Other Notation Conventions

Einstein Notation

Einstein summation notation is a  for representing summations of vectors over multiple dimensions. For example, consider the following expression:

y=\sum_{i=1}^3 a_ix^i = a_1x^2 + a_2x^2 + a_3x^3

(Note here that the upper indices should be treated as coordinate indices, not exponents, where the three x components correspond to the traditional x,y,z components of a traditional three dimensional coordinate system.)

This may be represented in Einstein notation as follows:

y = a_ix^i

The usual convention in general relativity is to use Greek letters as indices (\mu or \nu) for the the four space-time component indices 0, 1, 2, 3, and Latin letters (i, j, k) are used as indices for traditional Cartesian coordinates 1, 2, 3.

A lower index is used to represent vectors (column vectors, while an upper index is used to represent covectors (row vectors, or vectors in the dual space of the corresponding vector space). For example, if ei represents the basis vectors for the vector space V, then ei would represent the basis for the dual space V*:

\displaystyle a^ie_i = \begin{bmatrix}e_1&e_2&e_3\end{bmatrix}\begin{bmatrix}a^1\\a^2\\a^3\end{bmatrix}

\displaystyle b_ie^i = \begin{bmatrix}b_1&b_2&b_3\end{bmatrix}\begin{bmatrix}e^1\\e^2\\e^3\end{bmatrix}

For matrices, we can refer to the mth row and the nth column by writing Anm

An inner product of vectors v and u would be represented by viui. And outer product of these vectors would be represented by

A_j^i = u^iv_j

For tensors (which crop up everywhere in general relativity), the representation is as follows:

Covariant metric tensor: g_{\mu\nu} = g(e_{\mu}e_{\nu})
Contravariant metric tensor: g^{\mu\nu} = g(e^{\mu}e^{\nu})

Einstein notation – Wikipedia, the free encyclopedia

Feynman Slash Notation

Feynman slash notation is a shorthand (extending upon Einstein notation) for representing the application of the gamma matrices to a covector:

\displaystyle \not{p} = \gamma^\mu p_\nu

Feynman slash notation – Wikipedia, the free encyclopedia

Covariance and contravariance of vectors – Wikipedia, the free encyclopedia


One Response to Other Notation Conventions

  1. Pingback: A Century of General Relativity | Whiskey…Tango…Foxtrot?

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