Triangle on a sphere (Schutz, 6.10)

Click For Summary
SUMMARY

The discussion focuses on Exercise 6.10 from B.F. Schutz's "A First Course in General Relativity," which explores the properties of triangles on a sphere. It establishes that the sum of the interior angles of a spherical triangle exceeds 180°, and the rotation of a vector during parallel transport around such a triangle corresponds to this excess. Specifically, the total angle of rotation is calculated as 360° minus the excess angle (gamma), confirming the relationship between spherical geometry and vector rotation.

PREREQUISITES
  • Understanding of spherical geometry and great circles
  • Familiarity with the concept of parallel transport in differential geometry
  • Basic knowledge of vector rotation and angular measurements
  • Background in general relativity principles as outlined in Schutz's text
NEXT STEPS
  • Study the properties of geodesics in spherical geometry
  • Learn about parallel transport and its implications in curved spaces
  • Explore the relationship between curvature and angle excess in spherical triangles
  • Investigate applications of spherical geometry in general relativity
USEFUL FOR

Students of general relativity, mathematicians interested in geometry, and physicists exploring the implications of curvature in spacetime.

hellfire
Science Advisor
Messages
1,048
Reaction score
1
A first course in general relativity. B.F. Schutz. Exercise 6.10:
A straight line in a sphere is a great circle, and it is well known that the sum of interior angles of any triangle on a sphere whose sides are arcs of great circles exceeds 180°. Show that the amount by which a vector is rotated by parallel transport around such a triangle equals the excess of the sum of the angles over 180°.
How to proceed to prove this?
 
Physics news on Phys.org
hellfire said:
A first course in general relativity. B.F. Schutz. Exercise 6.10:
A straight line in a sphere is a great circle, and it is well known that the sum of interior angles of any triangle on a sphere whose sides are arcs of great circles exceeds 180°. Show that the amount by which a vector is rotated by parallel transport around such a triangle equals the excess of the sum of the angles over 180°.
How to proceed to prove this?
The first step is to recognize that since each side of the triangle is a great circle (geodesic), a vector parallel transported along that geodesic experiences no rotation. The rotation occurs only at each vertex where the vector is rotated by an angle equal to 180 less the angle made by the two sides of the triangle. So the total angle of rotation after returning to its starting point is (180-a1 + 180 - a2 + 180 - a3). But since we know that a1 + a2 + a3 = 180 + gamma (where gamma is the difference you are looking for):
[tex]\theta = 540 - (180 + \gamma)) = 360 - \gamma[/tex]

AM
 
Thank you Andrew. This was quite easier than I had supposed.
 

Similar threads

Graduate Curvature with different connections on the two-sphere
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Triangles on Spheres: Isosceles and Shortest Distance Inferences
  • · Replies 8 ·
Replies
8
Views
4K
Sum of the angles of a spherical triangle
  • · Replies 1 ·
Replies
1
Views
4K
High School Calc Excess Rad on Earth in Curved 3D Space | Feynman Lectures 6-2
  • · Replies 29 ·
Replies
29
Views
3K
Parallel Transport and Triangle Excess Angle
  • · Replies 1 ·
Replies
1
Views
4K
Graduate Finding Radius of Curvature of a Sphere Using Angle Excess
  • · Replies 3 ·
Replies
3
Views
3K
Graduate What Is a Lune on a Sphere and Its Key Properties?
  • · Replies 1 ·
Replies
1
Views
3K
MHB Find unknown angles of a triangle
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Does this spherical triangle exist?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Question about area on a sphere
  • · Replies 22 ·
Replies
22
Views
6K