<< Cosmo.CosmologyCourse %TOC% ---- ---+ with maxima <verbatim> /* FLRW metric: (c) CC-BY-SA B. Roukema (2011) */ load(ctensor); init_ctensor(); csetup(); 4; y; [t,r,theta,phi]; 1; 1; -c^2 ; /* t */ a^2 / (1 -k*r^2) ; /* r */ a^2 * r^2 ; /* theta */ a^2 * r^2 * (cos(theta))^2; /* phi */ /* Enter functional dependencies with the DEPENDS function or 'N' if none */ /* e.g. depends([a,d],x); */ /* depends([a],t); */ depends([a],t); y; lg; /* print the metric again (l= lower indices), use ug for the inverse */ einstein(true); /* "mixed" means e.g. G_a^b */ cdisplay(ein); /* print the Einstein tensor again */ ein[2,2] - ein[3,3]; /* both should equal p, but are they really the same? */ ein[2,2] - ein[4,4]; /* both should equal p, but are they really the same? */ /* define the stress-energy tensor */ T : matrix([-rho * c^2, 0,0,0], [0,p,0,0], [0,0,p,0], [0,0,0,p]); depends([rho,p],t); /* Friedmann equation */ ein[1,1] = 8* %pi * G/c^4 * T[1,1]; /* [1,1] component of G = 8 pi T */ -c^2/3 *lhs(%) = -c^2/3 * rhs(%), expand; /* simplify */ /* acceleration equation */ /* [2,2] component of G = 8 pi T, and subtract previous equation left- and right-hand sides, and factorise */ c^2 * ein[2,2] + lhs(%) = c^2 * 8* %pi * G /c^4 * T[2,2] + rhs(%), factor; /* simplify */ -lhs(%)/2 = -rhs(%)/2 ; </verbatim> -- Main.BoudRoukema - 04 Mar 2011
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28 Apr 2017,
BoudRoukema
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