From b8e4e8958d3884ea3c91ac9a8b99fb912ea44626 Mon Sep 17 00:00:00 2001 From: Christian Bauer Date: Thu, 22 Mar 2001 18:51:48 +0000 Subject: [PATCH] added another relativistic electrodynamics check (energy density and Poynting vector) --- check/exam_indexed.cpp | 39 +++++++++++++++++++++++++++++++++++++-- 1 file changed, 37 insertions(+), 2 deletions(-) diff --git a/check/exam_indexed.cpp b/check/exam_indexed.cpp index 8e80e4ec..e5a3772f 100644 --- a/check/exam_indexed.cpp +++ b/check/exam_indexed.cpp @@ -143,8 +143,10 @@ static unsigned symmetry_check(void) static unsigned edyn_check(void) { - // relativistic electrodynamics: check transformation laws of electric - // and magnetic fields by applying a Lorentz boost to the field tensor + // Relativistic electrodynamics + + // Test 1: check transformation laws of electric and magnetic fields by + // applying a Lorentz boost to the field tensor unsigned result = 0; @@ -202,6 +204,39 @@ static unsigned edyn_check(void) result += check_equal(By_p, gamma * (By + beta * Ez)); result += check_equal(Bz_p, gamma * (Bz - beta * Ey)); + // Test 2: check energy density and Poynting vector of electromagnetic field + + // Minkowski metric + matrix eta(4, 4); + eta.set(0, 0, 1); + eta.set(1, 1, -1); + eta.set(2, 2, -1); + eta.set(3, 3, -1); + + // Covariant field tensor + ex F_mu_nu = (indexed(eta, mu.toggle_variance(), rho.toggle_variance()) + * indexed(eta, nu.toggle_variance(), sigma.toggle_variance()) + * indexed(F, rho, sigma)).simplify_indexed(); + + // Energy-momentum tensor + ex T = (-indexed(eta, rho, sigma) * F_mu_nu.subs(s_nu == s_rho) + * F_mu_nu.subs(lst(s_mu == s_nu, s_nu == s_sigma)) + + indexed(eta, mu.toggle_variance(), nu.toggle_variance()) + * F_mu_nu.subs(lst(s_mu == s_rho, s_nu == s_sigma)) + * indexed(F, rho, sigma) / 4).simplify_indexed() / (4 * Pi); + + // Extract energy density and Poynting vector + ex E = T.subs(lst(s_mu == 0, s_nu == 0)).normal(); + ex Px = T.subs(lst(s_mu == 0, s_nu == 1)); + ex Py = T.subs(lst(s_mu == 0, s_nu == 2)); + ex Pz = T.subs(lst(s_mu == 0, s_nu == 3)); + + // Check results + result += check_equal(E, (Ex*Ex+Ey*Ey+Ez*Ez+Bx*Bx+By*By+Bz*Bz) / (8 * Pi)); + result += check_equal(Px, (Ez*By-Ey*Bz) / (4 * Pi)); + result += check_equal(Py, (Ex*Bz-Ez*Bx) / (4 * Pi)); + result += check_equal(Pz, (Ey*Bx-Ex*By) / (4 * Pi)); + return result; } -- 2.35.3