Graph, Wall, Tome: Difference between revisions

Graph, Wall, Tome (edit)

815 bytes added , 29 January 2020

20

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 ===== 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? ===