Catalogue of Spacetimes Thomas Mueller, Frank Grave Schwarzschild metric in Kruskal-Szekeres coordinates (T,X,theta,phi). Date: 27.04.2009 > 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(kruskSzek); 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,X,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 = > 4*rs^3/r(T,X)*exp(-r(T,X)/rs)*(-d[T]^2+d[X]^2)+r(T,X)^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, X, theta, phi] Metric: [ 3 ] [ 4 rs ] [- -------------------- , 0 , 0 , 0] [ r(T, X) ] [ r(T, X) exp(-------) ] [ rs ] [ ] [ 3 ] [ 4 rs ] g[a] [b] = [0 , -------------------- , 0 , 0] [ r(T, X) ] [ r(T, X) exp(-------) ] [ rs ] [ ] [ 2 ] [0 , 0 , r(T, X) , 0] [ ] [ 2 2] [0 , 0 , 0 , r(T, X) 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 kruskSzek (0.000000 sec.) Default spacetime = kruskSzek For the kruskSzek spacetime: Coordinates x(up) a x = [T, X, theta, phi] Line element 3 2 3 2 2 4 rs d T 4 rs d X ds = - -------------------- + -------------------- r(T, X) r(T, X) r(T, X) exp(-------) r(T, X) exp(-------) rs rs 2 2 2 2 2 + r(T, X) d theta + r(T, X) sin(theta) d phi makeg() completed. > grcalcd(Chr2): Calculated g(dn,dn,pdn) for kruskSzek (0.000000 sec.) Calculated Chr(dn,dn,dn) for kruskSzek (0.004000 sec.) Calculated detg for kruskSzek (0.000000 sec.) Calculated g(up,up) for kruskSzek (0.000000 sec.) Calculated Chr(dn,dn,up) for kruskSzek (0.004000 sec.) CPU Time = 0.008 For the kruskSzek spacetime: Christoffel symbol of the second kind (symmetric in first two in\ dices) /d \ |-- r(T, X)| (rs + r(T, X)) T \dT / Gamma[T T] = -1/2 --------------------------- rs r(T, X) /d \ |-- r(T, X)| (rs + r(T, X)) X \dX / Gamma[T T] = -1/2 --------------------------- rs r(T, X) /d \ |-- r(T, X)| (rs + r(T, X)) T \dX / Gamma[T X] = -1/2 --------------------------- rs r(T, X) /d \ |-- r(T, X)| (rs + r(T, X)) X \dT / Gamma[T X] = -1/2 --------------------------- rs r(T, X) d -- r(T, X) theta dT Gamma[T theta] = ---------- r(T, X) d -- r(T, X) phi dT Gamma[T phi] = ---------- r(T, X) /d \ |-- r(T, X)| (rs + r(T, X)) T \dT / Gamma[X X] = -1/2 --------------------------- rs r(T, X) /d \ |-- r(T, X)| (rs + r(T, X)) X \dX / Gamma[X X] = -1/2 --------------------------- rs r(T, X) d -- r(T, X) theta dX Gamma[X theta] = ---------- r(T, X) d -- r(T, X) phi dX Gamma[X phi] = ---------- r(T, X) 2 r(T, X) /d \ r(T, X) exp(-------) |-- r(T, X)| T rs \dT / Gamma[theta theta] = 1/4 ---------------------------------- 3 rs 2 r(T, X) /d \ r(T, X) exp(-------) |-- r(T, X)| X rs \dX / Gamma[theta theta] = -1/4 ---------------------------------- 3 rs phi cos(theta) Gamma[theta phi] = ---------- sin(theta) T Gamma[phi phi] = 2 r(T, X) 2 /d \ r(T, X) exp(-------) sin(theta) |-- r(T, X)| rs \dT / 1/4 ---------------------------------------------- 3 rs X Gamma[phi phi] = 2 r(T, X) 2 /d \ r(T, X) exp(-------) sin(theta) |-- r(T, X)| rs \dX / -1/4 ---------------------------------------------- 3 rs theta Gamma[phi phi] = -sin(theta) cos(theta) > eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(Chr2,[T,T,T]))); 2 2 / / 2 2 \\ X - T | | X - T || LambertW(-------) T |rs + rs |LambertW(-------) + 1|| exp(1) \ \ exp(1) // ----------------------------------------------------- / 2 2 \2 | X - T | 2 2 rs |LambertW(-------) + 1| (X - T ) \ exp(1) / > subs(LambertW((X^2-T^2)/exp(1))=(r/rs-1),%); / r \ rs |---- - 1| T (rs + r) \ rs / ------------------------ 2 2 2 r (X - T ) > subs(X^2-T^2=(r/rs-1)*exp(r/rs),%); rs T (rs + r) ------------- 2 r r exp(----) rs > diff(LambertW((X^2-T^2)/exp(1)),X); 2 2 X - T 2 LambertW(-------) X exp(1) --------------------------------- / 2 2 \ | X - T | 2 2 |LambertW(-------) + 1| (X - T ) \ exp(1) / > subs(LambertW((X^2-T^2)/exp(1))=(r/rs-1),%); / r \ 2 |---- - 1| rs X \ rs / ----------------- 2 2 r (X - T ) > subs(X^2-T^2=(r/rs-1)*exp(r/rs),%); 2 rs X ----------- r r exp(----) rs > diff(LambertW((X^2-T^2)/exp(1)),X,X); 2 2 2 2 4 %1 X 4 %1 X 4 %1 X -------------------- - -------------------- - ------------------- 2 2 2 2 3 2 2 2 2 2 2 (%1 + 1) (X - T ) (%1 + 1) (X - T ) (%1 + 1) (X - T ) 2 %1 + ------------------ 2 2 (%1 + 1) (X - T ) 2 2 X - T %1 := LambertW(-------) exp(1) > subs(LambertW((X^2-T^2)/exp(1))=(r/rs-1),%); / r \ 2 2 / r \2 3 2 / r \ 2 4 |---- - 1| rs X 4 |---- - 1| rs X 4 |---- - 1| rs X \ rs / \ rs / \ rs / ------------------- - -------------------- - ------------------ 2 2 2 2 3 2 2 2 2 2 2 r (X - T ) r (X - T ) r (X - T ) / r \ 2 |---- - 1| rs \ rs / + --------------- 2 2 r (X - T ) > subs(X^2-T^2=(r/rs-1)*exp(r/rs),%); 2 2 3 2 2 4 rs X 4 rs X 4 rs X ------------------------ - ------------- - ----------------------- / r \ 2 r 2 3 r 2 / r \ r 2 |---- - 1| r exp(----) r exp(----) |---- - 1| r exp(----) \ rs / rs rs \ rs / rs 2 rs + ----------- r r exp(----) rs > simplify(%); / 2 r 2 r 2 2\ - 2 |2 rs exp(- ----) X + 2 exp(- ----) X r rs - r | rs \ rs rs / r / 3 exp(- ----) / r rs / > grcalcd(Riemann); CPU Time = 0. For the kruskSzek spacetime: Covariant Riemann / / 2 \ 2 | |d | R [T X T X] = - 2 rs |rs |--- r(T, X)| r(T, X) | | 2 | \ \dX / / 2 \ |d | 2 /d \2 + |--- r(T, X)| r(T, X) - rs |-- r(T, X)| | 2 | \dX / \dX / / 2 \ / 2 \ |d | |d | 2 - rs |--- r(T, X)| r(T, X) - |--- r(T, X)| r(T, X) | 2 | | 2 | \dT / \dT / \ /d \2| / / 3 r(T, X) \ + rs |-- r(T, X)| | / |r(T, X) exp(-------)| \dT / | / \ rs / / / / 2 \ | |d | R [T theta T theta] = -1/2 |2 rs |--- r(T, X)| r(T, X) | | 2 | \ \dT / /d \2 /d \2 + rs |-- r(T, X)| + |-- r(T, X)| r(T, X) \dT / \dT / \ /d \2 /d \2 | + rs |-- r(T, X)| + |-- r(T, X)| r(T, X)|/rs \dX / \dX / | / / / 2 \ | | d | R [T theta X theta] = - |r(T, X) |----- r(T, X)| rs \ \dX dT / /d \ /d \ + |-- r(T, X)| |-- r(T, X)| rs \dT / \dX / \ /d \ /d \ | + |-- r(T, X)| |-- r(T, X)| r(T, X)|/rs \dT / \dX / / / / 2 \ 2 | |d | R [T phi T phi] = -1/2 sin(theta) |2 rs |--- r(T, X)| r(T, X) | | 2 | \ \dT / /d \2 /d \2 + rs |-- r(T, X)| + |-- r(T, X)| r(T, X) \dT / \dT / \ /d \2 /d \2 | + rs |-- r(T, X)| + |-- r(T, X)| r(T, X)|/rs \dX / \dX / | / / / 2 \ 2 | | d | R [T phi X phi] = - sin(theta) |r(T, X) |----- r(T, X)| rs \ \dX dT / /d \ /d \ + |-- r(T, X)| |-- r(T, X)| rs \dT / \dX / \ /d \ /d \ | + |-- r(T, X)| |-- r(T, X)| r(T, X)|/rs \dT / \dX / / / / 2 \ | |d | R [X theta X theta] = -1/2 |2 rs |--- r(T, X)| r(T, X) | | 2 | \ \dX / /d \2 /d \2 + rs |-- r(T, X)| + |-- r(T, X)| r(T, X) \dT / \dT / \ /d \2 /d \2 | + rs |-- r(T, X)| + |-- r(T, X)| r(T, X)|/rs \dX / \dX / | / / / 2 \ 2 | |d | R [X phi X phi] = -1/2 sin(theta) |2 rs |--- r(T, X)| r(T, X) | | 2 | \ \dX / /d \2 /d \2 + rs |-- r(T, X)| + |-- r(T, X)| r(T, X) \dT / \dT / \ /d \2 /d \2 | + rs |-- r(T, X)| + |-- r(T, X)| r(T, X)|/rs \dX / \dX / | / 2 2 / 3 R [theta phi theta phi] = 1/4 r(T, X) sin(theta) |4 rs \ r(T, X) /d \2 + r(T, X) exp(-------) |-- r(T, X)| rs \dT / r(T, X) /d \2\ / 3 - r(T, X) exp(-------) |-- r(T, X)| | / rs rs \dX / / / > eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(Riemann,[theta,phi,theta,phi]))): > simplify(subs(LambertW((X^2-T^2)/exp(1))=(r/rs-1),%)); / 2 r 2 2 2 r |-rs X r + exp(----) rs + rs T r + r exp(----) \ rs rs r \ 2 / 2 2 - 2 exp(----) rs r| (-1 + cos(theta) ) r / ((X - T ) rs) rs / / > simplify(subs(X^2=T^2+(r/rs-1)*exp(r/rs),%)); 2 sin(theta) rs r > grcalcd(Ricci); CPU Time = 0. For the kruskSzek spacetime: Covariant Ricci R(dn, dn) / / 2 \ / 2 \ | |d | |d | 2 R [T T] = -1/2 |rs |--- r(T, X)| r(T, X) + |--- r(T, X)| r(T, X) | | 2 | | 2 | \ \dX / \dX / /d \2 /d \2 + 2 |-- r(T, X)| r(T, X) + 3 rs |-- r(T, X)| \dX / \dT / / 2 \ /d \2 |d | + 2 |-- r(T, X)| r(T, X) + 3 rs |--- r(T, X)| r(T, X) \dT / | 2 | \dT / / 2 \ \ |d | 2 /d \2| / - |--- r(T, X)| r(T, X) + rs |-- r(T, X)| | / (rs | 2 | \dX / | / \dT / / 2 r(T, X) ) / / 2 \ | | d | R [T X] = - 2 |r(T, X) |----- r(T, X)| rs \ \dX dT / /d \ /d \ + |-- r(T, X)| |-- r(T, X)| rs \dT / \dX / \ /d \ /d \ | / 2 + |-- r(T, X)| |-- r(T, X)| r(T, X)| / (rs r(T, X) ) \dT / \dX / / / / / 2 \ / 2 \ | |d | |d | 2 R [X X] = -1/2 |rs |--- r(T, X)| r(T, X) + |--- r(T, X)| r(T, X) | | 2 | | 2 | \ \dT / \dT / /d \2 /d \2 + 2 |-- r(T, X)| r(T, X) + 3 rs |-- r(T, X)| \dT / \dX / / 2 \ /d \2 |d | + 2 |-- r(T, X)| r(T, X) + 3 rs |--- r(T, X)| r(T, X) \dX / | 2 | \dX / / 2 \ \ |d | 2 /d \2| / - |--- r(T, X)| r(T, X) + rs |-- r(T, X)| | / (rs | 2 | \dT / | / \dX / / 2 r(T, X) ) / | r(T, X) /d \2 R [theta theta] = 1/4 |r(T, X) exp(-------) |-- r(T, X)| | rs \dT / \ / 2 \ 2 r(T, X) |d | + r(T, X) exp(-------) |--- r(T, X)| rs | 2 | \dT / r(T, X) /d \2 - r(T, X) exp(-------) |-- r(T, X)| rs \dX / / 2 \ \ 2 r(T, X) |d | 3| / 3 - r(T, X) exp(-------) |--- r(T, X)| + 4 rs | / rs rs | 2 | | / \dX / / / 2 | r(T, X) /d \2 R [phi phi] = 1/4 sin(theta) |r(T, X) exp(-------) |-- r(T, X)| | rs \dT / \ / 2 \ 2 r(T, X) |d | + r(T, X) exp(-------) |--- r(T, X)| rs | 2 | \dT / r(T, X) /d \2 - r(T, X) exp(-------) |-- r(T, X)| rs \dX / / 2 \ \ 2 r(T, X) |d | 3| / 3 - r(T, X) exp(-------) |--- r(T, X)| + 4 rs | / rs rs | 2 | | / \dX / / > eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(Ricci,[phi,phi]))): > simplify(subs(LambertW((X^2-T^2)/exp(1))=r/rs-1,%)); / 2 r 2 r \ 2 - |rs X + rs exp(----) - rs T - exp(----) r| (-1 + cos(theta) ) \ rs rs / / 2 2 / ((X - T ) rs) / > simplify(subs(X^2=T^2+(r/rs-1)*exp(r/rs),%)); 0 > grcomponent(Ricci,[phi,phi]); / 2 | r(T, X) /d \2 1/4 sin(theta) |r(T, X) exp(-------) |-- r(T, X)| | rs \dT / \ / 2 \ 2 r(T, X) |d | + r(T, X) exp(-------) |--- r(T, X)| rs | 2 | \dT / r(T, X) /d \2 - r(T, X) exp(-------) |-- r(T, X)| rs \dX / / 2 \ \ 2 r(T, X) |d | 3| / 3 - r(T, X) exp(-------) |--- r(T, X)| + 4 rs | / rs rs | 2 | | / \dX / / > grcalcd(Ricciscalar); Calculated Ricciscalar for kruskSzek (0.000000 sec.) CPU Time = 0. For the kruskSzek spacetime: Ricci scalar / / 2 \ | r(T, X) 2 |d | R = 1/4 |-3 exp(-------) r(T, X) rs |--- r(T, X)| | rs | 2 | \ \dX / / 2 \ r(T, X) 3 |d | + exp(-------) r(T, X) |--- r(T, X)| rs | 2 | \dX / r(T, X) /d \2 + 3 exp(-------) r(T, X) rs |-- r(T, X)| rs \dT / / 2 \ r(T, X) 2 |d | + 3 exp(-------) r(T, X) rs |--- r(T, X)| rs | 2 | \dT / / 2 \ r(T, X) 3 |d | - exp(-------) r(T, X) |--- r(T, X)| rs | 2 | \dT / \ r(T, X) /d \2 4| / 4 - 3 exp(-------) r(T, X) rs |-- r(T, X)| + 8 rs | / (rs rs \dX / | / / 2 r(T, X) ) > eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(Ricciscalar))): > simplify(subs(LambertW((X^2-T^2)/exp(1))=r/rs-1,%)); / 2 r r 2\ 2 |rs X - exp(----) r + exp(----) rs - rs T | \ rs rs / ---------------------------------------------- 2 2 2 rs r (X - T ) > simplify(subs(X^2=T^2+(r/rs-1)*exp(r/rs),%)); 0 > grcalc(RiemSq): Created definition for R(dn,dn,up,up) Calculated R(dn,dn,up,up) for kruskSzek (0.004000 sec.) Calculated RiemSq for kruskSzek (0.000000 sec.) CPU Time = 0.024 > eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(RiemSq))): > simplify(subs(LambertW((X^2-T^2)/exp(1))=r/rs-1,%)); / 2 4 2 2 3 r 2 2 r 2 4 |rs X r - 2 X rs exp(----) r + 4 X rs exp(----) r \ rs rs 2 2 2 2 2 r 3 2 r 3 - 2 rs X r T - 2 X rs exp(----) r - 8 exp(---) rs r rs rs 2 r 2 2 2 r 4 2 r 4 + 8 exp(---) rs r + exp(---) r + 3 exp(---) rs rs rs rs r 3 2 2 r 2 2 + 2 rs exp(----) r T - 4 rs exp(----) r T rs rs 2 r 3 3 r 2 4 2 2\ / - 4 exp(---) rs r + 2 rs exp(----) r T + T rs r | / ( rs rs / / 2 2 2 2 6 rs (X - T ) r ) > simplify(subs(X^2=T^2+(r/rs-1)*exp(r/rs),%)); / 2 4 2 2 r 2 2 2 r 3 4 |rs X r + 2 exp(---) rs r - 6 exp(---) rs r \ rs rs 2 r 2 2 2 r 3 4 2 2 + 2 rs exp(----) r T + 2 exp(---) rs r - T rs r rs rs r 3 2 2 r 4 2 r 4\ - 2 rs exp(----) r T - exp(---) r + 3 exp(---) rs | rs rs rs / 2 r / 2 6 exp(- ---) / ((-r + rs) r ) rs / > simplify(subs(X^4=(T^2+(r/rs-1)*exp(r/rs))^2,%)); 2 12 rs ------ 6 r >