Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 457g 11 = ( 21 + 2a ) −11 , g33 = ( 21 + 2a ) −12 , anda 1 = exp(q 1 − q 2 ), a 2 = exp(q 2 − q 3 ).This space–time is of Petrov type I. The nonzero components of the energy–momentum tensor calculated in a Lorentz frame are (κ = 1)T 00 = −((g 11 ) 2 T 11 + T 22 + (g 33 ) 2 T 33 )= −4a 1 2 (g 11 ) 2 (a 1 2 − 1) + 4a 1 2 a 2 2 g 11 g 33 − 4a 2 2 (g 33 ) 2 (a 2 2 − 1),T 11 = 4a 1 2 (a 1 2 − 1), T 22 = −4a 1 2 a 2 2 g 11 g 33 ,T 33 = 4a 2 2 (a 2 2 − 1).3.16.3.2 Case IIHere we have the metric [Rosquist and Goliath (1997)]ds 2 = −(dq 0 ) 2 + g 11 (dq 1 ) 2 + (dq 2 ) 2 + (dq 3 ) 2 , whereg 11 = [ 1 + 2(a 2 1 + a 2 2 ) ] −1, a1 = exp( √ 1√3q 1 + 2 2 q2 ),a 2 = exp( 1 √2q 1 −√32 q2 ).This space–time is of Petrov type D. The nonzero components of theenergy–momentum tensor calculated in a Lorentz frame are (κ = 1)(T 00 = 12 e 2√ 2q 1 2g 11 4 − 2 sinh 2 ( √ 6q 2 ) + e −√ 2q 1 cosh( √ )6q 2 ) ,T 11 = −(g 11 ) 2 T 00 .3.16.3.3 Energy–Momentum TensorsIn both of the above cases, the energy–momentum tensor takes the form[Rosquist and Goliath (1997)]⎛ ⎞µ 0 0 0T αβ = ⎜ 0 p 1 0 0⎟⎝ 0 0 p 2 0 ⎠ ,0 0 0 p 3where µ ≡ T 00 is the energy density, and p i ≡ T ii , (i = 1, 2, 3) areanisotropic pressures. Such an energy–momentum tensor is physicallymeaningful if the weak energy condition [Hawking and Ellis (1973)]µ ≥ 0, µ + p i ≥ 0, (i = 1, 2, 3),