In differential geometry, the four-gradient (or 4-gradient) <math>\boldsymbol{\partial}</math> is the four-vector analogue of the gradient <math>\vec{\boldsymbol{\nabla</math> 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.
<math>c</math> indicates the speed of light in vacuum.
<math>\eta_{\mu\nu} = \operatorname{diag}[1,-1,-1,-1]</math> 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: <math>\mathbf{A} \cdot \mathbf{B}</math>, 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. <math>\vec{\mathbf{a \cdot \vec{\mathbf{b</math>. Most of the 3-space vector rules have analogues in four-vector mathematics.
- The Ricci calculus style can be used: <math>A^\mu \eta_{\mu\nu} B^\nu</math>, which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as <math>F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu</math>.
The Latin tensor index ranges in and represents a 3-space vector, e.g. <math>A^i = \left(a^1, a^2, a^3\right) = \vec{\mathbf{a</math>.
The Greek tensor index ranges in and represents a 4-vector, e.g. <math>A^\mu = \left(a^0, a^1, a^2, a^3\right) = \mathbf{A}</math>.
In SR physics, one typically uses a concise blend, e.g. <math>\mathbf{A} = \left(a^0, \vec{\mathbf{a\right)</math>, where <math>a^0</math> represents the temporal component and <math>\vec{\mathbf{a</math> represents the spatial 3-component.
Tensors in SR are typically 4D <math>(m,n)</math>-tensors, with <math>m</math> upper indices and <math>n</math> 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):
<math display="block">\mathbf{A} \cdot \mathbf{B} = A^\mu \eta_{\mu\nu} B^\nu = A_\nu B^\nu = A^\mu B_\mu = \sum_{\mu=0}^{3} a^\mu b_\mu = a^0 b^0 - \sum_{i=1}^{3} a^i b^i = a^0 b^0 - \vec{\mathbf{a \cdot \vec{\mathbf{b</math>
Definition
The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:
<math display="block">\dfrac{\partial}{\partial X^\mu} = \left(\partial_0,\partial_1,\partial_2,\partial_3\right) = \left(\partial_0,\partial_i\right) = \left(\frac{1}{c}\frac{\partial}{\partial t}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \partial_x,\partial_y,\partial_z\right) = \partial_\mu = {}_{,\mu}</math>
The comma in the last part above <math>{}_{,\mu}</math> implies the partial differentiation with respect to 4-position <math>X^\mu</math>.
The contravariant components are:
"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 <math>T^{\mu\nu}{}_{,\mu} = 0</math> in SR, then <math>T^{\mu\nu}{}_{;\mu} = 0</math> in GR.
On a (1,0)-tensor or 4-vector this would be:
Applying the 4-gradient to make an antisymmetric tensor, one gets:
<math display="block">F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu =
\begin{bmatrix}
0 & -E_x/c & -E_y/c & -E_z/c \\
E_x/c & 0 & -B_z & B_y \\
E_y/c & B_z & 0 & -B_x \\
E_z/c & -B_y & B_x & 0
\end{bmatrix}
</math>
where:
- Electromagnetic 4-potential <math>A^\mu = \mathbf{A} = \left(\frac{\phi}{c}, \vec{\mathbf{a\right)</math>, not to be confused with the 4-acceleration <math>\mathbf{A} = \gamma \left(c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u\right)</math>
- The electric scalar potential is <math>\phi</math>
- The magnetic 3-space vector potential is <math>\vec{\mathbf{a</math>
By applying the 4-gradient again, and defining the 4-current density as <math>J^{\beta} = \mathbf{J} = \left(c\rho, \vec{\mathbf{j\right)</math> one can derive the tensor form of the Maxwell equations:
<math display="block">\partial_{\alpha} F^{\alpha\beta} = \mu_o J^{\beta}</math>
<math display="block">\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0_{\alpha \beta \gamma}</math>
where the second line is a version of the Bianchi identity (Jacobi identity).
As a way to define the 4-wavevector
A wavevector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation
The 4-wavevector <math>K^\mu</math> is the 4-gradient of the negative phase <math>\Phi</math> (or the negative 4-gradient of the phase) of a wave in Minkowski Space: