b,d}{g}^{bd}{\mathrm{Ric}}_{bd}$, all sums from $0$ through $3$. These are well-defined smooth fields by finite contraction of the smooth components of $R^g$ and $g^{-1}$; basis independence follows from the trace and the intrinsic maps just defined. No symmetry identity or field equation is assumed here. Lowering the curvature output gives $R_{abcd}=g_{ae}{R}^e{}_{bcd}$, so the definitions agree with the full later curvature item, which proves the remaining symmetries and coordinate formulas.

• Construction and richer continuation. Clauses (a)-(c) define the two selecting conditions; the proof below independently constructs their unique common solution. The later nondegenerate-metric theorem proves the full extension to tensor derivatives, contraction compatibility and preservation of both metric pairings. No conclusion is assumed from the name of the class. The companion component lemma proves the chartwise equivalences.

• The two conditions constrain different data. In a chart, torsion freedom is the symmetry ${\mathrm{\Gamma }}^a{}_{bc}={\mathrm{\Gamma }}^a{}_{cb}$ of the connection coefficients, a condition on the connection alone, while $g$-compatibility is the differential identity ${\mathrm{\partial }}_a{g}_{bc}={\mathrm{\Gamma }}^d{}_{ab}{g}_{dc}+{\mathrm{\Gamma }}^d{}_{ac}{g}_{bd}$ relating the coefficients to the metric; both equivalences are proved in a companion lemma on this page. The framework adopts a metric-selected connection as part of its geometric structure; that adoption is a physical postulate of the classical model, recorded with the framework postulates, and the proof below establishes that the class has exactly one element. Its later extension theorem retains the richer tensor-derivative claims.