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Chapter 2: Covariance of Maxwell's Equations

Maxwell's four equations collapse into two elegant tensor equations that are manifestly Lorentz covariant. This reveals electromagnetism's intrinsic compatibility with special relativity.

Maxwell's Equations in Tensor Form

Inhomogeneous Equations

\( \partial_\mu F^{\mu\nu} = \mu_0 J^\nu \)

Contains Gauss's law and Ampère-Maxwell law

Homogeneous Equations (Bianchi Identity)

\( \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 \)

Contains Faraday's law and ∇·B = 0

Four-Potential A^μ

\( A^\mu = (\phi/c, \vec{A}) \)

and \( F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu \)

The four-potential unifies scalar and vector potentials. Gauge transformations: A^μ → A^μ + ∂^μχ leave F^μν unchanged.

Interactive Simulations

Covariant Maxwell Equations: Verifying Tensor Form

Python

script.py56 lines

import numpy as np

print("=" * 70)
print("COVARIANT MAXWELL EQUATIONS")
print("d_mu F^{mu nu} = J^nu  (inhomogeneous)")
print("d_mu *F^{mu nu} = 0    (homogeneous / Bianchi identity)")
print("=" * 70)

# Demonstrate that Maxwell's equations in tensor form
# are manifestly Lorentz covariant

# Example: plane EM wave propagating in x-direction
# E = E0 sin(kx - wt) y-hat
# B = B0 sin(kx - wt) z-hat, with B0 = E0/c

E0 = 1.0  # amplitude
k = 2 * np.pi  # wave number
omega = k  # omega = kc, c=1

print("\nPlane wave: E = E0 sin(kx-wt) y, B = E0 sin(kx-wt) z")
print(f"E0 = {E0}, k = {k:.4f}, omega = {omega:.4f}")
print(f"Phase velocity: omega/k = {omega/k:.4f} = c")

# Verify wave equation: d^2 F / dx^2 = (1/c^2) d^2 F / dt^2
# In natural units, this is d^2/dx^2 = d^2/dt^2
print("\nWave equation check at various (t, x) points:")
print(f"{'t':>6} {'x':>6} {'d2F/dx2':>12} {'d2F/dt2':>12} {'Equal?':>8}")
print("-" * 48)

dx = 0.001
dt = 0.001
for t in [0.0, 0.5, 1.0, 1.5]:
    for x in [0.0, 0.25, 0.5]:
        f = lambda t_, x_: E0 * np.sin(k*x_ - omega*t_)
        d2dx2 = (f(t,x+dx) - 2*f(t,x) + f(t,x-dx)) / dx**2
        d2dt2 = (f(t+dt,x) - 2*f(t,x) + f(t-dt,x)) / dt**2
        match = "YES" if abs(d2dx2 - d2dt2) < 0.1 else "NO"
        print(f"{t:6.2f} {x:6.2f} {d2dx2:12.4f} {d2dt2:12.4f} {match:>8}")

print("\n" + "=" * 70)
print("CONTINUITY EQUATION: d_mu J^mu = 0")
print("(Charge conservation in covariant form)")
print("=" * 70)

print("\nFor a point charge q moving at velocity v:")
print(f"{'v/c':>6} {'J^0 (rho)':>12} {'J^1 (jx)':>12} {'gamma':>8}")
print("-" * 42)
q = 1.0
for v in [0.0, 0.1, 0.3, 0.5, 0.7, 0.9]:
    gamma = 1.0 / np.sqrt(1.0 - v**2) if v > 0 else 1.0
    rho = q * gamma  # charge density (Lorentz contracted)
    jx = rho * v
    print(f"{v:6.2f} {rho:12.4f} {jx:12.4f} {gamma:8.4f}")
print("\nJ^mu = (rho*c, J) transforms as a four-vector!")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Maxwell Equations Covariance Check

Fortran

program.f9077 lines

program maxwell_covariance
  implicit none
  double precision :: pi, k, omega, E0
  double precision :: t, x, dx, dt
  double precision :: d2dx2, d2dt2, f_c, f_xp, f_xm, f_tp, f_tm
  integer :: i, j
  double precision :: v, gamma_val, rho, jx, q

pi = 4.0d0 * atan(1.0d0)
  E0 = 1.0d0
  k = 2.0d0 * pi
  omega = k  ! c = 1
  dx = 0.001d0
  dt = 0.001d0

print '(A)', repeat('=', 65)
  print '(A)', 'COVARIANT MAXWELL EQUATIONS'
  print '(A)', 'Wave equation: d^2F/dx^2 = d^2F/dt^2 (c=1)'
  print '(A)', repeat('=', 65)

print '(A6, A6, A14, A14, A10)', &
    't', 'x', 'd2F/dx2', 'd2F/dt2', 'Match?'
  print '(A)', repeat('-', 52)

do i = 0, 3
    do j = 0, 2
      t = dble(i) * 0.5d0
      x = dble(j) * 0.25d0

f_c  = E0 * sin(k*x - omega*t)
      f_xp = E0 * sin(k*(x+dx) - omega*t)
      f_xm = E0 * sin(k*(x-dx) - omega*t)
      f_tp = E0 * sin(k*x - omega*(t+dt))
      f_tm = E0 * sin(k*x - omega*(t-dt))

d2dx2 = (f_xp - 2.0d0*f_c + f_xm) / (dx**2)
      d2dt2 = (f_tp - 2.0d0*f_c + f_tm) / (dt**2)

if (abs(d2dx2 - d2dt2) < 0.1d0) then
        print '(F6.2, F6.2, F14.4, F14.4, A10)', &
          t, x, d2dx2, d2dt2, '  YES'
      else
        print '(F6.2, F6.2, F14.4, F14.4, A10)', &
          t, x, d2dx2, d2dt2, '  NO'
      end if
    end do
  end do

! Charge conservation: continuity equation
  print '(/,A)', repeat('=', 65)
  print '(A)', 'FOUR-CURRENT J^mu = (rho*c, J)'
  print '(A)', 'Continuity: d_mu J^mu = 0'
  print '(A)', repeat('=', 65)

q = 1.0d0
  print '(A8, A10, A12, A12, A12)', &
    'v/c', 'gamma', 'J^0(rho)', 'J^1(jx)', 'J_mu J^mu'
  print '(A)', repeat('-', 55)

do i = 0, 9
    v = dble(i) * 0.1d0
    if (v < 1.0d-15) then
      gamma_val = 1.0d0
    else
      gamma_val = 1.0d0 / sqrt(1.0d0 - v**2)
    end if
    rho = q * gamma_val
    jx = rho * v
    ! J_mu J^mu = -rho^2 + jx^2 = -q^2 (invariant)
    print '(F8.2, F10.4, F12.4, F12.4, F12.6)', &
      v, gamma_val, rho, jx, -rho**2 + jx**2
  end do
  print '(A)', 'J_mu J^mu = -q^2 is Lorentz invariant!'

end program maxwell_covariance

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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