jk22 said:I saw briefly
No, because winding around a closed curve several times is a different closed curve.jk22 said:I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve.
If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
jk22 said:if a vector is parallel transported along a CTC
I am going to guess what might be confusing you (always dangerous, but I'll risk it). I'm thinking you have in mind a vector field with a value at every spacetime point (event). You take one of these and parallel transport around a circuit. You get a different vector. Around a circuit again, you get still a different vector. But each of these transported vectors is a different vector than the original, not part of the original vector field at all. There is no multi-valued anything. Curvature is defined via a limiting operation for circuits that return to the starting event exactly once. Note, CTC's are not special at all, in this. In defining curvature at a point, you will consider spacelike circuits and timelike circuits, even mixed. It doesn't matter for defining curvature.jk22 said:Then how is the coming back at the same time as starting, it should but start again or time were stopped ?
Namely if a vector is parallel transported along a CTC is it coming back to its original state ?
I think I'm locked since the two alternatives are for me : If it is yes then there is no curvature, if it is no then the vector at ##(x^\mu)## has several values, hence multivalued ?
What am I thinking wrong here ?
The Riemann Tensor, also known as the Riemann Curvature Tensor, is a mathematical object used to describe the curvature of a manifold, such as a surface or space. It is defined by the components of the metric tensor and represents the change in direction of a vector as it is transported along a curved path.
Stoke's Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary of the surface. It is a generalization of the fundamental theorem of calculus and has applications in physics and engineering.
In general relativity, the Riemann Tensor is used to describe the curvature of spacetime. It is a key component in Einstein's field equations, which relate the curvature of spacetime to the distribution of matter and energy. The Riemann Tensor allows us to understand the behavior of objects in the presence of strong gravitational fields, such as those near black holes.
The winding number is a mathematical concept used in topology to describe the number of times a curve wraps around a point or another curve. It is a topological invariant, meaning it does not change under continuous deformations of the curve. The winding number is used to classify and distinguish different types of curves and surfaces in topology.
Stoke's Theorem can be used to calculate the winding number of a closed curve on a surface. The line integral along the boundary of the surface is equal to the surface integral of the curl of the vector field, which is related to the winding number. This relationship allows us to use Stoke's Theorem to solve problems in topology involving the winding number.