Relativistic Electromagnetism

The Relativistic Maxwell Equations

The transformation of electromagnetic field equations under a Lorentz boost in an arbitrary direction.

This essay derives the relativistic transformation of Maxwell's equations using multivariable calculus. After establishing the required differential identities, it transforms divergence and curl under a general Lorentz boost and applies the result to the vacuum field equations.

Prerequisites

Lorentz Transformations
Multivariable and vector calculus
Maxwell's equations in vacuum

I. Introduction
1. Definitions and Useful Formulas from Multivariate Calculus
II. Transformation of Differential Operators
1. Divergence in $F$ and $F^{'}$
2. Curl in $F$ and $F^{'}$
III. The Relativistic Maxwell Equations
1. Step 1: Divergence-Free Fields
2. Step 2: The Curl of the Triple Product

I. Introduction

In this segment, we derive the transformation of Maxwell's equations under a Lorentz boost in an arbitrary direction. Modern treatments almost invariably rely on the language of four-vectors and tensors, through which the relativistic covariance of Maxwell's equations follows naturally. While this formalism is undoubtedly the most elegant framework for relativistic electrodynamics, it often conceals the multivariable calculus underlying the transformation of the familiar differential operators. Our approach follows a different path. Inspired by Einstein's original derivation, we work entirely within the framework of classical multivariable calculus, extending the analysis from boosts along the coordinate axes to a Lorentz boost in an arbitrary direction. Although this approach is considerably longer than the tensor derivation, it exposes every intermediate step and makes explicit how the gradient, divergence, curl, and time derivative transform between inertial reference frames. Understanding these transformations in detail provides valuable intuition for the geometric ideas that the tensor formalism later encapsulates. We begin by introducing the multivariable calculus identities required throughout the chapter. We then derive the transformation laws for the differential operators under a Lorentz transformation. Finally, these results are combined to obtain the relativistic form of Maxwell's equations in a charge-free vacuum.

1. Definitions and Useful Formulas from Multivariate Calculus

We will consider a stationary system of reference where the coordinates of an event are represented by the four-vector

(x_{1}, x_{2}, x_{3}, t)

, and a moving frame where the same event has coordinates

x_{1}^{'}, x_{2}^{'}, x_{3}^{'}, t^{'}

. The fourth coordinate in both four-vectors has units of time, not space, as the mathematics of the Maxwell equations derivation keeps the notation considerably lighter throughout the derivation. A vector field

A

is a three-dimensional valued function acting on a four-vector,

A (x_{1}, x_{2}, x_{3}, t) A : R^{4} \to R^{3}

A few differential operators of

A

are extensively used in the derivation below and deserve to be explicitly defined to minimize confusion when the notation will get a little heavy, as it often does in the derivation of the relativistic Maxwell equations.

