Maxwell's equations in expanding background

Lorentz Transformation in Maxwell Equations for Slowly Moving Media, Xin-Li Sheng, Yang Li, Shi and Qun Wang.

Content

  1. Introduction
  2. Background: FLRW Metric
  3. Covariant Form of Maxwell's Equations
  4. Electromagnetic Field Tensor in FLRW Metric
  5. Evolution of Magnetic Fields
  6. Magnetic Field Constraints
  7. Homogeneous Maxwell Equations in an Expanding Universe
  8. Inhomogeneous Maxwell Equations in an Expanding Universe
  9. Derivation of Maxwell's equation in expanding background
  10. Reference

Introduction

Maxwell's equations are the foundation of classical electromagnetism, and in the context of an expanding universe, they must be modified to account for the effects of spacetime curvature described by general relativity. In this derivation, we will reframe Maxwell's equations in the context of an expanding universe, specifically using the metric of a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, which describes a homogeneous and isotropic universe.

Background: FLRW Metric

In cosmology, the expanding universe is often described using the FLRW metric, which in spherical coordinates \((t, r, \theta, \phi)\) is written as: \[ ds^2 = -c^2 dt^2 + a^2(t) \left( \frac{dr^2}{1 - kr^2} + r^2 d\theta^2 + r^2 \sin^2\theta \, d\phi^2 \right) \] Here:
  • \( t \) is cosmic time,
  • \( r, \theta, \phi \) are comoving spatial coordinates,
  • \( a(t) \) is the scale factor, which describes how distances between objects evolve over time,
  • \( k \) is the curvature parameter (\(k = 0\) for flat, \(k = +1\) for closed, \(k = -1\) for open universe).
In this metric, the scale factor \( a(t) \) encapsulates the expansion of the universe. The curvature \( k \) determines whether the universe is spatially flat, open, or closed.

Covariant Form of Maxwell's Equations

In curved spacetime, Maxwell's equations are written using tensor notation. The electromagnetic field is described by the electromagnetic field strength tensor \( F_{\mu\nu} \), which is related to the electric and magnetic fields. The covariant form of Maxwell's equations in curved spacetime is given as:
  1. Homogeneous Maxwell Equations (from the Bianchi identities): \[ \nabla_\alpha F_{\beta\gamma} + \nabla_\beta F_{\gamma\alpha} + \nabla_\gamma F_{\alpha\beta} = 0 \]
  2. Inhomogeneous Maxwell Equations (from the coupling with sources): \[ \nabla_\nu F^{\mu\nu} = \mu_0 J^\mu \] where \( \nabla_\mu \) is the covariant derivative, \( J^\mu \) is the four-current density, and \( \mu_0 \) is the vacuum permeability.

Electromagnetic Field Tensor in FLRW Metric

The electromagnetic field tensor \( F_{\mu\nu} \) is related to the electric and magnetic fields \( \mathbf{E} \) and \( \mathbf{B} \) in the following way: \[ F_{\mu\nu} = \begin{pmatrix} 0 & E_x & E_y & E_z \\ -E_x & 0 & B_z & -B_y \\ -E_y & -B_z & 0 & B_x \\ -E_z & B_y & -B_x & 0 \end{pmatrix} \] Here, the components \( E_x, E_y, E_z \) and \( B_x, B_y, B_z \) are the comoving electric and magnetic fields, which depend on time and space.

In the expanding universe, however, the physical electric and magnetic fields are scaled by the scale factor \( a(t) \) of the universe:

  • Physical electric field: \( \mathbf{E}_{\text{phys}} = a^{-2}(t) \mathbf{E} \),
  • Physical magnetic field: \( \mathbf{B}_{\text{phys}} = a^{-2}(t) \mathbf{B} \).
The factor \( a^{-2}(t) \) arises because the electric and magnetic fields are components of the field tensor, which behaves like a two-form and must be rescaled appropriately in curved spacetime.

Homogeneous Maxwell Equations in an Expanding Universe

The homogeneous Maxwell equations in curved spacetime are derived from the Bianchi identity: \[ \nabla_\alpha F_{\beta\gamma} + \nabla_\beta F_{\gamma\alpha} + \nabla_\gamma F_{\alpha\beta} = 0 \] This can be written in terms of the components of the electromagnetic fields \( \mathbf{E} \) and \( \mathbf{B} \) as: \[ \frac{\partial \mathbf{B}}{\partial t} + \nabla \times \mathbf{E} = 0 \] This equation remains valid in the FLRW metric without modification. It describes how the magnetic field evolves with time and space in the expanding universe.

The second homogeneous equation, \( \nabla \cdot \mathbf{B} = 0 \), also remains unchanged in the FLRW universe. The magnetic field lines remain divergence-free, which corresponds to the fact that there are no magnetic monopoles.

