Frobenius group

In differential geometry, the four-gradient is the four-vector analogue of the gradient from Gibbs-Heaviside vector calculus.

Definition

The covariant components compactly written in index notation are:

${\dfrac {\partial }{\partial x^{\alpha }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)=\partial _{\alpha }={}_{,\alpha }$

The comma in the last part above ${}_{,\alpha }$ implies the partial differentiation with respect to $x^{\alpha }$. This is not the same as a semi-colon, used for the covariant derivative.

The contravariant components are:

$\partial ^{\alpha }\ =g^{\alpha \beta }\partial _{\beta }=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-\nabla \right)$

where g^αβ is the metric tensor, which here has been chosen for flat spacetime with the metric signature (+,−,−,−).

Alternative symbols to $\partial _{\alpha }$ is $\Box$ or D.

Usage

The square of D is the four-Laplacian, which is called the d'Alembert operator:

$D\cdot D=\partial _{\alpha }\partial ^{\alpha }={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}$.

As it is the dot product of two four-vectors, the d'Alembertian is a Lorentz invariant scalar.

Occasionally, in analogy with the 3-dimensional notation, the symbols $\Box$ and $\Box ^{2}$ are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol $\Box$ is reserved for the d'Alembertian.

Derivation

In 3 dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to four dimensions should be:

$\partial ^{\alpha }\ =\left({\frac {\partial }{\partial t}},\nabla \right)$

incorrect

However, a line integral involves the application of the vector dot product, and when this is extended to four dimensional space-time, a change of sign is introduced to either the spacial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of space-time. In this article, we place a negative sign on the spatial co-ordinates. The factor of 1/c and −1 is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of four-gradient:

$\partial ^{\alpha }\ =\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-\nabla \right)$

correct

See also

• Ricci calculus
• Index notation
• Tensor
• Antisymmetric tensor
• Einstein notation
• Raising and lowering indices
• Abstract index notation
• Covariance and contravariance of vectors

References

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• S. Hildebrandt, "Analysis II" (Calculus II), ISBN 3-540-43970-6, 2003
• L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988
• J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley ISBN 0-471-30932-X