This comment was posted to reddit on Feb 03, 2016 at 11:48 am and was deleted within 14 minutes.

The [;d;]-dimensional generalisation of Pythagoras's theorem gives the squared distance between two points as [;\sum*{i=1} ^{n} dx_i^{2} ;] , where [; d\mathbf{x} ;] is the vector between two points. We can rewrite this squared distance as [;\sum{ij} g*{ij}dx

For a more general [; g;] , [; g\mathbf{v}\neq\mathbf{v};] so it's worth introducing a shorthand for each. In particular, we use a mixture of upper and lower indices, and summation over repeated indices (due to matrix multiplication or a generalisation thereof) occurs with one copy upper, one lower. For example, [; v^{i} = g^{{ij}} v*j,\,v_j =g*{jk}v^{k;]} with [;g^{{ij};]} the matrix inverse of [;g_{jk};].

Special relativity claims spacetime's geometry takes [; g_{00} = -1;] (here [; 0 ;] is the index for the time coordinate, while positive integers cover space coordinates). The [; \Lambda ;] are then Lorentz matrices. General relativity admits that [; g;] is a function of space and time, and the maths involved quickly becomes complicated. However, you can understand Einstein notation as long as you obey these rules:

(i) Never let the same index appear twice unless it's once upper, once lower;

(ii) When you need to move an index, use either [;g^{{ij};]} or [;g_{ij};] .