Framework And Geometry

Let $M$ be a smooth $n$-manifold (Smooth manifolds and their smooth charts), with smooth atlas $\mathcal{A}$ and maximal smooth atlas $[\mathcal{A}]$ (Smooth atlases, Smoothly compatible charts and the smoothness of Euclidean transition maps). 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 covariant $2$-tensor at $p$ is a bilinear map $A_p:T_{p}M\times T_{p}M\to \mathbb{R}$. It is symmetric when $A_p(v,w)=A_p(w,v)$ for all $v,w\in T_{p}M$, and nondegenerate when the only $v\in T_{p}M$ with $A_p(v,w)=0$ for all $w\in T_{p}M$ is $v=0$.

A smooth covariant $2$-tensor field on $M$ is a family $A=(A_p{)}_{p\in M}$ assigning to each $p$ a covariant $2$-tensor $A_p$ at $p$, such that for every chart $(U,x)$ of the maximal smooth atlas $[\mathcal{A}]$ the component functions

$A_{ab}(q):=A_q\text{\hspace{0.17em}⁣}(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^a}{\mid }_q,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^b}{\mid }_q),q\in U,$

are $C^{\mathrm{\infty }}$ on $U$. Here $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a{\mathrm{\mid }}_q$ is the coordinate derivation at $q$ (Derivations at a point and the tangent space, Coordinate derivations form a basis of the tangent space), the index ranges over the label set $I_n=\{1,\dots ,n\}$ (or the relabelled set $I_4=\{0,1,2,3\}$ when $n=4$), and smoothness of the real-valued functions $A_{ab}$ on the open set $U\subseteq M$ is the notion of $C^r$ and smooth maps between smooth manifolds applied to $M$ and $\mathbb{R}$.

For a coordinate tuple $z\in x(U)$ define the coordinate representative $A_{ab}^x(z):=A_{ab}(x^{-1}(z))$. When the argument is a coordinate tuple, $A_{ab}(z)$ abbreviates this representative in the named chart; when it is a point of $M$, it denotes the component function on the chart domain. Thus $A_{ab}(x(y^{-1}(u)))$ in a change-of-coordinates formula means $A_{ab}^x((x\circ y^{-1})(u))$, never evaluation of a function on $M$ at a tuple. For $n=4$, $x^a$ and ${\mathrm{\partial }}_a$ mean the original coordinate and derivation labelled $a+1$, respectively; this relabels every slot and sum. The label $0$ alone gives no coordinate vector a causal character.

Because the coordinate derivations at $q$ form a basis of $T_{q}M$ (Coordinate derivations form a basis of the tangent space), specifying $A$ is the same as specifying the matrix of component functions in each chart; Chart independence and the transformation law of a smooth covariant 2-tensor field ↗ proves that the smoothness requirement is independent of the chart used, so that the condition may be checked on any smooth atlas of $M$, and derives the transformation law relating components in two overlapping charts.

The field is symmetric when $A_p$ is symmetric for every $p\in M$, which in a chart is the identity $A_{ab}=A_{ba}$ of component functions; it is nondegenerate when $A_p$ is nondegenerate for every $p$, which in a chart is the nonvanishing at every point of the polynomial function $\det (A_{ab})$ of the components. The matrix of $A$ in a chart means the matrix $(A_{ab}(q))$ of its component functions at $q$.

Convention. In the four-dimensional case $n=4$ used in this library, indices are $a,b,c,d=0,1,2,3$ with $0$ the time-coordinate label when such a coordinate is chosen, and we write $A_{ab}$ for the components. A covariant $2$-tensor field is also called a $(0,2)$ tensor field.

Convention pointer. The abstract-index reading of the display $A_{ab}=A_q({\mathrm{\partial }}_a{\mathrm{\mid }}_q,{\mathrm{\partial }}_b{\mathrm{\mid }}_q)$ and the signature convention of this library are fixed for the Lorentzian case in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model on this page.

Let $M$ be a smooth $n$-manifold and let $A=(A_p{)}_{p\in M}$ be a family assigning to each $p$ a covariant $2$-tensor at $p$ (Smooth covariant 2-tensor fields on a smooth manifold). Suppose that the component functions of $A$ are $C^{\mathrm{\infty }}$ in the charts of one smooth atlas of $M$. Let $(U,x)$ and $(V,y)$ be overlapping smooth charts of the maximal smooth atlas and let $A_{cd}$ and $A_{ab}^{\mathrm{\prime }}$ denote their coordinate representatives in the charts $x$ and $y$ as specified in Smooth covariant 2-tensor fields on a smooth manifold. For $q\in U\cap V$ put $u=y(q)$ and write $x(u):=(x\circ y^{-1})(u)$. Then

$A_{ab}^{\mathrm{\prime }}(u)={\sum }_{c,d\in I_n}\frac{\mathrm{\partial }{x}^c}{\mathrm{\partial }{y}^a}(u)\frac{\mathrm{\partial }{x}^d}{\mathrm{\partial }{y}^b}(u)A_{cd}(x(u)),$

where $\mathrm{\partial }{x}^c\mathrm{/}\mathrm{\partial }{y}^a$ are the entries of the total derivative $D(x\circ y^{-1})(u)$; $I_n$ is the label set of the supplier, so in dimension four the sum runs from $0$ through $3$. In particular $A$ is a smooth covariant $2$-tensor field on $M$: its components are $C^{\mathrm{\infty }}$ in every chart of the maximal smooth atlas.

