Jump to content

Graph, Wall, Tome: Difference between revisions

815 bytes added ,  29 January 2020
Line 176: Line 176:


===== Putting it all together =====
===== Putting it all together =====
Now, moving to 4D, we can compute $$R_{\mu v}$$ as:
$$R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030}$$
$$R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130}$$
$$R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030}$$
etc.
Indexing i over all 4 component vectors / dimensions, we get:
$$R_{00} = \Sigma_{i} R^{i}_{0i0}$$
$$R_{10} = \Sigma_{i} R^{i}_{1i0}$$
$$R_{01} = \Sigma_{i} R^{i}_{0i1}$$
etc.
Summarizing on $$\mu$$, we get:
$$R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0}$$
$$R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1}$$
etc
Summarizing on $$v$$, we get:
$$R_{\mu v} = \Sigma_{i} R^{i}_{\mu iv}$$
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?


=== How do they relate? ===
=== How do they relate? ===
20

edits