Four-gradient

In differential geometry, the four-gradient (4-gradient) is the four-vector (4-vector) analogue of the gradient from Gibbs-Heaviside vector calculus.

In special relativity, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors.

Definition

The covariant components compactly written in tensor index notation are:[1]

\dfrac{\partial}{\partial x^\alpha} = \left(\frac{1}{c}\frac{\partial}{\partial t}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \vec{\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:[2]

\mathbf{\partial} = \partial^\alpha \ = g^{\alpha \beta} \partial_\beta = \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)

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

Alternative symbols to \partial_\alpha are \Box and D.

Usage

The four-gradient is used in a number of different ways in special relativity (SR):

Note that, throughout this article, the formulas are correct for Minkowski coordinates in SR, but may need to be modified for other coordinates.

Note also that there are alternate ways of writing the expressions:

\mathbf{\partial} \cdot \mathbf{X} is a four-vector style, which is typically more compact and can use dot notation, always using bold uppercase to represent the four-vector.
\partial^\mu \eta_{\mu\nu} X^\nu is a tensor index style, which is sometimes required in more complicated expressions, especially those involving tensors with more than one index, such as F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu.
The tensor contraction used in the Minkowski metric can go to either side:
A^\mu \eta_{\mu\nu} B^\nu = A_\nu B^\nu = A^\mu B_\mu = \sum_{ \nu \mathop =0..3}[a_\nu * b^\nu] = \sum_{ \mu \mathop =0..3}[a^\mu * b_\mu]

As a four-divergence

Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point.

The four-divergence of the four-position X^\mu gives the dimension of spacetime:

\mathbf{\partial} \cdot \mathbf{X} = \partial^\mu \eta_{\mu\nu} X^\nu = (\frac{\partial_t}{c},-\vec{\nabla})\cdot (ct,\vec{x}) = \frac{\partial_t}{c}(ct) + \vec{\nabla}\cdot \vec{x} = (\partial_t t) + (\partial_x x+\partial_y y+\partial_z z) = (1) + (3) = 4


The four-divergence of the four-current J^\mu = \rho_o U^\mu gives a conservation law - the conservation of charge:

\mathbf{\partial} \cdot \mathbf{J} = \partial^\mu \eta_{\mu\nu} J^\nu = (\frac{\partial_t}{c},-\vec{\nabla})\cdot (\rho c,\vec{j}) = \frac{\partial_t}{c}(\rho c) + \vec{\nabla}\cdot \vec{j} =\partial_t \rho + \vec{\nabla}\cdot \vec{j} = 0

This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density \partial_t \rho = -\vec{\nabla}\cdot \vec{j}.

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 four-divergence of the electromagnetic four-potential A^\mu is used in the Lorenz gauge condition:

\mathbf{\partial} \cdot \mathbf{A} = \partial^\mu \eta_{\mu\nu} A^\nu = (\frac{\partial_t}{c},-\vec{\nabla})\cdot (\frac{\phi}{c},\vec{a}) = \frac{\partial_t}{c}(\frac{\phi}{c}) + \vec{\nabla}\cdot \vec{a} =\frac{\partial_t \phi}{c^2} + \vec{\nabla}\cdot \vec{a} = 0

As a Jacobian matrix for the SR metric tensor

The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

The four-gradient \partial^\mu acting on the four-position X^\nu gives the SR Minkowski space metric \eta^{\mu\nu}. :

\mathbf{\partial} [\mathbf{X}] = \partial^\mu[X^\nu] = X^{\nu_,\mu} = (\frac{\partial_t}{c},-\vec{\nabla})[(ct,\vec{x})] = (\frac{\partial_t}{c},-\partial_x,-\partial_y,-\partial_z)[(ct,x,y,z)],
= \begin{bmatrix}\frac{\partial_t}{c} ct & \frac{\partial_t}{c} x & \frac{\partial_t}{c} y & \frac{\partial_t}{c} z \\ -\partial_x ct & -\partial_x x & -\partial_x y & -\partial_x z \\ -\partial_y ct & -\partial_y x & -\partial_y y & -\partial_y z \\ -\partial_z ct & -\partial_z x & -\partial_z y & -\partial_z z\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{bmatrix} = Diag[1,-1,-1,-1]
\mathbf{\partial} [\mathbf{X}] = \eta^{\mu\nu}

For the Minkowski metric, the components [\eta^{\mu\nu}] = [\eta_{\mu\nu}]

As part of the total proper time derivative

The scalar product of four-velocity U^\mu with the four-gradient gives the total derivative with respect to proper time \frac{d}{d\tau}:

\mathbf{U} \cdot \mathbf{\partial} =U^\mu \eta_{\mu\nu} \partial^\nu = \gamma(c,\vec{u}) \cdot (\frac{\partial_t}{c},-\vec{\nabla}) = \gamma (c \frac{\partial_t}{c} + \vec{u} \cdot \vec{\nabla} )= \gamma (\partial_t + \frac{dx}{dt} \partial_x + \frac{dy}{dt} \partial_y + \frac{dz}{dt} \partial_z) = \gamma \frac{d}{dt} = \frac{d}{d\tau}

The fact that \mathbf{U} \cdot \mathbf{\partial} is a Lorentz scalar invariant shows that the total derivative with respect to proper time \frac{d}{d\tau} is likewise a Lorentz scalar invariant.

