Tensor Formulation of Electromagnetism

In the previous section, we briefly reviewed electromagnetic theory and Maxwell's equations. In this section and the subsequent sections, we will explore a few different formulations of electromagnetism, which will help us understand the underlying structure of electromagnetism. The first formulation we will explore is the tensor formulation, which is particularly useful in the context of special relativity, as it makes relativistic invariance extremely clear.

Table of Contents

• Table of Contents
• Introduction
  • 4-Current
• Faraday Tensor
  • Continuity Equation in Tensor Form
• Maxwell's Equations in Tensor Form
• Summary and Next Steps

Introduction

Fundamentally, in a formulation of electromagnetism, we demand that the formulation provides:

1. A way to describe the electric and magnetic fields.
2. A way to describe how these fields interact with charged particles.
3. A way to describe how these fields evolve in time and space.

In the vector calculus formulation, we answer these questions as follows:

1. The electric field and magnetic field are described as vector fields.
2. The interaction of these fields with charged particles is described by the Lorentz force law, .
3. The evolution of these fields is described by Maxwell's equations, which are, here, a set of four partial differential equations.

However, this formulation is not particularly elegant, and it does not make the underlying structure of electromagnetism clear. The Lorentz invariance of Maxwell's equations is not immediately apparent, and the relationship between the electric and magnetic fields is not as clear as it could be. To address these issues, we can reformulate Maxwell's equations in a more elegant way using tensors and 4-vectors.

From the previous section we know that there are a few quantities that are useful in describing the electromagnetic field:

• The electric field , which describes the force per unit charge on a charged particle.
• The magnetic field , which describes the force per unit charge on a charged particle moving in a magnetic field.
• The scalar potential , which is related to the electric field.
• The vector potential , which is related to the magnetic field.

In relativistic physics, we often unify different quantities into a single mathematical object to simplify the equations and make the relationships between them clearer. For example, energy and momentum are unified into a single 4-momentum vector, which leads to a more elegant formulation of the laws of motion in special relativity. Likewise, we can unify the scalar and vector potentials into a single mathematical object known as the 4-potential :

As a 4-vector, it obeys the transformation rule

where is a Lorentz transformation matrix. This means that the 4-potential transforms in the same way as any other 4-vector under Lorentz transformations.

The dual of the 4-potential, denoted as , is defined as

where is the Minkowski metric tensor, which is used to lower indices of 4-vectors.

4-Current

Another important quantity in electromagnetism is the 4-current , which describes the flow of electric charge and current in spacetime. The charge density is the amount of charge per unit volume, and the current density is the flow of charge per unit area per unit time. In other words, we can consider as a spatial density of charge, and as a temporal density of charge. This motivates a unified description of charge and current in spacetime, leading to the definition of the 4-current:

Faraday Tensor

Next, consider the electric and magnetic fields in terms of the 4-potential. First, recall that the two fields are related to the potentials by

To write this more explicitly, we can consider the individual components of the electric and magnetic fields. First, we know that is just , and the vector potential has components , , and . Then, electric field components can be expressed as

Dividing by , we can rewrite these as

Similarly, the magnetic field components are given by

Notice that all six components of the electric and magnetic fields can be expressed in terms of the 4-potential and its derivatives. Notably, they all involve the format . This suggests that we can define a single mathematical object that encapsulates both the electric and magnetic fields. We can define a quantity , as

known as the Faraday tensor or electromagnetic field tensor. It is clear that this tensor is antisymmetric;

The components of the Faraday tensor can be expressed in terms of the electric and magnetic fields as follows:

As a (0, 2) tensor, it transforms under Lorentz transformations as

where is the Lorentz transformation matrix.

The contravariant form of the Faraday tensor, denoted as , is obtained by raising the indices using the Minkowski metric:

which can be evaluated as

The components of the electric and magnetic fields can be extracted from the Faraday tensor as follows:

where is the Levi-Civita symbol, which is used to express the cross product in a more compact form.

The dual tensor is defined as

It is the Hodge dual of the Faraday tensor and is useful in expressing the homogeneous Maxwell's equations. (If you are not familiar with the Hodge dual, it is a mathematical operation that takes a differential form and produces another differential form of the same degree, but in a different orientation. No need to worry about this for now, as we will not use it in this section.) Its matrix form is given by

Then, the components of the electric and magnetic fields can be extracted from the dual tensor as follows:

One can see that in the dual tensor, the roles of the electric and magnetic fields are "swapped" compared to the Faraday tensor.

