Wikipedia / Line element
Wikipedia / Line element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by .
Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved pseudo-Riemannian manifold with an appropriate metric tensor.
General formulation
Definition of the line element and arc length
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 where g is the metric tensor, · denotes inner product, and dq an infinitesimal displacement, we can define the arc length as the integral:
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
with the metric is even more easy to see in n-dimensional general curvilinear coordinates , where it is written as a symmetric rank 2 tensor
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.
Line elements in Euclidean space
Following are examples of how the line elements are found from the metric.
Cartesian coordinates
The simplest line element is in Cartesian coordinates - in which case the metric tensor is just the Kronecker delta (here i, j = 1, 2, 3 for space) or in matrix form (i denotes row, j
Orthogonal curvilinear coordinates
For all orthogonal coordinates the metric tensor is given by:
for i = 1, 2, 3 are scale factors
Some examples of line elements in these coordinates are below.
General curvilinear coordinates
)
The coordinate basis is a special type of basis that is regularly used in differential geometry.
Line elements in 4d spacetime
Minkowski spacetime
The Minkowski metric is: 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
Schwarzschild coordinates
(note the similitudes with the metric in 3D spherical polar coordinates).
General spacetime
The coordinate-independent definition of the square of the line element ds in spacetime is:
where for this case the indices and 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.
See also
References