Computing length in non-orthogonal bases
First, just describing the length of a vector on a curved space is hard. It is given by:
[math]\displaystyle{ Length^{squared} = g_{11}dX^{1}dX^{1} + g_{12}dX^{1}dX^{2} + g_{21}dX^{2}dX^{1} + g_{22}dX^{2}dX^{2} }[/math]
Some notes:
- This is not Pythagorean theorem, because [math]\displaystyle{ dX^{1} }[/math] and [math]\displaystyle{ dX^{2} }[/math] are not perpendicular.
- Instead, looks like a formula to get the diagonal from two opposite vertices in a parallelogram.
- If [math]\displaystyle{ dX^{1} }[/math] and [math]\displaystyle{ dX^{2} }[/math] are perpendicular, then [math]\displaystyle{ g_{12} }[/math] and [math]\displaystyle{ g_{21} }[/math] would be 0, and we would get [math]\displaystyle{ Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2} }[/math]
- See: the video @ 14m27s
Computing vector rotation due to parallel transport
Then, they show parallel transport when following a parallelogram, but over a curved 3D manifold. To compute the vector rotation by components, they show:
[math]\displaystyle{ dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312}) }[/math]
[math]\displaystyle{ dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312}) }[/math]
[math]\displaystyle{ dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312}) }[/math]
or, using [math]\displaystyle{ i }[/math] to summarize across all 3 components (difference vectors):
[math]\displaystyle{ dV^{i} = dX^{1}dX^{2} (V^{1}R^{i}_{112} + V^{2}R^{i}_{212} + V^{3}R^{i}_{312}) }[/math]
or , using [math]\displaystyle{ j }[/math] to index over all 3 components (original vector):
[math]\displaystyle{ dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}] }[/math]
See: the video @ 19m33s
Open questions:
- Why a parallelogram?
- How to properly overlay the parallelogram onto the 3d manifold, in order to do the parallel transport?
- How does this relate to the length computation above?
Putting it all together
Now, moving to 4D, we can compute [math]\displaystyle{ R_{\mu v} }[/math] as:
[math]\displaystyle{ R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030} }[/math]
[math]\displaystyle{ R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130} }[/math]
[math]\displaystyle{ R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030} }[/math]
etc.
Indexing i over all 4 component vectors / dimensions, we get:
[math]\displaystyle{ R_{00} = \Sigma_{i} R^{i}_{0i0} }[/math]
[math]\displaystyle{ R_{10} = \Sigma_{i} R^{i}_{1i0} }[/math]
[math]\displaystyle{ R_{01} = \Sigma_{i} R^{i}_{0i1} }[/math]
etc.
Summarizing on [math]\displaystyle{ \mu }[/math], we get:
[math]\displaystyle{ R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0} }[/math]
[math]\displaystyle{ R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1} }[/math]
etc
Summarizing on [math]\displaystyle{ v }[/math], we get:
[math]\displaystyle{ R_{\mu v} = \Sigma_{i} R^{i}_{\mu iv} }[/math]
Open questions:
- If we hadn't moved from 3D to 4D, what would this all have looked like?
- What does this have to do with the parallelogram?
- Why are there two indices?
