In the theory of

Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...

s, spherically symmetric spacetimes admit a family of nested round

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

s. In each of these spheres, every point can be carried to any other by an appropriate rotation about the centre of symmetry. There are several different types of coordinate chart that are ''adapted'' to this family of nested spheres, each introducing a different kind of distortion. The best known alternative is the Schwarzschild chart, which correctly represents distances within each sphere, but (in general) distorts radial distances and

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

s. Another popular choice is the isotropic chart, which correctly represents angles (but in general distorts both radial and transverse distances). A third choice is the Gaussian polar chart, which correctly represents radial distances, but distorts transverse distances and angles. There are other possible charts; the article on spherically symmetric spacetime describes a coordinate system with intuitively appealing features for studying infalling matter. In all cases, the nested geometric spheres are represented by coordinate spheres, so we can say that their ''roundness'' is correctly represented.

Definition

In a Gaussian polar chart (on a static spherically symmetric spacetime), the

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

(aka

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 ...

) takes the form :

g = -a(r)^2 \, dt^2 + dr^2 + b(r)^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right),

-\infty < t < \infty, \, r_0 < r < r_1, \, 0 < \theta < \pi, \, -\pi < \phi < \pi.

Depending on context, it may be appropriate to regard

a

and

b

as undetermined functions of the radial coordinate

r

. Alternatively, we can plug in specific functions (possibly depending on some parameters) to obtain an isotropic coordinate chart on a specific Lorentzian spacetime.

Applications

Gaussian charts are often less convenient than Schwarzschild or isotropic charts. However, they have found occasional application in the theory of static spherically symmetric perfect fluids.

Definition

Applications

See also