Let $M$ be a smooth four-manifold without boundary, so that $M$ is Hausdorff and second countable and carries a maximal smooth atlas (Smooth manifolds and their smooth charts). Let $g$ be a smooth covariant $2$-tensor field on $M$ in the sense of Smooth covariant 2-tensor fields on a smooth manifold. Then $g$ assigns to every $p\in M$ a bilinear form $g_p:T_{p}M\times T_{p}M\to \mathbb{R}$ (Derivations at a point and the tangent space).

(a) Lorentzian metric. The field $g$ is a Lorentzian metric of signature $(-,+,+,+)$ on $M$ when it is symmetric and nondegenerate, and at every $p\in M$ the pair $(T_{p}M,g_p)$ is a Lorentz vector space of signature $(-,+,+,+)$ in the sense of Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space: that is, $g_p$ has inertia $(3,1,0)$, so that some — equivalently every — basis of $T_{p}M$ in which the matrix of $g_p$ is diagonal has exactly one negative and three positive diagonal entries. The pair $(M,g)$ is a smooth Lorentzian four-manifold (or Lorentzian manifold of signature $(-,+,+,+)$). In a smooth chart $(U,x)$ the components of $g$ are the $C^{\mathrm{\infty }}$ functions $g_{ab}(q)=g_q({\mathrm{\partial }}_a{\mathrm{\mid }}_q,{\mathrm{\partial }}_b{\mathrm{\mid }}_q)$ of Smooth covariant 2-tensor fields on a smooth manifold, and $g_{ab}=g_{ba}$ by symmetry. We use indices $a,b,c,d=0,1,2,3$ and the Einstein summation convention, with components in a chart written as lower-case indexed symbols and equations between indexed fields understood basis-independently. These conventions are fixed once for this library and the physical reading of the symbols is recorded with the framework postulates, separately from this mathematical definition.

(b) Pointwise causal structure. For every $p\in M$ the vocabulary of Timelike, null, spacelike and causal vectors of a Lorentz vector space applies to the Lorentz vector space $(T_{p}M,g_p)$: a tangent vector $v\in T_{p}M$ is timelike when $g_p(v,v)<0$, null when $v\mathrm{\ne }0$ and $g_p(v,v)=0$, spacelike when $g_p(v,v)>0$, and causal when it is timelike or null. Every nonzero $v\in T_{p}M$ has exactly one causal character. The timelike set $T_p:=\{v\in T_{p}M:g_p(v,v)<0\}$ has exactly two connected components, each nonempty, open, convex and path-connected; each $g_p$-orthonormal basis $e_0,e_1,e_2,e_3$ of $T_{p}M$ satisfies $g_p(e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ for some ordering, so in particular nonzero orthonormal vectors exist at every point (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), The timelike set has exactly two convex components, described in orthonormal coordinates).

(c) Time orientation. A time orientation of $(M,g)$, also called a choice of future cone, is a smooth vector field $N$ on $M$ (A smooth vector field is a smooth section of the tangent bundle, whose tangent-bundle smooth structure uses 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) such that

$g_p(N_p,N_p)<0\text{for every }p\in M\mathrm{.}$

The function $p\mapsto g_p(N_p,N_p)$ is $C^{\mathrm{\infty }}$ because $g$ is a smooth covariant $2$-tensor field and $N$ a smooth field, so the condition is a pointwise strict inequality between a smooth function and $0$; since $g_p(N_p,N_p)<0$ implies $N_p\mathrm{\ne }0$, a time orientation is nowhere vanishing. A time-oriented smooth Lorentzian four-manifold is a triple $(M,g,N)$ consisting of a smooth Lorentzian four-manifold and a time orientation $N$.

Given a time orientation $N$ and $p\in M$, the vector $N_p$ is timelike, and we define a nonzero causal vector $v\in T_{p}M$ to be future-directed when $g_p(v,N_p)<0$ and past-directed when $g_p(v,N_p)>0$. The classification of nonzero causal vectors into exactly these two classes, and its agreement with the two components of the timelike set (the future component is the component $C_p^{+}$ containing $N_p$, and the future causal vectors form $\mathrm{cl}(C_p^{+})\setminus \{0\}$), is proved in Classification of future and past causal vectors by a timelike field ↗; in particular no nonzero causal $v$ satisfies $g_p(v,N_p)=0$, so the definition is exhaustive and mutually exclusive. Replacing $N$ by $-N$ reverses every future/past label, and two time orientations $N,N^{\mathrm{\prime }}$ determine the same future-directed causal vectors at every point where $g_p(N_p,N_p^{\mathrm{\prime }})<0$, while they determine opposite ones where $g_p(N_p,N_p^{\mathrm{\prime }})>0$, by Two time orientations agree or are opposite on each connected set ↗; that lemma also shows the sign of $p\mapsto g_p(N_p,N_p^{\mathrm{\prime }})$ is constant on every connected component of $M$.