•
The Jacobian on the spatial dimensions and time:
$\nabla A_{x, t} = \frac{\partial A _{1}}{\partial x _{1}} \frac{\partial A _{2}}{\partial x _{1}} \frac{\partial A _{3}}{\partial x _{1}} \frac{\partial A _{1}}{\partial x _{2}} \frac{\partial A _{2}}{\partial x _{2}} \frac{\partial A _{3}}{\partial x _{2}} \frac{\partial A _{1}}{\partial x _{3}} \frac{\partial A _{2}}{\partial x _{3}} \frac{\partial A _{3}}{\partial x _{3}} \frac{\partial A _{1}}{\partial t} \frac{\partial A _{2}}{\partial t} \frac{\partial A _{3}}{\partial t}$
•
The Jacobian on the spatial dimensions:
$\nabla A_{x} = \frac{\partial A _{1}}{\partial x _{1}} \frac{\partial A _{2}}{\partial x _{1}} \frac{\partial A _{3}}{\partial x _{1}} \frac{\partial A _{1}}{\partial x _{2}} \frac{\partial A _{2}}{\partial x _{2}} \frac{\partial A _{3}}{\partial x _{2}} \frac{\partial A _{1}}{\partial x _{3}} \frac{\partial A _{2}}{\partial x _{3}} \frac{\partial A _{3}}{\partial x _{3}}$
•
The product of the spatial Jacobian with a vector $v$ , typically the relative velocity between two inertial frames:
$\nabla_{x} A v = \frac{\partial A _{1}}{\partial x _{1}} v_{1} + \frac{\partial A _{1}}{\partial x _{2}} v_{2} + \frac{\partial A _{1}}{\partial x _{3}} v_{3} \frac{\partial A _{2}}{\partial x _{1}} v_{1} + \frac{\partial A _{2}}{\partial x _{2}} v_{2} + \frac{\partial A _{2}}{\partial x _{3}} v_{3} \frac{\partial A _{3}}{\partial x _{1}} v_{1} + \frac{\partial A _{3}}{\partial x _{2}} v_{2} + \frac{\partial A _{3}}{\partial x _{3}} v_{3} .$
•
The divergence of a vector field $A$ :
$\nabla \cdot A = \frac{\partial A _{1}}{\partial x _{1}} + \frac{\partial A _{2}}{\partial x _{2}} + \frac{\partial A _{3}}{\partial x _{3}}$
•
And finally the curl of a vector field $A$ :
$\nabla \times A = (\frac{\partial A _{3}}{\partial x _{2}} - \frac{\partial A _{2}}{\partial x _{3}}) e_{1} - (\frac{\partial A _{3}}{\partial x _{1}} - \frac{\partial A _{1}}{\partial x _{3}}) e_{2} + (\frac{\partial A _{2}}{\partial x _{1}} - \frac{\partial A _{1}}{\partial x _{2}}) e_{3}$
where $e_{1}$ , $e_{2}$ and $e_{3}$ are unit vectors along the three spatial axes.

A few identities from calculus are key to derive the relativistic Maxwell equations, and they are listed below:

•
The divergence and the curl of the cross product of a constant vector $v$ and a field $A$ :
Eq:Divergence_curl_cross_product
$\nabla \times (v \times A) \nabla \cdot (v \times A) = - \nabla_{x} A \cdot v + (\nabla \cdot A) v = - ⟨ v ∣ \nabla \times A ⟩$
•
The curl of the triple product of a constant vector $v$ with a field $A$ , which has two different expressions, the latter derived with the use of the equations above:
Eq:Curl_triple_product
$\nabla \times (v \times (v \times A)) \nabla \times (v \times (v \times A)) = \nabla \times (⟨ v ∣ A ⟩ v - v^{2} A) = - \nabla_{x} (v \times A) \cdot v + \nabla \cdot (v \times A) v = - v \times (\nabla_{x} A \cdot v) - ⟨ v ∣ \nabla \times A ⟩ v$

II. Transformation of Differential Operators

Before deriving the relativistic Maxwell equations, we first determine how the differential operators transform under a Lorentz transformation. We remind the reader of the definition,

γ = \frac{1}{1 - \frac{v ^{2}}{c ^{2}}} .

We consider a field

A

F

A = A (x_{1}, x_{2}, x_{3}, t) A : R^{4} \to R^{3}

which we compose with the inverse of the Lorentz transformation from

F^{'}

F

(see The Transformation of Time and Space ) obtaining a field

A^{'}

defined on the space

F^{'}

A^{'} A^{'} A^{'} = A \circ Λ (- v) = A^{'} (x^{'}, t^{'}) = A (Λ (- v) (x^{'}, t^{'})) : R^{4} \to R^{3}

Since the Maxwell equations are in

F

coordinates, we need to understand how the partial derivatives of the field

A^{'}

transform under the Lorentz transformation, which we are able to compute from the Jacobian chain rule,

\nabla_{x^{'}, t^{'}} A^{'} = \nabla_{x, t} A \cdot \nabla_{x^{'}, t^{'}} (Λ (- v) (x^{'}, t^{'}))

where

\nabla_{x^{'}, t^{'}} A^{'}

and

\nabla_{x, t} A

are the Jacobians of

A^{'}

and

A

respectively, computed in their own coordinate systems. The Lorentz transformation is linear, hence the Jacobian is

