Special Relativity | Lecture 8

The lecturer begins by expressing the intention to discuss the dynamics of electric and magnetic fields, specifically the field equations and their relation to charges. The conversation diverts to a historical account, mentioning a visit to Belgium and the experience of reading Einstein's first paper on the electrodynamics of moving objects. The paper's introduction is highlighted for its clarity in expressing Einstein's motivation, which is linked to the transformation properties of the electromagnetic field tensor. The lecturer suggests that the paper is worth reading for its historical and scientific significance.

The summary delves into the effects of magnetic fields on charged particles, referencing Einstein's contemplations on the forces involved when a charged particle moves through a magnetic field. It discusses the scenario of a particle moving along the x-axis under the influence of a magnetic field and the resulting force, known as the Lawrence force. The lecturer then contrasts this with the perspective of a moving magnet creating an electric field, emphasizing Einstein's insight that these seemingly different phenomena are fundamentally the same, leading to the concept of electric and magnetic fields being interconnected and transforming into each other under different frames of reference.

This section focuses on the mathematical representation of the electromagnetic field tensor, its anti-symmetric nature, and its components related to electric and magnetic fields. The lecturer outlines the tensor's relationship with the vector potential and how the magnetic field is associated with the curl of the vector potential. A comprehensive matrix representation of the tensor is provided, highlighting the spatial and temporal components, and the lecturer emphasizes the need to understand how this tensor transforms under different frames of reference.

The lecturer introduces the concept of Lorentz transformation, which is key to understanding how tensors, specifically the electromagnetic field tensor, transform between different frames of reference. The transformation rules for scalars, vectors, and tensors with indices are discussed. The Lorentz transformation matrix is derived and its components are explained in relation to the velocity of the moving frame. The goal is to apply these rules to find the new components of the electromagnetic field tensor in a frame moving along the x-axis.

The lecturer outlines Maxwell's equations, emphasizing that half of them are identities following from the definitions of electric and magnetic fields in terms of the vector potential. The divergence of the magnetic field is shown to be zero, implying the non-existence of magnetic monopoles. The time derivative of the magnetic field is related to the electric field, leading to another Maxwell equation. The lecturer also touches on the concept of charge density and current, which are key to the remaining Maxwell equations that are not identities and are derived from an action principle.

The concept of charge density (ρ) and current density (J) are explained in detail. Charge density is defined as the total charge within a volume divided by that volume, while current density is described as the amount of charge per unit area per unit time. The lecturer emphasizes the importance of these quantities in the context of Maxwell's equations, particularly in the conservation of charge. The idea that charge can only change within a region if it passes through the boundaries of that region is highlighted, leading to the local conservation of charge, which is a direct consequence of Maxwell's equations.

The lecturer aims to express Maxwell's equations in a relativistic notation to demonstrate their invariance under Lorentz transformations. The components of the electromagnetic field tensor (E and B) and the four-vector (ρ and J) are identified as key elements in this expression. The lecturer suggests that the set of four equations can be written in a tensor form that is invariant under Lorentz transformations, thus maintaining the symmetry between space and time. The challenge is to rewrite these equations in a way that clearly shows this invariance.

The lecturer concludes with a discussion on the Bianchi identity, a tensor equation that is a special case of an identity in general relativity. It is shown that the Bianchi identity, which is equivalent to certain Maxwell's equations, implies that the magnetic field has no divergence, and by extension, there are no magnetic charge densities or currents. The lecturer invites the audience to explore the Bianchi identity further and to understand its geometric meaning and implications in the context of electromagnetism.