16.13 For the line element ds² = (1+2Þ) dt² – (1 − 2Þ)R² (t) (dx² +dy² +dz²), show that, to first order in Þ, the pertur

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16 13 For The Line Element Ds 1 2th Dt 1 2th R T Dx Dy Dz Show That To First Order In Th The Pertur 1 (101.36 KiB) Viewed 14 times

16.13 For the line element ds² = (1+2Þ) dt² – (1 − 2Þ)R² (t) (dx² +dy² +dz²), show that, to first order in Þ, the perturbed parts of the connection coefficients take the form srº 0μ = όμΦ, srº ; = − R² (Ġ+4HO), ii 1 dri = 00 0,Ф, R² dri = iμ -dμÞ, where no sum over repeated i indices is implied and and the remaining perturbed coefficients either follow from symmetry or are zero. Hence show that the perturbed part of the Einstein tensor is given by 8G; = −23;($+HO), 8Gº = −2(7²Þ – 3HÒ — 3H²Þ), 8G = 2[Ï+4¢H+(2Ĥ+3H²)Þ],| where again no sum over repeated i indices is implied and the remaining entries either follow from symmetry or are zero.