Λ

, and the chain rule becomes, in coordinate form,

\frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} \frac{\partial A _{2}^{'}}{\partial x _{1}^{'}} \frac{\partial A _{3}^{'}}{\partial x _{1}^{'}} \frac{\partial A _{1}^{'}}{\partial x _{2}^{'}} \frac{\partial A _{2}^{'}}{\partial x _{2}^{'}} \frac{\partial A _{3}^{'}}{\partial x _{2}^{'}} \frac{\partial A _{1}^{'}}{\partial x _{3}^{'}} \frac{\partial A _{2}^{'}}{\partial x _{3}^{'}} \frac{\partial A _{3}^{'}}{\partial x _{3}^{'}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} \frac{\partial A _{2}^{'}}{\partial t ^{'}} \frac{\partial A _{3}^{'}}{\partial t ^{'}} = \frac{\partial A _{1}}{\partial x _{1}} \frac{\partial A _{2}}{\partial x _{1}} \frac{\partial A _{3}}{\partial x _{1}} \frac{\partial A _{1}}{\partial x _{2}} \frac{\partial A _{2}}{\partial x _{2}} \frac{\partial A _{3}}{\partial x _{2}} \frac{\partial A _{1}}{\partial x _{3}} \frac{\partial A _{2}}{\partial x _{3}} \frac{\partial A _{3}}{\partial x _{3}} \frac{\partial A _{1}}{\partial t} \frac{\partial A _{2}}{\partial t} \frac{\partial A _{3}}{\partial t} \cdot Λ (- v)

To get to the partial derivatives of

A

we need to invert the equation above,

\frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} \frac{\partial A _{2}^{'}}{\partial x _{1}^{'}} \frac{\partial A _{3}^{'}}{\partial x _{1}^{'}} \frac{\partial A _{1}^{'}}{\partial x _{2}^{'}} \frac{\partial A _{2}^{'}}{\partial x _{2}^{'}} \frac{\partial A _{3}^{'}}{\partial x _{2}^{'}} \frac{\partial A _{1}^{'}}{\partial x _{3}^{'}} \frac{\partial A _{2}^{'}}{\partial x _{3}^{'}} \frac{\partial A _{3}^{'}}{\partial x _{3}^{'}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} \frac{\partial A _{2}^{'}}{\partial t ^{'}} \frac{\partial A _{3}^{'}}{\partial t ^{'}} \cdot Λ (v) = \frac{\partial A _{1}}{\partial x _{1}} \frac{\partial A _{2}}{\partial x _{1}} \frac{\partial A _{3}}{\partial x _{1}} \frac{\partial A _{1}}{\partial x _{2}} \frac{\partial A _{2}}{\partial x _{2}} \frac{\partial A _{3}}{\partial x _{2}} \frac{\partial A _{1}}{\partial x _{3}} \frac{\partial A _{2}}{\partial x _{3}} \frac{\partial A _{3}}{\partial x _{3}} \frac{\partial A _{1}}{\partial t} \frac{\partial A _{2}}{\partial t} \frac{\partial A _{3}}{\partial t}

The bra-ket formalism helps us here once again to compute the product of the Jacobian matrices. Any row

j

\nabla_{x^{'}, t^{'}} A^{'} \cdot Λ (v)

is the vector

⟨ \frac{\partial A _{j}^{'}}{\partial x _{1}^{'}}, \frac{\partial A _{j}^{'}}{\partial x _{2}^{'}}, \frac{\partial A _{j}^{'}}{\partial x _{3}^{'}}, \frac{\partial A _{j}^{'}}{\partial t ^{'}} [\frac{γ}{v ^{2}} ∣ v, 0 ⟩ ⟨ v, 0 ∣ - γ ∣ v, 0 ⟩ ⟨ 0, 1 ∣ + (I - \frac{1}{v ^{2}} ∣ v, 0 ⟩ ⟨ v, 0 ∣) - \frac{γ}{c ^{2}} ∣ 0, 1 ⟩ ⟨ v, 0 ∣ + γ ∣ 0, 1 ⟩ ⟨ 0, 1 ∣]

