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The upper index position in x i is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see § Application below). Typically, (x 1 x 2 x 3) would be equivalent to the traditional (x y z). In general relativity, a common convention is that
Well-formulated expressions are constrained by the rules of Einstein summation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we can immediately see that an expression such as
Each index has one possible value per dimension of the underlying vector space. The number of indices equals the degree (or order) of the tensor. 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 ...
The Einstein tensor is a tensor of order 2 defined over pseudo-Riemannian manifolds.In index-free notation it is defined as =, where is the Ricci tensor, is the metric tensor and is the scalar curvature, which is computed as the trace of the Ricci tensor by = .
In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions x μ (τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates.
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time.In Albert Einstein's 1905 paper, On the Electrodynamics of Moving Bodies, the theory is presented as being based on just two postulates: [p 1] [1] [2]
The EFE describe how mass and energy (as represented in the stress–energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor). In abstract index notation, the EFE reads as follows: + = where is the Einstein tensor, is the cosmological constant, is the metric tensor, is the speed of light in vacuum and is ...
Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer.