Definition of the line element and arclength

The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement ${\displaystyle d\mathbf {q} }$^[2] (in pseudo Riemannian manifolds possibly negative [Clarification needed. what is possibly negative, the line element ds or the square of the line element?]) whose square root should be used for computing curve length:

$${\displaystyle ds^{2}=d\mathbf {q} \cdot d\mathbf {q} =g(d\mathbf {q} ,d\mathbf {q} )}$$

where g is the metric tensor, · denotes inner product, and dq an infinitesimal displacement on the (pseudo) Riemannian manifold. By parametrizing a curve ${\displaystyle q(\lambda )}$, we can define the arc length of the curve length of the curve between ${\displaystyle q(\lambda _{1})}$, and ${\displaystyle q(\lambda _{2})}$ as the integral:^[3]

$${\displaystyle s=\int _{\lambda _{1}}^{\lambda _{2}}d\lambda {\sqrt {\left|ds^{2}\right|}}=\int _{\lambda _{1}}^{\lambda _{2}}d\lambda {\sqrt {\left|g\left({\frac {dq}{d\lambda }},{\frac {dq}{d\lambda }}\right)\right|}}=\int _{\lambda _{1}}^{\lambda _{2}}d\lambda {\sqrt {\left|g_{ij}{\frac {dq^{i}}{d\lambda }}{\frac {dq^{j}}{d\lambda }}\right|}}}$$

To compute a sensible length of curves in pseudo Riemannian manifolds, it is best to assume that the infinitesimal displacements have the same sign everywhere. E.g. in physics the square of a line element along a timeline curve would (in the ${\displaystyle -+++}$ signature convention) be negative and the negative square root of the square of the line element along the curve would measure the proper time passing for an observer moving along the curve. From this point of view, the metric also defines in addition to line element the surface and volume elements etc.

Identification of the square of the line element with the metric tensor

Since ${\displaystyle d\mathbf {q} }$ is an arbitrary "square of the arc length", ${\displaystyle ds^{2}}$ completely defines the metric, and it is therefore usually best to consider the expression for ${\displaystyle ds^{2}}$ as a definition of the metric tensor itself, written in a suggestive but non tensorial notation:

$${\displaystyle ds^{2}=g}$$

This identification of the square of arc length ${\displaystyle ds^{2}}$ with the metric is even more easy to see in n-dimensional general curvilinear coordinates q = (q¹, q², q³, ..., qⁿ), where it is written as a symmetric rank 2 tensor^[3][4] coinciding with the metric tensor:

$${\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=g.}$$

Here the indices i and j take values 1, 2, 3, ..., n and Einstein summation convention is used. Common examples of (pseudo) Riemannian spaces include three-dimensional space (no inclusion of time coordinates), and indeed four-dimensional spacetime.

Cartesian coordinates

The simplest line element is in Cartesian coordinates - in which case the metric is just the Kronecker delta:

$${\displaystyle g_{ij}=\delta _{ij}}$$

(here i, j = 1, 2, 3 for space) or in matrix form (i denotes row, j denotes column):

$${\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$$

The general curvilinear coordinates reduce to Cartesian coordinates:

$${\displaystyle (q^{1},q^{2},q^{3})=(x,y,z)\,\Rightarrow \,d\mathbf {r} =(dx,dy,dz)}$$

so

$${\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=dx^{2}+dy^{2}+dz^{2}}$$

Orthogonal curvilinear coordinates

For all orthogonal coordinates the metric is given by:^[3]

$${\displaystyle [g_{ij}]={\begin{pmatrix}h_{1}^{2}&0&0\\0&h_{2}^{2}&0\\0&0&h_{3}^{2}\end{pmatrix}}}$$

where

$${\displaystyle h_{i}=\left|{\frac {\partial \mathbf {r} }{\partial q^{i}}}\right|}$$

for i = 1, 2, 3 are scale factors, so the square of the line element is:

$${\displaystyle ds^{2}=h_{1}^{2}(dq^{1})^{2}+h_{2}^{2}(dq^{2})^{2}+h_{3}^{2}(dq^{3})^{2}}$$

Some examples of line elements in these coordinates are below.^[2]

Coordinate system	(q1, q2, q3)	Metric	Line element
Cartesian	(x, y, z)	${\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}}$	${\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2}}$
Plane polars	(r, θ)	${\displaystyle [g_{ij}]={\begin{pmatrix}1&0\\0&r^{2}\\\end{pmatrix}}}$	${\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}}$
Spherical polars	(r, θ, φ)	${\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&r^{2}\sin ^{2}\theta \\\end{pmatrix}}}$	${\displaystyle ds^{2}=dr^{2}+r^{2}d\theta \ ^{2}+r^{2}\sin ^{2}\theta \ d\phi \ ^{2}}$
Cylindrical polars	(r, θ, z)	${\displaystyle [g_{ij}]={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&1\\\end{pmatrix}}}$	${\displaystyle ds^{2}=dr^{2}+r^{2}d\theta \ ^{2}+dz^{2}}$

Minkowskian spacetime

The Minkowski metric is:^[5][1]

$${\displaystyle [g_{ij}]=\pm {\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\\\end{pmatrix}}}$$

where one sign or the other is chosen, both conventions are used. This applies only for flat spacetime. The coordinates are given by the 4-position:

$${\displaystyle \mathbf {x} =(x^{0},x^{1},x^{2},x^{3})=(ct,\mathbf {r} )\,\Rightarrow \,d\mathbf {x} =(cdt,d\mathbf {r} )}$$

so the line element is:

$${\displaystyle ds^{2}=\pm (c^{2}dt^{2}-d\mathbf {r} \cdot d\mathbf {r} ).}$$

Schwarzschild coordinates

In Schwarzschild coordinates coordinates are ${\displaystyle \left(t,r,\theta ,\phi \right)}$, being the general metric of the form:

$${\displaystyle [g_{ij}]={\begin{pmatrix}-a(r)^{2}&0&0&0\\0&b(r)^{2}&0&0\\0&0&r^{2}&0\\0&0&0&r^{2}\sin ^{2}\theta \\\end{pmatrix}}}$$

(note the similitudes with the metric in 3D spherical polar coordinates).

so the line element is:

$${\displaystyle ds^{2}=-a(r)^{2}\,dt^{2}+b(r)^{2}\,dr^{2}+r^{2}\,d\theta ^{2}+r^{2}\sin ^{2}\theta \,d\phi ^{2}.}$$

General spacetime

The coordinate-independent definition of the square of the line element ds in spacetime is:^[1]

$${\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =g(d\mathbf {x} ,d\mathbf {x} )}$$

In terms of coordinates:

$${\displaystyle ds^{2}=g_{\alpha \beta }dx^{\alpha }dx^{\beta }}$$

where for this case the indices α and β run over 0, 1, 2, 3 for spacetime.

This is the spacetime interval - the measure of separation between two arbitrarily close events in spacetime. In special relativity it is invariant under Lorentz transformations. In general relativity it is invariant under arbitrary invertible differentiable coordinate transformations.