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# ps7 - Physics 523 Introduction to Relativity Homework 7 Due...

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Physics 523, Introduction to Relativity Homework 7 Due Tuesday, 13 th December 2011 Hans Bantilan Gravitational Plane Wave Consider an exact gravitational plane wave solution to the Einstein equations, expressed in double-null coordinates ( u, v, x, y ) as g ij dx i dx j = - ( dudv + dvdu ) + a 2 ( u ) dx 2 + b 2 ( u ) dy 2 . a. The calculation of Christoffel coefficients and Riemann tensor components associated with g ij is standard. The Ricci tensor associated with g ij can be expressed in terms of the functions a ( u ) and b ( u ) R ij = diag - a ( u ) a ( u ) - b ( u ) b ( u ) , 0 , 0 , 0 . b. The Einstein equations in vacuum can be written as R ij = 0 . By part a, we see that this is equivalent to the statement that a ( u ) a ( u ) + b ( u ) b ( u ) = 0 . c. From the conclusions of part b, we observe that the functions a ( u ) and b ( u ) can be determined in terms of a single arbitrary function f ( u ), such that a ( u ) = f ( u ) a ( u ) b ( u ) = - f ( u ) b ( u ) . We also observe that the second-order ordinary differential equations for a ( u ) and b ( u ) are of the most general form (ie: given some arbitrary f ( u ), any second-order ODE can be brought into this form). To

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