Four-gradient
Encyclopedia
The four-gradient is the four-vector
generalization of the gradient
:
and where gαβ is the metric tensor
, which here has been chosen for flat spacetime with the signature [ + - - - ]
and is sometimes also represented as D.
The square of D is the four-Laplacian, which is called the d'Alembert operator
:
.
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
and 
are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol 
is reserved for the d'Alembertian.
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. In addition to the change of sign, a factor of c must be introduced due to the different units being used to measure space and time. Adding these two corrections to the above expression gives the correct definition of four-gradient:
Four-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
generalization of the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
:
and where gαβ is the metric tensor
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
, which here has been chosen for flat spacetime with the signature [ + - - - ]
and is sometimes also represented as D.
The square of D is the four-Laplacian, which is called the d'Alembert operator
D'Alembert operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator , also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named for French mathematician and physicist Jean le Rond d'Alembert...
:
.As it is the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of two four-vectors, the d'Alembertian is a Lorentz invariant scalar.
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.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 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. In addition to the change of sign, a factor of c must be introduced due to the different units being used to measure space and time. Adding these two corrections to the above expression gives the correct definition of four-gradient: