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Why do the equations sometimes render as graphics, and sometimes as text?

Einstein Summation

An Einstein summation is a summation over an index that is implied without the use of a summation symbol. Authors who use it often say they are using the Einstein summation convention.

It is most often used in relativity when summing over an index that appears twice in a term, once as contravariant index and once as covariant index. In the following equation, on the left hand side we are doing an einstein summation, while on the right, we are explicitly showing the summation symbol:


 * $$ a_\mu b^\mu = \sum_{\mu = 0}^4 a_\mu b^\mu $$

In relativity, it is most common to use greek letters when the summation is taken from 0 to 3 inclusively, and to use latin letters when the summation is taken from 1 to 3 inclusively. We therefore have the following mathematical identity:


 * $$ a_\mu b^\mu = a_0 b^0 + a_i b^i $$

It is often said that as a rule, the summation only takes place if the index appears once as a contravariant index and once as a covariant index, but a lot of people break this rule and sum even when the index appears twice as the same kind. This rarely happens when the sum is over the 3 space dimensions and 1 time dimension of relativity. Thus, when using the standard metric of relativistic quantum theory we have:


 * $$ a_i b^i = - a^i b^i $$

The summation convention can also be used to display matrix multiplication such as:


 * $$ A_{ij} B_{jk} = C_{ik} $$

Sometimes we need to distribute over a factor before the summation can be clearly seen:


 * $$ a_\alpha (b^{\alpha \beta} - c_\beta d^\alpha) = a_\alpha b^{\alpha \beta} - a_\alpha c^\beta d^\alpha = e^\beta $$