Problem 2 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

On the \(n\)-sphere \(S^{n}\) find coordinates corresponding to (i) stereographic projection, (ii) spherical polars.

Step-by-Step Solution

Answer

The coordiantes corresponding to the stereographic projection are given by \(r=(1+z)/ (1-z), \varphi=\arctan (y/x),(1+z)=r\cos \theta\). And spherical polar coordinates are given by \(r = \sqrt{x^2+y^2+z^2}, \theta = \arccos (\frac{z}{\sqrt{x^2+y^2+z^2}}), \varphi = \arctan (\frac{y}{x})\).

Step 1: Stereographic Projection

Stereographic projection is a method of projecting points from the surface of a sphere onto a plane. We take a sphere of radius 1, without loss of generality, centered at the origin. The north pole N of the sphere is the point (0, 0, 1). Any point P on the sphere not equal to N determines a unique line in \(\mathbb{R}^3\), passing through N and P. This line intersects the plane z=0 in exactly one point P', which we call the stereographic projection of P.\nBy writing this explicitly and setting \(x = r \sin \theta \cos \varphi, y = r \sin \theta \sin \varphi, z = r \cos \theta\), where r is radius, \(\theta\) is the inclination angle and \(\varphi\) is the azimuthal angle, we find that \(r=(1+z)/ (1-z), \varphi=\arctan (y/x),(1+z)=r\cos \theta\).

Step 2: Spherical Polar Coordinates

To find the coordinates in spherical polars, we use the standard conversion formulae between Cartesian and spherical polar coordinates.\nFor a point P = (x, y, z) in Cartesian coordinates, the corresponding point in spherical polar coordinates is given by:\n\(r = \sqrt{x^2+y^2+z^2}\)\n\(\theta = \arccos (\frac{z}{\sqrt{x^2+y^2+z^2}})\)\n\(\varphi = \arctan (\frac{y}{x})\)\nwhere r is the radial distance, \(\theta\) is the polar angle, and \(\varphi\) is the azimuthal angle.

Key Concepts

Stereographic Projection

Imagine you have a globe and you want to make a flat map out of it without distortion. Stereographic projection is a mathematical method to achieve this by projecting a sphere's surface onto a plane. Here’s how it works:

• You have a sphere centered at the origin with a radius, typically 1 for simplicity.
• The north pole of the sphere is at (0, 0, 1).
• To project a point from the sphere’s surface to the plane, draw a line from the north pole through the point on the sphere.
• This line will intersect the plane at exactly one point. That point is the stereographic projection of the sphere's point.

The beauty of this projection is that it preserves angles, making it very useful in complex analysis and geometry. Mathematically, if a point on the sphere has coordinates \((x, y, z)\)in 3D, it transforms to a point on the plane according to specific formulas. These transformations help in analyzing spherical objects by studying them in a planar context, simplifying many complicated calculations.

Spherical Coordinates

Spherical coordinates are another way to describe points in three-dimensional space. They are especially useful when dealing with problems involving spheres or circular symmetry. Instead of using Cartesian coordinates (x, y, z), we use:

• Radial distance (r): The distance from the origin to the point. It’s equivalent to the radius in spherical geometry.
• Polar angle (\(\theta\)): The angle between the point and the z-axis. It's like measuring how "high" or "low" the point is on the sphere.
• Azimuthal angle (\(\varphi\)): The angle between the x-axis and the point's projection onto the x-y plane. It tells you how "around" the point is located.

Each point in space can be uniquely identified by these three numbers. To convert from Cartesian to spherical coordinates, you can use formulas:\[r = \sqrt{x^2 + y^2 + z^2}, \\theta = \arccos \left(\frac{z}{r}\right), \\varphi = \arctan \left(\frac{y}{x}\right)\]This system simplifies navigation and description of points in space, particularly when dealing with spherical symmetry.

n-Sphere

An \(n\)-sphere is a generalization of a sphere to \(n\) dimensions. The concept can be a bit challenging at first, but it essentially extends the idea of a circle (1-sphere) and a typical sphere (2-sphere) to higher dimensions.

• 1-sphere: A circle in 2D.
• 2-sphere: The usual sphere surface in 3D, like the Earth. It consists of all points equidistant from a central point.
• 3-sphere: A theoretical construct in 4D space.

For an \(n\)-sphere in \((n+1)\)-dimensional Euclidean space, all points have the same distance from a central point. This idea greatly aids in various branches of mathematics, including topology and geometry. In abstract mathematics, \(n\)-spheres are important because they help mathematicians understand and visualize spaces in higher dimensions, although they exist beyond our own visual perception.