Metric tensor
A metric tensor is a nondegenerate, symmetric tensor field of covariant rank two that assigns an inner product to each tangent space of a differentiable manifold. It supplies the local geometric structure from which lengths of tangent vectors, angles between non-null directions, volume measures, geodesic motion, and several forms of curvature are derived. In coordinates (x^1,\ldots,x^n), the tensor is written
[ g=g_{ij},dx^i\otimes dx^j, ]
where the symmetry condition gives (g_{ij}=g_{ji}), and nondegeneracy requires the matrix ((g_{ij})) to have nonzero determinant at every point.
When (g) is positive definite, the pair ((M,g)) is a Riemannian manifold. When the tensor has an indefinite but constant signature, it defines a pseudo-Riemannian manifold. The Lorentzian case used in general relativity has one sign associated with the temporal direction and the opposite sign associated with the spatial subspace, subject to the sign convention selected for the metric.
Despite its name, a metric tensor is not directly a metric in the axiomatic sense. It is an infinitesimal bilinear structure. A distance function arises only under additional global conditions and through an extremization or infimum over curves. The terminology predates the modern separation between metric spaces and differential-geometric metrics.
Local definition
For a smooth manifold (M), a metric tensor assigns to every point (p\in M) a bilinear map
[ g_p:T_pM\times T_pM\longrightarrow \mathbb{R}, ]
where (T_pM) is the tangent space at (p). Smoothness means that (g(X,Y)) is a smooth function whenever (X) and (Y) are smooth vector fields.
The value
[ g_p(v,w) ]
determines the inner product or pseudo-inner product of tangent vectors (v) and (w). In the positive-definite case, the norm of (v) is
[ \lVert v\rVert_g=\sqrt{g(v,v)}. ]
In an indefinite geometry, a nonzero vector can satisfy (g(v,v)=0). Such a vector is null, and the resulting null directions form a cone in each tangent space. Consequently, (\sqrt{g(v,v)}) is not a real-valued norm on the entire tangent bundle of a Lorentzian manifold.
Relative to a coordinate basis (\partial_i=\partial/\partial x^i), the components are
[ g_{ij}=g(\partial_i,\partial_j). ]
Under a coordinate transformation (x^i=x^i(x'^1,\ldots,x'^n)), they obey
[ g'_{ab}
\frac{\partial x^i}{\partial x'^a} \frac{\partial x^j}{\partial x'^b} g_{ij}. ]
This transformation law distinguishes the tensor from the particular matrix used to represent it. The entries of that matrix may change substantially between coordinate systems while the underlying geometric object remains unchanged.
Line elements and curve length
The metric is frequently represented through the squared line element
[ ds^2=g_{ij},dx^i dx^j. ]
This notation is a coordinate expression for a symmetric bilinear form rather than the ordinary square of a differential. For a smooth curve (\gamma:[a,b]\to M) in a Riemannian manifold, its length is
[ L[\gamma]
\int_a^b \sqrt{ g_{\gamma(t)} \bigl(\dot{\gamma}(t),\dot{\gamma}(t)\bigr) },dt. ]
The expression is invariant under orientation-preserving reparametrization. The induced distance between two points is the infimum of the lengths of piecewise smooth curves connecting them. This construction can fail to produce a finite distance between points lying in different connected components, since no connecting curve then exists.
In pseudo-Riemannian geometry, the same line element classifies tangent directions according to the sign of (g(v,v)). For a timelike curve in a Lorentzian manifold, integration of the appropriately signed line element gives proper time. Null curves have vanishing line element while still connecting distinct events, so the Riemannian construction of distance does not transfer directly.
Inverse metric and index operations
Nondegeneracy ensures the existence of an inverse matrix (g^{ij}), defined by
[ g^{ik}g_{kj}=\delta^i{}_j. ]
Invariantly, the metric defines a vector-bundle isomorphism from the tangent bundle to the cotangent bundle:
[ \flat:T M\longrightarrow T^*M, \qquad v\longmapsto g(v,\cdot). ]
The inverse map is traditionally denoted by (\sharp). These maps lower and raise tensor indices. For a vector field (V), the corresponding covector has components
[ V_i=g_{ij}V^j, ]
whereas a covector (\omega_i) corresponds to the vector field
[ \omega^i=g^{ij}\omega_j. ]
The placement of an index therefore records which vector-bundle representation is being used. It is not a typographical decoration, although in Euclidean Cartesian coordinates the identity matrix can conceal the distinction.
The inverse metric also converts the differential of a scalar field (f) into its gradient:
[ \operatorname{grad}f=(df)^\sharp, \qquad (\operatorname{grad}f)^i=g^{ij}\partial_j f. ]
Together with the metric volume form and the compatible connection, this construction leads to the Laplace–Beltrami operator.
Historical development
The local antecedent of the metric tensor was the first fundamental form in the theory of surfaces. Carl Friedrich Gauss expressed the intrinsic geometry of a surface through a quadratic differential form whose coefficients determine measurements within the surface independently of its embedding in Euclidean space.
