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.
Notation
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 and represents a 3-space vector, e.g. .
The Greek tensor index ranges in 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):
Definition
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 ).
The covariant derivative incorporates the 4-gradient plus spacetime curvature effects via the Christoffel symbols
The strong equivalence principle can be stated as:
"Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as the "comma to semi-colon rule".
So, for example, if in SR, then in GR.
On a (1,0)-tensor or 4-vector this would be:
On a (2,0)-tensor this would be:
Usage
The 4-gradient is used in a number of different ways in special relativity (SR):
Throughout this article the formulas are all correct for the flat spacetime Minkowski coordinates of SR, but have to be modified for the more general curved space coordinates of general relativity (GR).
As a 4-divergence and source of conservation laws
Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. Note that in this metric signature [+,−,−,−] the 4-Gradient has a negative spatial component. It gets canceled when taking the 4D dot product since the Minkowski Metric is Diagonal[+1,−1,−1,−1].
The 4-divergence of the 4-position gives the dimension of spacetime:
The 4-divergence of the 4-current density gives a conservation law – the conservation of charge:
This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density .
In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.
The 4-divergence of the 4-number flux (4-dust) is used in particle conservation:
This is a conservation law for the particle number density, typically something like baryon number density.
The 4-divergence of the electromagnetic 4-potential is used in the Lorenz gauge condition:
This is the equivalent of a conservation law for the EM 4-potential.
The 4-divergence of the transverse traceless 4D (2,0)-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).
The transverse condition
is the equivalent of a conservation equation for freely propagating gravitational waves.