Catalogue of Spacetimes Thomas Mueller, Frank Grave Schwarzschild metric in Schwarzschild coordinates (t,r,theta,phi). Date: 20.04.2009 > restart:grtw(); GRTensorII Version 1.79 (R4) 6 February 2001 Developed by Peter Musgrave, Denis Pollney and Kayll Lake Copyright 1994-2001 by the authors. Latest version available from: http://grtensor.phy.queensu.ca/ /home/muelleta/local/Grii/metrics > makeg(schwarzschild); Makeg 2.0: GRTensor metric/basis entry utility To quit makeg, type 'exit' at any prompt. Do you wish to enter a 1) metric [g(dn,dn)], 2) line element [ds], 3) non-holonomic basis [e(1)...e(n)], or 4) NP tetrad [l,n,m,mbar]? > 2: Enter coordinates as a LIST (eg. [t,r,theta,phi]): > [t,r,theta,phi]: Enter the line element using d[coord] to indicate differentials. (for example, r^2*(d[theta]^2 + sin(theta)^2*d[phi]^2) [Type 'exit' to quit makeg] ds^2 = > -(1-rs/r)*c^2*d[t]^2+d[r]^2/(1-rs/r)+r^2*(d[theta]^2+sin(theta)^2*d[phi]^2): If there are any complex valued coordinates, constants or functions for this spacetime, please enter them as a SET ( eg. { z, psi } ). Complex quantities [default={}]: > {}: The values you have entered are: Coordinates = [t, r, theta, phi] Metric: [ 2 ] [ 2 c rs ] [-c + ----- 0 0 0 ] [ r ] [ ] [ 1 ] [ 0 -------- 0 0 ] g[a] [b] = [ rs ] [ 1 - ---- ] [ r ] [ ] [ 2 ] [ 0 0 r 0 ] [ ] [ 2 2] [ 0 0 0 r sin(theta) ] You may choose to 0) Use the metric WITHOUT saving it, 1) Save the metric as it is, 2) Correct an element of the metric, 3) Re-enter the metric, 4) Add/change constraint equations, 5) Add a text description, or 6) Abandon this metric and return to Maple. > 0: Calculated ds for schwarzschild (0.000000 sec.) Default spacetime = schwarzschild For the schwarzschild spacetime: Coordinates x(up) a x = [t, r, theta, phi] Line element / 2 \ 2 2 | 2 c rs| 2 d r 2 2 ds = |-c + -----| d t + -------- + r d theta \ r / rs 1 - ---- r 2 2 2 + r sin(theta) d phi makeg() completed. > grcalcd(Chr2); Calculated detg for schwarzschild (0.000000 sec.) Calculated g(up,up) for schwarzschild (0.000000 sec.) Calculated g(dn,dn,pdn) for schwarzschild (0.004000 sec.) Calculated Chr(dn,dn,dn) for schwarzschild (0.000000 sec.) Calculated Chr(dn,dn,up) for schwarzschild (0.000000 sec.) CPU Time = 0.012 For the schwarzschild spacetime: Christoffel symbol of the second kind (symmetric in first two in\ dices) 2 r (r - rs) c rs Gamma[t t] = -------------- 3 2 r t rs Gamma[t r] = ------------ 2 r (r - rs) r rs Gamma[r r] = - ------------ 2 r (r - rs) theta Gamma[r theta] = 1/r phi Gamma[r phi] = 1/r r Gamma[theta theta] = -r + rs phi cos(theta) Gamma[theta phi] = ---------- sin(theta) r 2 Gamma[phi phi] = -(r - rs) sin(theta) theta Gamma[phi phi] = -sin(theta) cos(theta) > grcalcd(Riemann); Calculated R(dn,dn,dn,dn) for schwarzschild (0.000000 sec.) CPU Time = 0. For the schwarzschild spacetime: Covariant Riemann 2 c rs R [t r t r] = - ----- 3 r 2 (r - rs) c rs R [t theta t theta] = -------------- 2 2 r 2 2 (r - rs) c rs sin(theta) R [t phi t phi] = 1/2 -------------------------- 2 r rs R [r theta r theta] = - ---------- 2 (r - rs) 2 rs sin(theta) R [r phi r phi] = -1/2 -------------- r - rs 2 R [theta phi theta phi] = r sin(theta) rs > grcalcd(Ricci); Calculated R(dn,dn) for schwarzschild (0.004000 sec.) CPU Time = 0.004 For the schwarzschild spacetime: Covariant Ricci R(dn, dn) R [a] [b] = All components are zero > grcalcd(Ricciscalar); Calculated Ricciscalar for schwarzschild (0.000000 sec.) CPU Time = 0. For the schwarzschild spacetime: Ricci scalar R = 0 > grcalcd(RiemSq); Created definition for R(dn,dn,up,up) Calculated R(dn,dn,up,up) for schwarzschild (0.004000 sec.) Calculated RiemSq for schwarzschild (0.000000 sec.) CPU Time = 0.020 For the schwarzschild spacetime: Full Contraction of Riemann 2 12 rs K = ------ 6 r > >