(d) The inverse metric. Because $g_p$ is nondegenerate, it induces at each $p$ an isomorphism ${\mathrm{♭}}_p:T_{p}M\to T_p^{\ast }M$, $v\mapsto g_p(v,\cdot )$, and an inverse metric $g_p^{-1}$ on the dual space; the inverse metric, the musical isomorphisms and the components $g^{ab}$ are defined separately on this page, where the smoothness of those components is also proved. Neither construction uses any choice principle beyond the one already recorded in (c).

(e) Smooth timelike curves and their scalar normalization. Before using these quantities in the physical conventions, fix a positive real scaling parameter $c$ (with no physical meaning or dimension in this mathematical item). A smooth future timelike curve is a smooth map $\gamma :I\to M$ from a nonempty open real interval $I$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, $C^r$ and smooth maps between smooth manifolds) whose velocity derivation ${\gamma }^{\mathrm{\prime }}(s)\in T_{\gamma (s)}M$ (The velocity derivation of a smooth curve) satisfies $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})<0$ and $g({\gamma }^{\mathrm{\prime }},N)<0$ at every $s\in I$. In a chart its components are $d(x^a\circ \gamma )\mathrm{/}ds$, $a=0,1,2,3$. For a supplied base point $s_0\in I$ and additive constant ${\tau }_0\in \mathbb{R}$ define

$h(s):=\sqrt{-g_{\gamma (s)}({\gamma }^{\mathrm{\prime }}(s),{\gamma }^{\mathrm{\prime }}(s))},\tau (s):={\tau }_0+\frac{1}{c}{\int }_{s_0}^{s}h(r)dr,U(s):=\frac{c{\gamma }^{\mathrm{\prime }}(s)}{h(s)}\mathrm{.}$

The integral is oriented when $s<s_0$. The function $\tau$ is called proper time for this scaling and $U$ the four-velocity of the smooth curve; its increment times $c$ is the Lorentzian length. The well-definedness proof below shows ${\tau }^{\mathrm{\prime }}>0$, that $\tau :I\to \tau (I)$ is a smooth diffeomorphism, and that $U$ is the velocity of $\gamma \circ {\tau }^{-1}$. Thus $g(U,U)=-c^2$ and $U$ is future timelike. This introduces the smooth case before any physical unit reading; the later worldline definition extends it to its stated $C^k$, endpoint and piecewise-$C^1$ classes. It asserts no clock coupling, motion law or physical value of $c$.

Convention and scope. The physical reading of a smooth time-oriented Lorentzian four-manifold as a classical spacetime model, and the domain in which that reading is adopted, are recorded in the framework postulate post-gr-spacetime-and-classical-domain; the words spacetime and event carry no mathematical content beyond this definition, and this item asserts no claim about nature. The dimension $4$ and the absence of boundary are part of the data; nothing in the definition assumes that $M$ is nonempty or connected, and no completeness, extendibility or maximality property of $(M,g)$ is asserted.

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 $g_p$.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and let $g_p^{-1}$ and $g^{ab}$ be the inverse metric and its chart components as in The inverse metric, musical isomorphisms and index raising. Then:

1. For every smooth chart $(U,x)$ the component functions $g^{ab}:U\to \mathbb{R}$ are $C^{\mathrm{\infty }}$ on $U$.
2. For overlapping smooth charts $(U,x)$, $(V,y)$ the components satisfy the contravariant transformation law

$g^{\mathrm{\prime }ab}(u)={\sum }_{c,d=0}^3\frac{\mathrm{\partial }{y}^a}{\mathrm{\partial }{x}^c}(x(u))\frac{\mathrm{\partial }{y}^b}{\mathrm{\partial }{x}^d}(x(u))g^{cd}(x(u)),u\in y(U\cap V),$

where every component evaluated at a coordinate tuple is its coordinate representative, as in Smooth covariant 2-tensor fields on a smooth manifold, $x(u)$ denotes $(x\circ y^{-1})(u)$, and $\mathrm{\partial }{y}^a\mathrm{/}\mathrm{\partial }{x}^c$ are the entries of $D(y\circ x^{-1})$.

3. Consequently the pointwise bilinear forms $g_p^{-1}$ on $T_p^{\ast }M$ are presented by component functions that are smooth in every chart and glue by the contravariant law; the inverse metric is a well-defined smooth component family determined by $g$ alone.

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

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