How is isotropy of a space defined mathematically and physically?

Question

A definition question in gravity.

Answer 1

A bijective mapping of a metric manifold to itself f:M-->M which preserves the distance between pairs of neighboring points is called an isometry.

At each point P in an N dimensional space it's possible to choose a set S of N orthogonal tangent vectors. I think a space is isotropic at P if for every such pair of sets S and S' at P, there's an isometry of the whole space which takes S-->S'.

Likewise a space is isotropic (and homogeneous) everywhere if and only if for each pair P,S and P',S' (where P and P' are any two points and S and S' are sets of orthogonal vectors at P and P' respectively) there's an isometry which takes P-->P' and S-->S'.

The surface of revolution defined by rotating a function y=f(x) about the y axis is an example of a two dimensional space which is isotropic at the point where the surface intersects the y axis.

The surface of a sphere is everywhere isotropic. Any metric manifold which is isotropic everywhere also has a metric more or less like the surface of an sphere in N+1 dimensions, that is, characterized by a single curvature scalar except with a possibly different signature than a Euclidean sphere.

It's also possible for subspaces to be isotropic rather than the whole space. For example in general relativity, for a spherically symmetric field, in Schwarzchild coordinates, the two dimensional set of points r=constant, t= constant is isotropic.

Answer 2

Who said space was isotropic? Anywho, pick up a general relativity book and get your learn on. There's something called the metric (rank 2 square tensor with as many rows as spatial dimensions) it describes the curvature of space at a given point. In flat isotropic space the metric appears as an identity matrix. For the most part (excluding neutron stars and black holes), all calculations done using relativity assume the metric is the identity matrix.

Answer 3

a vanishing trace