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Mathematics for Physics

160 Constant Metric Tensor

A metric whose matrix entries do not depend on position that is used to compute distance when the geometry is the same at every point.

definition [d] (Constant Metric Tensor = Metric with Constant Components) A metric tensor \(g_{\mu\nu}\) whose components do not depend on the coordinates in a given chart:

  • \(\partial_{\lambda} g_{\mu\nu} = 0\) .

where

  • \(g_{\mu\nu}\) are the components of the metric tensor.
  • \(x^{\lambda}\) are the coordinates in the given chart.
  • \(\partial_{\lambda} = \partial / \partial x^{\lambda}\).

Note:

  • a standard flat example is the Euclidean metric \(\delta_{ij}\).
  • another standard flat example is the Minkowski metric \(\eta_{\mu\nu} = \operatorname{diag}(-1,1,1,1)\).
  • the opposite Minkowski signature is \(\eta_{\mu\nu} = \operatorname{diag}(1,-1,-1,-1)\).

definition [d] (Constant Metric Tensor = Metric with Constant Components) A metric with constant components in some chart,

  • \(\partial_{\lambda} g_{\mu\nu} = 0\) ,

so the Christoffel symbols built from \(g\) vanish:

  • \(\Gamma^{\rho}_{\ \mu\nu} = 0\) .

where

  • \(g_{\mu\nu}\) are the components of the metric tensor.
  • \(\partial_{\lambda} = \partial / \partial x^{\lambda}\).
  • \(\Gamma^{\rho}_{\ \mu\nu}\) are the Christoffel symbols of the metric.

Note:

  • vanishing \(\Gamma\) is the flat, inertial-coordinate case of Minkowski geometry.
  • the same holds for Euclidean geometry.
  • the prioritized sources treat this via constant flat metrics \(\delta\) and \(\eta\) rather than as a standalone named term.

160.1 Elementary Example

160.1.1 Simple

A constant metric has matrix entries that do not depend on position. Let \(g\) be the Euclidean metric on two axes.

\[ g = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

\[ g(e_{i},e_{j}) = \delta_{ij} \]

where

  • \(g\) is the constant metric tensor.
  • \(e_{1}, e_{2}\) are basis vectors.
  • \(\delta_{ij}\) equals \(1\) if \(i = j\) and equals \(0\) if \(i \neq j\).

160.1.2 General

In three dimensions the constant Euclidean metric is the \(3 \times 3\) identity matrix at every point.

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

\[ \partial_{\lambda} g_{ij} = 0 \]

where

  • \(g_{ij}\) are the constant components of \(g\).
  • \(I_{3}\) is the \(3 \times 3\) identity matrix.
  • \(\partial_{\lambda} g_{ij} = 0\) means the components do not depend on the coordinates.

160.2 References

  1. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — flat metrics with constant components; Minkowski and Euclidean cases.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — constant metrical matrix in orthogonal and Cartesian frames.
  3. Hassani, S. Mathematical Physics, 2nd ed. Springer. — metric tensor fields; flat limits.