- #1
vibhuav
- 43
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In trying to explain the concept of curved space, many books use the example of the surface of a sphere, which can be considered as a curved 2D space embedded in a higher dimensional, 3D space. I could derive, starting from ##a^2=x^2+y^2+z^2##, that the metric, or the line element, on the surface of the sphere - which we now consider to be 2D - is ##ds^2=\frac{a^2 {d\rho}^2}{a^2-\rho^2}+\rho^2 {d\phi}^2 ##. But here’s my question: in the equation for the metric, what are ##x(=\rho\cos\phi)## and ##y(=\rho\sin\phi)##? It makes sense that they are the original ##x## and ##y## from the 3D space, which are measurements in the external, third dimension, but I would rather have coordinates ##\it{on}## the 2D surface of the sphere. (Note the had we used ##\theta## and ##\phi##, they too are in the external 3D space.)
Some books use a tangent plane at the point in question on the surface of the sphere, and define a 2D ##x##-##y## Cartesian coordinate system on it, but - and this is my main question - I am not able to mathematically relate the original coordinates in 3D space to the coordinates of the tangent space. What am I missing?
While we are at it, does the tangent plane move around on the surface of the sphere for different points? And the line element on such a tangent plane ought to be projected on to the curved, 2D space. What is the math for this?
Some books use a tangent plane at the point in question on the surface of the sphere, and define a 2D ##x##-##y## Cartesian coordinate system on it, but - and this is my main question - I am not able to mathematically relate the original coordinates in 3D space to the coordinates of the tangent space. What am I missing?
While we are at it, does the tangent plane move around on the surface of the sphere for different points? And the line element on such a tangent plane ought to be projected on to the curved, 2D space. What is the math for this?