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Mathematics for Physics

178 Line Element

An expression for an infinitesimal squared distance from the metric that is used to compute length of a path.

definition [d] (Line Element = Infinitesimal Arc Length Squared) The quadratic form built from the metric components and the coordinate differentials:

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

where

  • \(ds^{2}\) is the line element.
  • \(g_{ij}\) are the components of the metric tensor.
  • \(dx^{i}, dx^{j}\) are coordinate differentials.
  • \(ds\) is the infinitesimal arc length.

Note:

  • also written \((ds)^{2} = g_{ij}\, du^{i}\, du^{j}\).
  • \(ds^{2}\) encodes lengths of infinitesimal displacements via the metric.

definition [d] (Line Element = Spacetime Interval Squared) The quadratic form built from the spacetime metric and the coordinate differentials:

  • \(ds^{2} = g_{\mu\nu}\, dx^{\mu}\, dx^{\nu}\) .

where

  • \(ds^{2}\) is the line element.
  • \(g_{\mu\nu}\) are the components of the spacetime metric.
  • \(dx^{\mu}, dx^{\nu}\) are spacetime coordinate differentials.
  • \(ds\) is the infinitesimal spacetime interval.

Note:

  • in Minkowski spacetime with \(\eta_{\mu\nu} = \operatorname{diag}(-1,\, 1,\, 1,\, 1)\), one has \(ds^{2} = -c^{2}\, dt^{2} + dx^{2} + dy^{2} + dz^{2}\).
  • the overall sign of \(ds^{2}\) follows the metric signature convention.

178.1 Elementary Example

178.1.1 Simple

The line element \(ds^{2}\) is the squared length from the metric on coordinate increments.

\[ ds^{2} = dx^{2} + dy^{2} \]

\[ g = \operatorname{diag}(1,1) \]

where

  • \(ds^{2}\) is the line element.
  • \(dx, dy\) are coordinate increments.
  • \(g\) is the metric matrix used to form \(ds^{2}\).

178.1.2 General

In three Euclidean dimensions the line element uses three squared increments.

\[ ds^{2} = dx^{2} + dy^{2} + dz^{2} \]

\[ g = \operatorname{diag}(1,1,1) \]

\[ ds^{2} = \sum_{i,j=1}^{3} g_{ij}\, dx^{i}\, dx^{j} \]

where

  • \(g_{ij}\) are the metric components.
  • \(dx^{1}, dx^{2}, dx^{3}\) may be written \(dx, dy, dz\).

178.2 References

  1. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — line element from the metric; \(ds^{2}=g_{ij}\,dx^{i}\,dx^{j}\).
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — \((ds)^{2}=g_{ij}\,du^{i}\,du^{j}\).
  3. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — spacetime line element \(ds^{2}=g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\).