So, for example, the four-velocity U^\mu is the proper-time derivative of the four-position X^\mu:

\frac{d}{d\tau} \mathbf{X} = (\mathbf{U} \cdot \mathbf{\partial})\mathbf{X} = \mathbf{U} \cdot \mathbf{\partial}[\mathbf{X}] = U^\alpha \cdot \eta^{\mu\nu} = U^\alpha \eta_{\alpha \nu} \eta^{\mu\nu} = U^\alpha \delta_\alpha^\mu = U^\mu = \mathbf{U}

or

\frac{d}{d\tau} \mathbf{X} = \gamma\frac{d}{dt} \mathbf{X} = \gamma\frac{d}{dt} (ct,\vec{x}) = \gamma (\frac{d}{dt}ct,\frac{d}{dt}\vec{x}) = \gamma (c,\vec{u}) = \mathbf{U}


Another example, the four-acceleration A^\mu is the proper-time derivative of the four-velocity U^\mu:

\frac{d}{d\tau} \mathbf{U} = (\mathbf{U} \cdot \mathbf{\partial})\mathbf{U} = \mathbf{U} \cdot \mathbf{\partial}[\mathbf{U}] = U^\alpha \eta_{\alpha\mu}\partial^\mu[U^\nu]
= U^\alpha \eta_{\alpha\mu}\begin{bmatrix} \frac{\partial_t}{c} \gamma c & \frac{\partial_t}{c} \gamma \vec{u} \\ -\vec{\nabla}\gamma c & -\vec{\nabla}\gamma \vec{u} \end{bmatrix} = U^\alpha \begin{bmatrix}\ \frac{\partial_t}{c} \gamma c & 0 \\ 0 & \vec{\nabla}\gamma \vec{u} \end{bmatrix}
= \gamma (c \frac{\partial_t}{c} \gamma c , \vec{u} \cdot \nabla\gamma \vec{u} )= \gamma (c \partial_t \gamma, \frac{d}{dt}[\gamma \vec{u}] ) = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} )= \mathbf{A}

or

\frac{d}{d\tau} \mathbf{U} =\gamma \frac{d}{dt} (\gamma c,\gamma \vec{u}) =\gamma (\frac{d}{dt}[\gamma c],\frac{d}{dt}[\gamma \vec{u}]) = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} ) = \mathbf{A}

As a way to define the Faraday electromagnetic tensor

The Faraday electromagnetic tensor is a mathematical object that describes the electromagnetic field in space-time of a physical system.

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}

where:

Electromagnetic four-potential A^\mu = \mathbf{A} = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right), not to be confused with the four-acceleration \mathbf{A} = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} )

\phi is the electric scalar potential, and \vec{\mathbf{a}} is the magnetic three-vector potential.

As a way to define the four-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 four-wavevector K^\mu is the four-gradient of the negative phase \Phi (or the negative four-gradient of the phase) of a wave in Minkowski Space

K^\mu = \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \mathbf{\partial} [-\Phi]= -\mathbf{\partial} [\Phi]

This is mathematically equivalent to the definition of the phase of a wave as:

-\Phi = \mathbf{K} \cdot \mathbf{X} = \omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}

where:

four-position \mathbf{X} = (ct, \vec{\mathbf{x}})

\omega is the temporal angular frequency and \vec{\mathbf{k}} is the spatial 3-wavevector.

As the d'Alembertian operator

The square of \mathbf{\partial} is the four-Laplacian, which is called the d'Alembert operator:

\mathbf{\partial} \cdot \mathbf{\partial} = \partial^\mu \cdot \partial^\nu = \partial^\mu \eta_{\mu\nu} \partial^\nu = \partial_\nu \partial^\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 = \left(\frac{\partial_t}{c}\right)^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 four-gradient and d'Alembertian respectively. More commonly however, the symbol \Box is reserved for the d'Alembertian.

Some examples of the four-gradient as used in the d'Alembertian follow:

In the Klein-Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs_boson):

[(\mathbf{\partial} \cdot \mathbf{\partial}) + \left(\frac {m_o c}{\hbar}\right)^2]\psi = [\left(\frac{\partial_t^2}{c^2} - \vec{\nabla}^2\right) + \left(\frac {m_o c}{\hbar}\right)^2] \psi = 0

In the wave equation for the electromagnetic field:

(\mathbf{\partial} \cdot \mathbf{\partial}) A^{\alpha} = 0 {in vacuum}
(\mathbf{\partial} \cdot \mathbf{\partial}) A^\alpha = \mu_0 J^\alpha {with a four-current source}

where:

Electromagnetic four-potential A^{\alpha} = \left(\frac{\phi}{c},\vec{a}\right) is an electromagnetic vector potential
Four-current J^{\alpha} = (\rho c,\vec{j}) is an electromagnetic current density

In the four-dimensional version of Green's_function:

(\mathbf{\partial} \cdot \mathbf{\partial}) G(x-x') = \delta^4(x-x')

As a component of the Schrödinger relations in quantum mechanics

The four-gradient is connected with quantum mechanics. The relation between the four-momentum \mathbf{P} and the four-gradient \mathbf{\partial} gives the Schrödinger QM relations.

\mathbf{P} = \left(\frac{E}{c},\vec{p}\right) = i\hbar \mathbf{\partial} = i\hbar \left(\frac{\partial_t}{c},-\vec{\nabla}\right)

The temporal component gives: E = i\hbar \partial_t

The spatial components give: \vec{p} = -i\hbar \vec{\nabla}

Derivation

In three 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}, \vec{\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 spatial 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 coordinates (the time-positive Metric convention \eta^{\mu\nu}=Diag[1,-1,-1,-1]). The factor of (1/c) is to keep the correct unit dimensionality {1/[length]} for all components of the four-vector and the (−1) is to keep the four-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}, -\vec{\nabla} \right)    correct

See also

References

  1. ↑ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2
  2. ↑ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2
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