* Here we test manipulations on GiNaC's indexed objects. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
return 0;
}
-static unsigned delta_check(void)
+static unsigned check_equal_simplify(const ex &e1, const ex &e2, const scalar_products &sp)
+{
+ ex e = simplify_indexed(e1, sp) - e2;
+ if (!e.is_zero()) {
+ clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
+ << e << " instead of 0" << endl;
+ return 1;
+ }
+ return 0;
+}
+
+static unsigned delta_check()
{
// checks identities of the delta tensor
return result;
}
-static unsigned metric_check(void)
+static unsigned metric_check()
{
// checks identities of the metric tensor
return result;
}
-static unsigned symmetry_check(void)
+static unsigned epsilon_check()
+{
+ // checks identities of the epsilon tensor
+
+ unsigned result = 0;
+
+ symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma"), s_tau("tau");
+ symbol d("d");
+ varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4), tau(s_tau, 4);
+ varidx mu_co(s_mu, 4, true), nu_co(s_nu, 4, true), rho_co(s_rho, 4, true), sigma_co(s_sigma, 4, true), tau_co(s_tau, 4, true);
+
+ // antisymmetry
+ result += check_equal(lorentz_eps(mu, nu, rho, sigma) + lorentz_eps(sigma, rho, mu, nu), 0);
+
+ // convolution is zero
+ result += check_equal(lorentz_eps(mu, nu, rho, nu_co), 0);
+ result += check_equal(lorentz_eps(mu, nu, mu_co, nu_co), 0);
+ result += check_equal_simplify(lorentz_g(mu_co, nu_co) * lorentz_eps(mu, nu, rho, sigma), 0);
+
+ // contraction with symmetric tensor is zero
+ result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), mu_co, nu_co), 0);
+ result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), nu_co, sigma_co, rho_co), 0);
+ result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) * indexed(d, nu_co), 0);
+ result += check_equal_simplify(lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu) * indexed(d, nu_co), 0);
+ ex e = lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) - lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu);
+ result += check_equal_simplify(e, 0);
+
+ // contractions of epsilon tensors
+ result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * lorentz_eps(mu_co, nu_co, rho_co, sigma_co), -24);
+ result += check_equal_simplify(lorentz_eps(tau, nu, rho, sigma) * lorentz_eps(mu_co, nu_co, rho_co, sigma_co), -6 * delta_tensor(tau, mu_co));
+
+ return result;
+}
+
+DECLARE_FUNCTION_2P(symm_fcn)
+REGISTER_FUNCTION(symm_fcn, set_symmetry(sy_symm(0, 1)));
+DECLARE_FUNCTION_2P(anti_fcn)
+REGISTER_FUNCTION(anti_fcn, set_symmetry(sy_anti(0, 1)));
+
+static unsigned symmetry_check()
{
// check symmetric/antisymmetric objects
unsigned result = 0;
- symbol s_i("i"), s_j("j"), s_k("k");
- idx i(s_i, 3), j(s_j, 3), k(s_k, 3);
- symbol A("A");
- ex e, e1, e2;
+ idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3), l(symbol("l"), 3);
+ symbol A("A"), B("B"), C("C");
+ ex e;
+
+ result += check_equal(indexed(A, sy_symm(), i, j), indexed(A, sy_symm(), j, i));
+ result += check_equal(indexed(A, sy_anti(), i, j) + indexed(A, sy_anti(), j, i), 0);
+ result += check_equal(indexed(A, sy_anti(), i, j, k) - indexed(A, sy_anti(), j, k, i), 0);
+ e = indexed(A, sy_symm(), i, j, k) *
+ indexed(B, sy_anti(), l, k, i);
+ result += check_equal_simplify(e, 0);
+ e = indexed(A, sy_symm(), i, i, j, j) *
+ indexed(B, sy_anti(), k, l); // GiNaC 0.8.0 had a bug here