where

I

is the identity operator, and reads,

\frac{γ}{v ^{2}} ⟨ \nabla_{x^{'}} A_{j}^{'} ∣ v ⟩ ⟨ v, 0 ∣ - γ ⟨ \nabla_{x^{'}} A_{j}^{'} ∣ v ⟩ ⟨ 0, 1 ∣ + ⟨ \nabla_{x^{'}} A_{j}^{'}, 0 - \frac{1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{j}^{'} ∣ v ⟩ ⟨ v, 0 ∣ - \frac{γ}{c ^{2}} \frac{\partial A _{j}^{'}}{\partial t ^{'}} ⟨ v, 0 ∣ + γ \frac{\partial A _{j}^{'}}{\partial t ^{'}} ⟨ 0, 1 ∣

The spatial gradient of any component of the field

A

computed at

(x, t) = Λ (- v) (x^{'}, t^{'})

is,

Eq:Gradient_in_F

\nabla_{x} A_{j} = \nabla_{x^{'}} A_{j}^{'} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{j}^{'} ∣ v ⟩ ⟨ v ∣ - \frac{γ}{c ^{2}} \frac{\partial A _{j}^{'}}{\partial t ^{'}} ⟨ v ∣

and the time derivative is

Eq:Time_derivative_in_F

\frac{\partial A _{j}}{\partial t} = - γ ⟨ \nabla_{x^{'}} A_{j}^{'} ∣ v ⟩ + γ \frac{\partial A _{j}^{'}}{\partial t ^{'}}

where,

\nabla_{x^{'}} A_{j}^{'} \nabla_{x} A_{j} = ⟨ \frac{\partial A _{j}^{'}}{\partial x _{1}^{'}}, \frac{\partial A _{j}^{'}}{\partial x _{2}^{'}}, \frac{\partial A _{j}^{'}}{\partial x _{3}^{'}} = ⟨ \frac{\partial A _{j}}{\partial x _{1}}, \frac{\partial A _{j}}{\partial x _{2}}, \frac{\partial A _{j}}{\partial x _{3}}

Equations gradient in F and time derivative in F provide the transformation law for the first-order partial derivatives of any vector field. We now use these expressions to derive the transformation laws for the differential operators appearing in Maxwell's equations.

1. Divergence in $F$ and $F^{'}$

We now use gradient in F to compute the divergence of the field

A

F

\frac{\partial A _{1}}{\partial x _{1}} \frac{\partial A _{2}}{\partial x _{2}} \frac{\partial A _{3}}{\partial x _{3}} = \frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{1}^{'} ∣ v ⟩ v_{1} - \frac{γ}{c ^{2}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} v_{1} = \frac{\partial A _{2}^{'}}{\partial x _{2}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{2}^{'} ∣ v ⟩ v_{2} - \frac{γ}{c ^{2}} \frac{\partial A _{2}^{'}}{\partial t ^{'}} v_{2} = \frac{\partial A _{3}^{'}}{\partial x _{3}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{3}^{'} ∣ v ⟩ v_{3} - \frac{γ}{c ^{2}} \frac{\partial A _{3}^{'}}{\partial t ^{'}} v_{3}

To derive an expression for the field divergence, we expand the scalar products in the equation above and group the terms on the partial derivative along each coordinate. We will work out the details for the first equation, as the others are identically derived by just changing the indexes.

\frac{\partial A _{1}}{\partial x _{1}} = \frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{1}^{'} ∣ v ⟩ v_{1} - \frac{γ}{c ^{2}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} v_{1} = \frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} + \frac{γ - 1}{v ^{2}} (\frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} v_{1} + \frac{\partial A _{1}^{'}}{\partial x _{2}^{'}} v_{2} + \frac{\partial A _{1}^{'}}{\partial x _{3}^{'}} v_{3}) v_{1} - \frac{γ}{c ^{2}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} v_{1} = (1 + \frac{γ - 1}{v ^{2}} v_{1}^{2}) \frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} + \frac{γ - 1}{v ^{2}} (\frac{\partial ( A _{1}^{'} v _{1} )}{\partial x _{2}^{'}} v_{2} + \frac{\partial ( A _{1}^{'} v _{1} )}{\partial x _{3}^{'}} v_{3}) - \frac{γ}{c ^{2}} \frac{\partial ( A _{1}^{'} v _{1} )}{\partial t ^{'}}