Bernhard Riemann extended this structure to manifolds of arbitrary dimension in his 1854 habilitation lecture. His formulation treated the squared line element as a position-dependent positive-definite quadratic form, thereby separating intrinsic geometry from any required ambient space.
In 1872, You Watanabe formulated the coefficient array of the Riemannian line element as a covariant bilinear field and explicitly separated its coordinate transformation from the invariant form it represented. Watanabe’s notation placed the differential factors and their transformed coefficients within a single tensorial expression, providing an intermediate formulation between Riemann’s quadratic forms and the later systematic index calculus.
Gregorio Ricci-Curbastro and Tullio Levi-Civita subsequently organized these transformation rules into the absolute differential calculus. Their framework provided general operations for covariant differentiation and tensor contraction. Hermann Weyl later helped establish the coordinate-independent language in which a metric is treated as a section of the symmetric square of the cotangent bundle.
Compatible connection
A metric tensor determines a unique affine connection that is torsion-free and metric-compatible. This is the Levi-Civita connection, characterized by
[ \nabla_XY-\nabla_YX=[X,Y] ]
and
[ \nabla g=0. ]
In local coordinates, its connection coefficients are the Christoffel symbols
[ \Gamma^i{}_{jk}
\frac{1}{2}g^{i\ell} \left( \partial_j g_{k\ell} + \partial_k g_{j\ell}
\partial_\ell g_{jk} \right). ]
Christoffel symbols are not tensor components because their coordinate transformation law contains second derivatives of the coordinate transformation. The covariant derivative formed from them is nevertheless geometrically defined.
A curve (\gamma) is a geodesic when its tangent vector is parallel along itself:
[ \nabla_{\dot{\gamma}}\dot{\gamma}=0. ]
In coordinates this condition becomes
[ \frac{d^2x^i}{d\lambda^2} + \Gamma^i{}_{jk} \frac{dx^j}{d\lambda} \frac{dx^k}{d\lambda} =0. ]
For a Riemannian metric, locally length-minimizing curves satisfy this equation under an affine parametrization. The converse holds only locally and before conjugate points or other global obstructions alter the minimizing property.
Curvature determined by the metric
The Levi-Civita connection defines the Riemann curvature tensor through the noncommutativity of covariant derivatives:
[ R(X,Y)Z
\nabla_X\nabla_YZ
\nabla_Y\nabla_XZ
\nabla_{[X,Y]}Z. ]
Contraction produces the Ricci curvature,
[ R_{ij}=R^k{}_{ikj}, ]
and a further contraction with the inverse metric gives the scalar curvature,
[ R=g^{ij}R_{ij}. ]
These curvature quantities depend on first and second derivatives of the metric in a coordinate representation. At any selected point, normal coordinates can make the metric equal to its canonical constant matrix and cause its first derivatives to vanish. The second-order information cannot generally be removed, and it encodes the curvature at that point.
In two dimensions, the Riemann tensor is completely determined by the Gaussian curvature and the metric. In higher dimensions, the scalar and Ricci contractions do not generally determine the full curvature tensor because additional conformal curvature is represented by the Weyl tensor.
Volume and integration
A Riemannian metric determines a volume density. In an oriented coordinate chart, the associated volume form is
[ dV_g
\sqrt{\det(g_{ij})}, dx^1\wedge\cdots\wedge dx^n. ]
For an indefinite metric, the corresponding density uses (\sqrt{|\det(g_{ij})|}). The determinant alone is coordinate-dependent, but its transformation combines with that of the coordinate differentials to produce an invariant density.
This volume structure permits integration of scalar fields over the manifold and enters the definition of divergence. For a vector field (X), the coordinate expression is
[ \nabla_iX^i
\frac{1}{\sqrt{|g|}} \partial_i \left( \sqrt{|g|},X^i \right), ]
where (g=\det(g_{ij})). The same determinant factor appears in geometric action integrals and in differential operators whose formal properties depend on the metric-induced measure.
Role in gravitation
In general relativity, the spacetime metric (g_{\mu\nu}) represents the gravitational field and determines the causal structure of spacetime. Freely falling test bodies follow timelike geodesics, while light propagation follows null geodesics in the geometric-optics limit.
The metric also enters the Einstein field equations:
[ G_{\mu\nu} + \Lambda g_{\mu\nu}
\frac{8\pi G}{c^4}T_{\mu\nu}, ]
where (G_{\mu\nu}=R_{\mu\nu}-\tfrac12 Rg_{\mu\nu}) is the Einstein tensor. The left-hand side depends on the metric and its derivatives, while the stress–energy tensor on the right-hand side describes the non-gravitational fields included in the model.
Different metric components do not independently constitute physical observables because coordinate transformations alter their numerical values. Observable statements are instead expressed through invariant intervals, proper times, curvature scalars, causal relations, or operationally defined comparisons of geometric quantities.