+ result += check_equal_simplify(e, e);
+
+ symmetry R = sy_symm(sy_anti(0, 1), sy_anti(2, 3));
+ e = indexed(A, R, i, j, k, l) + indexed(A, R, j, i, k, l);
+ result += check_equal(e, 0);
+ e = indexed(A, R, i, j, k, l) + indexed(A, R, i, j, l, k);
+ result += check_equal(e, 0);
+ e = indexed(A, R, i, j, k, l) - indexed(A, R, j, i, l, k);
+ result += check_equal(e, 0);
+ e = indexed(A, R, i, j, k, l) + indexed(A, R, k, l, j, i);
+ result += check_equal(e, 0);
+
+ e = indexed(A, i, j);
+ result += check_equal(symmetrize(e) + antisymmetrize(e), e);
+ e = indexed(A, sy_symm(), i, j, k, l);
+ result += check_equal(symmetrize(e), e);
+ result += check_equal(antisymmetrize(e), 0);
+ e = indexed(A, sy_anti(), i, j, k, l);
+ result += check_equal(symmetrize(e), 0);
+ result += check_equal(antisymmetrize(e), e);
+
+ e = (indexed(A, sy_anti(), i, j, k, l) * (indexed(B, j) * indexed(C, k) + indexed(B, k) * indexed(C, j)) + indexed(B, i, l)).expand();
+ result += check_equal_simplify(e, indexed(B, i, l));
+
+ result += check_equal(symm_fcn(0, 1) + symm_fcn(1, 0), 2*symm_fcn(0, 1));
+ result += check_equal(anti_fcn(0, 1) + anti_fcn(1, 0), 0);
+ result += check_equal(anti_fcn(0, 0), 0);
+
+ return result;
+}
+
+static unsigned scalar_product_check()
+{
+ // check scalar product replacement
+
+ unsigned result = 0;
- result += check_equal(indexed(A, indexed::symmetric, i, j), indexed(A, indexed::symmetric, j, i));
- result += check_equal(indexed(A, indexed::antisymmetric, i, j) + indexed(A, indexed::antisymmetric, j, i), 0);
- result += check_equal(indexed(A, indexed::antisymmetric, i, j, k) - indexed(A, indexed::antisymmetric, j, k, i), 0);
+ idx i(symbol("i"), 3), j(symbol("j"), 3);
+ symbol A("A"), B("B"), C("C");
+ ex e;
+
+ scalar_products sp;
+ sp.add(A, B, 0); // A and B are orthogonal
+ sp.add(A, C, 0); // A and C are orthogonal
+ sp.add(A, A, 4); // A^2 = 4 (A has length 2)
+
+ e = (indexed(A + B, i) * indexed(A + C, i)).expand(expand_options::expand_indexed);
+ result += check_equal_simplify(e, indexed(B, i) * indexed(C, i) + 4, sp);
+ e = indexed(A, i, i) * indexed(B, j, j); // GiNaC 0.8.0 had a bug here
+ result += check_equal_simplify(e, e, sp);
return result;
}
-static unsigned edyn_check(void)
+static unsigned edyn_check()
{
- // 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;
// Lorentz transformation matrix (boost along x axis)
matrix L(4, 4);
- L.set(0, 0, gamma);
- L.set(0, 1, -beta*gamma);
- L.set(1, 0, -beta*gamma);
- L.set(1, 1, gamma);
- L.set(2, 2, 1);
- L.set(3, 3, 1);
+ L(0, 0) = gamma;
+ L(0, 1) = -beta*gamma;
+ L(1, 0) = -beta*gamma;
+ L(1, 1) = gamma;
+ L(2, 2) = 1; L(3, 3) = 1;
// Electromagnetic field tensor
- matrix F(4, 4);
- F.set(0, 1, -Ex);
- F.set(1, 0, Ex);
- F.set(0, 2, -Ey);
- F.set(2, 0, Ey);
- F.set(0, 3, -Ez);
- F.set(3, 0, Ez);
- F.set(1, 2, -Bz);
- F.set(2, 1, Bz);
- F.set(1, 3, By);
- F.set(3, 1, -By);
- F.set(2, 3, -Bx);
- F.set(3, 2, Bx);
+ matrix F(4, 4, lst(
+ 0, -Ex, -Ey, -Ez,
+ Ex, 0, -Bz, By,
+ Ey, Bz, 0, -Bx,
+ Ez, -By, Bx, 0
+ ));
// Indices
symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
// Apply transformation law of second rank tensor
ex e = (indexed(L, mu, rho.toggle_variance())
- * indexed(L, nu, sigma.toggle_variance())
- * indexed(F, rho, sigma)).simplify_indexed();
+ * indexed(L, nu, sigma.toggle_variance())
+ * indexed(F, rho, sigma)).simplify_indexed();
