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In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.
Print/export Download as PDF; ... Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when ...
The text here uses Einstein notation in which summation over repeated indices is assumed. Two types of derivatives are used: Partial derivatives are denoted either by the operator ∂ i {\displaystyle \partial _{i}} or by subscripts preceded by a comma.
The equation for the mass shell is also often written in terms of the four-momentum; in Einstein notation with metric signature (+,−,−,−) and units where the speed of light =, as =. In the literature, one may also encounter p μ p μ = − m 0 2 {\displaystyle p^{\mu }p_{\mu }=-m_{0}^{2}} if the metric signature used is (−,+,+,+).
For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions ...
A discrete version of the Einstein–Hilbert action is obtained by considering so-called deficit angles of these blocks, a zero deficit angle corresponding to no curvature. This novel idea finds application in approximation methods in numerical relativity and quantum gravity , the latter using a generalisation of Regge calculus.
In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...
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