Metric tensor

The metric tensor is a fundamental geometric object in differential geometry and the mathematical formulation of physical theories such as general relativity. It provides a way to measure lengths of curves, angles between vectors, and volumes on smooth manifolds by defining an inner product on the tangent space at each point.

Definition

Let $M$ be a smooth $n$-dimensional manifold. A metric tensor $g$ on $M$ is a smooth, symmetric, non‑degenerate (0,2)-tensor field, $$ g \in \Gamma\bigl(T^{}M\otimes T^{}M\bigr), $$ such that for each point $p\in M$ the bilinear form $$ g_{p} : T_{p}M \times T_{p}M \rightarrow \mathbb{R} $$ is an inner product on the tangent space $T_{p}M$. The signature of $g$ (the numbers of positive and negative eigenvalues) classifies the type of geometry; for example, a Riemannian metric has signature $(+,\dots,+)$, while a Lorentzian metric, used in relativity, has signature $(- ,+,\dots,+)$.

Local Coordinate Expression

In a local coordinate chart $(x^{1},\dots,x^{n})$, the metric tensor can be written as $$ g = g_{ij}(x),dx^{i}\otimes dx^{j}, $$ where the components $g_{ij}=g_{ji}$ form a symmetric matrix varying smoothly with the coordinates. The line element, representing the infinitesimal squared distance between neighboring points, is $$ ds^{2}=g_{ij},dx^{i}dx^{j}. $$

Fundamental Properties

Symmetry: $g_{ij}=g_{ji}$.
Non‑degeneracy: The matrix $(g_{ij})$ is invertible at every point; its inverse is denoted $g^{ij}$.
Compatibility with Levi‑Civita connection: The unique torsion‑free affine connection $ abla$ satisfying $ abla g =0$ is called the Levi‑Civita connection. This connection defines covariant differentiation of tensors consistent with the metric.
Raising and lowering indices: Using $g_{ij}$ and its inverse $g^{ij}$, one can convert between covariant and contravariant components of tensors (e.g., $v^{i}=g^{ij}v_{j}$ and $v_{i}=g_{ij}v^{j}$).

Geometric Quantities Derived from the Metric

Length of a curve: For a smooth curve $\gamma:[a,b]\to M$, $$ L(\gamma)=\int_{a}^{b}\sqrt{g_{\gamma(t)}\bigl(\dot\gamma(t),\dot\gamma(t)\bigr)},dt. $$
Angle between vectors: For $u,v\in T_{p}M$, $$ \cos\theta=\frac{g_{p}(u,v)}{\sqrt{g_{p}(u,u),g_{p}(v,v)}}. $$
Volume element: The metric induces a natural volume form $\mathrm{d}\mu_g = \sqrt{|\det(g_{ij})|},dx^{1}\wedge\cdots\wedge dx^{n}$.

Applications

Riemannian Geometry: The study of manifolds endowed with a positive‑definite metric tensor, leading to concepts such as geodesics, curvature, and theorems of global analysis (e.g., the Hopf–Rinow theorem).
General Relativity: Spacetime is modeled as a four‑dimensional Lorentzian manifold $(M,g)$. The Einstein field equations relate the metric tensor to the stress‑energy tensor, governing gravitational dynamics.
Gauge Theories and Field Theory: Metric tensors appear in constructing invariant actions, defining Hodge duals, and coupling fields to curved backgrounds.
Continuum Mechanics: Strain and stress tensors are often expressed relative to an underlying material metric.

Historical Notes

The concept of a metric tensor originated in the 19th‑century work of Bernhard Riemann, who generalized the notion of distance in his seminal lecture “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (1854). Subsequent formalization was provided by Gregorio Ricci‑Curbastro and Tullio Levi‑Civita in the development of tensor calculus. In the early 20th century, Albert Einstein adopted the Lorentzian metric as the central geometric object of his theory of general relativity.

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