Continuity Equation in Tensor Form

In the context of electromagnetism, the conservation of charge is expressed through the continuity equation. In the vector calculus formulation, the continuity equation is given by

As previously stated, and are parts of the 4-current . As such, is just , and is just . Thus, we can rewrite the continuity equation as

Recall that this is just the 4-divergence of the 4-current . As you can see, tensor notation allows us to express the continuity equation in a more compact and elegant form.

Maxwell's Equations in Tensor Form

Finally, we can express Maxwell's equations in tensor form.

We can classify Maxwell's equations into homogeneous and inhomogeneous equations. A homogeneous differential equation is one whose terms all involve the function being solved for, while an inhomogeneous differential equation has terms that do not involve the function being solved for. For example, the equation is a homogeneous differential equation, while the equation is an inhomogeneous differential equation.

Gauss's law, , and the Ampère-Maxwell law, , are inhomogeneous equations. The other two equations, Faraday's law of induction, , and Gauss's law for magnetism, , are homogeneous equations. We can express both homogeneous equations as one tensor equation, and likewise for the inhomogeneous equations.

First, notice that the divergence of the electric field can be expressed as

Since the electric field is related to the Faraday tensor by (we can see this by looking at the components of the Faraday tensor), we can rewrite this as

We can switch from the spatial index to the temporal index because the th component of the Faraday tensor is zero, so it does not contribute to the divergence.

The curl of the magnetic field can be expressed as

where we have just used the formula for a cross product but plugged in for one of the vectors. Next, plugging in , we can rewrite this as

Next, we can use the identity

to rewrite this as

The time derivative of the electric field can be expressed as

To put these together, we summarize the substitutions that we have made so far:

• (from the definition of the 4-current)
• (the spatial components of the 4-current)

Gauss's law becomes

We can divide both sides by to get

And since , we can rewrite this as

Similarly, the Ampère-Maxwell law becomes

Placing on the left-hand side, we get

Then, relabeling to in the term, we get

We can merge the two terms on the left into a single term, as and are spatial and temporal indices, respectively. They combine to a single index;

Finally, we can combine Equations and . The former describes the derivatives of , while the latter describes the derivatives of . The and indices are once again spatial and temporal indices, respectively, so we can combine them into a single index. Thus, we can write the two equations as

---

Next, consider the homogeneous equations—Gauss's law for magnetism and Faraday's law of induction. By homogeneous, we mean that the equations do not have any source terms, or terms that do not involve the electromagnetic field.

Just like before, we will replace all the terms in the equations with their tensor equivalents. This time, however, we will be leveraging the dual tensor . This is because the roles of the electric and magnetic fields are "swapped" in this derivation as compared to the previous one. As such, using the dual tensor allows the derivation to stay consistent with the structure of the equations.

The divergence of the magnetic field can be expressed as

Then, using , we can rewrite this as

Given that the divergence of the magnetic field is zero (from Gauss's law for magnetism), we can write this as

Next, we will express Faraday's law in terms of the dual tensor. The curl of the electric field can be expressed as

The time derivative of the magnetic field can be expressed as

Therefore, Faraday's law becomes

or

As and are spatial and temporal indices, respectively, we can combine them into a single index;

Once again, we can combine Equations and . The former describes the derivatives of , while the latter describes the derivatives of . The and indices are once again spatial and temporal indices, respectively, so we can combine them into a single index. Thus, we can write the two equations as

Summary and Next Steps

Thus we have fully expressed electromagnetism in terms of tensors.

Maxwell's Equations in Tensor Formulation: The inhomogeneous and homogeneous Maxwell's equations can be expressed in tensor notation as follows:

where:

The conservation of charge is expressed through the continuity equation

In this section, we have reformulated Maxwell's equations in a more elegant way using tensors and 4-vectors. This formulation makes the underlying structure of electromagnetism clearer and highlights the Lorentz invariance of the equations. In the next section, we will explore the Clifford algebra formulation of Maxwell's equations, which provides an alternative perspective on electromagnetism.