Exact Solution Of Einstein's Equation To The Gravitational Field Of ...

Ozean Journal of Applied Sciences 1(1), 2008Beside these three equations the functions f 1 , f2,f3must fulfill also the equation of the determinant2f 0 f1f2= 1(20d)1 ∂f01 ∂f11 ∂f2i.e. + + = 0.f0∂x1f1∂x1f2∂x1From (20c) we can get∂ ⎛ 1 ∂f0 ⎞ 1 ∂f1∂f0x⎜1 f0x⎟ =(20e)∂ ⎝ ∂ 1 ⎠ f1f0∂x1∂x1Now integrating on both sides we get∂ ⎛ 1 ∂f⎞ ⎛⎜0 1⎟ = ∫⎜∂x1⎝ f0∂x1⎠ ⎝ f1f1 ∂f01 ∂f0∂=f ∂xf f ∂x∫ ∂ x∫011∂f10110∂f∂x11( f )1∂f∂x0∴ = af1, where a is integration constantf0∂x1Adding (20a) and (20e) and with help of (20d), we get∂∂x1∂⇒∂x⎛ 1 ∂f⎜⎝ f1∂x1∂⇒ − 2∂x11⎛ 1 ∂f⎜⎝ f1∂x11 ∂f+f ∂x11⎛ 1⎜⎝ f201+f001⎞ 1⎟ =⎠ f1f001∂f∂x⎟ ⎞⎠11∂f0⎞ ⎛ 1 ∂f⎟ =⎜∂x1⎠ ⎝ f2∂x∂f2⎞ ⎛ 1 ∂f⎟ = 3⎜∂x1⎠ ⎝ f2∂xBy integrating on both sides, we get21⎞⎟⎠2∂f∂x∂ ⎛ 1 ∂f⎞ ⎛ 1 ∂f⎞2 dx∫ ⎜ 2⎟ = ∫ ⎜ 2dx13∂x⎟ 1 ⎝ f2∂x1⎠ ⎝ f2∂x1⎠− 1⎛ 1 ∂f2⎞ ⎛ 1 ∂f2⎞− 2 ⎜⎟ = 3⎜⎟∫ dx⎝ f2∂x1⎠ ⎝ f2∂x1⎠1 ∂f2f2∂x1⇒ − 2 = 3x+ c2 1⎛ 1 ∂f⎞2⎜f x⎟⎝ 2∂1 ⎠1 3 c⇒ = x1+⎛ 1 ∂f2 ⎞ 2 2⎜f2x⎟⎝ ∂ 1 ⎠Where c is integration constantBy integrating once more⇒1221⎞⎟⎠01221 ⎛ 1 ∂f1⎞+2⎜⎟⎝ f1∂x1⎠1 ⎛ 1 ∂f+2⎜⎝ f1∂x112⎛ 1 ∂f+⎜⎝ f2∂x1 ∂f+f ∂x001⎞⎟⎠221⎞⎟⎠21 ⎛+2⎜⎝1f0∂f∂x01⎞⎟⎠235