The divergence of the field in

F

is,

\frac{\partial A _{1}}{\partial x _{1}} + \frac{\partial A _{2}}{\partial x _{2}} + \frac{\partial A _{3}}{\partial x _{3}} = (1 + \frac{γ - 1}{v ^{2}} v_{1}^{2}) \frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} + \frac{γ - 1}{v ^{2}} (\frac{\partial ( A _{1}^{'} v _{1} )}{\partial x _{2}^{'}} v_{2} + \frac{\partial ( A _{1}^{'} v _{1} )}{\partial x _{3}^{'}} v_{3}) - \frac{γ}{c ^{2}} \frac{\partial ( A _{1}^{'} v _{1} )}{\partial t ^{'}} + (1 + \frac{γ - 1}{v ^{2}} v_{2}^{2}) \frac{\partial A _{2}^{'}}{\partial x _{2}^{'}} + \frac{γ - 1}{v ^{2}} (\frac{\partial ( A _{2}^{'} v _{2} )}{\partial x _{1}^{'}} v_{1} + \frac{\partial ( A _{2}^{'} v _{2} )}{\partial x _{3}^{'}} v_{3}) - \frac{γ}{c ^{2}} \frac{\partial ( A _{2}^{'} v _{2} )}{\partial t ^{'}} + (1 + \frac{γ - 1}{v ^{2}} v_{3}^{2}) \frac{\partial A _{3}^{'}}{\partial x _{3}^{'}} + \frac{γ - 1}{v ^{2}} (\frac{\partial ( A _{3}^{'} v _{3} )}{\partial x _{1}^{'}} v_{1} + \frac{\partial ( A _{3}^{'} v _{3} )}{\partial x _{2}^{'}} v_{2}) - \frac{γ}{c ^{2}} \frac{\partial ( A _{3}^{'} v _{3} )}{\partial t ^{'}}

By grouping the terms by the components of the field

A

, we obtain the divergence of the field

A

F

\frac{\partial A _{1}}{\partial x _{1}} + \frac{\partial A _{2}}{\partial x _{2}} + \frac{\partial A _{3}}{\partial x _{3}} = \frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} + \frac{\partial A _{2}^{'}}{\partial x _{2}^{'}} + \frac{\partial A _{3}^{'}}{\partial x _{3}^{'}} + \frac{γ - 1}{v ^{2}} (\frac{\partial A _{1}^{'}}{\partial x _{1}^{'}} v_{1} + \frac{\partial A _{1}^{'}}{\partial x _{2}^{'}} v_{2} + \frac{\partial A _{1}^{'}}{\partial x _{3}^{'}} v_{3}) v_{1} + \frac{γ - 1}{v ^{2}} (\frac{\partial A _{2}^{'}}{\partial x _{1}^{'}} v_{1} + \frac{\partial A _{2}^{'}}{\partial x _{2}^{'}} v_{2} + \frac{\partial A _{2}^{'}}{\partial x _{3}^{'}} v_{3}) v_{2} + \frac{γ - 1}{v ^{2}} (\frac{\partial A _{3}^{'}}{\partial x _{1}^{'}} v_{1} + \frac{\partial A _{3}^{'}}{\partial x _{2}^{'}} v_{2} + \frac{\partial A _{3}^{'}}{\partial x _{3}^{'}} v_{3}) v_{3} - \frac{γ}{c ^{2}} \frac{\partial ( A _{1}^{'} v _{1} + A _{2}^{'} v _{2} + A _{3}^{'} v _{3} )}{\partial t ^{'}}

We get the compact equation for the transformation of the divergence of

A

A^{'}

\nabla \cdot A = \nabla \cdot A^{'} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A^{'} v ∣ v ⟩ - \frac{γ}{c ^{2}} \frac{\partial ⟨ A ^{'} ∣ v ⟩}{\partial t ^{'}}

