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Concept
Four-gradient
Summary
In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus.
In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors.
This article uses the (+ − − −) metric signature.
SR and GR are abbreviations for special relativity and general relativity respectively.
indicates the speed of light in vacuum.
is the flat spacetime metric of SR.
There are alternate ways of writing four-vector expressions in physics:
The four-vector style can be used: , which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. . Most of the 3-space vector rules have analogues in four-vector mathematics.
The Ricci calculus style can be used: , which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as .
The Latin tensor index ranges in {1, 2, 3}, and represents a 3-space vector, e.g. .
The Greek tensor index ranges in {0, 1, 2, 3}, and represents a 4-vector, e.g. .
In SR physics, one typically uses a concise blend, e.g. , where represents the temporal component and represents the spatial 3-component.
Tensors in SR are typically 4D -tensors, with upper indices and lower indices, with the 4D indicating 4 dimensions = the number of values each index can take.
The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):
The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:
The comma in the last part above implies the partial differentiation with respect to 4-position .
The contravariant components are:
Alternative symbols to are and D (although can also signify as the d'Alembert operator).
In GR, one must use the more general metric tensor and the tensor covariant derivative (not to be confused with the vector 3-gradient ).
Official source
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