Finding Radius of Curvature of a Sphere Using Angle Excess

[pervect]

On the surface of a sphere, we can find the radius of cuvature of the sphere by:

angle excess / area = 1/ r_s^2

http://en.wikipedia.org/w/index.php?title=Angle_excess&oldid=543583039

If we use triangles, for instance, the angle excess is the sum of the angles of the triangle minus 180 degrees.

Can we use this basic idea to define the sectional curvature of a plane in terms that are relatively layman-friendly, and leverage this up to a fuller explanation of the Riemann curvature tensor?

T seems to me it's that "angle excess" is the same basic idea as talking about the parallel transport of a vector around a closed path, but expressed in simpler language.

 

[Dale]

That seems like a pedagogically good approach, but I don't know of a treatment which does it that way.

 

[WannabeNewton]

Off of the top of my head, the closest thing I can think of relating geodesic triangles to constant sectional curvature is the famous Toponogov's theorem: http://en.wikipedia.org/wiki/Sectional_curvature#Toponogov.27s_theorem

 

[Jonathan Scott]

I've always been interested in the relationship of the angular defect (or angular deficit) to the fraction enclosed of a sphere-like surface, noting that by Descartes Theorem the total defect always adds up to 4pi on something similar to an ordinary sphere. For an unevenly curved surface, this remains exact, while the calculation of the effective radius of the spherical surface from the angular defect is just an approximation.

It seemed to me that the angular defect was like a conserved quantity, and therefore that it might be like the distribution of mass in the universe. However, in the 3D case, allocating the solid angle defect in this way gives additional extrinsic curvature proportional to sqrt(m)/r, not to m/r^2 as in Newtonian gravity.

Somewhat surprisingly, if you assume that the solid angle defect for the whole universe corresponds approximately to the estimated mass of the universe, of the order of 10^54 kg, then you find that the constant of proportionality for the extrinsic curvature in this model happens to be equal to the MOND acceleration parameter, and the corresponding acceleration matches the MOND law.

(Note however that this model relates to spatial curvature, not space-time curvature, and the acceleration would therefore only affect slow-moving objects if some additional assumptions were made, for example that the usual relationship of space to time curvature applies locally as in Einstein's field equations).

 

[sardar]

Yes, the concept of angle excess can be used to define the sectional curvature of a plane in a more accessible way. In basic terms, sectional curvature measures the deviation from Euclidean geometry in a specific plane within a curved space. It is calculated by dividing the angle excess of a triangle in that plane by the area of the triangle, as shown in the formula in the provided link.

This idea can then be expanded to explain the Riemann curvature tensor, which is a mathematical object that describes the curvature of a space in all directions. It is essentially a way to generalize the concept of sectional curvature to a higher-dimensional space.

Using the concept of angle excess, we can understand that the Riemann curvature tensor measures the deviation from Euclidean geometry in all possible planes within a curved space. This allows us to fully describe the curvature of a space and understand its properties.

In summary, the concept of angle excess provides a simpler way to understand and calculate sectional curvature, which can then be used to explain the more complex concept of the Riemann curvature tensor.