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Mathematics for Physics

181 Metric Tensor

A mapping that assigns an inner product on the tangent space at each point that is used to compute distance and length on a curved domain.

definition [d] (Metric Tensor = Riemannian Metric = Metrical Matrix = First Fundamental Form) A symmetric, covariant second-rank tensor field \(G\) giving a bilinear map on tangent vectors,

  • \(G(v,w) = \langle v, w \rangle = \sum_{i,j} g_{ij}\, v^{i}\, w^{j}\) ,

with local components \(g_{ij}(x) = \left\langle \partial/\partial x^{i},\, \partial/\partial x^{j} \right\rangle\).

where

  • \(G\) is the metric tensor field.
  • \(v, w\) are tangent vectors.
  • \(g_{ij}\) are the components of \(G\) in local coordinates.
  • \(v^{i}, w^{j}\) are the components of \(v\) and \(w\).
  • \(\langle \cdot,\, \cdot \rangle\) is the metric bilinear form.
  • \(x\) are local coordinates.
  • \(\partial / \partial x^{i}\) are the coordinate basis vectors.

Note:

  • \(G\) is symmetric: \(g_{ij} = g_{ji}\).
  • the components are differentiable functions of the coordinates.
  • Riemannian if positive-definite; indefinite Lorentzian metrics are allowed in relativity.

definition [d] (Metric Tensor = Riemannian Metric = Metrical Matrix = First Fundamental Form) A symmetric second-order tensor \(g_{ij}\) that fixes the square of infinitesimal arc length,

  • \((ds)^{2} = g_{ij}\, du^{i}\, du^{j}\) ,

and raises and lowers indices via its inverse \(g^{ik}\):

  • \(V_{i} = g_{ij}\, V^{j}\) ,
  • \(V^{i} = g^{ij}\, V_{j}\) .

where

  • \(g_{ij}\) are the covariant metric components.
  • \(g^{ik}\) are the contravariant inverse-metric components.
  • \(du^{i}\) are coordinate differentials.
  • \(ds\) is the infinitesimal arc length.
  • \(V^{i}\) are contravariant components of a vector.
  • \(V_{i}\) are covariant components of a vector.
  • \(\delta^{i}_{\ j}\) is the Kronecker delta.

Note:

  • in spacetime, \(ds\) is the infinitesimal spacetime interval.
  • \(g^{ik} g_{kj} = \delta^{i}_{\ j}\).
  • in flat Minkowski spacetime, \(g_{\mu\nu} = \eta_{\mu\nu}\).

181.1 Elementary Example

181.1.1 Simple

A metric \(g\) pairs two vectors to a scalar. On two basis vectors the values form a \(2 \times 2\) matrix.

\[ g : V \times V \rightarrow \mathbb{R} \]

\[ V = \{ e_{1},\ e_{2} \} \]

\[ (g_{ij}) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

where

  • \(g\) is the metric tensor.
  • \(g_{ij} = g(e_{i},e_{j})\) are its components.

181.1.2 General

On \(\mathbb{R}^{3}\) the Euclidean metric is the \(3 \times 3\) identity, and \(g(u,v)\) is the ordinary dot product.

\[ (g_{ij}) = I_{3} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

\[ g(u,v) = \sum_{i,j=1}^{3} g_{ij}\, u^{i}\, v^{j} = u \cdot v \]

where

  • \(I_{3}\) is the \(3 \times 3\) identity matrix.
  • \(u^{i}, v^{j}\) are components of \(u, v\) in \(\mathbb{R}^{3}\).

181.2 References

  1. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — metric as symmetric covariant \(2\)-tensor; \(g_{ij}=\langle\partial_{i},\partial_{j}\rangle\); first fundamental form.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — \((ds)^{2}=g_{ij}\,du^{i}\,du^{j}\); metrical matrix.
  3. Hassani, S. Mathematical Physics, 2nd ed. Springer. — type-\((0,2)\) tensor fields; nondegenerate metrics.
  4. Reed, M., & Simon, B. Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press. — bilinear / sesquilinear forms in the functional-analytic setting.