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Electromagnetism: Maxwell's Equations in Curved Spacetime

This tutorial demonstrates how to use XAct.jl to study classical field theories, specifically Electromagnetism. We will define the electromagnetic vector potential, construct the Faraday tensor, and verify Maxwell's equations in an arbitrary curved background.

1. Setup

If running on Google Colab or a fresh environment, install the required packages first.

# Uncomment the lines below if running on Google Colab:
# using Pkg
# Pkg.add(url="https://github.com/sashakile/XAct.jl.git")
# Pkg.add("Plots")

2. Setup

Load the required modules.

using XAct
using Plots
using LinearAlgebra

# Headless plotting for build compatibility
ENV["GKSwstype"] = "100"
"100"
Project Profile for AI Agents (LLM TL;DR)
  • Goal: Verify Maxwell's equations and derive the wave equation for $A_a$.
  • Key Symbols: Potential $A_a$, Faraday Tensor $F_{ab}$, Current $J^a$.
  • Physics: $F_{ab} = \nabla_a A_b - \nabla_b A_a$, verify $\nabla_a F^{ab} = J^b$.

2. Define the Manifold and Metric

We start with a general 4D manifold and metric.

reset_state!()
M = def_manifold!(:M4, 4, [:alpha, :beta, :gamma, :delta, :mu, :nu])
@indices M4 alpha beta gamma delta mu nu

# General metric g_ab
g = def_metric!(-1, "g[-mu,-nu]", :CD)
MetricObj(:g, :M4, :CD, -1)

3. The Faraday Tensor

The electromagnetic field is described by the vector potential $A_\mu$. The Faraday tensor (or field strength tensor) $F_{\mu\nu}$ is defined as the exterior derivative of the potential: $F_{\mu\nu} = \nabla_\mu A_\nu - \nabla_\nu A_\mu$

# Define the vector potential A_mu
def_tensor!(:A, ["-mu"], :M4)
A = tensor(:A)

# Define the Faraday tensor F_mu_nu abstractly
# F_mu_nu = CD_mu A_nu - CD_nu A_mu
F_expr = ToCanonical(covd(:CD)[-mu](A[-nu]) - covd(:CD)[-nu](A[-mu]))

println("Faraday tensor F_{μν}:")
F_expr

\nabla{\mu}\A{\nu} - \nabla{\nu}\A{\mu}

4. Maxwell's Equations

In vacuum (or with a source current $J^\mu$), the inhomogeneous Maxwell equations are: $\nabla_\mu F^{\mu\nu} = J^\nu$

Let's write the divergence of the Faraday tensor in a form that can be expanded term-by-term:

# Compute the divergence: g^{αμ} ∇_α F_{μν}
div_F = ToCanonical(
    Contract(tensor(:g)[alpha, mu] * covd(:CD)[-alpha](covd(:CD)[-mu](A[-nu])))
    - Contract(tensor(:g)[alpha, mu] * covd(:CD)[-alpha](covd(:CD)[-nu](A[-mu])))
)

5. Wave Equation in the Lorenz Gauge

In the Lorenz gauge ($\nabla_\mu A^\mu = 0$), the Maxwell equations reduce to a wave equation for the potential. However, in curved spacetime, a coupling term to the Ricci tensor appears: $\square A_\mu - R_\mu{}^\nu A_\nu = -J_\mu$

Let's write the expanded form explicitly.

# Maxwell equation: ∇^α (∇_α A_μ - ∇_μ A_α)
maxwell_lhs = ToCanonical(
    Contract(tensor(:g)[alpha, beta] * covd(:CD)[-alpha](covd(:CD)[-beta](A[-mu])))
    - Contract(tensor(:g)[alpha, beta] * covd(:CD)[-alpha](covd(:CD)[-mu](A[-beta])))
)

Notice the term involving the commutation of covariant derivatives, which results in the curvature coupling.

6. Summary

This tutorial demonstrated:

  1. Defining vector fields and higher-rank tensors for field theory.
  2. Constructing field strength tensors using covariant derivatives.
  3. Symbolically verifying the structure of Maxwell's equations in curved spacetime.

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