If we consider a divergence-free field

A

F

, the equation above gives an important condition on the divergence of

A^{'}

Eq:Divergence_in_Fprime

\nabla \cdot A^{'} = - \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A^{'} v ∣ v ⟩ + \frac{γ}{c ^{2}} \frac{\partial ⟨ A ^{'} ∣ v ⟩}{\partial t ^{'}}

2. Curl in $F$ and $F^{'}$

The other operator needed for the Maxwell equations is the curl,

(\nabla \times A)_{1} (\nabla \times A)_{2} (\nabla \times A)_{3} = \frac{\partial A _{3}}{\partial x _{2}} - \frac{\partial A _{2}}{\partial x _{3}} = \frac{\partial A _{3}^{'}}{\partial x _{2}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{3}^{'} ∣ v ⟩ v_{2} - \frac{γ}{c ^{2}} \frac{\partial A _{3}^{'}}{\partial t ^{'}} v_{2} - \frac{\partial A _{2}^{'}}{\partial x _{3}^{'}} - \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{2}^{'} ∣ v ⟩ v_{3} + \frac{γ}{c ^{2}} \frac{\partial A _{2}^{'}}{\partial t ^{'}} v_{3} = - \frac{\partial A _{3}}{\partial x _{1}} + \frac{\partial A _{1}}{\partial x _{3}} = - \frac{\partial A _{3}^{'}}{\partial x _{1}^{'}} - \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{3}^{'} ∣ v ⟩ v_{1} + \frac{γ}{c ^{2}} \frac{\partial A _{3}^{'}}{\partial t ^{'}} v_{1} + \frac{\partial A _{1}^{'}}{\partial x _{3}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{1}^{'} ∣ v ⟩ v_{3} - \frac{γ}{c ^{2}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} v_{3} = \frac{\partial A _{2}}{\partial x _{1}} - \frac{\partial A _{1}}{\partial x _{2}} = \frac{\partial A _{2}^{'}}{\partial x _{1}^{'}} + \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{2}^{'} ∣ v ⟩ v_{1} - \frac{γ}{c ^{2}} \frac{\partial A _{2}^{'}}{\partial t ^{'}} v_{1} - \frac{\partial A _{1}^{'}}{\partial x _{2}^{'}} - \frac{γ - 1}{v ^{2}} ⟨ \nabla_{x^{'}} A_{1}^{'} ∣ v ⟩ v_{2} + \frac{γ}{c ^{2}} \frac{\partial A _{1}^{'}}{\partial t ^{'}} v_{2}

which is in vector form easily recognisable as,

Eq:Curl_in_F

\nabla \times A = \nabla \times A^{'} + \frac{γ - 1}{v ^{2}} v \times (\nabla_{x^{'}} A^{'} v) - \frac{γ}{c ^{2}} \frac{\partial v \times A ^{'}}{\partial t ^{'}}

III. The Relativistic Maxwell Equations

We consider an electric field

E

and a magnetic field

B

in a region of the system of reference

F

free of charges and currents. The Maxwell equations in this case are

Eq:Maxwell_in_F

\frac{\partial B}{\partial t} \frac{\partial E}{\partial t} \nabla \cdot B \nabla \cdot E = - \nabla \times E = c^{2} \nabla \times B = 0 = 0

The fields

B^{'}

and

E^{'}

, obtained by applying the composition of the Lorentz transformation are,

B^{'} E^{'} = B^{'} (x^{'}, t^{'}) = B (Λ (- v) (x^{'}, t^{'})) = E^{'} (x^{'}, t^{'}) = E (Λ (- v) (x^{'}, t^{'}))

and fulfill the equations obtained by applying directly the Lorentz transformation to equations maxwell in F (see section Transformation of Differential Operators ),

Eq:Maxwell_transformed

- γ \nabla_{x^{'}} B^{'} v + γ \frac{\partial B ^{'}}{\partial t ^{'}} - \frac{γ}{c ^{2}} \nabla_{x^{'}} E^{'} v + \frac{γ}{c ^{2}} \frac{\partial E ^{'}}{\partial t ^{'}} = - \nabla \times E^{'} - \frac{γ - 1}{v ^{2}} v \times (\nabla_{x^{'}} E^{'} v) + \frac{γ}{c ^{2}} \frac{\partial v \times E ^{'}}{\partial t ^{'}}, = \nabla \times B^{'} + \frac{γ - 1}{v ^{2}} v \times (\nabla_{x^{'}} B^{'} v) - \frac{γ}{c ^{2}} \frac{\partial v \times B ^{'}}{\partial t ^{'}} .