Inhomogeneous Maxwell Equations in an Expanding Universe

The inhomogeneous Maxwell equations in curved spacetime are written as: \[ \nabla_\nu F^{\mu\nu} = \mu_0 J^\mu \] Expanding this, we get two sets of equations, one for the electric field and one for the magnetic field.
  • Gauss's Law for Electricity: The \( \mu = 0 \) component of the above equation gives the generalized Gauss's law: \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \] where \( \rho \) is the comoving charge density. In an expanding universe, this equation remains the same, but the physical electric field is related to the comoving electric field by the scale factor, so: \[ \nabla \cdot \left( a^2(t) \mathbf{E}_{\text{phys}} \right) = \frac{\rho}{\epsilon_0} \] This means that the electric field decreases in magnitude as the universe expands.
  • Ampère's Law with Maxwell's Correction: The \( \mu = i \) components give Ampère's law, modified to account for the expansion: \[ \nabla \times \mathbf{B} - \frac{\partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J} \] In an expanding universe, this becomes: \[ \nabla \times \left( a^2(t) \mathbf{B}_{\text{phys}} \right) - a^2(t) \frac{\partial \mathbf{E}_{\text{phys}}}{\partial t} = \mu_0 \mathbf{J} \] where \( \mathbf{J} \) is the comoving current density. Similar to the electric field, the physical current density is related to the comoving current density by \( \mathbf{J}_{\text{phys}} = a^{-3}(t) \mathbf{J} \).
Summary
The Maxwell equations in an expanding FLRW universe can be summarized as follows:
  1. Gauss's Law: \[ \nabla \cdot \left( a^2(t) \mathbf{E}_{\text{phys}} \right) = \frac{\rho}{\epsilon_0} \]
  2. Faraday's Law: \[ \frac{\partial \mathbf{B}_{\text{phys}}}{\partial t} + \nabla \times \mathbf{E}_{\text{phys}} = 0 \]
  3. Gauss's Law for Magnetism: \[ \nabla \cdot \mathbf{B}_{\text{phys}} = 0 \]
  4. Ampère's Law: \[ \nabla \times \mathbf{B}_{\text{phys}} - \frac{\partial \mathbf{E}_{\text{phys}}}{\partial t} = \mu_0 a^{-3}(t) \mathbf{J}_{\text{phys}} \]

Implications of Maxwell's Equations in an Expanding Universe:

In an expanding universe, several important features arise from these equations:
  • The electric and magnetic fields decrease with the expansion of the universe as \( a^{-2}(t) \). This reflects the dilution of energy density as the universe expands.
  • The current density also dilutes with expansion as \( a^{-3}(t) \), meaning that electromagnetic sources become weaker as the universe grows.
  • Faraday rotation, which depends on the interaction of electromagnetic waves with magnetic fields in plasma, will also evolve with time in the expanding universe, providing a tool for probing cosmic magnetic fields in the large-scale structure of the universe.

Derivation of Maxwell's equation in expanding background

To derive the homogeneous Maxwell equations in an expanding universe using the Bianchi identity, we need to first understand the basic mathematical structure. The homogeneous Maxwell equations can be expressed in terms of the electromagnetic field strength tensor \(F_{\mu\nu}\), and the Bianchi identity is a statement about the properties of the curvature of spacetime. It naturally leads to the form of Maxwell's homogeneous equations (such as Faraday's law and the condition \( \nabla \cdot \mathbf{B} = 0 \)).

The Bianchi Identity

The Bianchi identity in curved spacetime is written as: \[ \nabla_\alpha F_{\beta\gamma} + \nabla_\beta F_{\gamma\alpha} + \nabla_\gamma F_{\alpha\beta} = 0 \] This identity holds in any spacetime and is a generalization of the fact that the curl of a gradient vanishes. In flat spacetime, this is related to the fact that the electric and magnetic fields are derivable from a potential.
  • \(F_{\mu\nu}\) is the electromagnetic field tensor.
  • \( \nabla_\alpha \) denotes the covariant derivative in curved spacetime.
The Bianchi identity is the geometric foundation of the homogeneous Maxwell equations, which describe the absence of magnetic monopoles and the relationship between the changing magnetic field and the electric field (i.e., Faraday's law). The Bianchi identity provides a natural mathematical foundation for the homogeneous Maxwell equations, which in an expanding universe are modified by the presence of the scale factor \(a(t)\). The physical electric and magnetic fields dilute as \(a^{-2}(t)\), but the form of the equations remains very similar to that in flat spacetime. These equations are crucial for understanding the behavior of electromagnetic fields in cosmological contexts, such as the generation and evolution of cosmic magnetic fields in the early universe.

References

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