Electromagnetic four-potential

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The electromagnetic four-potential is a covariant four-vector defined in volt·seconds/meter (and in maxwell/centimeter in parentheses) as

$A_{\alpha} = \left(- \frac{\phi}{c}, \vec A \right) \qquad \left(A_{\alpha} = (- \phi, \vec A)\right)$

in which $f$ is the electrical potential, and $\vec A$ is the magnetic potential, a vector potential.

The electric and magnetic fields associated with these four-potentials are:

$\vec{E} = -\vec{\nabla} \phi - \frac{\partial \vec{A}}{\partial t} \qquad \left( -\vec{\nabla} \phi - \frac{1}{c} \frac{\partial \vec{A}}{\partial t} \right)$

$\vec{B} = \vec{\nabla} \times \vec{A}$

It is useful to group the potentials together in this form because $A a$ is a covariant vector. This means that it transforms in the same way as the gradient of a scalar, e.g. $\frac{\partial \psi}{\partial x^{\alpha}}\,,$ under arbitrary curvilinear coordinate transformations. So, for example, the inner product

$A_{\alpha} g^{\alpha \beta} A_{\beta} = |\vec{A}|^2 -\frac{\phi^2}{c^2} \qquad \left(A_{\alpha} g^{\alpha \beta} A_{\beta} \, = |\vec{A}|^2 - \phi^2 \right)$

is the same in every inertial frame of reference.

Often, physicists employ the Lorenz gauge condition $\partial_{\alpha} A^{\alpha} = 0$ in an inertial frame of reference to simplify Maxwell's equations as:

$\Box A_{\alpha} = -\mu_0 \eta_{\alpha \beta} J^{\beta} \qquad \left( \Box A_{\alpha} = -\frac{4 \pi}{c} \eta_{\alpha \beta} J^{\beta} \right)$

where $J^{\beta} \,$ are the components of the four-current,

and

$\Box = \nabla^2 -\frac{1}{c^2} \frac{\partial^2} {\partial t^2}$ is the d'Alembertian operator.

In terms of the scalar and vector potentials, this last equation becomes:

$\Box \phi = -\frac{\rho}{\epsilon_0} \qquad \left(\Box \phi = -4 \pi \rho \right)$

$\Box \vec{A} = -\mu_0 \vec{j} \qquad \left( \Box \vec{A} = -\frac{4 \pi}{c} \vec{j} \right)$

For a given charge and current distribution, $\rho(\vec{x},t)$ and $\vec{j}(\vec{x},t)$ , the solutions to these equations in SI units are

$\phi (\vec{x}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}$

$\vec A (\vec{x}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\vec{j}( \vec{x}^\prime, \tau)}{ \left| \vec{x} - \vec{x}^\prime \right|}$ ,

where $\tau = t - \frac{\left|\vec{x}-\vec{x}'\right|}{c}$ is the retarded time. This is sometimes also expressed with $\rho(\vec{x}',\tau)=[\rho(\vec{x}',t)]$ , where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary.

When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying as $r - 2$ (the induction field) and a component decreasing as $r - 1$ (the radiation field).

References

Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-19-853952-5.
Jackson, J D (1999). Classical Electrodynamics (3rd). New York: Wiley. ISBN ISBN 0-471-30932-X.

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