Since the equations at hand do not conform to the standard form of Maxwell's equations, the fields denoted by

B^{'}

and

E^{'}

do not represent the transformed electric and magnetic fields we seek. Substantial effort is required to manipulate these equations into the correct form that yields the desired fields. We outline below the steps required to derive the relativistic Maxwell equations in a vacuum, starting with the equations above and manipulating them to obtain the correct form.

1. Step 1: Divergence-Free Fields

Taking the scalar product of equations maxwell transformed with the velocity

v

gives

- γ ⟨ \nabla_{x^{'}} B^{'} v ∣ v ⟩ + γ \frac{\partial ⟨ B ^{'} ∣ v ⟩}{\partial t ^{'}} - \frac{γ}{c ^{2}} ⟨ \nabla_{x^{'}} E^{'} v ∣ v ⟩ + \frac{γ}{c ^{2}} \frac{\partial ⟨ E ^{'} ∣ v ⟩}{\partial t ^{'}} = - ⟨ \nabla \times E^{'} ∣ v ⟩, = ⟨ \nabla \times B^{'} ∣ v ⟩ .

Together with the divergence-free condition divergence in Fprime , these equations give useful expressions for the divergences of

B^{'}

and

E^{'}

\nabla \cdot B^{'} \nabla \cdot E^{'} = \frac{γ - 1}{γ v ^{2}} (\frac{\partial ⟨ B ^{'} ∣ v ⟩}{\partial t ^{'}} - ⟨ \nabla \times E^{'} ∣ v ⟩), = \frac{γ - 1}{γ v ^{2}} (\frac{\partial ⟨ E ^{'} ∣ v ⟩}{\partial t ^{'}} + c^{2} ⟨ \nabla \times B^{'} ∣ v ⟩) .

The terms

\nabla_{x^{'}} B^{'} v

and

\nabla_{x^{'}} E^{'} v

on the left-hand side of maxwell transformed can be substituted using divergence curl cross product . Using also the expressions for the divergences of

B^{'}

and

E^{'}

just obtained, we find

Eq:Divergence_free_fields

\frac{\partial}{\partial t ^{'}} (γ B^{'} - \frac{γ - 1}{v ^{2}} ⟨ B^{'} ∣ v ⟩ v - \frac{γ}{c ^{2}} v \times E^{'}) \frac{\partial}{\partial t ^{'}} (γ E^{'} - \frac{γ - 1}{v ^{2}} ⟨ E^{'} ∣ v ⟩ v + γ v \times B^{'}) = - γ \nabla \times (v \times B^{'}) - \nabla \times E^{'} - \frac{γ - 1}{v ^{2}} ⟨ \nabla \times E^{'} ∣ v ⟩ v - \frac{γ - 1}{v ^{2}} v \times (\nabla_{x^{'}} E^{'} v), = - γ \nabla \times (v \times E^{'}) + \nabla \times (c^{2} B^{'}) + \frac{γ - 1}{v ^{2}} ⟨ \nabla \times (c^{2} B^{'}) ∣ v ⟩ v + \frac{γ - 1}{v ^{2}} v \times (\nabla_{x^{'}} (c^{2} B^{'}) v) .

We are nearing the solution. It is now necessary to evaluate the curl of the triple cross product.

2. Step 2: The Curl of the Triple Product

Equations divergence free fields bring us closer to the solution. On the left-hand side, we have the partial derivatives with respect to

t^{'}

of two fields. On the right-hand side, several terms are already expressed as curls, but the following terms are not:

- \frac{γ - 1}{v ^{2}} ⟨ \nabla \times E^{'} ∣ v ⟩ v - \frac{γ - 1}{v ^{2}} v \times (\nabla_{x^{'}} E^{'} v) .

