4-gradient

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4-gradient

From the use of the Minkowski metric, converting between contravariant and covariant


xμημμxμ


Where we have already defined the covariant term,

xμ=[x0x1x2x3]

and the contravariant term

xμ=[x0x1x2x3]


From quantum mechanics we know that partial differential is a linear operator. Following the rules of matrix multiplication this implies that the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector, and the derivative with respect to a covariant coordinate transforms as a contravariant vector.

μ=xμ


μ[x0x1x2x3]=[txyz]=[t]


μ=xμ


μ[x0x1x2x3]=[txyz]=[t]


Since it is an operator, the dot product of two partial differentials yields an operator known as the D'Alembert operator.

μμ=[x0x1x2x3][x0x1x2x3]=2t22


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