// Extract transformed electric and magnetic fields
ex Ex_p = e.subs(lst(mu == 1, nu == 0)).normal();
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
+ ex eta = diag_matrix(lst(1, -1, -1, -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;
+}
+
+static unsigned spinor_check()
+{
+ // check identities of the spinor metric
+
+ unsigned result = 0;
+
+ symbol psi("psi");
+ spinidx A(symbol("A")), B(symbol("B")), C(symbol("C")), D(symbol("D"));
+ ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
+ ex e;
+
+ e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
+ result += check_equal_simplify(e, 2);
+ e = spinor_metric(A_co, B_co) * spinor_metric(B, A);
+ result += check_equal_simplify(e, -2);
+ e = spinor_metric(A_co, B_co) * spinor_metric(A, C);
+ result += check_equal_simplify(e, delta_tensor(B_co, C));
+ e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
+ result += check_equal_simplify(e, -delta_tensor(A_co, C));
+ e = spinor_metric(A_co, B_co) * spinor_metric(C, A);
+ result += check_equal_simplify(e, -delta_tensor(B_co, C));
+ e = spinor_metric(A, B) * indexed(psi, B_co);
+ result += check_equal_simplify(e, indexed(psi, A));
+ e = spinor_metric(A, B) * indexed(psi, A_co);
+ result += check_equal_simplify(e, -indexed(psi, B));
+ e = spinor_metric(A_co, B_co) * indexed(psi, B);
+ result += check_equal_simplify(e, -indexed(psi, A_co));
+ e = spinor_metric(A_co, B_co) * indexed(psi, A);
+ result += check_equal_simplify(e, indexed(psi, B_co));
+ e = spinor_metric(D, A) * spinor_metric(A_co, B_co) * spinor_metric(B, C) - spinor_metric(D, A_co) * spinor_metric(A, B_co) * spinor_metric(B, C);
+ result += check_equal_simplify(e, 0);
+
+ return result;
+}
+
+static unsigned dummy_check()
+{
+ // check dummy index renaming/repositioning
+
+ unsigned result = 0;
+
+ symbol p("p"), q("q");
+ idx i(symbol("i"), 3), j(symbol("j"), 3), n(symbol("n"), 3);
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
+ ex e;
+
+ e = indexed(p, i) * indexed(q, i) - indexed(p, j) * indexed(q, j);
+ result += check_equal_simplify(e, 0);
+
+ e = indexed(p, i) * indexed(p, i) * indexed(q, j) * indexed(q, j)
+ - indexed(p, n) * indexed(p, n) * indexed(q, j) * indexed(q, j);
+ result += check_equal_simplify(e, 0);
+
+ e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu, nu.toggle_variance());
+ result += check_equal_simplify(e, 0);
+
+ e = indexed(p, mu.toggle_variance(), nu, mu) * indexed(q, i)
+ - indexed(p, mu, nu, mu.toggle_variance()) * indexed(q, i);
+ result += check_equal_simplify(e, 0);
+
+ e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu.toggle_variance(), nu);
+ result += check_equal_simplify(e, 0);
+ e = indexed(p, mu.toggle_variance(), mu) - indexed(p, nu, nu.toggle_variance());
+ result += check_equal_simplify(e, 0);
+
return result;
}
-unsigned exam_indexed(void)
+unsigned exam_indexed()
{
unsigned result = 0;
result += delta_check(); cout << '.' << flush;
result += metric_check(); cout << '.' << flush;
+ result += epsilon_check(); cout << '.' << flush;
result += symmetry_check(); cout << '.' << flush;
+ result += scalar_product_check(); cout << '.' << flush;
result += edyn_check(); cout << '.' << flush;
+ result += spinor_check(); cout << '.' << flush;
+ result += dummy_check(); cout << '.' << flush;
if (!result) {
cout << " passed " << endl;
git://www.ginac.de / ginac.git / blobdiff