The contravariant components are:
Alternative symbols to are and D.
Occasionally, in analogy with the 3-dimensional notation, the symbols and are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol is reserved for the d'Alembertian.
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:
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:
- Ricci calculus
- Index notation
- Antisymmetric tensor
- Einstein notation
- Raising and lowering indices
- Abstract index notation
- Covariance and contravariance of vectors