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The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear ...
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study.The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, ... where g is the metric tensor (see below).
Specifying a metric tensor is part of the definition of any Lorentzian manifold. The simplest way to define this tensor is to define it in compatible local coordinate charts and verify that the same tensor is defined on the overlaps of the domains of the charts. In this article, we will only attempt to define the metric tensor in the domain of ...
Rewriting the metric in spherical coordinates reduces four coordinates to three coordinates. The radial coordinate is written as a product of a comoving coordinate, r, and a time dependent scale factor R(t). The resulting metric can be written in several forms.
is the Einstein tensor, G is the Newtonian constant of gravitation, g ab is the metric tensor, and R (scalar curvature) is the trace of the Ricci curvature tensor. The stress–energy tensor is composed of the stress–energy from particles, but also stress–energy from the electromagnetic field. This generates the nonlinearity.
Conventionally, one uses spherical coordinates = (,,,), to write the metric (the line element). Several coordinate charts are possible; these include: Schwarzschild coordinates; Isotropic coordinates, in which light cones are round, and thus useful for studying null dusts.