The key to the solution is the curl of the triple cross product, given by curl triple product :

\nabla \times (v \times (v \times E^{'})) = \nabla \times (⟨ v ∣ E^{'} ⟩ v - v^{2} E^{'}) = - v \times (\nabla_{x^{'}} E^{'} v) - ⟨ v ∣ \nabla \times E^{'} ⟩ v .

Substitution into divergence free fields finally yields equations having the same differential-operator form as maxwell in F , but now expressed in the coordinates of

F^{'}

Eq:Pre_final_maxwell

\frac{\partial}{\partial t ^{'}} (γ B^{'} - (γ - 1) ⟨ B^{'} ∣ \frac{v}{v} ⟩ \frac{v}{v} - \frac{γ}{c ^{2}} v \times E^{'}) \frac{\partial}{\partial t ^{'}} (γ E^{'} - (γ - 1) ⟨ E^{'} ∣ \frac{v}{v} ⟩ \frac{v}{v} + γ v \times B^{'}) = - \nabla \times (γ E^{'} - (γ - 1) ⟨ E^{'} ∣ \frac{v}{v} ⟩ \frac{v}{v} + γ v \times B^{'}), = c^{2} \nabla \times (γ B^{'} - (γ - 1) ⟨ B^{'} ∣ \frac{v}{v} ⟩ \frac{v}{v} - \frac{γ}{c ^{2}} v \times E^{'}) .

Hence, by the principle of relativity, the electric and magnetic fields measured in

F^{'}

must be

B_{F^{'}} E_{F^{'}} = ϕ (v) (γ B^{'} - (γ - 1) ⟨ B^{'} ∣ \frac{v}{v} ⟩ \frac{v}{v} - \frac{γ}{c ^{2}} v \times E^{'}), = ϕ (v) (γ E^{'} - (γ - 1) ⟨ E^{'} ∣ \frac{v}{v} ⟩ \frac{v}{v} + γ v \times B^{'}),

where

ϕ (v)

is a scalar function of the boost velocity. The possible values of

ϕ (v)

can be constrained by requiring that a boost by

v

followed by the inverse boost

- v

reproduce the original fields:

B E = ϕ (- v) (γ B_{F^{'}} - (γ - 1) ⟨ B_{F^{'}} ∣ \frac{v}{v} ⟩ \frac{v}{v} + \frac{γ}{c ^{2}} v \times E_{F^{'}}), = ϕ (- v) (γ E_{F^{'}} - (γ - 1) ⟨ E_{F^{'}} ∣ \frac{v}{v} ⟩ \frac{v}{v} - γ v \times B_{F^{'}}) .

This gives

ϕ (- v) ϕ (v) = 1.

By spatial isotropy, the scalar

ϕ

can depend only on the magnitude of the boost velocity, and therefore

ϕ (- v) = ϕ (v) .

Consequently,

ϕ (v)^{2} = 1.

The transformed fields must approach the original fields as

v \to 0

v \to 0 lim B_{F^{'}} v \to 0 lim ϕ (v) = B, = 1.

Continuity therefore selects

ϕ (v) = 1.

The final transformation laws for the electric and magnetic fields are

Eq:Relativistic_maxwell

B_{F^{'}} E_{F^{'}} = γ B - (γ - 1) ⟨ B ∣ \frac{v}{v} ⟩ \frac{v}{v} - \frac{γ}{c ^{2}} v \times E, = γ E - (γ - 1) ⟨ E ∣ \frac{v}{v} ⟩ \frac{v}{v} - γ v \times B .

Contents

I. Introduction

1. Definitions and Useful Formulas from Multivariate Calculus

II. Transformation of Differential Operators

1. Divergence in F and F′

2. Curl in F and F′

III. The Relativistic Maxwell Equations

1. Step 1: Divergence-Free Fields

2. Step 2: The Curl of the Triple Product

1. Divergence in $F$ and $F^{'}$

2. Curl in $F$ and $F^{'}$