Question

3. An almost-flat spacetime has metric in coordinates (zo, х, x2,T3) = (ct, x, y, z), where ηοβ = diag(-1, 1,1 , 1) is the Minkowski metric and the perturbation has is small, with hasl1. Let hue huv Σημι,h, where h y®ßhaß. The Einstein field equation becomes in the absence of matter and omitting terms of order ha. Consider a change of coordinates frorn zor to r/a-r" + ξα, in which the functions ξα are comparable in size to the has. What are the components hoa of the tensor hag after this transformation? Show that the E can be chosen to makepho0 and that, with this choice, the field equation becomes the wave equation, The partial derivatives in this equation are with respect to the old" coordinates ", not the "new" 'a. Comment on the significance of this. [10]

Answer 1

g_alpha beta = eta_alpha beta + h_alpha beta therefore After coordinate transformation, x_alpha rightarrow x^'_alpha = x_alpha + xi_alpha g_alpha beta rightarrow g^'_alpha beta = g_alpha beta + nabla_alpha xi_beta + nabla_beta xi_alpha when, nabla_alpha is covariant derivative with respect to g_alpha beta nabla_alpha xi_alpha = partial differential_beta + 1/2 (-h_alpha beta, mu + h_alpha beta + h_mu beta, alpha)xi_mu nabla_beta xi_alpha = partial differential_beta xi_alpha + 1/2 (-h_beta alpha, mu + h_mu beta, alpha + h_mu alpha, beta)xi_mu therefore h^'_alpha beta = h_alpha beta + partial differential_alpha xi_alpha + partial differential_beta xi_alpha + (-h_alpha beta, mu + h_mu alpha beta + h_mu beta, alpha)xi_mu equation (1) h^' = h + 2(partial differential middot xi) + (-h_mu + 2 h^alpha_mu, alpha)xi_mu equation (2) \r\n therefore we need to choose xi_alpha such that partial differential_p h^bar^' lambda P = 0 partial differential p(h^' lambda p - 1/2 eta lambda p h^') = 0 Substituting (1) & (2) we get, partial differential_p