Riemann tensor and Einstein equation with Sage

This Sage worksheet is devoted to the computation of the Riemann tensor of a given metric and to the writing of the Einstein equation. It has been developed with the version 4.7.1 of Sage.

© Eric Gourgoulhon 16 Dec. 2011

This is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (version 2) as published by the Free Software Foundation.

Parameters of the computation

Manifold dimension, coordinates and metric tensor

Pick a space below or add your own choice

Sphere S2:

{{{id=1| n = 2 ; var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [th, ph] ; # spherical coordinates (theta, phi) var('r') ; # sphere radius g = matrix( [[r^2,0], [0, (r*sin(th))^2]] ) ; g # S^2 metric /// }}}

Euclidean space R3:

{{{id=59| n = 3 ; var('r'); var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [r, th, ph] ; # spherical coordinates (r, theta, phi) g = matrix( [[1,0,0],[0,r^2,0],[0,0,(r*sin(th))^2]] ) ; g # Euclidean metric /// }}}

Hypersphere S3:

{{{id=61| n = 3 ; var('ch', latex_name="\\chi"); var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [ch, th, ph] ; # coordinates (chi, theta, phi) g = matrix( [[1,0,0],[0,(sin(ch))^2,0],[0,0,(sin(ch)*sin(th))^2]] ) ; g # S^3 metric /// }}}

Hyperbolic space H3:

{{{id=64| n = 3 ; var('ro', latex_name="\\rho"); var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [ro, th, ph] ; # coordinates (rho, theta, phi) var('b') ; # constant scale factor g = matrix( [[b^2,0,0], [0,(b*sinh(ro))^2,0], [0,0,(b*sinh(ro)*sin(th))^2]] ) ; g # H^3 metric /// }}}

Friedmann-Lemaître-Robertson-Walker spacetime:

{{{id=2| n = 4 ; var('t, r') ; var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [t, r, th, ph] ; # coordinates (t, r, theta, phi) var('K') ; # curvature parameter of the t = const hypersurfaces a = function('a', t) ; # scale factor a(t) g = matrix( [[-1,0,0,0], [0,a^2/(1-K*r^2), 0, 0], [0, 0, (a*r)^2, 0], [0, 0, 0, (a*r*sin(th))^2 ]] ) ; g # FLRW metric /// }}}

Schwarzschild spacetime:

{{{id=106| n = 4 ; var('t, r') ; var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [t, r, th, ph] ; # Schwarzschild coordinates (t, r, theta, phi) var('m') ; # mass parameter g = matrix( [[-(1-2*m/r),0,0,0], [0,1/(1-2*m/r), 0, 0], [0, 0, r^2, 0], [0, 0, 0, (r*sin(th))^2 ]] ) ; g /// }}}

Kerr spacetime:

{{{id=110| n = 4 ; var('t, r') ; var('th', latex_name="\\theta"); var('ph', latex_name="\\phi") ; x = [t, r, th, ph] ; # Boyer-Lindquist coordinates (t, r, theta, phi) var('m, a') ; # mass and angular momentum parameter rho2 = r^2 + (a*cos(th))^2 ; Delta = r^2 - 2*m*r + a^2 ; g = matrix( [[-(1-2*m*r/rho2),0,0,-2*a*m*r*(sin(th))^2/rho2], [0,rho2/Delta, 0, 0], \ [0, 0, rho2, 0], [-2*a*m*r*(sin(th))^2/rho2, 0, 0, (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*(sin(th))^2 ]] ) ; g /// }}} {{{id=125| # ginv1 = matrix( [[-((r^2+a^2)*rho2+2*m*a^2*r*(sin(th))^2)/rho2/Delta,0,0, -2*m*a*r/(rho2*Delta)], [0,Delta/rho2,0,0],\ # [0,0,1/rho2,0], [-2*m*a*r/(rho2*Delta),0,0,(Delta-(a*sin(th))^2)/(rho2*Delta*(sin(th))^2)]] ) /// }}}

Energy-momentum tensor

Vacuum:

{{{id=95| Tener = zero_matrix(ZZ, n) /// }}}

Perfect fluid for a FLRW cosmological model:

{{{id=79| var('rho, p') ; Tener = matrix( [[rho, 0, 0, 0], [0, p*a^2 / (1-K*r^2), 0, 0], \ [0, 0, p*(a*r)^2, 0], [0, 0, 0, p*(a*r*sin(th))^2]] ) ; Tener /// }}}

Cosmological constant

{{{id=104| with_lambda = False # determines whether a cosmological constant is to be used in the Einstein equation /// }}}

Start of the computation

Inverse metric

{{{id=6| ginv0 = g.inverse() ; ginv = matrix( [[ginv0[i,j].simplify_full() for j in range(n)] for i in range(n)] ) /// }}}

Christoffel symbols

{{{id=9| Chr0 = [[[ sum( ginv[i][l]/2 * ( diff(g[l][k],x[j]) + diff(g[j][l],x[k]) - diff(g[j][k],x[l]) ) for l in range(n) ) for k in range(n) ] for j in range(n) ] for i in range(n) ] ; Chr = [[[ Chr0[i][j][k].simplify_full() for k in range(n) ] for j in range(n) ] for i in range(n) ] /// }}} {{{id=28| Chr /// }}}

Riemann tensor

{{{id=27| Riem0 = [[[[ diff(Chr[i][j][l],x[k]) - diff(Chr[i][j][k],x[l]) \ + sum( Chr[i][m][k] * Chr[m][j][l] - Chr[i][m][l] * Chr[m][j][k] for m in range(n) ) \ for l in range(n) ] for k in range(n) ] for j in range(n) ] for i in range(n) ] /// }}} {{{id=30| Riem = [[[[ Riem0[i][j][k][l].simplify_full() for l in range(n) ] for k in range(n) ] for j in range(n) ] for i in range(n) ] /// }}} {{{id=31| Riem /// }}}

Ricci tensor

{{{id=32| Ric0 = [[ sum( Riem[k][i][k][j] for k in range(n) ) for j in range(n) ] for i in range(n) ] ; Ric = matrix( [[ Ric0[i][j].simplify_full() for j in range(n) ] for i in range(n) ] ) /// }}} {{{id=38| Ric /// }}}

Scalar curvature

{{{id=34| Rscal0 = sum( sum( ginv[i][j] * Ric[i][j] for j in range(n) ) for i in range(n) ) ; Rscal = Rscal0.simplify_full() ; Rscal /// }}}

Check whether the space is maximally symmetric

{{{id=39| max_sym0 = [[[[ Riem[i][j][k][l] - Rscal/(n*(n-1)) * ( identity_matrix(n)[i,k] * g[j][l] - identity_matrix(n)[i,l] * g[j][k] ) \ for l in range(n) ] for k in range(n) ] for j in range(n) ] for i in range(n) ] /// }}} {{{id=44| max_sym = [[[[ max_sym0[i][j][k][l].simplify_full() for l in range(n) ] for k in range(n) ] for j in range(n) ] for i in range(n) ] /// }}} {{{id=40| max_sym # should vanish for a maximally symmetric space /// }}}

Einstein equation

{{{id=76| var('Lamb', latex_name="\Lambda") ; if with_lambda : Lambda = Lamb else : Lambda = 0 /// }}} {{{id=75| var('G') ; Einst0 = Ric + (Lambda - Rscal/2) * g - 8*pi*G * Tener ; Einst = matrix( [[ Einst0[i][j].simplify_full() for j in range(n) ] for i in range(n) ] ) /// }}} {{{id=97| Einst /// }}} {{{id=100| Einst[0][0] /// }}} {{{id=103| Einst[1][1] /// }}}