In the theory of

Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

spherically symmetric spacetime In physics, spherically symmetric spacetimes are commonly used to obtain analytic and numerical solutions to Einstein's field equations in the presence of radially moving matter or energy. Because spherically symmetric spacetimes are by definition ...

s admit a family of nested round

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

s. In each of these spheres, every point can be carried to any other by an appropriate rotation about the center of symmetry. There are several different types of coordinate chart which 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 is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

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

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: * 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 In mathem ...

(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