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pasmith

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Riemannian manifolds have a concept of a metric tensor [itex]g_{ij}[/itex], so that the element of arclength [itex]ds[/itex] is given by [tex]

ds^2 = g_{ij}dx^i dx^j[/tex] in terms of generalized coordinates [itex]x_1, \dots, x_n[/itex], and the distance between two points is then the infimum of [itex]\int_C \frac{ds}{dt}\,dt = \int_C \sqrt{g_{ij}\dot x^i \dot x^j}\,dt[/itex] over all continuous curves [itex]C[/itex] between those two points.

Thus in spherical polar coordinates with [itex]x^1 = r[/itex], [itex]x^2 = \theta[/itex] and [itex]x^3 = \phi[/itex] the euclidean metric is given by [tex]

\begin{array}{ccc}

g_{11} = 1 & g_{12} = 0 & g_{13} = 0 \\

g_{21} = 0 & g_{22} = r^2 & g_{23} = 0 \\

g_{31} = 0 & g_{32} = 0 & g_{33} = r^2 \sin^2 \theta

\end{array}

[/tex] so that the arclength element is given by [tex]

ds^2 = dr^2 + r^2 \sin^2 \theta d\phi^2 + r^2 d\theta^2.

[/tex] Introductory texts on general relativity (such as Foster & Nightingale) should discuss this for the case of the (pseudo-)Riemannian geometry of space-time.

ds^2 = g_{ij}dx^i dx^j[/tex] in terms of generalized coordinates [itex]x_1, \dots, x_n[/itex], and the distance between two points is then the infimum of [itex]\int_C \frac{ds}{dt}\,dt = \int_C \sqrt{g_{ij}\dot x^i \dot x^j}\,dt[/itex] over all continuous curves [itex]C[/itex] between those two points.

Thus in spherical polar coordinates with [itex]x^1 = r[/itex], [itex]x^2 = \theta[/itex] and [itex]x^3 = \phi[/itex] the euclidean metric is given by [tex]

\begin{array}{ccc}

g_{11} = 1 & g_{12} = 0 & g_{13} = 0 \\

g_{21} = 0 & g_{22} = r^2 & g_{23} = 0 \\

g_{31} = 0 & g_{32} = 0 & g_{33} = r^2 \sin^2 \theta

\end{array}

[/tex] so that the arclength element is given by [tex]

ds^2 = dr^2 + r^2 \sin^2 \theta d\phi^2 + r^2 d\theta^2.

[/tex] Introductory texts on general relativity (such as Foster & Nightingale) should discuss this for the case of the (pseudo-)Riemannian geometry of space-time.

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